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a/0NFKT4oBgHgl3EQfNi2Q/content/tmp_files/2301.11755v1.pdf.txt b/0NFKT4oBgHgl3EQfNi2Q/content/tmp_files/2301.11755v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..a9196091ededd561eb0b6c0b07d2088aa8742ab4 --- /dev/null +++ b/0NFKT4oBgHgl3EQfNi2Q/content/tmp_files/2301.11755v1.pdf.txt @@ -0,0 +1,2123 @@ +Time-frequency metrology with two single-photon states: phase space picture and the +Hong-Ou-Mandel interferometer +´Eloi Descamps1,2, Arne Keller2,3, P´erola Milman2 +1D´epartement de Physique de l’´Ecole Normale Sup´erieure - PSL, 45 rue d’Ulm, 75005 Paris, France +2Universit´ee Paris Cit´e, CNRS, Laboratoire Mat´eriaux et Ph´enom`enes Quantiques, 75013 Paris, France and +3D´epartement de Physique, Universit´e Paris-Saclay, 91405 Orsay Cedex, France +(Dated: January 30, 2023) +We use time-frequency continuous variables as the standard framework to describe states of light +in the subspace of individual photons occupying distinguishable auxiliary modes. We adapt to this +setting the interplay between metrological properties and the phase space picture already extensively +studied for quadrature variables. We also discuss in details the Hong-Ou-Mandel interferometer, +which was previously shown to saturate precision limits, and provide a general formula for the co- +incidence probability of a generalized version of this experiment. From the obtained expression, we +systematically analyze the optimality of this measurement setting for arbitrary unitary transforma- +tions applied to each one of the input photons. As concrete examples, we discuss transformations +which can be represented as translations and rotations in time-frequency phase space for some +specific states. +PACS numbers: +I. +INTRODUCTION +Much has been discovered since the first proposals to +use quantum systems in metrology. +From the role of +entanglement [1–4] to the one of modes, for pure and +noisy systems and measurements, several main results +have been established, and the most important one is +the fact that quantum mechanical protocols can provide +a better scaling in precision with the number of probes +than classical ones. Nevertheless, much still remains to +be done, in particular concerning the application and the +adaptation of such results to specific physical configura- +tions. Of practical importance, for instance, is the issue +of finding measurement strategies that lead to the opti- +mal calculated limits, and this is far from being obvious +for general states. +Another relevant problem concerns +adapting the general principles to physical constraints, +as energy or temperature limits and thresholds [5, 6]. +Those are the main issues of this paper: in one hand, we +deeply study the conditions for optimality of a specific +measurement set-up and on the other hand, we consider a +specific physical system, consisting of individual photons, +for measuring time and frequency related parameters. +In order to measure a given parameter κ one performs +an experiment producing different outcomes x with asso- +ciated probabilities Pκ(x) and build an unbiased estima- +tor K such that κ = ⟨K⟩κ is recovered. Here the index κ +means that we take the average for the probability distri- +bution Pκ. The Cram´er-Rao bound (CRB) [7] imposes a +limit on the precision of parameter estimation: +δκ ≥ +1 +√ +NF +, +(1) +where, δκ is the standard deviation in the estimation of +κ: δκ = +� +Varκ(K), N is the number of independent +measurements which were performed to estimate κ and +F is the quantity known as the Fisher information (FI), +defined by : F = +� +dx +1 +Pκ(x) +� +∂Pκ(x) +∂κ +�2 +. +In a quantum setting, one can use as a probe a quan- +tum state |ψ⟩ which can evolve under the action of +an operator ˆU(κ) = e−iκ ˆ +H generated by an Hamilto- +nian ˆH. +By optimizing the precision over all possible +quantum measurements of a parameter κ, one obtains a +bound, called the quantum Cram´er-Rao bound (QCRB) +[8] which reads: +δκ ≥ +1 +√NQ, +(2) +where Q is a quantity known as the quantum Fisher +information (QFI) which for pure states and uni- +tary +evolutions +(as +the +ones +considered +in +the +present +paper), +is +equal +to +Q += +4(∆ ˆH)2, +with +(∆ ˆH)2 = ⟨ψ(κ)| ˆH2 |ψ(κ)⟩ − ⟨ψ(κ)| ˆH |ψ(κ)⟩2. +The FI indicates the precision of a given measurement, +whereas the QFI is the maximum precision obtainable +with any measurement. For a given setting, we can thus +compute both quantities (FI and QFI) to have an idea if +the measurement is optimal (QFI=FI) or not (QFI>FI). +Determining the QFI is a mathematical task much +easier than finding a physical experimental set-up that +reaches it. In quantum optical systems, several propos- +als and implementations exist where the QFI is indeed +achieved [4, 9–11], and one example where this is possi- +ble is the Hong-Ou-Mandel (HOM) experiment [12–15]. +In this experiment, one focus on simple physical systems +composed of two photons occupying distinguishable spa- +tial modes with a given spectral distributions. This state +is a particular example of a state defined in the single +photon subspace (where each mode is populated by at +most one photon), in which a general pure state that can +arXiv:2301.11755v1 [quant-ph] 27 Jan 2023 + +2 +be expanded as: +|ψ⟩ = +� +dω1 · · · dωnF(ω1, · · · , ωn) |ω1, · · · , ωn⟩ . +(3) +In this formula, the indexes 1,2, ..n, label different aux- +iliary degrees of freedom (as for instance polarization or +the propagation direction). The state |ω1, · · · , ωn⟩ is a +pure state where each photon propagating in the mode +α is exactly at the frequency ωα. The spectral function +F also known as the joint spectral amplitude (JSA) is +normalized to one: +� +|F(ω1, ..., ωn)|2dω1...dωn = 1. +In this setting one can introduce time and frequency +operators for each mode α: ˆωα and ˆtα. They correspond +respectively to the generators of time and frequency +shifts of the photon in the mode labeled by α. +An +important property of these operators is that, in the +considered single photon subspace they satisfy the +commutation relation [ˆωα, ˆtβ] = iδα,β analogous to the +one observed for the quadrature operators ˆXα and ˆPα. +Notice that we are using throughout this paper dimen- +sionless operators, which are relative to particular time +and frequency scales of the associated implementation. +For a more complete description of the time frequency +continuous variables one can refer to Appendix A and +to [16]. +Previous works on quantum metrology using the +electromagnetic field quadratures or particles’ posi- +tion and momentum have shown how the phase space +(x1, · · · , xn, p1, · · · , pn) can provide not only insight but +also an elegant geometrical picture of the measurement +precision [16–18]. Indeed the QFI can also be defined in +terms of the Bures distance [19] s(|ψ(κ)⟩ , |ψ(κ + dκ)⟩): +Q = 4( s(|ψ(κ)⟩,|ψ(κ+dκ)⟩) +dκ +)2. In the case of pure states, +this distance is simply expressed in terms of the overlap +s(|ψ⟩ , |φ⟩) = +� +2(1 − |⟨φ|ψ⟩|). Since the overlap of two +states can be computed as the overlap of their respective +Wigner function, one can interpret the QFI as a measure +of how much the Wigner function must be shifted so as it +becomes orthogonal to the initial one. A consequence of +this is that the maximum precision of a measurement can +be seen geometrically on the Wigner function, by looking +at their typical size of variation in the direction of an evo- +lution [17]. Since in the case of single photon states one +can also define a time-frequency phase space associated +to the variables (τ1, · · · , τn, ϕ1, · · · , ϕn), it is natural to +investigate wether the same type of interpretation makes +sense in this context. +The present paper purposes are thus twofold: in the +first place, we provide general conditions for the HOM to +saturate precision limits using time-frequency (TF) vari- +ables. For such, we consider arbitrary evolution operators +acting on TF variables of single photons. In second place, +we provide a phase-space picture and interpretation of +the QFI for this type of system. Indeed, as shown in [20], +there is an analogy between the quadrature phase space +and the TF phase space from which metrological proper- +ties of time and frequency states can be inferred. Never- +theless, in the present case, photons have both spectral +classical wave-like properties and quantum particle-like +ones. Interpreting from a quantum perspective both the +role of the spectral distribution and of collective quantum +properties as entanglement in the single photon subspace +has shown to demand taking a different perspective on +the TF phase space [21]. Having this in mind, we in- +vestigate how relevant examples of evolution operators, +taken from the universal set of continuous variables quan- +tum gates, can be implemented and represented in phase +space, as well as the precision reached when one measures +them using the HOM experiment. We’ll concentrate on +single-mode Gaussian operations, analogously to what +was done in [5], even though we provide a general for- +mula for any transformation. +This paper is organized as follows: +In Section II +we provide a description of the TF phase space and +introduce the states we’ll discuss in details as well as +their representation. In Section III we discuss the HOM +experiment and the conditions for it to reach optimal +precision limits. Finally, in Sections IV and V we discuss +two different Gaussian operations in phase space as well +as their implementation and the associated precision +reached in the HOM experiment. +II. +TIME FREQUENCY PHASE SPACE +We consider pure two-photon states which can be writ- +ten in the form: |ψ⟩ = +� +dω1dω2F(ω1, ω2) |ω1, ω2⟩. The +Wigner function in variables (τ1, τ2, ϕ1, ϕ2) of such states +can be defined as +W|ψ⟩(τ1, τ2, ϕ1, ϕ2) = +� +dω1dω2e2i(ω1τ1+ω2τ2)F(ϕ1 + ω1, ϕ2 + ω2)F ∗(ϕ1 − ω1, ϕ2 − ω2). +(4) +Evolutions generated by ˆωα and ˆtα (α = 1, 2) correspond +to translations in phase space: +We−iˆ +ω1κ|ψ⟩(τ1, τ2, ϕ1, ϕ2) = W|ψ⟩(τ1 − κ, τ2, ϕ1, ϕ2), +(5a) +We−iˆt1κ|ψ⟩(τ1, τ2, ϕ1, ϕ2) = W|ψ⟩(τ1, τ2, ϕ1 − κ, ϕ2), +(5b) +and analogously for ˆω2 and ˆt2. + +3 +Using the QFI formulation based on the Bures +distance, we can safely state that the precision of +a measurement device is related to its capability of +distinguishing between an initial state |ψ(κ)⟩ and a state +|ψ(κ + dκ)⟩ that has evolved according to a parameter +κ. This precision is then directly related to how small +the parameter dκ should be such that these two states +can be distinguished i.e. the overlap |⟨ψ(κ)|ψ(κ + dκ)⟩| +gets close to zero. This can be also elegantly interpreted +using the overlap of the two states’s respective Wigner +functions, that describe trajectories in the phase space +that are governed by the interaction Hamiltonian and +the parameter dκ. +To gain some familiarity with the studied problem +we start with the case of a single-photon state |ψ⟩ = +� +dωS(ω) |ω⟩. Although using this type of state is not +current in metrology, this simpler case can be seen as a +building block and will help understanding the role of the +spectrum in the present configuration. +For a single photon, the Wigner function is defined +as: W(τ, ϕ) = +� +dωe2iωτS(ϕ + ω)S∗(ϕ − ω). In the case +of a Gaussian state |ψG⟩ with spectral wave function +SG(ω) = +e +− ω2 +4σ2 +(2πσ2)1/4 its Wigner function is also Gaussian: +WG(τ, ϕ) = exp +� +−2σ2τ 2 − ϕ2 +2σ2 +� +. It is characterized by +its width in the orthogonal directions τ and ϕ: 1/2σ and +σ respectively. +An evolution generated by ˆω corresponds to a trans- +lation in the direction τ in phase space. The associated +measurement precision is given by the smallest value of +dκ such that the initial Wigner function is almost orthog- +onal to the translated one in the corresponding direction. +Since the width of the Wigner function in the direction +of evolution is proportional to 1/σ, we have dκ ∼ 1/σ +leading to a QFI of the order of Q ∼ σ2. Alternatively if +one considers the generator ˆt, the associated width of the +state will be σ leading to a QFI of the order of Q ∼ 1/σ2. +We thus remark that the estimated QFI depends on the +width of the state in phase space in the direction of evo- +lution. We notice as well the similarities and differences +with the quadrature phase space case: even though the +relation between the phase space geometrical properties +and metrological interest are common to both variables, +in the case of quadrature they are related to some abso- +lute quantum resource dependent quantity, the number +of photons of the state. In the present case, the single +photon spectrum is a classical resource and its width can +only set a relative size scale in phase space. +It is interesting to notice that this type of interpreta- +tion is also possible for classical fields, as studied in [22– +24]. In this classical context, the electromagnetic field +amplitude replaces the function F and one can also re- +late spectral metrological properties to the phase space +structures. Nevertheless, as discussed in [21], this picture +is merely associated to classical metrological properties +of single mode fields (their spectrum) and no interesting +scaling can be observed in this context. As a matter of +fact, the classical single mode field and the single photon +phase space can be mapped into one another. +In the present paper, the multi-modal character of the +quantum field is an essential ingredient for the discussion +of the quantum metrological advantage, since it is a con- +sequence of the multi-photon state. We will see in par- +ticular how these two features (spectral and particle-like) +of the considered single photon subspace are combined in +the QFI. +The situation is different and richer for bi-photon +states, since the phase space is of dimension 4. +One +can thus imagine different directions of translation as +for instance the ones generated by operators ˆω1, ˆω2, +ˆω1 − ˆω2, . . . Then, optimizing the measurement precision +involves, for a given spectral distribution, choosing a di- +rection of evolution for which the Wigner function of the +state has the smallest scale structures. This direction, as +we’ll see, will depend on the number of photons, and can +display a non-classical scaling. +III. +THE HOM AS A MEASUREMENT DEVICE +A. +The setup +In the setup proposed by Hong, Ou and Mandel [25] +two photons impinge into a balanced beam splitter (BS), +each one of them from a different port, as represented on +figure 1. By measuring the output of the beam-splitter +using single-photon detectors we can compute the prob- +ability of obtaining coincidences (when the two photons +exit the BS by different paths) or anti-coincidences (when +they bunch and exit the BS at the same path). +FIG. 1: Schematic representation of HOM experiment. +Since its original proposal and implementation, many +modifications and adaptations were made to the HOM +set-up, which was shown to be very versatile to reveal +different aspects of quantum optics using two-photon in- +terference [26]: it can be used to witness particle [27] and + +A +BS +B4 +spectral [28] entanglement, to saturate precision bounds +on time delay measurements[12, 13] or to directly mea- +sure the Wigner function of the incoming state [29, 30]. +We’re interested in quantum metrological tasks, so +we’ll start by discussing the results obtained in [12], +where the authors provided experimental evidence that +the HOM device can saturate precision limits on time +measurements. To achieve this result, the authors con- +sidered the initial state: +|ψU⟩ = +1 +√ +2 +� +dΩf(Ω) +� ��ω0 +1 + Ω, ω0 +2 − Ω +� +− +��ω0 +2 + Ω, ω0 +1 − Ω +� � +, +(6) +where ω0 +1 and ω0 +2 are the central frequencies of the pho- +tons. Due to the energy conservation and to the phase- +matching conditions, the support of the JSA associated +to (6) is the line ω1 + ω2 = 0 in the plane (ω1, ω2). It +is anti-diagonal in the plane (ω1, ω2) and infinitely thin +along the diagonal direction ω− = ω1 − ω2. Adding a +delay in the arm 1 of the HOM interferometer corre- +sponds to an evolution generated by the operator ˆω1, +corresponding to a translation κ in the τ1 direction. The +QFI is simply calculated as: Q = 4∆(ˆω1)2. After the +beam-splitter, the measurement can lead to two out- +comes: coincidence or anti-coincidence, with probability +Pc and Pa, respectively. +The FI is thus expressed as: +F = +1 +Pc +� ∂Pc +∂κ +�2 + +1 +Pa +� ∂Pa +∂κ +�2. The authors of [12] thus +showed that using the input state (6) in the HOM inter- +ferometer, the two quantities F and Q are the same. +In [13] the HOM interferometer was also used and +shown to lead to the QFI in a two-parameter estimation +experiment. Finally, in [14] biphoton states were classi- +fied as metrological resources according to their spectral +width, still in the situation where the HOM experiment +is used as a measurement apparatus. +B. +Generalization: the HOM as an optimal +measurement device for quantum metrology with +biphotons +We now make a general description of the HOM ex- +periment as a parameter estimation device and try to +understand and determine when it corresponds to an op- +timal measurement strategy. In [13], the authors tackle a +part of this problem by studying the HOM as a measure- +ment apparatus for two parameter estimation by estab- +lishing conditions on frequency correlation states. In this +reference, the authors restrict themselves to time delay +evolutions. +In the present paper, we are interested in studying any +evolution that can be described by a two photon unitary +|ψ(κ)⟩ = ˆU(κ) |ψ⟩ = e−i ˆ +Hκ |ψ⟩ (see figure 2). We will see +that under a symmetry assumption on the JSA of the +state, it is possible to obtain an explicit formula for the +FI, and this formula can be used to compute at a glance +if the measurement setup considered is optimal or not. +FIG. 2: HOM setup where we apply a general gate ˆU +before the BS. +For any input state |ψ⟩, the QFI will then be expressed +as: +Q = 4∆( ˆH)2. +(7) +On the other hand, one can show that the coincidence +probability is: +Pc = 1 +2(1 − ⟨ψ| ˆU † ˆS ˆU |ψ⟩). +(8) +(see Appendix B) where we introduced the hermitian +swap operators ˆS whose action on the states is given +by ˆS |ω1, ω2⟩ = |ω2, ω1⟩. Furthermore we can compute +the associated FI. If the state |ψ⟩ is symmetric or anti- +symmetric (i.e. ˆS |ψ⟩ = ± |ψ⟩) the FI at κ = 0 it is given +by: +F = ∆( ˆH − ˆS ˆH ˆS)2. +(9) +(see Appendix B). This means that under the symme- +try assumption on the JSA, comparing the QFI and +the FI is done simply by comparing the variance of +two different operators, mainly: +2 ˆH and +ˆH − ˆS ˆH ˆS. +Equation (9) implies that if [ ˆH, ˆS] = 0, then F = 0 +and no information can be obtained about κ from the +measurements. +However, if { ˆH, ˆS} = 0 then F = Q +since ˆS ˆH ˆS = − ˆS2 ˆH = − ˆH. +In this last case, the +measurement strategy is optimal. In [31], general con- +ditions for reaching the QFI were also obtained in the +context of amplitude correlation measurements. These +conditions are based on a quantum state’s symmetry +under (unphysical) path exchange. +The previous calculations form a simple tool that can +be applied to different evolution Hamiltonians ˆH. We’ll +now discuss examples taken from the universal set of +quantum gates in continuous variables: translations (gen- +erated by operator ˆωα’s) and rotations (generated by +ˆH = (ˆω2 +ˆt2)/2). These gates have already been studied +in [5] in the case of quadrature or position and momen- +tum. In the present physical configuration, they corre- +spond to the free evolution of single photons in free space + +A +U +2 +BS +B5 +(translations) or in a dispersive medium, as for instance +an optical fiber combined to time lenses (rotation). +IV. +TIME-FREQUENCY PHASE-SPACE +TRANSLATIONS +A. +Different types of translations +Since we’re considering two-photon states, translations +can be represented by any linear combination of the cor- +responding operators, that is : ˆH = αˆω1+βˆω2+γˆt1+δˆt2. +To illustrate our results we choose to focus on the four +operators ˆω1, ˆω2 and ˆω± = ˆω1 ± ˆω2, since they are the +most easily implemented in HOM experiment. +Notice +that ˆω± are collective operators acting in both input +photons while ˆω1,2 act in a single photon only. +If we consider a state which is (anti-)symmetric and +separable in the variables ω± = ω1 ± ω2, we can write: +|ψ⟩ = +1 +√ +2 +� +dω+dω−f(ω+)g(ω−) +���� +ω+ + ω− +2 +, ω+ − ω− +2 +� +, +(10) +with g satisfying g(−ω) = ±g(ω) and the functions g +and f being normalized to one. +The specific form of +each function is related to the phase-matching conditions +and the energy conservation of the two-photon generation +process and this type of state can be experimentally pro- +duced in many set-ups [32, 33]. Using equations (7) and +(9) we can compute the QFI and FI associated to each +type of evolution: +• For ˆH = ˆω1, we get Q = ∆(2ˆω1)2 = ∆(ˆω++ˆω−)2 = +∆(ˆω−)2 + ∆(ˆω+)2, while F = ∆(ˆω−)2. Thus this +situation is optimal only if ∆(ˆω+)2 = 0, which was +the case for the state |ψU⟩ of Eq. (6) used in [12]). +We obtain the same type of result for ˆω2. +• For ˆH = ˆω+, Q = 4∆(ˆω+)2, while F = ∆(ˆω+ − +ˆω+)2 = 0. +In this situation the precision of the +measurement is zero, and the reason for that is that +variables ω+ cannot be measured using the HOM +experiment (we notice that [ˆω+, ˆS] = 0). +• For ˆH = ˆω−, we get Q = 4∆(ˆω−)2, while F = +∆(ˆω− + ˆω−)2 = 4∆(ˆω−)2. This time we have F = +Q, which means that the measurement is optimal. +In this case, we have that {ˆω−, ˆS} = 0. +We now illustrate these general expressions and inter- +pret them using different quantum states and their phase +space representations. +B. +Example: Gaussian and Schr¨odinger cat-like +state +To illustrate our point we discuss as an example two +states |ψG⟩ and |ψC⟩ that can be expressed in the form +of equation (10). For |ψG⟩, f and g are Gaussians: +fG(ω+) = e +− +(ω+−ωp)2 +4σ2 ++ +(2πσ2 ++)1/4 +gG(ω−) = +e +− +ω2 +− +4σ2 +− +(2πσ2 +−)1/4 , +(11) +where σ± is the width of the corresponding function and +ωp is a constant, which is also the photon’s central fre- +quency. As for state |ψC⟩, it can be seen as the general- +ization of (6). We consider f to be Gaussian and g to be +the sum of two Gaussians: +fC(ω+) = fG(ω+) +gC(ω−) = +1 +√ +2 +� +gG(ω− + ∆/2) − gG(ω− − ∆/2) +� +, +(12) +where ∆ is the distance between the two Gaussian peaks +of gC. We assume that the two peaks are well separated: +∆ ≫ σ−. +Consequently, gC is approximately normal- +ized to one. We can verify that with these definitions +the function gG is even while gC is odd by exchange of +variables ω1 and ω2. We first compute the variances for +both states (table I) and then apply the formula (7) and +(9). +State +|ψG⟩ +|ψC⟩ +∆(ˆω1)2 or ∆(ˆω2)2 +1 +4σ2 ++ + 1 +4σ2 +− +1 +16∆2 + 1 +4σ2 ++ + 1 +4σ2 +− +(∆ˆω+)2 +σ2 ++ +σ2 ++ +(∆ˆω−)2 +σ2 +− +1 +4∆2 + σ2 +− +TABLE I: Variance of various time translation operators +for states |ψG⟩ and |ψC⟩. See Appendix C for details. +So for the case of an evolution generated by ˆω1, for +|ψG⟩ we obtain: +Q = σ2 ++ + σ2 +− +F = σ2 +−, +(13) +while for |ψC⟩ we have: +Q = 1 +4∆2 + σ2 ++ + σ2 +− +F = 1 +4∆2 + σ2 +−. +(14) +We thus see that time precision using the HOM measure- +ment and the quantum state evolution generated by ˆω1 is +optimal only if the parameter σ+ is negligible compared +to ∆ or σ−. This is exactly the case for the state (6) +where σ+ = 0. +In addition, we see that there is a difference between +the QFI associated to |ψC⟩ and |ψG⟩ involving the param- +eter ∆. This difference can be interpreted, as discussed +in [14], as a spectral effect. In this reference, the spectral +width is considered as a resource, and for a same spectral +width state |ψC⟩ has a larger variance than state |ψG⟩. +Nevertheless, as discussed in [21], this effect has a clas- +sical spectral engineering origin and choosing to use one +rather than the other depends on the experimentalists +constraints. + +6 +(a) Projection on the plane τ−, +ω− +(b) Projection on the plane τ1, +ω1 +FIG. 3: Wigner function of the cat-like state |ψC⟩ +projected in different variables. +C. +Interpretation of translations in the +time-frequency phase space +We now discuss the dependency of precision on the +direction of translation. For such, we can consider the +Wigner function associated to a JSA which is separable +in the ω± variables. +Its Wigner function will also be +separable on these variables: +W(τ1, τ2, ϕ1, ϕ2) = W+(τ+, ϕ+)W−(τ−, ϕ−), +(15) +where the phase space variables τ± and ϕ± are defined +as: ϕ± = +ϕ1±ϕ2 +2 +and τ± = τ1 ± τ2. Even though the +Wigner function W+ (resp. W−) can be associated to +the one of a single variable (ω+ (ω−)) and spectral wave +function f (resp. g), it displays some differences with the +single photon one. This fact is well illustrated in Fig. 3. +For state |ψC⟩, according to (15) the projection of the +Wigner function W− in the plane τ−, φ− of the phase +space can be represented as show in Figure 3 (a). We see +that it is composed of two basic shapes: two Gaussian +peaks and an oscillation pattern in between. Figure 3 (b) +represents another way to project this very same Wigner +function onto the plane τ1, φ1 of the phase space. One can +observe that in this case the distance between the peaks +is larger than in the previous representation by a factor +of 2. As precision is directly related to the size of the +Wigner function structures in phase space, we observe +that the interference fringes are closer apart in the phase +space associated to the minus variable than in the one +associated to mode 1. Thus, the precision in parameter +estimation will be better using ˆω− as the generator of the +evolution than when using ˆω1. This phase space based +observations explain well the result of the computation +of the QFI: +4∆(ˆω1)2 = ∆(ˆω−)2. +(16) +with the assumption that σ+ ≪ ∆, σ−. +The reason for the appearance of a factor 2 difference +in fringe spacing for the Wigner function associated to +variable ω− is the fact that it is a collective variable, +and translations in the phase space associated to these +variables are associated to collective operators, acting on +both input photons (instead of a single one, as is the case +of translations generated by operator ˆω1, for instance). +Thus, one can observe, depending on the biphoton quan- +tum state (i.e., for some types of frequency entangled +states), a scaling depending on the number of particles (in +this case, two). As analyzed in [21] for general single pho- +ton states composed of n individual photons, we have for +frequency separable states a scaling corresponding to the +shot-noise one (i.e., proportional to √n). A Heisenberg- +like scaling (proportional to n) can be achieved for non- +physical maximally frequency correlated states, and con- +sidering a physical non-singular spectrum leads to a non- +classical scaling in between the shot-noise and the Heisen- +berg limit. +Experimentally, such collective translation can be im- +plemented by adding a delay of τ in arm 1 and of −τ in +arm 2. Notice that this situation is different from cre- +ating a delay of 2τ in only one arm, even though both +situations lead to the same experimental results in the +particular context of the HOM experiment. +V. +TIME-FREQUENCY PHASE SPACE +ROTATIONS +We now move to the discussion of the phase space ro- +tations. For this, we’ll start by providing some intuition +by discussing in first place the single photon (or single +mode) situation. In this case, time-frequency phase space +rotations are generated by the operators ˆR = 1 +2(ˆω2 + ˆt2). +As previously mentioned, we consider here dimension- +less observables. Physically, time-frequency phase space +rotations correspond to performing a fractional Fourier +transform of the JSA. While for transverse variable of +single photons the free propagation or a combination of +lenses can be used for implementing this type of oper- +ation [34, 35], in the case of time and frequency this +transformation corresponds to the free propagation in a +dispersive medium [36–40] combined to temporal lenses +[41–43]. +A. +Single mode rotations +In this Section, we compute the QFI associated to +a rotation +ˆR for a single photon, single mode state +using the variance of this operator for different states +|ψ⟩ = +� +dωS(ω) |ω⟩. As for the translation, this simpler +configuration is used as a tool to better understand the +two photon case. + +1.5 +1.0 +0.5 +0.0 +T +-0.5 +-1.0 +1.5 +-10 +-5 +0 +5 +101.5 +1.0 +0.5 +0.0 +T +-0.5 +-1.0 +1.5 +-5 +-10 +0 +5 +107 +1. +Gaussian state: +We start by discussing a single-photon Gaussian state +at central frequency ω0 and spectral width σ: +|ψG(ω0)⟩ = +1 +(2πσ2)1/4 +� +dωe− (ω−ω0)2 +4σ2 +|ω⟩ . +(17) +For this state, we have that: +∆( ˆR)2 = σ2ω2 +0 + 1 +8 +� 1 +4σ4 + 4σ4 − 2 +� +. +(18) +Eq. (18) has two types of contributions that we can in- +terpret: +• The first term σ2ω2 +0 corresponds to the distance in +phase space (ω0) of the center of the distribution, to +the origin of the phase space (ω = 0, τ = 0), times +the width of the state σ in the direction of rotation +(see Figure 4 (a)). This term is quite intuitive. The +Wigner function of a state which is rotated by an +angle θ = 1/2σω0 has an overlap with the Wigner +function of the initial one which is close to zero. +• The term +1 +4σ4 + 4σ4 − 2 reaches 0 as a minimum +when σ = +1 +√ +2. For this value the Wigner function +is perfectly rotationally symmetric. +Its meaning +can be intuitively understood if we consider that +ω0 = 0, so that this term becomes the only con- +tribution to the variance(see Figure 4 (b)). In this +case, we are implementing a rotation around the +center of the state. If the state is fully symmetric +then this rotation has no effect, and the variance +is 0. Only in the case where the distributions ro- +tational symmetry is broken we obtain a non zero +contribution. +2. +Schr¨odinger cat-like state centered at the origin (ω = 0): +We now consider the superposition of two Gaussian +states: +��ψ0 +C +� += +1 +√ +2(|ψG(∆/2)⟩ − |ψG(−∆/2)⟩). +(19) +This state is of course non physical as a single-photon +state, since it contains negative frequencies. +However, +since it can be be well defined using collective variables +(as for instance ω−) for a two or more photons state, +we still discuss it. Assuming that the two peaks are well +separated (∆ ≫ σ), we can ignore the terms proportional +to e− ∆2 +8σ2 , and this leads to: +∆( ˆR)2 = 1 +8 +� 1 +4σ4 + 4σ4 − 2 +� ++ 1 +4∆2σ2. +(20) +We see that there is no clear metrological advantage +when using this state compared to the Gaussian state: +the quantity ∆/2 plays the same role as ω0. This can +be understood geometrically once again, with the help +of the Wigner function. +We see in Figure 4 (c) how +the considered state evolves under a rotation. +In this +situation the interference fringes are rotated around +their center so even though they display a small scale +structure, they are moved only by a small amount, +resulting in a non significant precision improvement. +3. +Schr¨odinger cat-like state centered at any frequency: +We can now discuss the state formed by the superpo- +sition of two Gaussian states whose peaks are at frequen- +cies ω0 − ∆/2 and ω0 + ∆/2, and with the same spectral +width σ as previously considered: +|ψC⟩ = +1 +√ +2 +� +|ψG(ω0 + ∆/2)⟩ − |ψG(ω0 − ∆/2)⟩ +� +. (21) +Still under the assumption of a large separation between +the two central frequencies (∆ ≫ σ), we obtain: +∆( ˆR)2 = 1 +8 +� 1 +4σ4 + 4σ4 − 2 +� ++ 1 +4∆2(σ2 + ω2 +0) + σ2ω2 +0. +(22) +We can notice that by setting ω0 = 0 we recover the +variance corresponding to the same state rotated around +its center. Nevertheless, in the present case ω0 ̸= 0, and +we have two additional terms: σ2ω2 +0 and ∆2ω2 +0/4. Both +terms can be interpreted as a product of the state’s dis- +tance to the origin and its structure in phase space. How- +ever, while the first one is simply the one corresponding +to the Gaussian state, the second one is a product of the +states’ distance to the origin and its small structures in +phase space, created by the interference between the two +Gaussian states (see Figure 4 (d)). The interference pat- +tern is thus rotated by an angle θ corresponding to an arc +of length ω0θ, and since the distance between the fringes +is of order ∆, if θ ∼ 1/ω0∆ (corresponding to the term +∆2ω2 +0/4 in the expression of the variance) the rotated +state is close to orthogonal to the initial one. +In all this section, we have considered rotations about +the time and frequency origin of the phase space. Never- +theless, it is of course possible to displace this origin and +consider instead rotations about different points of the +TF phase space. In this case, for a rotation around an +arbitrary point τ0 and ϕ0, the generator would be given +by (ˆω − ϕ0)2/2 + (ˆt − τ0)2/2. +B. +Different types of rotations +We now move to the case of two single photons +(biphoton states). As for the case of translations, there +are many possible variables and can consider rotations +in different planes of the phase space: +ˆR1, +ˆR2, +ˆR±, + +8 +(a) Gaussian state centered at +ω0. For θω0 ∼ 1/2σ the initial +state and the rotate one are +distinguishable. +(b) Gaussian state centered at +the origin. The rotated state will +be distinguishable from the +initial one only in the absence of +rotational symmetry. +(c) Superposition of two Gaussian +states (cat-like state) centered the +origin. The small structures of +the fringes do not play a relevant +role since they are only moved by +a small distance under rotation. +(d) Superposition of two +Gaussian states (cat-like state) +centered at ω0. The fringes play +an important role, since with +θω0 ∼ 1/∆, the two states are +nearly orthogonal. +FIG. 4: Schematic representation of the Wigner +function of various states under rotation. The ellipses +represent the typical width of Gaussians. The doted +lines represent the rotated states. +ˆR1 ± ˆR2 . . . where ˆR1 = +1 +2(ˆω2 +1 + ˆt2 +1) (and similarly for +ˆR2) and ˆR± = 1 +4(ˆω2 +± + ˆt2 +±) (recall that ˆω± = ˆω1 ± ˆω2 and +ˆt± = ˆt1 ± ˆt2). For all these operators we can as before +apply the general formula for the QFI and of the FI to +the corresponding HOM measurement. The results are +displayed in table II. +Operator +QFI +FI +ˆR1 +4∆( ˆR1)2 +∆( ˆR1 − ˆR2)2 +ˆR± +4∆( ˆR±)2 +0 +ˆR1 + ˆR2 +4∆( ˆR1 + ˆR2)2 +0 +ˆR1 − ˆR2 +4∆( ˆR1 − ˆR2)2 +4∆( ˆR1 − ˆR2)2 +TABLE II: QFI and FI of various rotation operators. +We see that the only two situations where the HOM +can indeed be useful as a measurement device for metro- +logical applications are ˆR1 and ˆR1 − ˆR2. The reason for +that is the symmetry of ˆR± and ˆR1+ ˆR2, which commute +with the swap operator ˆS. As for ˆR1, it corresponds to +the rotation of only one of the photons and may not be +the optimal strategy. Finally, ˆR1 − ˆR2 corresponds to +the simultaneous rotation in opposite directions of both +photons sent into the two different input spatial modes. +As ˆR1 − ˆR2 anti-commutes with ˆS then we can affirm +that the HOM measurement is optimal for this type of +evolution. +C. +QFI and FI computation with Gaussian and +cat-like state +We now compute the QFI and FI using the variance +of ˆR1 and ˆR1 − ˆR2 calculated for states |ψG⟩ and |ψC⟩. +For |ψG⟩: +We have: +∆( ˆR1)2 = 1 +32 +�� 1 +σ2 ++ ++ 1 +σ2 +− +�2 ++ (σ2 ++ + σ2 +−)2 − 8 +� ++ 1 +16ω2 +p(σ2 ++ + σ2 +−) +∆( ˆR1 − ˆR2) = 1 +4 +� +1 +σ2 ++σ2 +− ++ σ2 ++σ2 +− − 2 +� ++ 1 +4σ2 +−ω2 +p. (23) +For |ψC⟩: +We have: +∆( ˆR1)2 = 1 +32 +�� 1 +σ2 ++ ++ 1 +σ2 +− +�2 ++ (σ2 ++ + σ2 +−)2 − 8 +� ++ 1 +64(4ω2 +p + ∆2)(σ2 ++ + σ2 +−) ++ 1 +64∆2ω2 +p + ∆2 +128 +� 1 +σ2 +− ++ σ2 +− +� +∆( ˆR1 − ˆR2) = 1 +4 +� +1 +σ2 ++σ2 +− ++ σ2 ++σ2 +− − 2 +� ++ 1 +4σ2 +−ω2 +p. (24) +We notice that for both states 4∆( ˆR1)2 ≥ ∆( ˆR1− ˆR2)2, +meaning that the measurement of a rotation imple- +mented in only one mode using the HOM is not an +optimal measurement. +Experimentally realizing an evolution generated by ˆR1 +is easier than implementing the one associated to ˆR1− ˆR2. +Furthermore we see that for the Gaussian state |ψG⟩ a +dominant term is ω2 +pσ2 +− which appears with the same +factor in 4∆( ˆR1)2 and ∆( ˆR1 − ˆR2)2, meaning that one +could perform a measurement which although not opti- +mal would be pretty efficient. The same applies to the +Schr¨odinger cat-like state |ψC⟩ where one dominant term +is ∆2ω2 +p. +D. +Phase space interpretation +We now provide a geometrical interpretation of the +previous results. +If we consider that σ− ≫ σ+ in + +T +0 +6 +1 +2g +03T +个 +6 +1 +2g7 +0T +039 +the case of a Gaussian state or ∆ ≫ σ+ in the case +of a Schr¨odinger cat-like state, the projection of the +Wigner function on the plane corresponding to collective +minus variables (τ−, φ−) is the one presenting a relevant +phase space structure. +Thus, it would be interesting +to consider, as in the case of translations, that these +states are manipulated using operators acting on modes +associated to this collective variable. +A na¨ıve guess +would then trying to apply the rotation operator ˆR−. +However it comes with many difficulties. +Indeed it +first poses an experimental problem, since this rotation +corresponds to a non-local action which would be very +hard to implement. In addition, the HOM is not able to +measure such evolution. Finally, it turns out that this is +not the operator with the greatest QFI. This fact can be +understood by taking a more careful look at the Wigner +function of the considered states. The Wigner function +for separable states can be factorized as the product +of two Wigner functions defined in variables plus and +minus, and we have that W+ is the Wigner function +of a Gaussian state centered at ωp (corresponding to +the situation (a) in Figure (4). As for W−, it is either +the Wigner function of a Gaussian state or the one +associated to a superposition of two Gaussian states +centered around zero (corresponding to the situation +(b) and (c) in Figure 4). +The QFI increases with the +distance of the states to the rotation point. +For this +reason, states |ψG⟩ and |ψC⟩ under a rotation using ˆR−, +do not lead to a high QFI. +A higher QFI is obtained using rotations around a +point which is far away from the center of the state. In +this case, the QFI displays a term which is proportional +to the distance from the center of rotation squared di- +vided by the width of the state squared. +Both terms +ω2 +pσ2 +− and ∆2ω2 +p which were dominant in the expression +of the variance of ˆR1 and ˆR1 − ˆR2 can be interpreted +as such. +This means that the rotation ˆR1, whose ac- +tion is not easily seen in the variables plus and minus, +can be interpreted as a rotation which moves W− around +the distance ωp from the origin of the TF phase space +(ω = 0). +For both states then, the main numerical contribution +to the QFI comes from a classical effect, related to the +intrinsic resolution associated to the central (high) fre- +quency of the field. In general, in phase space rotations, +both in the quadrature and in the TF configuration, the +distance from the phase space origin plays an important +role. While in the quadrature configuration this distance +has a physical meaning that can be associated both to the +phase space structure and to the number of probes. In +the case of TF phase space, the distance from the origin +and the phase space scaling are independent. In partic- +ular, the distance from the origin can be considered as a +classical resource that plays no role on the scaling with +the number of probes. +E. +A discussion on scaling properties of rotations +The different types of FT phase space rotations have +different types of interpretation in terms of scaling. The +combined rotations of the type ˆR1 ± ˆR2, for instance, +can be generalized to an n photon set-up through oper- +ators as ˆR = �n +i αi ˆRi, with αi = ±1. In this situation, +we have that rotation operators are applied individually +and independently to each one of the the n photons. In +this case, we can expect, in first place, a collective (clas- +sical) effect, coming simply from the fact that we have n +probes (each photon). In addition, it is possible to show +that a Heisenberg-like scaling can be obtained by con- +sidering states which are maximally mode entangled in a +mode basis corresponding to the eigenfunctions of oper- +ators ˆRi. Indeed, for each photon (the i-th one), we can +define a mode basis such that ˆRi |φk⟩i = (k + 1/2) |φk⟩i, +with |φk⟩i = +1 +√ +2kk! +1 +π1/4 +� +dωe− ω2 +2 Hk(ω) |ω⟩i with Hk(ω) +being the k-th Hermite polynomial associated to the i- +th photon. For a maximally entangled state in this mode +basis, i.e. , a state of the type |φ⟩ = �∞ +k=0 Ak +�n +i=1 |φk⟩i, +(where we recall that the subscript i refers to each pho- +ton and k to the rotation eigenvalues) the ˆR eigenvalues +behave as random classical variables and we can show +that the QFI scales as n2. +As for rotations of the type ˆR±, they cannot be de- +composed as independently acting on each photon, but +consist of entangling operators that can be treated ex- +actly as ˆR1 and ˆR2 but using variables ω± = ω1 ± ω2 +instead of ω1 and ω2. +We can also compute the scal- +ing of operators as ˆJ = � +Ωβ ˆRΩβ where Ωβ = �n +i αiωi, +αi = ±1 and β is one of the 2n−1 ways to define a collec- +tive variable using the coefficients αi. For such, we can +use the same techniques as in the previous paragraph but +for the collective variables Ωβ. Nevertheless, the exper- +imental complexity of producing this type of evolution +and the entangled states reaching the Heisenberg limit is +such that we’ll omit this discussion here. +VI. +CONCLUSION +We have extensively analyzed a quantum optical set- +up, the HOM interferometer, in terms of its quantum +metrological properties. We provided a general formula +for the coincidence probability of this experiment which +led to a general formula for the associated FI. We used +this formula to analyze different types of evolution and +showed when it is possible to reach the QFI in this set- +up. In particular, we made a clear difference between col- +lective quantum effects that contribute to a better than +classical precision scaling and classical only effects, asso- +ciated to single mode spectral properties. We then briefly +discussed the general scaling properties of the QFI asso- +ciated to the studied operators. +Our results provide a complete recipe to optimize the +HOM experiment with metrological purposes. They rely + +10 +on the symmetry properties of quantum states that are +revealed by the HOM interferometer. An interesting per- +spective is to generalize this type of reasoning for differ- +ent set-ups where different symmetries play a role on the +measurement outputs. +Acknowledgements +The French gouvernement through the action France +2030 from Agence Nationale de la Recherche, reference +“ANR-22-PETQ-0006” provided financial support to this +work. We thanks Nicolas Fabre for fruitful discussions +and comments on the manuscript. +[1] Daniel Braun, Gerardo Adesso, Fabio Benatti, Roberto +Floreanini, Ugo Marzolino, Morgan W. Mitchell, and +Stefano Pirandola. +Quantum-enhanced measurements +without entanglement. +Rev. Mod. Phys., 90:035006, +Sep 2018. 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URL: https:// +link.aps.org/doi/10.1103/PhysRevLett.94.073601, +doi:10.1103/PhysRevLett.94.073601. +[44] C. +Fabre +and +N. +Treps. +Modes +and +states +in +quantum +optics. +Rev. +Mod. +Phys., +92(3):035005, +2020. +URL: +https://link.aps.org/doi/10.1103/ +RevModPhys.92.035005, +doi:10.1103/RevModPhys.92. +035005. +Appendix A: Time frequency formalism +In quantum mechanics, light is described with the help of modes [44], representing the various physical properties +a photon can have: frequency, position, spectral shape, wave vector, polarization... Mathematically we associate +to each mode α a creation and annihilation operators ˆa† +α and ˆaα which satisfy the familiar bosonic commutation +relation [ˆaα, ˆa† +β] = δα,β. The quantum states are then obtained by acting with the creation operators on the vacuum +|vac⟩, which can be interpreted as adding a photon in the corresponding mode. +In time frequency continuous variables we look at modes parameterized by the frequency [16]. We will thus adapt +the terminology: for us a mode will correspond to all physical parameter needed to describe a photon excluding +the frequency (position, wave vector, polarization...). In the following we will look at interferometers, and thus the +parameter α will describe in which arm the photon is propagating. We will thus describe single photon states in +a given mode α with frequency ω with the help of a creation operator acting on the vacuum state: ˆa† +α(ω). In this +situation the commutation relation is written as +[ˆaα(ω), ˆa† +β(ω′)] = δ(ω − ω′)δα,β, +(A1) +the other commutation relations (between two creation or two annihilation operators) vanishing. It’s useful to intro- +duce the conjugated temporal variable t, by the use of the Fourier transform: +ˆaα(t) = +1 +√ +2π +� +dωˆaα(ω)e−iωt. +(A2) +We can verify that the creation and annihilation operators in the temporal domain verify the same commutation +relation as the one in the spectral domain: [ˆaα(t), ˆa† +β(t′)] = δ(t − t′)δα,β. +a. +States in time-frequency variables +The creation operators allow to define general single photon states on a single mode via: +|ψ⟩ = +� +dωS(ω)ˆa†(ω) |vac⟩ = +� +dωS(ω) |ω⟩ . +(A3) +The spectrum S(ω) is the Fourier transform of the time of arrival distribution and it can be recovered from the +state S(ω) = ⟨ω|ψ⟩. If we are interested in a collection of n single photons states in n different modes, we can work +with the state: +|ψ⟩ = +� +dω1 · · · dωnF(ω1, · · · , ωn)ˆa† +1(ω1) · · · ˆa† +n(ωn) |vac⟩ = +� +dω1 · · · dωnF(ω1, · · · , ωn) |ω1, · · · , ωn⟩ , +(A4) +where the spectral function F is normalised to one: +� +|F(ω1, ω2|2dω1dω2 = 1. + +13 +b. +Time-frequency operators +We can introduce two very useful operators as follows: +ˆtα = +� +dt tˆa† +α(t)ˆaα(t) +ˆωα = +� +dω ωˆa† +α(ω)ˆaα(ω). +(A5) +The fundamental property of these operators is the fact that they verify the familiar commutation relation on the +subspace of single photons: +[ˆωα, ˆtα] = i. +(A6) +More precisely, we have the general result: +[ˆωα, ˆtα] = i +∞ +� +−∞ +dωˆa† +α(ω)ˆaα(ω) = i ˆNα, +(A7) +where the operator ˆNα count the number of photon operator in the mode α. +The action of the both operators ˆω and ˆt can be computed on the JSA and we have: +ˆω : S(ω) �→ ωS(ω) +ˆt : S(ω) �→ −i∂ωS(ω). +(A8) +Appendix B: Appendix: Derivation of equations (8) and (9) +a. +Equation (8) +To show equation (8) we start with the state before the BS: +ˆU |ψ⟩ = +� +dω1dω2F(ω1, ω2) |ω1, ω2⟩ . +(B1) +The usual balanced BS relation reads: +|ω1⟩1 |ω2⟩2 �→ 1 +2 +� +|ω1⟩1 |ω2⟩1 − |ω1⟩1 |ω2⟩2 + |ω1⟩2 |ω2⟩1 − |ω1⟩2 |ω2⟩2 +� +. +(B2) +To be able to use it, we introduce two mode changing operators ˆT1 and ˆT2 defined by: +ˆT1 |ω1⟩1 |ω2⟩2 = |ω1⟩1 |ω2⟩1 +ˆT2 |ω1⟩1 |ω2⟩2 = |ω1⟩2 |ω2⟩2 . +(B3) +With these definition the BS splitter relation is equivalent to applying the operator: +1 +2( ˆT1 − ˆ1 + ˆS − ˆT2), +(B4) +where ˆS is the swap operator, defined as ˆS |ω1, ω2⟩ = |ω2, ω1⟩ So the state coming out of the BS is: +|ψout⟩ = 1 +2 +� +dω1dω2F(ω1, ω2) +� +ˆT1 ˆU − ˆU + ˆS ˆU − ˆT2 ˆU +� +|ω1, ω2⟩ . +(B5) +If we do selection on coincidence, we only keep the part of the state with one photon in each mode. We get the state: +|ψfin⟩ = −1 +2 +� +dω1dω2F(ω1, ω2) +� +ˆU − ˆS ˆU +� +|ω1, ω2⟩ +(B6a) += 1 +2 +� +ˆS ˆU − ˆU +� +|ψ⟩ . +(B6b) + +14 +We can finally compute the coincidence probability by taking the norm square of |ψfin⟩: +Pc = ⟨ψfin|ψfin⟩ +(B7a) += 1 +4 ⟨ψ| +� +ˆU † − ˆU † ˆS +�� +ˆU − ˆS ˆU +� +|ψ⟩ +(B7b) += 1 +4 ⟨ψ| +� +ˆU † ˆU +���� +=1 +−2 ˆU † ˆS ˆU + ˆU † ˆS ˆS ˆU +� �� � += ˆU † ˆU=1 +� +|ψ⟩ +(B7c) += 1 +2 +� +1 − ⟨ψ| ˆU † ˆS ˆU |ψ⟩ +� +. +(B7d) +b. +Equation (9) +The expression for Q is a direct consequence of the expression of the QFI for pure state. +The proof of the expression of F is a little bit more involved. We have to compute: +FI(κ) = 1 +Pc +�∂Pc +∂κ +�2 ++ 1 +Pa +�∂Pa +∂κ +�2 +. +(B8) +We have seen the expression of the (anti)-coincidence probability Pc and Pa that depends on ⟨ψ| ˆU † ˆS ˆU |ψ⟩. If we +make the assumption that the state |ψ⟩ is either symmetric or anti-symmetric we known that we have: ⟨ψ| ˆU † ˆS ˆU |ψ⟩ = +± ⟨ψ| ˆU † ˆS ˆU ˆS |ψ⟩ = ⟨ψ| ˆV (κ) |ψ⟩ where we denote ˆV (κ) = ˆU † ˆS ˆU ˆS = eiκ ˆ +He−iκ ˆS ˆ +H ˆS. We first start by expanding this +scalar product up to the second order in κ, using the short hand notation ⟨·⟩ = ⟨ψ| · |ψ⟩. +⟨ψ| ˆV (κ) |ψ⟩ = +� +eiκ ˆ +He−iκ ˆS ˆ +H ˆS� +(B9a) +≃ +�� +1 + iκ ˆH − κ2 +2 +ˆH2�� +1 − iκ ˆS ˆH ˆS − κ2 +2 ( ˆS ˆH ˆS)2�� +(B9b) += +� +1 + iκ ˆH − iκ ˆS ˆH ˆS − κ2 +2 +ˆH2 − κ2 +2 ( ˆS ˆH ˆS)2 + κ ˆH ˆS ˆH ˆS +� +(B9c) +Since the state |ψ⟩ is (anti)-symmetric, for any operators ˆG, we have +� +ˆS ˆG +� += ± +� +ˆG +� += +� +ˆG ˆS +� +, which allows some +simplifications. += 1 − κ2 +2 +� � +ˆH2� ++ +� +( ˆS ˆH ˆS)2� +− +� +ˆH ˆS ˆH ˆS +� +− +� +ˆS ˆH ˆS ˆH +� � +(B9d) += 1 − κ2 +2 +� +( ˆH − ˆS ˆH ˆS)2� +(B9e) += 1 − κ2 +2 ∆( ˆH − ˆS ˆH ˆS)2 +(B9f) +Since thanks to the symmetry of |ψ⟩, +� +ˆH − ˆS ˆH ˆS +� += +� +ˆH − ˆH ˆS2� += 0 + +15 +By defining ˆG = ˆH − ˆS ˆH ˆS it remains to compute the FI: +FI(κ = 0) = 1 +Pc +�∂Pc +∂κ +�2 ++ 1 +Pa +�∂Pa +∂κ +�2 +(B10a) += +1 +4Pc +� +κ∆( ˆG)2�2 ++ +1 +4Pa +� +κ∆( ˆG)2�2 +(B10b) += κ2∆( ˆG)4 +4 +� 1 +Pc ++ 1 +Pa +� +(B10c) += κ2∆( ˆG)4 +4 +Pa + Pc +PcPa +(B10d) += κ2∆( ˆG)4 +4 +4 +� +1 + ⟨ψ| ˆV (κ) |ψ⟩ +� � +1 − ⟨ψ| ˆV (κ) |ψ⟩ +� +(B10e) += κ2∆( ˆG)4 +1 +1 − ⟨ψ| ˆV (κ) |ψ⟩2 +(B10f) += κ2∆( ˆG)4 +1 +κ2∆( ˆG)2 +(B10g) += ∆( ˆG)2 +(B10h) +It is interesting to note that the computation of the Fisher information is singular. Indeed for the HOM interfer- +ometer around κ = 0 the derivative of the probabilities vanishes ∂κPc,a = 0, while one of the two probability (Pc +if the state is symmetric or Pa if its anti-symmetric) is also equal to zero. We thus obtain here the FI at zero by +computing it at κ ̸= 0 and taking the limit. As a result we see that the FI is proportional to the second derivative +of the coincidence probability. This means that for such a measurement what is important is the curvature of the +probability peak/dip. +Appendix C: Appendix: Details on the computation of the various variances +To compute explicitly the various variances of this paper on the two states |ψG⟩ and |ψG⟩ one can note that +since these states are separable in the variables ω±, if we consider two operators ˆH+ and ˆH− which are respectively +functions of ˆω+ and ˆt+ or ˆω− and ˆt− we have: +� +ˆH+ ˆH− +� += +� +ˆH+ +� � +ˆH− +� +. Where for a fixed state |ψ⟩, +� +ˆH +� += ⟨ψ| ˆH |ψ⟩. +In order to compute any variance, one only has to compute some expectation values. By expanding and using +the independence property from above, one only need to compute as building block expectation value of the form: +� +ˆωk +±ˆtl +± +� +. Indeed we can use the commutation relation to reorder any product such that the frequency operators are +on the left of the time operators. One has to pay attention that due to the choice of normalisation in the definition of +ˆω± = ˆω1 ± ˆω2 and ˆt± = ˆt1 ± ˆt2 we have [ˆω±, ˆt±] = 2i. Such expectation values can be obtained systematically using +a software (here we used Mathematica), we have the following values: + +16 +Operator +Variable + +Variable − for |ψG⟩ +Variable − for |ψC⟩ +ˆω +ωp +0 +0 +ˆω2 +ω2 +p + σ2 ++ +σ2 +− +σ2 +− + 1 +4∆2 +ˆω3 +3σ2 ++ωp + ω3 +p +0 +0 +ˆω4 +3σ4 ++ + 6σ2 ++ω2 +p + ω4 +p +3σ4 +− +3σ4 +− + 3 +2σ2 +−∆2 + +1 +16∆4 +ˆt +0 +0 +0 +ˆt2 +1 +σ2 ++ +1 +σ2 +− +1 +σ2 +− +ˆt3 +0 +0 +0 +ˆt4 +3 +σ4 ++ +3 +σ4 +− +3 +σ4 +− +ˆωˆt +i +i +i +ˆω2ˆt +2iωp +0 +0 +ˆωˆt2 +ωp +σ2 ++ +0 +0 +ˆω2ˆt2 +ω2 +p +σ2 ++ − 1 +−1 +∆2 +4σ2 +− − 1 +TABLE III: Expectation values of the various product of plus and minus operators on the states |ψG⟩ and |ψC⟩. + diff --git a/0NFKT4oBgHgl3EQfNi2Q/content/tmp_files/load_file.txt b/0NFKT4oBgHgl3EQfNi2Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..057756c5f0e7486b93b380216232820f7dffc00f --- /dev/null +++ b/0NFKT4oBgHgl3EQfNi2Q/content/tmp_files/load_file.txt @@ -0,0 +1,1153 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf,len=1152 +page_content='Time-frequency metrology with two single-photon states: phase space picture and the Hong-Ou-Mandel interferometer ´Eloi Descamps1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Arne Keller2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' P´erola Milman2 1D´epartement de Physique de l’´Ecole Normale Sup´erieure - PSL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 45 rue d’Ulm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 75005 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' France 2Universit´ee Paris Cit´e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' CNRS,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Laboratoire Mat´eriaux et Ph´enom`enes Quantiques,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 75013 Paris,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' France and 3D´epartement de Physique,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Universit´e Paris-Saclay,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 91405 Orsay Cedex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' France (Dated: January 30,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 2023) We use time-frequency continuous variables as the standard framework to describe states of light in the subspace of individual photons occupying distinguishable auxiliary modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We adapt to this setting the interplay between metrological properties and the phase space picture already extensively studied for quadrature variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We also discuss in details the Hong-Ou-Mandel interferometer, which was previously shown to saturate precision limits, and provide a general formula for the co- incidence probability of a generalized version of this experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' From the obtained expression, we systematically analyze the optimality of this measurement setting for arbitrary unitary transforma- tions applied to each one of the input photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' As concrete examples, we discuss transformations which can be represented as translations and rotations in time-frequency phase space for some specific states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' PACS numbers: I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' INTRODUCTION Much has been discovered since the first proposals to use quantum systems in metrology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' From the role of entanglement [1–4] to the one of modes, for pure and noisy systems and measurements, several main results have been established, and the most important one is the fact that quantum mechanical protocols can provide a better scaling in precision with the number of probes than classical ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Nevertheless, much still remains to be done, in particular concerning the application and the adaptation of such results to specific physical configura- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Of practical importance, for instance, is the issue of finding measurement strategies that lead to the opti- mal calculated limits, and this is far from being obvious for general states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Another relevant problem concerns adapting the general principles to physical constraints, as energy or temperature limits and thresholds [5, 6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Those are the main issues of this paper: in one hand, we deeply study the conditions for optimality of a specific measurement set-up and on the other hand, we consider a specific physical system, consisting of individual photons, for measuring time and frequency related parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In order to measure a given parameter κ one performs an experiment producing different outcomes x with asso- ciated probabilities Pκ(x) and build an unbiased estima- tor K such that κ = ⟨K⟩κ is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Here the index κ means that we take the average for the probability distri- bution Pκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The Cram´er-Rao bound (CRB) [7] imposes a limit on the precision of parameter estimation: δκ ≥ 1 √ NF , (1) where, δκ is the standard deviation in the estimation of κ: δκ = � Varκ(K), N is the number of independent measurements which were performed to estimate κ and F is the quantity known as the Fisher information (FI), defined by : F = � dx 1 Pκ(x) � ∂Pκ(x) ∂κ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In a quantum setting, one can use as a probe a quan- tum state |ψ⟩ which can evolve under the action of an operator ˆU(κ) = e−iκ ˆ H generated by an Hamilto- nian ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' By optimizing the precision over all possible quantum measurements of a parameter κ, one obtains a bound, called the quantum Cram´er-Rao bound (QCRB) [8] which reads: δκ ≥ 1 √NQ, (2) where Q is a quantity known as the quantum Fisher information (QFI) which for pure states and uni- tary evolutions (as the ones considered in the present paper), is equal to Q = 4(∆ ˆH)2, with (∆ ˆH)2 = ⟨ψ(κ)| ˆH2 |ψ(κ)⟩ − ⟨ψ(κ)| ˆH |ψ(κ)⟩2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The FI indicates the precision of a given measurement, whereas the QFI is the maximum precision obtainable with any measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For a given setting, we can thus compute both quantities (FI and QFI) to have an idea if the measurement is optimal (QFI=FI) or not (QFI>FI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Determining the QFI is a mathematical task much easier than finding a physical experimental set-up that reaches it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In quantum optical systems, several propos- als and implementations exist where the QFI is indeed achieved [4, 9–11], and one example where this is possi- ble is the Hong-Ou-Mandel (HOM) experiment [12–15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this experiment, one focus on simple physical systems composed of two photons occupying distinguishable spa- tial modes with a given spectral distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This state is a particular example of a state defined in the single photon subspace (where each mode is populated by at most one photon), in which a general pure state that can arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='11755v1 [quant-ph] 27 Jan 2023 2 be expanded as: |ψ⟩ = � dω1 · · · dωnF(ω1, · · · , ωn) |ω1, · · · , ωn⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (3) In this formula, the indexes 1,2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='.n, label different aux- iliary degrees of freedom (as for instance polarization or the propagation direction).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The state |ω1, · · · , ωn⟩ is a pure state where each photon propagating in the mode α is exactly at the frequency ωα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The spectral function F also known as the joint spectral amplitude (JSA) is normalized to one: � |F(ω1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=', ωn)|2dω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='dωn = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this setting one can introduce time and frequency operators for each mode α: ˆωα and ˆtα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' They correspond respectively to the generators of time and frequency shifts of the photon in the mode labeled by α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' An important property of these operators is that, in the considered single photon subspace they satisfy the commutation relation [ˆωα, ˆtβ] = iδα,β analogous to the one observed for the quadrature operators ˆXα and ˆPα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Notice that we are using throughout this paper dimen- sionless operators, which are relative to particular time and frequency scales of the associated implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For a more complete description of the time frequency continuous variables one can refer to Appendix A and to [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Previous works on quantum metrology using the electromagnetic field quadratures or particles’ posi- tion and momentum have shown how the phase space (x1, · · · , xn, p1, · · · , pn) can provide not only insight but also an elegant geometrical picture of the measurement precision [16–18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Indeed the QFI can also be defined in terms of the Bures distance [19] s(|ψ(κ)⟩ , |ψ(κ + dκ)⟩): Q = 4( s(|ψ(κ)⟩,|ψ(κ+dκ)⟩) dκ )2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In the case of pure states, this distance is simply expressed in terms of the overlap s(|ψ⟩ , |φ⟩) = � 2(1 − |⟨φ|ψ⟩|).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Since the overlap of two states can be computed as the overlap of their respective Wigner function, one can interpret the QFI as a measure of how much the Wigner function must be shifted so as it becomes orthogonal to the initial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' A consequence of this is that the maximum precision of a measurement can be seen geometrically on the Wigner function, by looking at their typical size of variation in the direction of an evo- lution [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Since in the case of single photon states one can also define a time-frequency phase space associated to the variables (τ1, · · · , τn, ϕ1, · · · , ϕn), it is natural to investigate wether the same type of interpretation makes sense in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The present paper purposes are thus twofold: in the first place, we provide general conditions for the HOM to saturate precision limits using time-frequency (TF) vari- ables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For such, we consider arbitrary evolution operators acting on TF variables of single photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In second place, we provide a phase-space picture and interpretation of the QFI for this type of system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Indeed, as shown in [20], there is an analogy between the quadrature phase space and the TF phase space from which metrological proper- ties of time and frequency states can be inferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Never- theless, in the present case, photons have both spectral classical wave-like properties and quantum particle-like ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Interpreting from a quantum perspective both the role of the spectral distribution and of collective quantum properties as entanglement in the single photon subspace has shown to demand taking a different perspective on the TF phase space [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Having this in mind, we in- vestigate how relevant examples of evolution operators, taken from the universal set of continuous variables quan- tum gates, can be implemented and represented in phase space, as well as the precision reached when one measures them using the HOM experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We’ll concentrate on single-mode Gaussian operations, analogously to what was done in [5], even though we provide a general for- mula for any transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This paper is organized as follows: In Section II we provide a description of the TF phase space and introduce the states we’ll discuss in details as well as their representation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In Section III we discuss the HOM experiment and the conditions for it to reach optimal precision limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Finally, in Sections IV and V we discuss two different Gaussian operations in phase space as well as their implementation and the associated precision reached in the HOM experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' TIME FREQUENCY PHASE SPACE We consider pure two-photon states which can be writ- ten in the form: |ψ⟩ = � dω1dω2F(ω1, ω2) |ω1, ω2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The Wigner function in variables (τ1, τ2, ϕ1, ϕ2) of such states can be defined as W|ψ⟩(τ1, τ2, ϕ1, ϕ2) = � dω1dω2e2i(ω1τ1+ω2τ2)F(ϕ1 + ω1, ϕ2 + ω2)F ∗(ϕ1 − ω1, ϕ2 − ω2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (4) Evolutions generated by ˆωα and ˆtα (α = 1, 2) correspond to translations in phase space: We−iˆ ω1κ|ψ⟩(τ1, τ2, ϕ1, ϕ2) = W|ψ⟩(τ1 − κ, τ2, ϕ1, ϕ2), (5a) We−iˆt1κ|ψ⟩(τ1, τ2, ϕ1, ϕ2) = W|ψ⟩(τ1, τ2, ϕ1 − κ, ϕ2), (5b) and analogously for ˆω2 and ˆt2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 3 Using the QFI formulation based on the Bures distance, we can safely state that the precision of a measurement device is related to its capability of distinguishing between an initial state |ψ(κ)⟩ and a state |ψ(κ + dκ)⟩ that has evolved according to a parameter κ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This precision is then directly related to how small the parameter dκ should be such that these two states can be distinguished i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' the overlap |⟨ψ(κ)|ψ(κ + dκ)⟩| gets close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This can be also elegantly interpreted using the overlap of the two states’s respective Wigner functions, that describe trajectories in the phase space that are governed by the interaction Hamiltonian and the parameter dκ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' To gain some familiarity with the studied problem we start with the case of a single-photon state |ψ⟩ = � dωS(ω) |ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Although using this type of state is not current in metrology, this simpler case can be seen as a building block and will help understanding the role of the spectrum in the present configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For a single photon, the Wigner function is defined as: W(τ, ϕ) = � dωe2iωτS(ϕ + ω)S∗(ϕ − ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In the case of a Gaussian state |ψG⟩ with spectral wave function SG(ω) = e − ω2 4σ2 (2πσ2)1/4 its Wigner function is also Gaussian: WG(τ, ϕ) = exp � −2σ2τ 2 − ϕ2 2σ2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' It is characterized by its width in the orthogonal directions τ and ϕ: 1/2σ and σ respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' An evolution generated by ˆω corresponds to a trans- lation in the direction τ in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The associated measurement precision is given by the smallest value of dκ such that the initial Wigner function is almost orthog- onal to the translated one in the corresponding direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Since the width of the Wigner function in the direction of evolution is proportional to 1/σ, we have dκ ∼ 1/σ leading to a QFI of the order of Q ∼ σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Alternatively if one considers the generator ˆt, the associated width of the state will be σ leading to a QFI of the order of Q ∼ 1/σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We thus remark that the estimated QFI depends on the width of the state in phase space in the direction of evo- lution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We notice as well the similarities and differences with the quadrature phase space case: even though the relation between the phase space geometrical properties and metrological interest are common to both variables, in the case of quadrature they are related to some abso- lute quantum resource dependent quantity, the number of photons of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In the present case, the single photon spectrum is a classical resource and its width can only set a relative size scale in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' It is interesting to notice that this type of interpreta- tion is also possible for classical fields, as studied in [22– 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this classical context, the electromagnetic field amplitude replaces the function F and one can also re- late spectral metrological properties to the phase space structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Nevertheless, as discussed in [21], this picture is merely associated to classical metrological properties of single mode fields (their spectrum) and no interesting scaling can be observed in this context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' As a matter of fact, the classical single mode field and the single photon phase space can be mapped into one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In the present paper, the multi-modal character of the quantum field is an essential ingredient for the discussion of the quantum metrological advantage, since it is a con- sequence of the multi-photon state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We will see in par- ticular how these two features (spectral and particle-like) of the considered single photon subspace are combined in the QFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The situation is different and richer for bi-photon states, since the phase space is of dimension 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' One can thus imagine different directions of translation as for instance the ones generated by operators ˆω1, ˆω2, ˆω1 − ˆω2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Then, optimizing the measurement precision involves, for a given spectral distribution, choosing a di- rection of evolution for which the Wigner function of the state has the smallest scale structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This direction, as we’ll see, will depend on the number of photons, and can display a non-classical scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' THE HOM AS A MEASUREMENT DEVICE A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The setup In the setup proposed by Hong, Ou and Mandel [25] two photons impinge into a balanced beam splitter (BS), each one of them from a different port, as represented on figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' By measuring the output of the beam-splitter using single-photon detectors we can compute the prob- ability of obtaining coincidences (when the two photons exit the BS by different paths) or anti-coincidences (when they bunch and exit the BS at the same path).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 1: Schematic representation of HOM experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Since its original proposal and implementation, many modifications and adaptations were made to the HOM set-up, which was shown to be very versatile to reveal different aspects of quantum optics using two-photon in- terference [26]: it can be used to witness particle [27] and A BS B4 spectral [28] entanglement, to saturate precision bounds on time delay measurements[12, 13] or to directly mea- sure the Wigner function of the incoming state [29, 30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We’re interested in quantum metrological tasks, so we’ll start by discussing the results obtained in [12], where the authors provided experimental evidence that the HOM device can saturate precision limits on time measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' To achieve this result, the authors con- sidered the initial state: |ψU⟩ = 1 √ 2 � dΩf(Ω) � ��ω0 1 + Ω, ω0 2 − Ω � − ��ω0 2 + Ω, ω0 1 − Ω � � , (6) where ω0 1 and ω0 2 are the central frequencies of the pho- tons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Due to the energy conservation and to the phase- matching conditions, the support of the JSA associated to (6) is the line ω1 + ω2 = 0 in the plane (ω1, ω2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' It is anti-diagonal in the plane (ω1, ω2) and infinitely thin along the diagonal direction ω− = ω1 − ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Adding a delay in the arm 1 of the HOM interferometer corre- sponds to an evolution generated by the operator ˆω1, corresponding to a translation κ in the τ1 direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The QFI is simply calculated as: Q = 4∆(ˆω1)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' After the beam-splitter, the measurement can lead to two out- comes: coincidence or anti-coincidence, with probability Pc and Pa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The FI is thus expressed as: F = 1 Pc � ∂Pc ∂κ �2 + 1 Pa � ∂Pa ∂κ �2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The authors of [12] thus showed that using the input state (6) in the HOM inter- ferometer, the two quantities F and Q are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In [13] the HOM interferometer was also used and shown to lead to the QFI in a two-parameter estimation experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Finally, in [14] biphoton states were classi- fied as metrological resources according to their spectral width, still in the situation where the HOM experiment is used as a measurement apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Generalization: the HOM as an optimal measurement device for quantum metrology with biphotons We now make a general description of the HOM ex- periment as a parameter estimation device and try to understand and determine when it corresponds to an op- timal measurement strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In [13], the authors tackle a part of this problem by studying the HOM as a measure- ment apparatus for two parameter estimation by estab- lishing conditions on frequency correlation states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this reference, the authors restrict themselves to time delay evolutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In the present paper, we are interested in studying any evolution that can be described by a two photon unitary |ψ(κ)⟩ = ˆU(κ) |ψ⟩ = e−i ˆ Hκ |ψ⟩ (see figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We will see that under a symmetry assumption on the JSA of the state, it is possible to obtain an explicit formula for the FI, and this formula can be used to compute at a glance if the measurement setup considered is optimal or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 2: HOM setup where we apply a general gate ˆU before the BS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For any input state |ψ⟩, the QFI will then be expressed as: Q = 4∆( ˆH)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (7) On the other hand, one can show that the coincidence probability is: Pc = 1 2(1 − ⟨ψ| ˆU † ˆS ˆU |ψ⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (8) (see Appendix B) where we introduced the hermitian swap operators ˆS whose action on the states is given by ˆS |ω1, ω2⟩ = |ω2, ω1⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Furthermore we can compute the associated FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' If the state |ψ⟩ is symmetric or anti- symmetric (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' ˆS |ψ⟩ = ± |ψ⟩) the FI at κ = 0 it is given by: F = ∆( ˆH − ˆS ˆH ˆS)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (9) (see Appendix B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This means that under the symme- try assumption on the JSA, comparing the QFI and the FI is done simply by comparing the variance of two different operators, mainly: 2 ˆH and ˆH − ˆS ˆH ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Equation (9) implies that if [ ˆH, ˆS] = 0, then F = 0 and no information can be obtained about κ from the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' However, if { ˆH, ˆS} = 0 then F = Q since ˆS ˆH ˆS = − ˆS2 ˆH = − ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this last case, the measurement strategy is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In [31], general con- ditions for reaching the QFI were also obtained in the context of amplitude correlation measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' These conditions are based on a quantum state’s symmetry under (unphysical) path exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The previous calculations form a simple tool that can be applied to different evolution Hamiltonians ˆH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We’ll now discuss examples taken from the universal set of quantum gates in continuous variables: translations (gen- erated by operator ˆωα’s) and rotations (generated by ˆH = (ˆω2 +ˆt2)/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' These gates have already been studied in [5] in the case of quadrature or position and momen- tum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In the present physical configuration, they corre- spond to the free evolution of single photons in free space A U 2 BS B5 (translations) or in a dispersive medium, as for instance an optical fiber combined to time lenses (rotation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' TIME-FREQUENCY PHASE-SPACE TRANSLATIONS A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Different types of translations Since we’re considering two-photon states, translations can be represented by any linear combination of the cor- responding operators, that is : ˆH = αˆω1+βˆω2+γˆt1+δˆt2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' To illustrate our results we choose to focus on the four operators ˆω1, ˆω2 and ˆω± = ˆω1 ± ˆω2, since they are the most easily implemented in HOM experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Notice that ˆω± are collective operators acting in both input photons while ˆω1,2 act in a single photon only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' If we consider a state which is (anti-)symmetric and separable in the variables ω± = ω1 ± ω2, we can write: |ψ⟩ = 1 √ 2 � dω+dω−f(ω+)g(ω−) ���� ω+ + ω− 2 , ω+ − ω− 2 � , (10) with g satisfying g(−ω) = ±g(ω) and the functions g and f being normalized to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The specific form of each function is related to the phase-matching conditions and the energy conservation of the two-photon generation process and this type of state can be experimentally pro- duced in many set-ups [32, 33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Using equations (7) and (9) we can compute the QFI and FI associated to each type of evolution: For ˆH = ˆω1, we get Q = ∆(2ˆω1)2 = ∆(ˆω++ˆω−)2 = ∆(ˆω−)2 + ∆(ˆω+)2, while F = ∆(ˆω−)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Thus this situation is optimal only if ∆(ˆω+)2 = 0, which was the case for the state |ψU⟩ of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (6) used in [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We obtain the same type of result for ˆω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For ˆH = ˆω+, Q = 4∆(ˆω+)2, while F = ∆(ˆω+ − ˆω+)2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this situation the precision of the measurement is zero, and the reason for that is that variables ω+ cannot be measured using the HOM experiment (we notice that [ˆω+, ˆS] = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For ˆH = ˆω−, we get Q = 4∆(ˆω−)2, while F = ∆(ˆω− + ˆω−)2 = 4∆(ˆω−)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This time we have F = Q, which means that the measurement is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this case, we have that {ˆω−, ˆS} = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We now illustrate these general expressions and inter- pret them using different quantum states and their phase space representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Example: Gaussian and Schr¨odinger cat-like state To illustrate our point we discuss as an example two states |ψG⟩ and |ψC⟩ that can be expressed in the form of equation (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For |ψG⟩, f and g are Gaussians: fG(ω+) = e − (ω+−ωp)2 4σ2 + (2πσ2 +)1/4 gG(ω−) = e − ω2 − 4σ2 − (2πσ2 −)1/4 , (11) where σ± is the width of the corresponding function and ωp is a constant, which is also the photon’s central fre- quency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' As for state |ψC⟩, it can be seen as the general- ization of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We consider f to be Gaussian and g to be the sum of two Gaussians: fC(ω+) = fG(ω+) gC(ω−) = 1 √ 2 � gG(ω− + ∆/2) − gG(ω− − ∆/2) � , (12) where ∆ is the distance between the two Gaussian peaks of gC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We assume that the two peaks are well separated: ∆ ≫ σ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Consequently, gC is approximately normal- ized to one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We can verify that with these definitions the function gG is even while gC is odd by exchange of variables ω1 and ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We first compute the variances for both states (table I) and then apply the formula (7) and (9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' State |ψG⟩ |ψC⟩ ∆(ˆω1)2 or ∆(ˆω2)2 1 4σ2 + + 1 4σ2 − 1 16∆2 + 1 4σ2 + + 1 4σ2 − (∆ˆω+)2 σ2 + σ2 + (∆ˆω−)2 σ2 − 1 4∆2 + σ2 − TABLE I: Variance of various time translation operators for states |ψG⟩ and |ψC⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' See Appendix C for details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' So for the case of an evolution generated by ˆω1, for |ψG⟩ we obtain: Q = σ2 + + σ2 − F = σ2 −, (13) while for |ψC⟩ we have: Q = 1 4∆2 + σ2 + + σ2 − F = 1 4∆2 + σ2 −.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (14) We thus see that time precision using the HOM measure- ment and the quantum state evolution generated by ˆω1 is optimal only if the parameter σ+ is negligible compared to ∆ or σ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This is exactly the case for the state (6) where σ+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In addition, we see that there is a difference between the QFI associated to |ψC⟩ and |ψG⟩ involving the param- eter ∆.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This difference can be interpreted, as discussed in [14], as a spectral effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this reference, the spectral width is considered as a resource, and for a same spectral width state |ψC⟩ has a larger variance than state |ψG⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Nevertheless, as discussed in [21], this effect has a clas- sical spectral engineering origin and choosing to use one rather than the other depends on the experimentalists constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 6 (a) Projection on the plane τ−, ω− (b) Projection on the plane τ1, ω1 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 3: Wigner function of the cat-like state |ψC⟩ projected in different variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Interpretation of translations in the time-frequency phase space We now discuss the dependency of precision on the direction of translation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For such, we can consider the Wigner function associated to a JSA which is separable in the ω± variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Its Wigner function will also be separable on these variables: W(τ1, τ2, ϕ1, ϕ2) = W+(τ+, ϕ+)W−(τ−, ϕ−), (15) where the phase space variables τ± and ϕ± are defined as: ϕ± = ϕ1±ϕ2 2 and τ± = τ1 ± τ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Even though the Wigner function W+ (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' W−) can be associated to the one of a single variable (ω+ (ω−)) and spectral wave function f (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' g), it displays some differences with the single photon one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This fact is well illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For state |ψC⟩, according to (15) the projection of the Wigner function W− in the plane τ−, φ− of the phase space can be represented as show in Figure 3 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We see that it is composed of two basic shapes: two Gaussian peaks and an oscillation pattern in between.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Figure 3 (b) represents another way to project this very same Wigner function onto the plane τ1, φ1 of the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' One can observe that in this case the distance between the peaks is larger than in the previous representation by a factor of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' As precision is directly related to the size of the Wigner function structures in phase space, we observe that the interference fringes are closer apart in the phase space associated to the minus variable than in the one associated to mode 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Thus, the precision in parameter estimation will be better using ˆω− as the generator of the evolution than when using ˆω1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This phase space based observations explain well the result of the computation of the QFI: 4∆(ˆω1)2 = ∆(ˆω−)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (16) with the assumption that σ+ ≪ ∆, σ−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The reason for the appearance of a factor 2 difference in fringe spacing for the Wigner function associated to variable ω− is the fact that it is a collective variable, and translations in the phase space associated to these variables are associated to collective operators, acting on both input photons (instead of a single one, as is the case of translations generated by operator ˆω1, for instance).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Thus, one can observe, depending on the biphoton quan- tum state (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=', for some types of frequency entangled states), a scaling depending on the number of particles (in this case, two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' As analyzed in [21] for general single pho- ton states composed of n individual photons, we have for frequency separable states a scaling corresponding to the shot-noise one (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=', proportional to √n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' A Heisenberg- like scaling (proportional to n) can be achieved for non- physical maximally frequency correlated states, and con- sidering a physical non-singular spectrum leads to a non- classical scaling in between the shot-noise and the Heisen- berg limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Experimentally, such collective translation can be im- plemented by adding a delay of τ in arm 1 and of −τ in arm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Notice that this situation is different from cre- ating a delay of 2τ in only one arm, even though both situations lead to the same experimental results in the particular context of the HOM experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' TIME-FREQUENCY PHASE SPACE ROTATIONS We now move to the discussion of the phase space ro- tations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For this, we’ll start by providing some intuition by discussing in first place the single photon (or single mode) situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this case, time-frequency phase space rotations are generated by the operators ˆR = 1 2(ˆω2 + ˆt2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' As previously mentioned, we consider here dimension- less observables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Physically, time-frequency phase space rotations correspond to performing a fractional Fourier transform of the JSA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' While for transverse variable of single photons the free propagation or a combination of lenses can be used for implementing this type of oper- ation [34, 35], in the case of time and frequency this transformation corresponds to the free propagation in a dispersive medium [36–40] combined to temporal lenses [41–43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Single mode rotations In this Section, we compute the QFI associated to a rotation ˆR for a single photon, single mode state using the variance of this operator for different states |ψ⟩ = � dωS(ω) |ω⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' As for the translation, this simpler configuration is used as a tool to better understand the two photon case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='5 10 5 0 5 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 T 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='5 5 10 0 5 107 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Gaussian state: We start by discussing a single-photon Gaussian state at central frequency ω0 and spectral width σ: |ψG(ω0)⟩ = 1 (2πσ2)1/4 � dωe− (ω−ω0)2 4σ2 |ω⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (17) For this state, we have that: ∆( ˆR)2 = σ2ω2 0 + 1 8 � 1 4σ4 + 4σ4 − 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (18) Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (18) has two types of contributions that we can in- terpret: The first term σ2ω2 0 corresponds to the distance in phase space (ω0) of the center of the distribution, to the origin of the phase space (ω = 0, τ = 0), times the width of the state σ in the direction of rotation (see Figure 4 (a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This term is quite intuitive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The Wigner function of a state which is rotated by an angle θ = 1/2σω0 has an overlap with the Wigner function of the initial one which is close to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The term 1 4σ4 + 4σ4 − 2 reaches 0 as a minimum when σ = 1 √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For this value the Wigner function is perfectly rotationally symmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Its meaning can be intuitively understood if we consider that ω0 = 0, so that this term becomes the only con- tribution to the variance(see Figure 4 (b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this case, we are implementing a rotation around the center of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' If the state is fully symmetric then this rotation has no effect, and the variance is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Only in the case where the distributions ro- tational symmetry is broken we obtain a non zero contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Schr¨odinger cat-like state centered at the origin (ω = 0): We now consider the superposition of two Gaussian states: ��ψ0 C � = 1 √ 2(|ψG(∆/2)⟩ − |ψG(−∆/2)⟩).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (19) This state is of course non physical as a single-photon state, since it contains negative frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' However, since it can be be well defined using collective variables (as for instance ω−) for a two or more photons state, we still discuss it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Assuming that the two peaks are well separated (∆ ≫ σ), we can ignore the terms proportional to e− ∆2 8σ2 , and this leads to: ∆( ˆR)2 = 1 8 � 1 4σ4 + 4σ4 − 2 � + 1 4∆2σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (20) We see that there is no clear metrological advantage when using this state compared to the Gaussian state: the quantity ∆/2 plays the same role as ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This can be understood geometrically once again, with the help of the Wigner function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We see in Figure 4 (c) how the considered state evolves under a rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this situation the interference fringes are rotated around their center so even though they display a small scale structure, they are moved only by a small amount, resulting in a non significant precision improvement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Schr¨odinger cat-like state centered at any frequency: We can now discuss the state formed by the superpo- sition of two Gaussian states whose peaks are at frequen- cies ω0 − ∆/2 and ω0 + ∆/2, and with the same spectral width σ as previously considered: |ψC⟩ = 1 √ 2 � |ψG(ω0 + ∆/2)⟩ − |ψG(ω0 − ∆/2)⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (21) Still under the assumption of a large separation between the two central frequencies (∆ ≫ σ), we obtain: ∆( ˆR)2 = 1 8 � 1 4σ4 + 4σ4 − 2 � + 1 4∆2(σ2 + ω2 0) + σ2ω2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (22) We can notice that by setting ω0 = 0 we recover the variance corresponding to the same state rotated around its center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Nevertheless, in the present case ω0 ̸= 0, and we have two additional terms: σ2ω2 0 and ∆2ω2 0/4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Both terms can be interpreted as a product of the state’s dis- tance to the origin and its structure in phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' How- ever, while the first one is simply the one corresponding to the Gaussian state, the second one is a product of the states’ distance to the origin and its small structures in phase space, created by the interference between the two Gaussian states (see Figure 4 (d)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The interference pat- tern is thus rotated by an angle θ corresponding to an arc of length ω0θ, and since the distance between the fringes is of order ∆, if θ ∼ 1/ω0∆ (corresponding to the term ∆2ω2 0/4 in the expression of the variance) the rotated state is close to orthogonal to the initial one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In all this section, we have considered rotations about the time and frequency origin of the phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Never- theless, it is of course possible to displace this origin and consider instead rotations about different points of the TF phase space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this case, for a rotation around an arbitrary point τ0 and ϕ0, the generator would be given by (ˆω − ϕ0)2/2 + (ˆt − τ0)2/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Different types of rotations We now move to the case of two single photons (biphoton states).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' As for the case of translations, there are many possible variables and can consider rotations in different planes of the phase space: ˆR1, ˆR2, ˆR±, 8 (a) Gaussian state centered at ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For θω0 ∼ 1/2σ the initial state and the rotate one are distinguishable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (b) Gaussian state centered at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The rotated state will be distinguishable from the initial one only in the absence of rotational symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (c) Superposition of two Gaussian states (cat-like state) centered the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The small structures of the fringes do not play a relevant role since they are only moved by a small distance under rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (d) Superposition of two Gaussian states (cat-like state) centered at ω0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The fringes play an important role, since with θω0 ∼ 1/∆, the two states are nearly orthogonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 4: Schematic representation of the Wigner function of various states under rotation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The ellipses represent the typical width of Gaussians.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The doted lines represent the rotated states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' ˆR1 ± ˆR2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' where ˆR1 = 1 2(ˆω2 1 + ˆt2 1) (and similarly for ˆR2) and ˆR± = 1 4(ˆω2 ± + ˆt2 ±) (recall that ˆω± = ˆω1 ± ˆω2 and ˆt± = ˆt1 ± ˆt2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For all these operators we can as before apply the general formula for the QFI and of the FI to the corresponding HOM measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The results are displayed in table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Operator QFI FI ˆR1 4∆( ˆR1)2 ∆( ˆR1 − ˆR2)2 ˆR± 4∆( ˆR±)2 0 ˆR1 + ˆR2 4∆( ˆR1 + ˆR2)2 0 ˆR1 − ˆR2 4∆( ˆR1 − ˆR2)2 4∆( ˆR1 − ˆR2)2 TABLE II: QFI and FI of various rotation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We see that the only two situations where the HOM can indeed be useful as a measurement device for metro- logical applications are ˆR1 and ˆR1 − ˆR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The reason for that is the symmetry of ˆR± and ˆR1+ ˆR2, which commute with the swap operator ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' As for ˆR1, it corresponds to the rotation of only one of the photons and may not be the optimal strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Finally, ˆR1 − ˆR2 corresponds to the simultaneous rotation in opposite directions of both photons sent into the two different input spatial modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' As ˆR1 − ˆR2 anti-commutes with ˆS then we can affirm that the HOM measurement is optimal for this type of evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' QFI and FI computation with Gaussian and cat-like state We now compute the QFI and FI using the variance of ˆR1 and ˆR1 − ˆR2 calculated for states |ψG⟩ and |ψC⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For |ψG⟩: We have: ∆( ˆR1)2 = 1 32 �� 1 σ2 + + 1 σ2 − �2 + (σ2 + + σ2 −)2 − 8 � + 1 16ω2 p(σ2 + + σ2 −) ∆( ˆR1 − ˆR2) = 1 4 � 1 σ2 +σ2 − + σ2 +σ2 − − 2 � + 1 4σ2 −ω2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (23) For |ψC⟩: We have: ∆( ˆR1)2 = 1 32 �� 1 σ2 + + 1 σ2 − �2 + (σ2 + + σ2 −)2 − 8 � + 1 64(4ω2 p + ∆2)(σ2 + + σ2 −) + 1 64∆2ω2 p + ∆2 128 � 1 σ2 − + σ2 − � ∆( ˆR1 − ˆR2) = 1 4 � 1 σ2 +σ2 − + σ2 +σ2 − − 2 � + 1 4σ2 −ω2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (24) We notice that for both states 4∆( ˆR1)2 ≥ ∆( ˆR1− ˆR2)2, meaning that the measurement of a rotation imple- mented in only one mode using the HOM is not an optimal measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Experimentally realizing an evolution generated by ˆR1 is easier than implementing the one associated to ˆR1− ˆR2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Furthermore we see that for the Gaussian state |ψG⟩ a dominant term is ω2 pσ2 − which appears with the same factor in 4∆( ˆR1)2 and ∆( ˆR1 − ˆR2)2, meaning that one could perform a measurement which although not opti- mal would be pretty efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The same applies to the Schr¨odinger cat-like state |ψC⟩ where one dominant term is ∆2ω2 p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Phase space interpretation We now provide a geometrical interpretation of the previous results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' If we consider that σ− ≫ σ+ in T 0 6 1 2g 03T 个 6 1 2g7 0T 039 the case of a Gaussian state or ∆ ≫ σ+ in the case of a Schr¨odinger cat-like state, the projection of the Wigner function on the plane corresponding to collective minus variables (τ−, φ−) is the one presenting a relevant phase space structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Thus, it would be interesting to consider, as in the case of translations, that these states are manipulated using operators acting on modes associated to this collective variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' A na¨ıve guess would then trying to apply the rotation operator ˆR−.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' However it comes with many difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Indeed it first poses an experimental problem, since this rotation corresponds to a non-local action which would be very hard to implement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In addition, the HOM is not able to measure such evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Finally, it turns out that this is not the operator with the greatest QFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This fact can be understood by taking a more careful look at the Wigner function of the considered states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The Wigner function for separable states can be factorized as the product of two Wigner functions defined in variables plus and minus, and we have that W+ is the Wigner function of a Gaussian state centered at ωp (corresponding to the situation (a) in Figure (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' As for W−, it is either the Wigner function of a Gaussian state or the one associated to a superposition of two Gaussian states centered around zero (corresponding to the situation (b) and (c) in Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The QFI increases with the distance of the states to the rotation point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For this reason, states |ψG⟩ and |ψC⟩ under a rotation using ˆR−, do not lead to a high QFI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' A higher QFI is obtained using rotations around a point which is far away from the center of the state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this case, the QFI displays a term which is proportional to the distance from the center of rotation squared di- vided by the width of the state squared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Both terms ω2 pσ2 − and ∆2ω2 p which were dominant in the expression of the variance of ˆR1 and ˆR1 − ˆR2 can be interpreted as such.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This means that the rotation ˆR1, whose ac- tion is not easily seen in the variables plus and minus, can be interpreted as a rotation which moves W− around the distance ωp from the origin of the TF phase space (ω = 0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For both states then, the main numerical contribution to the QFI comes from a classical effect, related to the intrinsic resolution associated to the central (high) fre- quency of the field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In general, in phase space rotations, both in the quadrature and in the TF configuration, the distance from the phase space origin plays an important role.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' While in the quadrature configuration this distance has a physical meaning that can be associated both to the phase space structure and to the number of probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In the case of TF phase space, the distance from the origin and the phase space scaling are independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In partic- ular, the distance from the origin can be considered as a classical resource that plays no role on the scaling with the number of probes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' A discussion on scaling properties of rotations The different types of FT phase space rotations have different types of interpretation in terms of scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The combined rotations of the type ˆR1 ± ˆR2, for instance, can be generalized to an n photon set-up through oper- ators as ˆR = �n i αi ˆRi, with αi = ±1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this situation, we have that rotation operators are applied individually and independently to each one of the the n photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this case, we can expect, in first place, a collective (clas- sical) effect, coming simply from the fact that we have n probes (each photon).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In addition, it is possible to show that a Heisenberg-like scaling can be obtained by con- sidering states which are maximally mode entangled in a mode basis corresponding to the eigenfunctions of oper- ators ˆRi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Indeed, for each photon (the i-th one), we can define a mode basis such that ˆRi |φk⟩i = (k + 1/2) |φk⟩i, with |φk⟩i = 1 √ 2kk!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 1 π1/4 � dωe− ω2 2 Hk(ω) |ω⟩i with Hk(ω) being the k-th Hermite polynomial associated to the i- th photon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For a maximally entangled state in this mode basis, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' , a state of the type |φ⟩ = �∞ k=0 Ak �n i=1 |φk⟩i, (where we recall that the subscript i refers to each pho- ton and k to the rotation eigenvalues) the ˆR eigenvalues behave as random classical variables and we can show that the QFI scales as n2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' As for rotations of the type ˆR±, they cannot be de- composed as independently acting on each photon, but consist of entangling operators that can be treated ex- actly as ˆR1 and ˆR2 but using variables ω± = ω1 ± ω2 instead of ω1 and ω2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We can also compute the scal- ing of operators as ˆJ = � Ωβ ˆRΩβ where Ωβ = �n i αiωi, αi = ±1 and β is one of the 2n−1 ways to define a collec- tive variable using the coefficients αi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' For such, we can use the same techniques as in the previous paragraph but for the collective variables Ωβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Nevertheless, the exper- imental complexity of producing this type of evolution and the entangled states reaching the Heisenberg limit is such that we’ll omit this discussion here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' CONCLUSION We have extensively analyzed a quantum optical set- up, the HOM interferometer, in terms of its quantum metrological properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We provided a general formula for the coincidence probability of this experiment which led to a general formula for the associated FI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We used this formula to analyze different types of evolution and showed when it is possible to reach the QFI in this set- up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In particular, we made a clear difference between col- lective quantum effects that contribute to a better than classical precision scaling and classical only effects, asso- ciated to single mode spectral properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We then briefly discussed the general scaling properties of the QFI asso- ciated to the studied operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Our results provide a complete recipe to optimize the HOM experiment with metrological purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' They rely 10 on the symmetry properties of quantum states that are revealed by the HOM interferometer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' An interesting per- spective is to generalize this type of reasoning for differ- ent set-ups where different symmetries play a role on the measurement outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Acknowledgements The French gouvernement through the action France 2030 from Agence Nationale de la Recherche, reference “ANR-22-PETQ-0006” provided financial support to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We thanks Nicolas Fabre for fruitful discussions and comments on the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' [1] Daniel Braun, Gerardo Adesso, Fabio Benatti, Roberto Floreanini, Ugo Marzolino, Morgan W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Mitchell, and Stefano Pirandola.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1103/ RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='035006, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1103/RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='90.' 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+page_content=' Labont´e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Quantum-limited deter- mination of refractive index difference by means of en- tanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' npj Quantum Inf, (8):58, 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' URL: https://doi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='org/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1038/s41534-022-00567-7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' [38] Keisuke Goda, Daniel R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Solli, Kevin K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Tsia, and Bahram Jalali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Theory of amplified dispersive fourier transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' A, 80:043821, Oct 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' URL: https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1103/PhysRevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='043821, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1103/PhysRevA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='043821.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' [39] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Goda and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Jalali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Dispersive fourier transforma- 12 tion for fast continuous single-shot measurements.' metadata={'source': 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shaping of entangled photons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=', 94:073601, Feb 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' URL: https:// link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='073601, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1103/PhysRevLett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='94.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='073601.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' [44] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Fabre and N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Treps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Modes and states in quantum optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Mod.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=', 92(3):035005, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' URL: https://link.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='aps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='org/doi/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1103/ RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='035005, doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1103/RevModPhys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='92.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 035005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Appendix A: Time frequency formalism In quantum mechanics, light is described with the help of modes [44], representing the various physical properties a photon can have: frequency, position, spectral shape, wave vector, polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Mathematically we associate to each mode α a creation and annihilation operators ˆa† α and ˆaα which satisfy the familiar bosonic commutation relation [ˆaα, ˆa† β] = δα,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The quantum states are then obtained by acting with the creation operators on the vacuum |vac⟩, which can be interpreted as adding a photon in the corresponding mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In time frequency continuous variables we look at modes parameterized by the frequency [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We will thus adapt the terminology: for us a mode will correspond to all physical parameter needed to describe a photon excluding the frequency (position, wave vector, polarization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In the following we will look at interferometers, and thus the parameter α will describe in which arm the photon is propagating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We will thus describe single photon states in a given mode α with frequency ω with the help of a creation operator acting on the vacuum state: ˆa† α(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In this situation the commutation relation is written as [ˆaα(ω), ˆa† β(ω′)] = δ(ω − ω′)δα,β, (A1) the other commutation relations (between two creation or two annihilation operators) vanishing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' It’s useful to intro- duce the conjugated temporal variable t, by the use of the Fourier transform: ˆaα(t) = 1 √ 2π � dωˆaα(ω)e−iωt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (A2) We can verify that the creation and annihilation operators in the temporal domain verify the same commutation relation as the one in the spectral domain: [ˆaα(t), ˆa† β(t′)] = δ(t − t′)δα,β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' States in time-frequency variables The creation operators allow to define general single photon states on a single mode via: |ψ⟩ = � dωS(ω)ˆa†(ω) |vac⟩ = � dωS(ω) |ω⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (A3) The spectrum S(ω) is the Fourier transform of the time of arrival distribution and it can be recovered from the state S(ω) = ⟨ω|ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' If we are interested in a collection of n single photons states in n different modes, we can work with the state: |ψ⟩ = � dω1 · · · dωnF(ω1, · · · , ωn)ˆa† 1(ω1) · · · ˆa† n(ωn) |vac⟩ = � dω1 · · · dωnF(ω1, · · · , ωn) |ω1, · · · , ωn⟩ , (A4) where the spectral function F is normalised to one: � |F(ω1, ω2|2dω1dω2 = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' 13 b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Time-frequency operators We can introduce two very useful operators as follows: ˆtα = � dt tˆa† α(t)ˆaα(t) ˆωα = � dω ωˆa† α(ω)ˆaα(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (A5) The fundamental property of these operators is the fact that they verify the familiar commutation relation on the subspace of single photons: [ˆωα, ˆtα] = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (A6) More precisely, we have the general result: [ˆωα, ˆtα] = i ∞ � −∞ dωˆa† α(ω)ˆaα(ω) = i ˆNα, (A7) where the operator ˆNα count the number of photon operator in the mode α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The action of the both operators ˆω and ˆt can be computed on the JSA and we have: ˆω : S(ω) �→ ωS(ω) ˆt : S(ω) �→ −i∂ωS(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (A8) Appendix B: Appendix: Derivation of equations (8) and (9) a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Equation (8) To show equation (8) we start with the state before the BS: ˆU |ψ⟩ = � dω1dω2F(ω1, ω2) |ω1, ω2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (B1) The usual balanced BS relation reads: |ω1⟩1 |ω2⟩2 �→ 1 2 � |ω1⟩1 |ω2⟩1 − |ω1⟩1 |ω2⟩2 + |ω1⟩2 |ω2⟩1 − |ω1⟩2 |ω2⟩2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (B2) To be able to use it, we introduce two mode changing operators ˆT1 and ˆT2 defined by: ˆT1 |ω1⟩1 |ω2⟩2 = |ω1⟩1 |ω2⟩1 ˆT2 |ω1⟩1 |ω2⟩2 = |ω1⟩2 |ω2⟩2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (B3) With these definition the BS splitter relation is equivalent to applying the operator: 1 2( ˆT1 − ˆ1 + ˆS − ˆT2), (B4) where ˆS is the swap operator, defined as ˆS |ω1, ω2⟩ = |ω2, ω1⟩ So the state coming out of the BS is: |ψout⟩ = 1 2 � dω1dω2F(ω1, ω2) � ˆT1 ˆU − ˆU + ˆS ˆU − ˆT2 ˆU � |ω1, ω2⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (B5) If we do selection on coincidence, we only keep the part of the state with one photon in each mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We get the state: |ψfin⟩ = −1 2 � dω1dω2F(ω1, ω2) � ˆU − ˆS ˆU � |ω1, ω2⟩ (B6a) = 1 2 � ˆS ˆU − ˆU � |ψ⟩ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (B6b) 14 We can finally compute the coincidence probability by taking the norm square of |ψfin⟩: Pc = ⟨ψfin|ψfin⟩ (B7a) = 1 4 ⟨ψ| � ˆU † − ˆU † ˆS �� ˆU − ˆS ˆU � |ψ⟩ (B7b) = 1 4 ⟨ψ| � ˆU † ˆU ���� =1 −2 ˆU † ˆS ˆU + ˆU † ˆS ˆS ˆU � �� � = ˆU † ˆU=1 � |ψ⟩ (B7c) = 1 2 � 1 − ⟨ψ| ˆU † ˆS ˆU |ψ⟩ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (B7d) b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Equation (9) The expression for Q is a direct consequence of the expression of the QFI for pure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' The proof of the expression of F is a little bit more involved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We have to compute: FI(κ) = 1 Pc �∂Pc ∂κ �2 + 1 Pa �∂Pa ∂κ �2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' (B8) We have seen the expression of the (anti)-coincidence probability Pc and Pa that depends on ⟨ψ| ˆU † ˆS ˆU |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' If we make the assumption that the state |ψ⟩ is either symmetric or anti-symmetric we known that we have: ⟨ψ| ˆU † ˆS ˆU |ψ⟩ = ± ⟨ψ| ˆU † ˆS ˆU ˆS |ψ⟩ = ⟨ψ| ˆV (κ) |ψ⟩ where we denote ˆV (κ) = ˆU † ˆS ˆU ˆS = eiκ ˆ He−iκ ˆS ˆ H ˆS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We first start by expanding this scalar product up to the second order in κ, using the short hand notation ⟨·⟩ = ⟨ψ| · |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' ⟨ψ| ˆV (κ) |ψ⟩ = � eiκ ˆ He−iκ ˆS ˆ H ˆS� (B9a) ≃ �� 1 + iκ ˆH − κ2 2 ˆH2�� 1 − iκ ˆS ˆH ˆS − κ2 2 ( ˆS ˆH ˆS)2�� (B9b) = � 1 + iκ ˆH − iκ ˆS ˆH ˆS − κ2 2 ˆH2 − κ2 2 ( ˆS ˆH ˆS)2 + κ ˆH ˆS ˆH ˆS � (B9c) Since the state |ψ⟩ is (anti)-symmetric, for any operators ˆG, we have � ˆS ˆG � = ± � ˆG � = � ˆG ˆS � , which allows some simplifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' = 1 − κ2 2 � � ˆH2� + � ( ˆS ˆH ˆS)2� − � ˆH ˆS ˆH ˆS � − � ˆS ˆH ˆS ˆH � � (B9d) = 1 − κ2 2 � ( ˆH − ˆS ˆH ˆS)2� (B9e) = 1 − κ2 2 ∆( ˆH − ˆS ˆH ˆS)2 (B9f) Since thanks to the symmetry of |ψ⟩,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ˆH − ˆS ˆH ˆS ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ˆH − ˆH ˆS2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='= 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='By defining ˆG = ˆH − ˆS ˆH ˆS it remains to compute the FI: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='FI(κ = 0) = 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='Pc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='�∂Pc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='∂κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='Pa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='�∂Pa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='∂κ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='(B10a) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='4Pc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='κ∆( ˆG)2�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='4Pa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='κ∆( ˆG)2�2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='(B10b) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='= κ2∆( ˆG)4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='� 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='Pc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='+ 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='Pa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='(B10c) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='= κ2∆( ˆG)4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='Pa + Pc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='PcPa ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='(B10d) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='= κ2∆( ˆG)4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1 + ⟨ψ| ˆV (κ) |ψ⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1 − ⟨ψ| ˆV (κ) |ψ⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='(B10e) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='= κ2∆( ˆG)4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1 − ⟨ψ| ˆV (κ) |ψ⟩2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='(B10f) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='= κ2∆( ˆG)4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='κ2∆( ˆG)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='(B10g) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='= ∆( ˆG)2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='(B10h) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='It is interesting to note that the computation of the Fisher information is singular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Indeed for the HOM interfer- ometer around κ = 0 the derivative of the probabilities vanishes ∂κPc,a = 0, while one of the two probability (Pc if the state is symmetric or Pa if its anti-symmetric) is also equal to zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' We thus obtain here the FI at zero by computing it at κ ̸= 0 and taking the limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' As a result we see that the FI is proportional to the second derivative of the coincidence probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' This means that for such a measurement what is important is the curvature of the probability peak/dip.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Appendix C: Appendix: Details on the computation of the various variances To compute explicitly the various variances of this paper on the two states |ψG⟩ and |ψG⟩ one can note that since these states are separable in the variables ω±, if we consider two operators ˆH+ and ˆH− which are respectively functions of ˆω+ and ˆt+ or ˆω− and ˆt− we have: � ˆH+ ˆH− � = � ˆH+ � � ˆH− � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Where for a fixed state |ψ⟩, � ˆH � = ⟨ψ| ˆH |ψ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' In order to compute any variance, one only has to compute some expectation values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' By expanding and using the independence property from above, one only need to compute as building block expectation value of the form: � ˆωk ±ˆtl ± � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Indeed we can use the commutation relation to reorder any product such that the frequency operators are on the left of the time operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' One has to pay attention that due to the choice of normalisation in the definition of ˆω± = ˆω1 ± ˆω2 and ˆt± = ˆt1 ± ˆt2 we have [ˆω±, ˆt±] = 2i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' Such expectation values can be obtained systematically using a software (here we used Mathematica),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content=' we have the following values: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='Operator ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='Variable + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='Variable − for |ψG⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='Variable − for |ψC⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ˆω ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ωp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ˆω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='p + σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='− + 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='4∆2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ˆω3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='3σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='+ωp + ω3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ˆω4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='3σ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='+ + 6σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='+ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='p + ω4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='3σ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='3σ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='− + 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='2σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='−∆2 + ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='16∆4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ˆt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ˆt2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ˆt3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ˆt4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='σ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='σ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='σ4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='− ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ˆωˆt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='i ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ˆω2ˆt ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='2iωp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ˆωˆt2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ωp ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ˆω2ˆt2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='ω2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='p ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='+ − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='−1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='∆2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='4σ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='− − 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} +page_content='TABLE III: Expectation values of the various product of plus and minus operators on the states |ψG⟩ and |ψC⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/0NFKT4oBgHgl3EQfNi2Q/content/2301.11755v1.pdf'} diff --git a/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf b/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf new file mode 100644 index 0000000000000000000000000000000000000000..507b40d96c98622a8ecb011dbafc3375467d84d0 --- /dev/null +++ b/0dFIT4oBgHgl3EQf3CsM/content/2301.11379v1.pdf @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:16ed4c1a7014e17a039c02a7b4caaef5a1d13a85923ca127c88b09befb95bec1 +size 635058 diff --git a/0dFIT4oBgHgl3EQf3CsM/vector_store/index.faiss b/0dFIT4oBgHgl3EQf3CsM/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..173aaeba3413c9ce9e4b473a47669931582dc2ff --- 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Bayesian networks (BNs) are a probabilistic graphical model +widely used for representing expert knowledge and reasoning under un- +certainty. Traditionally, they are based on directed acyclic graphs that +capture dependencies between random variables. However, directed cy- +cles can naturally arise when cross-dependencies between random vari- +ables exist, e.g., for modeling feedback loops. Existing methods to deal +with such cross-dependencies usually rely on reductions to BNs without +cycles. These approaches are fragile to generalize, since their justifica- +tions are intermingled with additional knowledge about the application +context. In this paper, we present a foundational study regarding seman- +tics for cyclic BNs that are generic and conservatively extend the cycle- +free setting. First, we propose constraint-based semantics that specify +requirements for full joint distributions over a BN to be consistent with +the local conditional probabilities and independencies. Second, two kinds +of limit semantics that formalize infinite unfolding approaches are intro- +duced and shown to be computable by a Markov chain construction. +1 +Introduction +A Bayesian network (BN) is a probabilistic graphical model representing a set of +random variables and their conditional dependencies. BNs are ubiquitous across +many fields where reasoning under uncertainties is of interest [10]. Specifically, +a BN is a directed acyclic graph with the random variables as nodes and edges +manifesting conditional dependencies, quantified by conditional probability ta- +bles (CPTs). The probability of any random variable can then be deduced by +the CPT entries along all its predecessors. Here, these probabilities are indepen- +dent of all variables that are no (direct or transitive) predecessors in the graph. +Acyclicity is hence crucial and commonly assumed to be rooted in some sort of +causality [23]. A classical use of BNs is in expert systems [22] where BNs ag- +gregate statistical data obtained by several independent studies. In the medical +⋆ This work was partially supported by the DFG in projects TRR 248 (CPEC, +see https://perspicuous-computing.science, project ID 389792660) and EXC 2050/1 +(CeTI, project ID 390696704, as part of Germany’s Excellence Strategy), and the +Key-Area Research and Development Program Grant 2018B010107004 of Guang- +dong Province. + +2 +C. Baier et al. +X +Y +X=T +X=F +F +s1 +1 − s1 +T +s2 +1 − s2 +Y +X +Y=T +Y=F +F +t1 +1 − t1 +T +t2 +1 − t2 +Fig. 1: A cyclic GBN with CPTs for X and Y +domain, e.g., they can capture the correlation of certain symptoms, diseases, and +human factors [11,15,26]. +Imagine for instance an expert system for supporting diagnosis of Covid- +19, harvesting multiple clinical studies. One study might have investigated the +percentage of patients who have been diagnosed with fever also having Covid- +19, while another study in turn might have investigated among the Covid-19 +patients whether they have fever, too. Clearly, both studies investigate the de- +pendency between fever and Covid-19, but under different conditions. Fever may +weaken the immune system and could increase the risk of a Covid-19 infection, +while Covid-19 itself has fever as a symptom. In case there is uniform knowledge +about “which symptom was first” in each of the constituent studies, then dy- +namic Bayesian networks (DBNs) [19] could be used as a model for the expert +system, breaking the interdependence of fever and Covid-19 through a prece- +dence relation. However, this implies either to rely only on studies where these +temporal dependencies are clearly identified or to introduce an artificial notion +of time that might lead to spurious results [18]. A naive encoding into the BN +framework always yields a graph structure that contains cycles, as is the case in +our small example shown in Fig. 1 where X and Y stand for the random variables +of diagnosing Covid-19 and fever, respectively. +That cycles might be unavoidable has already been observed in seminal pa- +pers such as [22,15]. But acyclicity is crucial for computing the joint probability +distribution of a BN, and thereby is a prerequisite for, e.g., routine inference +tasks. Existing literature that considers cycles in BNs mainly recommends re- +ducing questions on the probability values to properties in acyclic BNs. For +instance, in [11] nodes are collapsed towards removing cycles, while [22] suggests +to condition on each value combination on a cycle, generating a decomposition +into tree-like BNs and then averaging over the results to replace cycles. Some- +times, application-specific methods that restructure the cyclic BN towards an +acyclic BN by introducing additional nodes [26,8] or by unrolling cycles up to a +bounded depth [17,2] have been reported to give satisfactory results. Other ap- +proaches either remove edges that have less influence or reverse edges on cycles +(see, e.g., [10]). However, such approaches are highly application dependent and +hinge on knowledge about the context of the statistical data used to construct +the BN. Furthermore, as already pointed out by [30], they usually reduce the +solution space of families of joint distributions to a single one, or introduce so- +lutions not consistent with the CPTs of the original cyclic BN. While obviously +many practitioners have stumbled on the problem how to treat cycles in BNs + +On the Foundations of Cycles in Bayesian Networks +3 +and on the foundational question “What is the meaning of a cyclic BN?”, there +is very little work on the foundations of Bayesian reasoning with cycles. +In this paper, we approach this question by presenting general semantics for +BNs with cycles, together with algorithms to compute families of joint distri- +butions for such BNs. First, we investigate how the two main constituents of +classical BNs, namely consistency with the CPTs and independencies induced +by the graph structure, influence the joint distributions in the presence of cycles. +This leads to constraints semantics for cyclic BNs that comprise all those joint +distributions respecting the constraints, being either a single uniquely defined +one, none, or infinitely many distributions. Second, we present semantics that +formalize unfolding approaches and depend on the choice of a cutset, a set of +random variables that break every cycle in a cyclic BN. Intuitively, such cutsets +form the seams along which feedback loops can be unraveled. These semantics +are defined in terms of the limit (or limit average) of a sequence of distribu- +tions at descending levels in the infinite unfolding of the BN. We show that +the same semantics can be defined using a Markov chain construction and sub- +sequent long-run frequency analysis, which enables both precise computation +of the semantics and deep insights in the semantics’ behavior. Among others, +an immediate result is that the family of distributions induced with respect to +the limit semantics is always non-empty. As we will argue, the limit semantics +have obvious relations to a manifold of approaches that have appeared in the +literature, yet they have not been spelled out and studied explicitly. +1.1 +Notation +Let V be a set of Boolean random variables4 over the domain B = {F, T}. We +usually denote elements of V by X, Y, or Z. An assignment over V is a function +b: V → B which we may specify through set notation, e.g., b = {X=T, Y=F} for +b(X) = T and b(Y ) = F, or even more succinctly as XY . The set of all possible +assignments over V is denoted by Asg(V). We write bU for the restriction of b to a +subset U ⊆ V, e.g., b{X} = {X=T}, and may omit set braces, e.g., bX,Y = b{X,Y }. +A distribution over a set S is a function µ: S → [0, 1] where � +s∈S µ(s) = 1. +The set of all distributions over S is denoted by Dist(S). For |S| = n, µ will +occasionally be represented as a vector of size n for some fixed order on S. In +the following, we are mainly concerned with distributions over assignments, that +is distributions µ ∈ Dist(Asg(V)) for some set of random variables V. Each +such distribution µ induces a probability measure (also called µ) on 2Asg(V). +Thus, for a set of assignments φ ⊆ Asg(V), we have µ(φ) = � +b∈φ µ(b). We +are often interested in the probability of a partial assignment d ∈ Asg(U) on a +subset U ⊊ V of variables, which is given as the probability of the set of all full +4 We use Boolean random variables for simplicity of representation, an extension of +the proposed semantics over random variables with arbitrary finite state spaces is +certainly possible. + +4 +C. Baier et al. +assignments b ∈ Asg(V) that agree with d on U. As a shorthand, we define +µ(d) := µ +� +{b ∈ Asg(V) : bU = d} +� += +� +b∈Asg(V) +s.t. bU =d +µ(b). +The special case µ(X=T) is called the marginal probability of X. The restriction of +µ ∈ Dist(Asg(V)) to U, denoted µ|U ∈ Dist(Asg(U)), is given by µ|U(d) := µ(d). +For a set W disjoint from V and ν ∈ Dist(Asg(W)), the product distribution of +µ and ν is given by (µ ⊗ ν)(c) := µ(cV) · ν(cW) for every c ∈ Asg(V ∪ W). µ is +called a Dirac distribution if µ(b) = 1 for some assignment b ∈ Asg(V) and thus +µ(c) = 0 for all other assignments c ̸= b. A Dirac distribution derived from a +given assignment b is denoted by Dirac(b). +Graph Notations. For a graph G = ⟨V, E⟩ with nodes V and directed edges +E ⊆ V × V, we may represent an edge (X, Y ) ∈ E as X → Y if E is clear from +context. Pre(X) := {Y ∈ V : Y → X} denotes the set of parents of a node X ∈ V, +and Post∗(X) := {Y ∈ V : X → · · · → Y } is the set of nodes reachable from +X. A node X is called initial if Pre(X) = ∅, and Init(G) is the set of all nodes +initial in G. A graph G is strongly connected if each node in V is reachable from +every other node. A set of nodes D is a strongly connected component (SCC) of +G if all nodes in D can reach each other and D is not contained in another SCC, +and a bottom SCC (BSCC) if no node in V \ D can be reached from D. +Markov Chains. A discrete-time Markov chain (DTMC) is a tuple M = ⟨S, P⟩ +where S is a finite set of states and P: S × S → [0, 1] a function such that +P(s, ·) ∈ Dist(S) for all states s ∈ S. The underlying graph GM = ⟨S, E⟩ is +defined by E = {(s, t) ∈ S × S : P(s, t) > 0}. The transient distribution πι +n ∈ +Dist(S) at step n is defined through the probability πι +n(s) to be in state s after +n steps if starting with initial state distribution ι. It satisfies (in matrix-vector +notation) πι +n = ι · Pn. We are also interested in the long-run frequency of state +occupancies when n tends to infinity, defined as the Ces`aro limit lrfι : S → [0, 1]: +lrfι(s) := +lim +n→∞ +1 +n + 1 +n +� +i=0 +πι +n(s). +(LRF) +This limit always exists and corresponds to the long-run fraction of time spent +in each state [12]. The limit probability limn→∞ πι +n is arguably more intuitive as +a measure of the long-run behavior, but may not exist (due to periodicity). In +case of existence, it agrees with the Ces`aro limit lrfι. If GM forms an SCC, the +limit is independent of the choice of ι and the superscript can be dropped. We +denote this limit by lrfM. +2 +Generalized Bayesian Networks +We introduce generalized Bayesian networks (GBNs) as a BN model that does +not impose acyclicity and comes with a distribution over initial nodes. + +On the Foundations of Cycles in Bayesian Networks +5 +Definition 1 (Generalized BN). A GBN B is a tuple ⟨G, P, ι⟩ where +– G = ⟨V, E⟩ is a directed graph with nodes V and an edge relation E ⊆ V × V, +– P is a function that maps all non-initial nodes X ∈ V\Init(G) paired with +each of their parent assignments b ∈ Asg(Pre(X)) to a distribution +P(X, b): Asg +� +{X} +� +→ [0, 1], +– ι is a distribution over the assignments for the initial nodes Init(G), i.e., +ι ∈ Dist +� +Asg(Init(G)) +� +. +The distributions P(X, b) have the same role as the entries in a conditional +probability table (CPT) for X in classical BNs: they specify the probability for +X=T or X=F depending on the assignments of the predecessors of X. To this +end, for X ∈ V\Init(G) and b ∈ Asg(Pre(X)), we also write Pr(X=T | b) for +P(X, b)(X=T). In the literature, initial nodes are often assigned a marginal prob- +ability via a CPT as well, assuming independence of all initial nodes. Differently, +in our definition of GBNs, it is possible to specify an arbitrary distribution ι over +all initial nodes. If needed, P can be easily extended to initial nodes by setting +P(X, ∅) := ι|{X} for all X ∈ Init(G). Hence, classical BNs arise as a special +instance of GBNs where the graph G is acyclic and initial nodes are pairwise +independent. In that case, the CPTs given by P are a compact representation of +a single unique full joint distribution dist BN(B) over all random variables X ∈ V. +For every assignment b ∈ Asg(V), we can compute dist BN(B)(b) by the so-called +chain rule: +dist BN(B)(b) := ι +� +bInit(G) +� +· +� +X∈V\Init(G) +Pr +� +bX | bPre(X) +� +. +(CR) +In light of the semantics introduced later on, we define the standard BN-semantics +of an acyclic GBN B as the set �B�BN := {distBN(B)}, and �B�BN := ∅ if B con- +tains cycles. +The distribution dist BN(B) satisfies two crucial properties: First, it is con- +sistent with the CPT entries given by P and the distribution ι, and second, it +observes the independencies encoded in the graph G. In fact, those two properties +are sufficient to uniquely characterize distBN(B). We briefly review the notion of +independence and formally define CPT consistency later on in Section 3. +Independence. Any full joint probability distribution µ ∈ Dist(Asg(V)) may in- +duce a number of conditional independencies among the random variables in V. +For X, Y, and Z disjoint subsets of V, the random variables in X and Y are +independent under µ given Z if the conditional probability of each assignment +over the nodes in X given an assignment for Z is unaffected by further condi- +tioning on any assignment of Y. Formally, the set Indep(µ) contains the triple +(X, Y, Z) iff for all a ∈ Asg(X), b ∈ Asg(Y), and c ∈ Asg(Z), we have +µ(a | b, c) = µ(a | c) +or +µ(b, c) = 0. +We also write (X ⊥ Y | Z) for (X, Y, Z) ∈ Indep(µ) and may omit the set +brackets of X, Y, and Z. + +6 +C. Baier et al. +d-separation. For classical BNs, the graph topology encodes independencies that +are necessarily satisfied by any full joint distribution regardless of the CPT +entries. Given two random variables X and Y as well as a set of observed variables +Z, then X and Y are conditionally independent given Z if the corresponding +nodes in the graph are d-separated given Z [6]. To establish d-separation, all +simple undirected paths5 between X and Y need to be blocked given Z. Let W +denote such a simple path W0, W1, . . . , Wk with W0 = X, Wk = Y, and either +Wi → Wi+1 or Wi ← Wi+1 for all i < k. Then W is blocked given Z if and only +if there exists an index i, 0 < i < k, such that one of the following two conditions +holds: (1) Wi is in Z and is situated in a chain or a fork in W, i.e., +– Wi−1 → Wi → Wi+1 (forward chain) +– Wi−1 ← Wi ← Wi+1 (backward chain) +and +Wi ∈ Z, +– Wi−1 ← Wi → Wi+1 (fork) +(2) Wi is in a collider and neither Wi nor any descendant of Wi is in Z, i.e., +– Wi−1 → Wi ← Wi+1 (collider) +and +Post∗(Wi) ∩ Z = ∅. +Two sets of nodes X and Y are d-separated given a third set Z if for each X ∈ X +and Y ∈ Y, X and Y are d-separated given Z. Notably, the d-separation criterion +is applicable also in presence of cycles [28]. For a graph G = ⟨V, E⟩ of a GBN, +we define the set d-sep(G) as +d-sep(G) := +� +(X, Y, Z) ∈ (2V)3 : X and Y are d-separated given Z +� +. +For acyclic Bayesian networks it is well known that the independencies ev- +ident from the standard BN semantics’ distribution include the independencies +derived from the graph. That is, for acyclic GBNs B̸⟳ = ⟨G, P, ι⟩ where all initial +nodes are pairwise independent under ι, we have +d-sep(G) ⊆ Indep +� +dist BN(B̸⟳) +� +. +For an arbitrary initial distribution, the above relation does not necessarily +hold. However, we can still find a set of independencies that are necessarily +observed by the standard BN semantics and thus act as a similar lower bound. +We do so by assuming the worst case, namely that each initial node depends on +every other initial node under ι. Formally, given a graph G = ⟨V, E⟩, we define a +closure operation Close(·) as follows and compute the set d-sep +� +Close(G) +� +: +Close(G) := +� +V, E ∪ {(A, B) for A, B ∈ Init(G), A ̸= B} +� +. +Lemma 1. Let B̸⟳ = ⟨G, P, ι⟩ be an acyclic GBN. Then +d-sep +� +Close(G) +� +⊆ Indep +� +distBN(B̸⟳) +� +. +As intuitively expected, the presence of cycles in G generally reduces the +number of graph independencies, though note that also in strongly connected +graphs independencies may exist. For example, if G is a four-node cycle with +nodes W, X, Y, and Z, then d-sep(G) = +� +(W ⊥ Y | X, Z), (X ⊥ Z | W, Y ) +� +. +5 A path is simple if no node occurs twice in the path. “Undirected” in this context +means that edges in either direction can occur along the path. + +On the Foundations of Cycles in Bayesian Networks +7 +3 +Constraints Semantics +For classical acyclic BNs there is exactly one distribution that agrees with all +CPTs and satisfies the independencies encoded in the graph. This distribution +can easily be constructed by means of the chain rule (CR). For cyclic GBNs, +applying the chain rule towards a full joint distribution is not possible in general, +as the result is usually not a valid probability distribution. Still, we can look for +distributions consistent with a GBN’s CPTs and the independencies derived +from its graph. Depending on the GBN, we will see that there may be none, +exactly one, or even infinitely many distributions fulfilling these constraints. +3.1 +CPT-consistency +We first provide a formal definition of CPT consistency in terms of linear con- +straints on full joint distributions. +Definition 2 (Strong and weak CPT-consistency). Let B be a GBN with +nodes V and X ∈ V. Then µ is called strongly CPT-consistent for X in B (or +simply CPT-consistent) if for all c ∈ Asg(Pre(X)) +µ(X=T, c) = µ(c) · Pr(X=T | c). +(Cpt) +We say that µ is weakly CPT-consistent for X in B if +µ(X=T) = +� +c∈Asg(Pre(X)) +µ(c) · Pr(X=T | c). +(wCpt) +Intuitively, the constraint (Cpt) is satisfied for µ if the conditional proba- +bility µ(X=T | c) equals the entry in the CPT for X under assignment c, i.e., +µ(X=T | c) = Pr(X=T | c). In the weak case (wCpt), only the resulting marginal +probability of X needs to agree with the CPTs. +Definition 3 (Cpt and wCpt semantics). +For a GBN B = ⟨G, P, ι⟩, the +CPT-semantics �B�Cpt is the set of all distributions µ ∈ Dist(Asg(V)) where +µ|Init(G) = ι and µ is CPT-consistent for every node X ∈ V\Init(G). The weak +CPT-semantics �B�wCpt is defined analogously. +Clearly, we have �B�Cpt ⊆ �B�wCpt for all B. The next example shows that +depending on the CPT values, the set �B�Cpt may be empty, a singleton, or of +infinite cardinality. +Example 1. To find CPT-consistent distributions for the GBN from Fig. 1, we +construct a system of linear equations whose solutions form distributions µ ∈ +Dist +� +Asg({X, Y }) +� +, represented as vectors in the space [0, 1]4: + + + + + + + + +s1 +0 +s1−1 +0 +0 +s2 +0 +s2−1 +t1 +t1−1 +0 +0 +0 +0 +t2 +t2−1 +1 +1 +1 +1 + + + + + + + + +· + + + + + +µXY +µXY +µXY +µXY + + + + + = + + + + + + + + +0 +0 +0 +0 +1 + + + + + + + + + +8 +C. Baier et al. +where, e.g., µXY abbreviates µ(X=T, Y=F). The first line of the matrix states +the (Cpt) constraint for node X and the parent assignment c = {Y=F}: +0 = s1 · µXY + 0 · µXY + (s1−1) · µXY + 0 · µXY +µXY += (µXY + µXY ) · s1 +µXY += µY · Pr(X=T | Y=F) +µ(X=T, c) = µ(c) · Pr(X=T | c). +Analogously, the following three rows encode the CPT constraints for X, Y, +and their remaining parent assignments. The last row ensures that solutions are +indeed probability distributions satisfying � +c µ(c) = 1. +The number of solutions for the system now depends on the CPT entries s1, +s2, t1, and t2. For s1 = t2 = 0 and s2 = t1 = 1, no solution exists as the first +four equations require µ(b) = 0 for all b ∈ Asg({X, Y }), while the last equation +ensures µXY + µXY + µXY + µXY = 1. For s1 = t1 = 0 and s2 = t2 = 1, all +distributions with µXY = 1 − µXY and µXY = µXY = 0 are solutions. Finally, +e.g., for s1 = t1 = 3/4 and s2 = t2 = 1/2, there is exactly one solution with +µXY = 1/10 and µ(b) = 3/10 for the other three assignments. +3.2 +Independence-consistency +We extend Cpt semantics with a set of independencies that need to be observed +by all induced distributions. +Definition 4 (Cpt-I semantics). For a GBN B = ⟨G, P, ι⟩ and a set of inde- +pendencies I, the CPT-I semantics �B�Cpt-I is defined as the set of all CPT- +consistent distributions µ for which I ⊆ Indep(µ) holds. +Technically, the distributions in �B�Cpt-I have to fulfill the following polynomial +constraints in addition to the CPT-consistency constraints: +µ(b) · µ(bW) = µ(b{X}∪W) · µ(bU∪W) +(Cpt-I) +for each independence (X ⊥ U | W) ∈ I with variable X∈V and sets of variables +U, W ⊆ V, and for each assignment b ∈ Asg({X} ∪ U ∪ W). Note that in case +µ(bW) > 0, (Cpt-I) is equivalent to the constraint µ(bX | bU∪W) = µ(bX | bW). +We can now formally state the alternative characterization of the standard +BN semantics as the unique CPT-consistent distribution that satisfies the d- +separation independencies of the graph. For each classical BN B with acyclic +graph G and I = d-sep(G), we have �B�BN = {distBN(B)} = �B�Cpt-I. Thus, the +Cpt-I semantics provides a conservative extension of the standard BN semantics +to GBNs with cycles. However, in practice, its use is limited since there might be +no distribution that satisfies all constraints. In fact, the case where �B�Cpt-I = ∅ +is to be expected for most cyclic GBNs, given that the resulting constraint +systems tend to be heavily over-determined. +The next section introduces semantics that follow a more constructive ap- +proach. We will see later on in Section 5.1 that the families of distributions +induced by these semantics are always non-empty and usually singletons. + +On the Foundations of Cycles in Bayesian Networks +9 +X +Y +Z +Fig. 2: The graph of a strongly connected GBN +X0 +Y0 +Z0 +X1 +Y1 +Z1 +X2 +Y2 +Z2 +... +(a) Unfolding along all nodes +Z0 +X1 +Y1 +Z1 +X2 +Y2 +Z2 +... +(b) Unfolding along the Z nodes +Fig. 3: Two infinite unfoldings of the graph in Fig. 2 +4 +Limit and Limit Average Semantics +We first develop the basic ideas underling the semantics by following an example, +before giving a formal treatment in Section 4.2. +4.1 +Intuition +Consider the GBN B whose graph G is depicted in Fig. 2. One way to get rid +of the cycles is to construct an infinite unfolding of B as shown in Fig. 3a. In +this new graph G∞, each level contains a full copy of the original nodes and +corresponds to some n ∈ N. For any edge X → Y in the original graph, we +add edges Xn → Yn+1 to G∞, such that each edge descends one level deeper. +Clearly any graph constructed in this way is acyclic, but this fact alone does +not aid in finding a matching distribution since we dearly bought it by giving +up finiteness. However, we can consider what happens when we plug in some +initial distribution µ0 over the nodes X0, Y0, and Z0. Looking only at the first +two levels, we then get a fully specified acyclic BN by using the CPTs given +by P for X1, Y1, and Z1. For this sub-BN, the standard BN semantics yields +a full joint distribution over the six nodes from X0 to Z1, which also induces +a distribution µ1 over the three nodes at level 1. This procedure can then be +repeated to construct a distribution µ2 over the nodes X2, Y2, and Z2, and, +more generally, to get a distribution µn+1 given a distribution µn. Recall that + +10 +C. Baier et al. +each of those distributions can be viewed as vector of size 23. Considering the +sequence µ0, µ1, µ2, . . . , the question naturally arises whether a limit exists, i.e., +a distribution/vector µ such that +µ += +lim +n→∞ µn. +Example 2. Consider the GBN from Fig. 1 with CPT entries s1 = t2 = 1 and +s2 = t1 = 0, which intuitively describe the contradictory dependencies “X iff not +Y ” and “Y iff X”. For any initial distribution µ0 = ⟨e f g h⟩, the construction +informally described above yields the following sequence of distributions µn: +µ0 = + + + + + +e +f +g +h + + + + +, µ1 = + + + + + +f +h +e +g + + + + +, µ2 = + + + + + +h +g +f +e + + + + +, µ3 = + + + + + +g +e +h +f + + + + +, µ4 = + + + + + +e +f +g +h + + + + +, . . . +As µ4 = µ0, the sequence starts to cycle infinitely between the first four distribu- +tions. The series converges for e = f = g = h = 1/4 (in which case the sequence +is constant), but does not converge for any other initial distribution. +The example shows that the existence of the limit depends on the given initial +distribution. In case no limit exists because some distributions keep repeating +without ever converging, it is possible to determine the limit average (or Ces`aro +limit) of the sequence: +˜µ += +lim +n→∞ +1 +n + 1 +n +� +i=0 +µi. +The limit average has three nice properties: First, if the regular limit µ exists, +then the limit average ˜µ exists as well and is identical to µ. Second, in our use +case, ˜µ in fact always exists for any initial distribution µ0. And third, as we +will see in Section 5, the limit average corresponds to the long-run frequency of +certain Markov chains, which allows us both to explicitly compute and to derive +important properties of the limit distributions. +Example 3. Continuing Ex. 2, the limit average of the sequence µ0, µ1, µ2, . . . is +the uniform distribution ˜µ = ⟨1/4 1/4 1/4 1/4⟩, regardless of the choice of µ0. +Before we formally define the infinite unfolding of GBNs and the resulting +limit semantics, there is one more observation to be made. To ensure that the +unfolded graph G∞ is acyclic, we redirected every edge of the GBN B to point +one level deeper, resulting in the graph displayed in Fig. 3a. As can be seen +in Fig. 3b, we also get an acyclic unfolded graph by only redirecting the edges +originating in the Z nodes to the next level and keeping all other edges on the +same level. The relevant property is to pick a set of nodes such that for each +cycle in the original GBN B, at least one node in the cycle is contained in the +set. We call such sets the cutsets of B. + +On the Foundations of Cycles in Bayesian Networks +11 +Definition 5 (Cutset). Let B be an GBN with graph G = ⟨V, E⟩. A subset +C ⊆ V is a cutset for B if every cycle in G contains at least one node from C. +Example 4. The GBN in Fig. 2 has the following cutsets: {Y }, {Z}, {X, Y }, +{X, Z}, {Y, Z}, and {X, Y, Z}. Note that {X} does not form a cutset as no +node from the cycle Y → Z → Y is contained. +So far we implicitly used the set V of all nodes for the unfolding, which always +trivially forms a cutset. The following definitions will be parameterized with a +cutset, as the choice of cutsets influences the resulting distributions as well as +the time complexity. +4.2 +Formal Definition +Let Vn := {Xn : X ∈ V} denote the set of nodes on the nth level of the unfolding +in G∞. For C ⊆ V a cutset of the GBN, the subset of cutset nodes on that level +is given by Cn := {Xn ∈ Vn : X ∈ C}. Then a distribution γn ∈ Dist(Asg(Cn)) for +the cutset nodes in Cn suffices to get a full distribution µn+1 ∈ Dist(Asg(Vn+1)) +over all nodes on the next level, n + 1: We look at the graph fragment Gn+1 of +G∞ given by the nodes Cn ∪ Vn+1 and their respective edges. In this fragment, +the cutset nodes are initial, so the cutset distribution γn can be combined with +the initial distribution ι to act as new initial distribution. For the nodes in +Vn+1, the corresponding CPTs as given by P can be used, i.e., Pn(Xn, ·) = +P(X, ·) for Xn ∈ Vn. Putting everything together, we obtain an acyclic GBN +Bn+1 = ⟨Gn+1, Pn+1, ι ⊗ γn⟩. However, GBNs constructed in this way for each +level n > 0 are all isomorphic and only differ in the given cutset distribution γ. +For simplicity and in light of later use, we thus define a single representative +GBN Dissect(B, C, γ) that represents a dissection of B along a given cutset C, +with ι ⊗ γ as initial distribution. +Definition 6 (Dissected GBN). +Let B = ⟨G, P, ι⟩ be a GBN with graph +G = ⟨V, E⟩ and C ⊆ V a cutset for B with distribution γ ∈ Dist(Asg(C)). Then, +the C-dissected GBN Dissect(B, C, γ) is the acyclic GBN ⟨GC, PC, ι ⊗ γ⟩ with +graph GC = ⟨V ∪ C′, EC⟩ where +– C′ := {X′ : X ∈ C} extends V by fresh copies of all cutset nodes; +– incoming edges to nodes in C are redirected to their copies, i.e., +EC := +� +(X, Y ′) : (X, Y ) ∈ E, Y ∈ C +� +∪ +� +(X, Y ) : (X, Y ) ∈ E, Y /∈ C +� +; +– the function PC uses the CPT entries given by P for the cutset nodes as +entries for their copies and the original entries for all other nodes, i.e., we +have PC(Y ′, a) = P(Y, a) for each node Y ′ ∈ C′ and parent assignment a ∈ +Asg(Pre(Y ′)), and PC(X, b) = P(X, b) for X ∈ V\C and b ∈ Asg(Pre(X)). +Fig. 4 shows two examples of dissections on the GBN of Fig. 2. As any dissected +GBN is acyclic by construction, the standard BN semantics yields a full joint +distribution over all nodes in V ∪ C′. We restrict this distribution to the nodes + +12 +C. Baier et al. +X +Y +Z +X′ +Y ′ +Z′ +(a) Cutset C = {X, Y, Z} +Z +X +Y +Z′ +(b) Cutset C = {Z} +Fig. 4: Dissections of the GBN in Fig. 2 for two cutsets +in (V \ C) ∪ C′, as those are the ones on the “next level” of the unfolding, while +re-identifying the cutset node copies with the original nodes to get a distribution +over V. Formally, we define the distribution Next(B, C, γ) for each assignment +b ∈ Asg(V) as +Next(B, C, γ)(b) := dist BN +� +Dissect(B, C, γ) +� +(b′) +where the assignment b′ ∈ Asg +� +(V\C) ∪ C′� +is given by b′(X) = b(X) for all +X ∈ V\C and b′(Y ′) = b(Y ) for all Y ∈ C. In the unfolded GBN, this allows +us to get from a cutset distribution γn to the next level distribution µn+1 = +Next(B, C, γn). The next cutset distribution γn+1 is then given by restricting the +full distribution to the nodes in C, i.e., γn+1 = Next(B, C, γn)|C.6 Vice versa, a +cutset distribution γ suffices to recover the full joint distribution over all nodes +V. Again using the standard BN semantics of the dissected GBN, we define the +distribution Extend(B, C, γ) ∈ Dist(Asg(V)) as +Extend(B, C, γ) := dist BN +� +Dissect(B, C, γ) +��� +V. +With these definitions at hand, we can formally define the limit and limit +average semantics described in the previous section. +Definition 7 (Limit and limit average semantics). Let B be a GBN over +nodes V with cutset C. The limit semantics of B w.r.t. C is the partial function +Lim(B, C, ·) : Dist +� +Asg(C) +� +⇀ Dist +� +Asg(V) +� +from initial cutset distributions γ0 to full distributions µ = Extend(B, C, γ) where +γ = lim +n→∞ γn +and +γn+1 = Next(B, C, γn)|C. +The set �B�Lim-C is given by the image of Lim(B, C, ·), i.e., +�B�Lim-C := {Lim(B, C, γ0) : γ0 ∈ Dist(Asg(C)) s.t. Lim(B, C, γ0) is defined}. +The limit average semantics of B w.r.t. C is the partial function +LimAvg(B, C, ·) : Dist +� +Asg(C) +� +⇀ Dist +� +Asg(V) +� +6 Recall that we may view distributions as vectors which allows us to equate distribu- +tions over different but isomorphic domains. + +On the Foundations of Cycles in Bayesian Networks +13 +X=T +Y=T +X=T +Y=F +X=F +Y=T +X=F +Y=F +3/8 +3/8 +1/8 +1/8 +1/2 +1/2 +3/4 +1/4 +1 +X=T +Y=T +X=T +Y=F +X=F +Y=T +X=F +Y=F +Fig. 5: A cutset Markov chain for a cutset C = {X, Y } +from γ0 to distributions µ = Extend(B, C, γ) where +γ = lim +n→∞ +1 +n + 1 +n +� +i=0 +γn +and +γn+1 = Next(B, C, γn)|C. +The set �B�LimAvg-C is likewise given by the image of LimAvg(B, C, ·). +We know that the limit average coincides with the regular limit if the lat- +ter exists, so for every initial cutset distribution γ0, we have Lim(B, C, γ0) = +LimAvg(B, C, γ0) if Lim(B, C, γ0) is defined. Thus, �B�Lim-C ⊆ �B�LimAvg-C. +5 +Markov Chain Semantics +While we gave some motivation for the limit and limit average semantics, their +definitions do not reveal an explicit way to compute their member distributions. +In this section we introduce the (cutset) Markov chain semantics which offers +explicit construction of distributions and is shown to coincide with the limit +average semantics. It further paves the way for proving several properties of +both limit semantics in Section 5.1. +At the core of the cutset Markov chain semantics lies the eponymous cut- +set Markov chain which captures how probability mass flows from one cutset +assignment to the others. To this end, the Dirac distributions corresponding to +each assignment are used as initial distributions in the dissected GBN. With the +Next function we then get a new distribution over all cutset assignments, and +the probabilities assigned by this distribution are used as transition probabilities +for the Markov chain. +Definition 8 (Cutset Markov chain). +Let B be a GBN with cutset C. The +cutset Markov chain CMC(B, C) = ⟨Asg(C), P⟩ w.r.t. B and C is a DTMC where +the transition matrix P is given for all cutset assignments b, c ∈ Asg(C) by +P(b, c) := Next +� +B, C, Dirac(b) +� +(c). +Example 5. Fig. 5 shows the cutset Markov chain for the GBN from Fig. 1 with +CPT entries s1 = 1/4, s2 = 1, t1 = 1/2, t2 = 0, and cutset C = {X, Y }. Exem- +plarily, the edge at the bottom from assignment b = {X=F, Y=F} to assignment + +14 +C. Baier et al. +c = {X=F, Y=T} with label 3/8 is derived as follows: +P(b, c) += +Next +� +B, C, Dirac(b) +� +(c) = dist BN +� +Dissect(B, C, Dirac(b)) +� +(c′) += +� +a∈Asg(VC) +s.t. c′⊆a +dist BN +� +Dissect(B, C, Dirac(b)) +� +(a) += +� +a∈Asg(VC) +s.t. c′⊆a +Dirac(b)(aX,Y ) · Pr(X′=F | aY ) · Pr(Y ′=T | aX) += +Pr(X′=F | Y=F) · Pr(Y ′=T | X=F) = (1 − s1) · t1 += +3/8. +Note that in the second-to-last step, in the sum over all full assignments a which +agree with the partial assignment c′, only the assignment which also agrees with +b remains as for all other assignments we have Dirac(b)(aX,Y ) = 0. +Given a cutset Markov chain with transition matrix P and an initial cutset +distribution γ0, we can compute the uniquely defined long-run frequency distri- +bution lrfγ0 (see Section 1.1). Then the Markov chain semantics is given by the +extension of this distribution over the whole GBN. +Definition 9 (Markov chain semantics). +Let B be a GBN over nodes V +with a cutset C ⊆ V and cutset Markov chain CMC(B, C) = ⟨Asg(C), P⟩. Then +the Markov chain semantics of B w.r.t. C is the function +MCS(B, C, ·) : Dist +� +Asg(C) +� +→ Dist +� +Asg(V) +� +from cutset distributions γ0 to full distributions µ = Extend(B, C, lrfγ0) where +lrfγ0 = +lim +n→∞ +1 +n+1 +n +� +i=0 +γi +and +γi+1 = γi · P. +The set �B�MC-C is defined as the image of MCS(B, C, ·). +In the following lemma, we give four equivalent characterizations of the long- +run frequency distributions of the cutset Markov chain. +Lemma 2. Let B be a GBN with cutset C, cutset distribution γ ∈ Dist(Asg(C)), +and M = ⟨Asg(C), P⟩ the cutset Markov chain CMC(B, C). Then the following +statements are equivalent: +(a) γ = γ · P. +(b) There exists γ0 ∈ Dist(Asg(C)) such that for γi+1 = γi · P, we have +γ = +lim +n→∞ +1 +n+1 +n +� +i=0 +γi. +(c) γ belongs to the convex hull of the long-run frequency distributions lrfD of +the bottom SCCs D of M. +(d) γ = Next(B, C, γ)|C. + +On the Foundations of Cycles in Bayesian Networks +15 +Following Lemma 2, we can equivalently define the cutset Markov chain se- +mantics as the set of extensions of all stationary distributions for P: +�B�MC-C := +� +Extend(B, C, γ) : γ ∈ Dist +� +Asg(C) +� +s.t. γ = γ · P +� +. +Example 6. Continuing Ex. 5, there is a unique stationary distribution γ with +γ = γ · P for the cutset Markov chain in Fig. 5: γ = ⟨48/121 18/121 40/121 15/121⟩. +As in this case the cutset C = {X, Y } equals the set of all nodes V, we have +Extend(B, C, γ) = γ and thus �B�MC-{X,Y } = {γ}. +As shown by Lemma 2, the behavior of the Next function is captured by +multiplication with the transition matrix P. Both the distributions in the limit +average semantics and the long-run frequency distributions of the cutset Markov +chain are defined in terms of a Ces`aro limit, the former over the sequence of +distributions obtained by repeated application of Next, the latter by repeated +multiplication with P. Thus, both semantics are equivalent. +Theorem 1. Let B be a GBN. Then for any cutset C of B and initial distribution +γ0 ∈ Dist(Asg(C)), we have +MCS(B, C, γ0) = LimAvg(B, C, γ0). +We know that Lim(B, C, γ0) is not defined for all initial distributions γ0. +However, the set of all limits that do exist contains exactly the distributions +admitted by the Markov chain and limit average semantics. +Lemma 3. Let B be a GBN. Then for any cutset C of B, we have +�B�MC-C = �B�LimAvg-C = �B�Lim-C. +5.1 +Properties +By the equivalences established in Theorem 1 and Lemma 3, we gain profound +insights about the limit and limit average distributions by Markov chain analysis. +As every finite-state Markov chain has at least one stationary distribution, it +immediately follows that �B�MC-C—and thus �B�LimAvg-C and �B�Lim-C—is always +non-empty. Further, if the cutset Markov chain is irreducible, i.e., the graph +is strongly connected, the stationary distribution is unique and �B�MC-C is a +singleton. The existence of the limit semantics for a given initial distribution γ0 +hinges on the periodicity of the cutset Markov chain. +Example 7. We return to Example 2 and construct the cutset Markov chain +CMC(B, C) = ⟨Asg(C), P⟩ for the (implicitly used) cutset C = {X, Y }: +X=T +Y=T +X=T +Y=F +X=F +Y=T +X=F +Y=F +1 +1 +1 +1 + +16 +C. Baier et al. +The chain is strongly connected and has a period of length 4, which explains the +observed behavior that for any initial distribution γ0, we got the sequence +γ0, γ1, γ2, γ3, γ0, γ1, . . . +This sequence obviously converges only for initial distributions that are station- +ary, i.e., if we have γ0 = γ0 · P. +The following lemma summarizes the implications that can be drawn from +close inspection of the cutset Markov chain. +Lemma 4 (Cardinality). +Let B be a GBN with cutset C and cutset Markov +chain CMC(B, C) = ⟨Asg(C), P⟩. Further, let k > 0 denote the number of bottom +SCCs D1, . . . , Dk of CMC(B, C). Then +1. the cardinality of the cutset Markov chain semantics is given by +���B�MC-C +�� = +� +1 +if k = 1, +∞ +if k > 1; +2. Lim(B, C, γ0) is defined for all γ0 ∈ Dist(Asg(C)) if all Di are aperiodic; +3. Lim(B, C, γ) is only defined for stationary distributions γ with γ = γ · P if +Di is periodic for any 1 ⩽ i ⩽ k. +A handy sufficient (albeit not necessary) criterion for both aperiodicity and +the existence of a single bottom SCC in the cutset Markov chain is the absence +of zero and one entries in the CPTs and the initial distribution of a GBN. +Definition 10 (Smooth GBNs). +A GBN B = ⟨G, P, ι⟩ is called smooth iff +all CPT entries as given by P and all values in ι are in the open interval ]0, 1[. +Lemma 5. Let B be a smooth GBN and C a cutset of B. Then the graph of the +cutset Markov chain CMC(B, C) is a complete digraph. +Corollary 1. The limit semantics of a smooth GBN B is a singleton for every +cutset C of B and Lim(B, C, γ0) is defined for all γ0 ∈ Dist(Asg(C)). +As noted in [14], one rarely needs to assign a probability of zero (or, con- +versely, of one) in real-world applications; and doing so in cases where some +event is extremely unlikely but not impossible is a common modeling error. This +observation gives reason to expect that most GBNs encountered in practice are +smooth, and their semantics is thus, in a sense, well-behaved. +5.2 +Relation to Constraints Semantics +We take a closer look at how the cutset semantics relates to the CPT-consistency +semantics defined in Section 3. CPTs of nodes outside cutsets remain unaffected +in the dissected BNs from which the Markov chain semantics is computed. Since +there are cyclic GBNs for which no CPT-consistent distribution exists (cf. Ex- +ample 1) while Markov chain semantics always yields at least one solution due +to Lemma 4, it cannot be expected that cutset nodes are necessarily CPT- +consistent. However, they are always weakly CPT-consistent. + +On the Foundations of Cycles in Bayesian Networks +17 +Lemma 6. Let B be a GBN over nodes V, C ⊆ V a cutset for B, and µ ∈ +�B�MC-C. Then µ is strongly CPT-consistent for all nodes in V\C and weakly +CPT-consistent for the nodes in C. +The lemma shows a way to find fully CPT consistent distributions: Consider +there is a distribution µ ∈ �B�MC-C ∩ �B�MC-D for two disjoint cutsets C and D. +Then by Lemma 6 the nodes in V \ C and V \ D are CPT consistent, so in fact +µ is CPT consistent. In general, we get the following result. +Lemma 7. Let B be a GBN over nodes V and C1, . . . , Ck cutsets of B s.t. for +each node X ∈ V there is an i ∈ {1, . . . , k} with X /∈ Ci. Then +� +0⩽i⩽k +�B�MC-Ci ⊆ �B�Cpt. +We take a look at which independencies are necessarily observed by the +distributions in �B�MC-C. Let γ ∈ Dist(Asg(C)) be the cutset distribution and +let G[C] denote the graph of Dissect(B, C, γ) restricted to the nodes in V such +that the cutset nodes in C are initial. Then by Lemma 1, the d-separation in- +dependencies of the closure of G[C] hold in all distributions µ ∈ �B�MC-C, i.e., +d-sep +� +Close(G[C]) +� +⊆ Indep(µ). The next lemma states that any Cpt-consistent +distribution that satisfies these independence constraints for some cutset C also +belongs to �B�MC-C. +Lemma 8. Let B be a GBN with cutset C and IC = Close(G[C]). Then we have +�B�Cpt-IC ⊆ �B�MC-C. +Combining Lemma 7 and Lemma 8 yields the following equivalence. +Corollary 2. For a GBN B with cutsets C1, . . . , Ck as in Lemma 7 and the +independence set I = � +0⩽i⩽k Close(G[Ci]), we have +� +0⩽i⩽k +�B�MC-Ci = �B�Cpt-I. +5.3 +Overview +Fig. 6 gives an overview of the relations between all proposed semantics. Boxes +represent the set of distributions induced by the respective semantics and arrows +stand for set inclusion. For the non-trivial inclusions the arrows are annotated +with the respective lemma or theorem. As an example, Cpt→wCpt states that +�B�Cpt ⊆ �B�wCpt holds for all GBNs B. The three semantics in the top row +parameterized with a cutset C and a distribution γ stand for the singleton set +containing the respective function applied to γ, i.e., �B�Lim-C-γ = {Lim(B, C, γ)}. +� +C MC-C stands for the intersection of the Markov chain semantics for various +cutsets as in Lemma 7, and the incoming arrow from Cpt-IC holds for the set +of independencies IC as in Lemma 8. + +18 +C. Baier et al. +Lim-C-γ +LimAvg-C-γ +MC-C-γ +Lim-C +LimAvg-C +MC-C +� +C MC-C +Cpt-IC +Cpt +wCpt-IC +wCpt +C.2 +L.6 +L.7 +L.8 +L.3 +L.3 +T.1 +Lim-C-γ +LimAvg-C-γ +MC-C-γ +Lim-C +LimAvg-C +MC-C +� +C MC-C +Cpt-IC +Cpt +wCpt-IC +wCpt +Fig. 6: Relations between different variations of limit, limit average, and Markov +chain semantics (blue) as well as strong and weak CPT-consistency semantics +(yellow resp. orange) +6 +Related Work +That cycles in a BN might be unavoidable when learning its structure is well +known for more than 30 years [15,22]. During the learning process of BNs, cy- +cles might even be favorable as demonstrated in the context of gene regulatory +networks where cyclic structures induce monotonic scores [32]. That work only +discusses learning algorithms, but does not deal with evaluating the joint dis- +tribution of the resulting cyclic BNs. In most applications, however, cycles have +been seen as a phenomenon to be avoided to ease the computation of the joint +distribution in BNs. By an example BN comprising a single isolated cycle, [30] +showed that reversing or removing edges to avoid cycles may reduce the solution +space from infinitely many joint distributions that are (weakly) consistent with +the CPTs to a single one. In this setting, our results on weak CPT-semantics +also provide that wCpt cannot express conditions on the relation of variables +like implications or mutual exclusion. This is rooted in the fact that the solution +space of weak CPT-semantics always contains at least one full joint distribu- +tion with pairwise independent variables. An example where reversing edges led +to satisfactory results has been considered in [3], investigating the impact of +reinforced defects by steel corrosion in concrete structures. +Unfolding cycles up to a bounded depth has been applied in the setting of +a robotic sensor system by [2]. In their use case, only cycles of length two may +appear, and only the nodes appearing on the cycles are implicitly used as cutset +for the unfolding. In [13], the set of all nodes is used for unfolding (correspond- +ing to a cutset C = V in our setting) and subsequent limit construction, but +restricted to cases where the limit exists. +There have been numerous variants of BNs that explicitly or implicitly ad- +dress cyclic dependencies. Dynamic Bayesian networks (DBNs) [19] extend BNs +by an explicit notion of discrete time steps that could break cycles through + +On the Foundations of Cycles in Bayesian Networks +19 +timed ordering of random variables. Cycles in BNs could be translated to the +DBN formalism by introducing a notion of time, e.g., following [13]. Our cutset +approach is orthogonal, choosing a time-abstract view on cycles and treating +them as stabilizing feedback loops. Learning DBNs requires “relatively large +time-series data” [32] and thus, may be computationally demanding. In [18] ac- +tivator random variables break cycles in DBNs to circumvent spurious results in +DBN reasoning when infinitesimal small time steps would be required. +Causal BNs [23] are BNs that impose a meaning on the direction of an +edge in terms of causal dependency. Several approaches have been proposed +to extend causal BNs for modeling feedback loops. In [25], an equilibrium se- +mantics is sketched that is similar to our Markov chain semantics, albeit based +on variable oderings rather than cutsets. Determining independence relations, +Markov properties, and joint distributions are central problems addressed for +cyclic causal BNs [2,5,20,24,29]. Markov properties and joint distributions for +extended versions of causal BNs have been considered recently, e.g., in directed +graphs with hyperedges (HEDGes) [5] and cyclic structural causal models (SCMs) +[2]. Besides others, they show that in presence of cycles, there might be multiple +solutions for a joint distribution or even no solution at all [7]. While we consider +all random variables to be observable, the latter approaches focus on models +with latent variables. Further, while our focus in this paper is not on causality, +our approach is surely also applicable to causal BNs with cycles. +Recursive relational Bayesian networks (RRBNs) [9] allow representing prob- +abilistic relational models where the random variables are given by relations over +varying domains. The resulting first-order dependencies can become quite com- +plex and may contain cycles, though semantics are given only for the acyclic +cases by the construction of corresponding standard BNs. +Bayesian attack graphs (BAGs) [16] are popular to model and reason about +security vulnerabilities in computer networks. Learned graphs and thus their +BN semantics frequently contain cycles, e.g., when using the tool MulVAL [21]. +In [27], “handling cycles correctly” is identified as “a key challenge” in security +risk analysis. Resolution methods for cyclic patterns in BAGs [1,4,17,31] are +mainly based on context-specific security considerations, e.g., to break cycles by +removing edges. The semantic foundations for cyclic BNs laid in this paper do +not require graph manipulations and decouple the probability theoretic basis +from context-specific properties. +7 +Conclusion +This paper has developed a foundational perspective on the semantics of cycles in +Bayesian networks. Constraint-based semantics provide a conservative extension +of the standard BN semantics to the cyclic setting. While conceptually impor- +tant, their practical use is limited by the fact that for many GBNs, the induced +constraint system is unsatisfiable. On the other hand, the two introduced limit +semantics echo in an abstract and formal way what practitioners have been devis- +ing across a manifold of domain-specific situations. In this abstract perspective, + +20 +C. Baier et al. +cutsets are the ingredients that enable a controlled decoupling of dependencies. +The appropriate choice of cutsets is where, in our view, domain-specific knowl- +edge is confined to enter the picture. Utilizing the constructively defined Markov +chain semantics, we established key results relating and demarcating the differ- +ent semantic notions and showed that for the ubiquitous class of smooth GBNs +a unique full joint distribution always exists. +References +1. 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The idea is to show that the dependencies of every possible BN structure +for the initial distribution ι are covered by the closure operation. Let the graph +G⋆ +ι = ⟨Init(G), E⋆⟩ be a DAG that is an I-map for ι, i.e., d-sep(G⋆ +ι ) ⊆ Indep(ι). +Then ι factorizes according to G⋆ +ι , that is for every assignment b ∈ Asg(Init(G)), +we have +ι(b) = +� +X∈Init(G) +ι +� +bX | bPreG⋆ι (X) +� +. +Now consider the BN B⋆ +̸⟳ with graph G⋆ = ⟨V, E ∪ E⋆⟩ where we add the edges +of G⋆ +ι to G. The CPTs for the nodes in V \ Init(G) are given by P whereas the +new CPTs (according to the structure in G⋆ +ι ) for the nodes in Init(G) are derived +from ι. Then for every assignment c ∈ Asg(V): +distBN(B⋆ +̸⟳)(c) = +� +X∈V +Pr +� +cX | cPre(X) +� += +� +X∈Init(G) +ι +� +cX | cPreG⋆ι (X) +� +· +� +X∈V\Init(G) +Pr +� +cX | cPre(X) +� += ι +� +cInit(G) +� +· +� +X∈V\Init(G) +Pr +� +cX | cPre(X) +� += distBN(B̸⟳)(c). +As B⋆ +̸⟳ is a regular BN without an initial distribution, we have d-sep(G⋆) ⊆ +Indep(dist BN(B⋆ +̸⟳)). +We proceed to show d-sep(Close(G)) ⊆ d-sep(G⋆). Let (X ⊥ Y | Z) ∈ +d-sep(Close(G)). Then each path from X to Y in Close(G) is blocked by the +nodes in Z. As Close(G) contains all possible edges between the nodes Init(G) +but G⋆ only a subset thereof, it is clear that each path in G⋆ also exists in +Close(G). Thus, there cannot be an unblocked path from X to Y given Z in G⋆ +either, so (X ⊥ Y | Z) ∈ d-sep(G⋆). Altogether, we have +d-sep +� +Close(G) +� +⊆ d-sep(G⋆) ⊆ Indep +� +dist BN(B⋆ +̸⟳) +� += Indep +� +distBN(B̸⟳) +� +. +⊓⊔ +Lemma 2. Let B be a GBN with cutset C, cutset distribution γ ∈ Dist(Asg(C)), +and M = ⟨Asg(C), P⟩ the cutset Markov chain CMC(B, C). Then the following +statements are equivalent: +(a) γ = γ · P. + +On the Foundations of Cycles in Bayesian Networks +23 +(b) There exists γ0 ∈ Dist(Asg(C)) such that for γi+1 = γi · P, we have +γ = +lim +n→∞ +1 +n+1 +n +� +i=0 +γi. +(c) γ belongs to the convex hull of the long-run frequency distributions lrfD of +the bottom SCCs D of M. +(d) γ = Next(B, C, γ)|C. +Proof. (a) =⇒ (b): If we have γ = γ · P, then statement (b) is obtained by +considering γ0 = γ, as then γi = γ for all i. +(b) =⇒ (c): The proof of the implication relies on the following standard facts +about finite-state Markov chains. Given a BSCC D and an arbitrary distribution +ν0 ∈ Dist(Asg(D)), the distribution lrfD agrees with the Ces`aro limit of the +sequence (νi)i⩾0 where νi+1 = νi · PD and PD denotes the restriction of P to +assignments on D. That is, +lrfD = lim +n→∞ +1 +n+1 +n +� +i=0 +νi. +Vice versa, for γ0 ∈ Dist(Asg(C)) and γi+1 = γi · P, then the Ces`aro limit γ +of the sequence (γi)i⩾0 has the form +γ = +� +D +λ(D) · lrfD +where D ranges over all BSCCs of M, λ(D) is the probability for reaching D in M +with the initial distribution γ0, and all vectors lrfD are padded with zero entries +to range over the whole state space. In particular, γ is a convex combination +of the distributions lrfD as 0 ⩽ λ(D) ⩽ 1 and � +D λ(D) = 1 (because every +finite-state Markov chain almost surely reaches a BSCC). +(c) =⇒ (a): Suppose γ = � +D λ(D)·lrfD where 0 ⩽ λ(D) ⩽ 1, � +D λ(D) = 1, +and each lrfD is padded appropriately as before. Then: +γ · P = +� +D +λ(D) · lrfD · P = +� +D +λ(D) · lrfD = γ +where we use the fact that lrfD = lrfD · P. +(a) ⇐⇒ (d): Because γ can be represented as convex combination of Dirac +distributions as γ = � +c∈Asg(C) γ(c) · Dirac(c), we know: +Next(B, C, γ) = +� +c∈Asg(C) +γ(c) · Next +� +B, C, Dirac(c) +� +. +As P(c, b) = Next +� +B, C, Dirac(c) +� +(b) for any assignment b ∈ Asg(C), and assum- +ing γ = γ · P, we get +Next(B, C, γ)(b) = +� +c∈Asg(C) +γ(c) · P(c, b) = (γ · P)(b) = γ(b). +Conversely, assuming Next(B, C, γ)|C = γ, we yield γ = γ · P. +⊓⊔ + +24 +C. Baier et al. +Lemma 3. Let B be a GBN. Then for any cutset C of B, we have +�B�MC-C = �B�LimAvg-C = �B�Lim-C. +Proof. We have �B�MC-C = �B�LimAvg-C by Theorem 1 and know �B�Lim-C ⊆ +�B�LimAvg-C, so it remains to show �B�MC-C ⊆ �B�Lim-C. Let µ ∈ �B�MC-C. Then +there exists a cutset distribution γ s.t. µ = Extend(B, C, γ). We need to show +there exists an initial distribution γ0 ∈ Dist(Asg(C)) such that γ = limn→∞ γi +where γi+1 = Next(B, C, γi)|C. Let us choose γ0 = γ. Then we know γ0 = +Next(B, C, γ0)|C by Lemma 2, so γi = γ0 for all i ∈ N. Thus, γ = limi→∞ γi and +therefore µ ∈ �B�Lim-C. +⊓⊔ +Lemma 4 (Cardinality). Let B be a GBN with cutset C and cutset Markov +chain CMC(B, C) = ⟨Asg(C), P⟩. Further, let k > 0 denote the number of bottom +SCCs D1, . . . , Dk of CMC(B, C). Then +1. the cardinality of the cutset Markov chain semantics is given by +���B�MC-C +�� = +� +1 +if k = 1, +∞ +if k > 1; +2. Lim(B, C, γ0) is defined for all γ0 ∈ Dist(Asg(C)) if all Di are aperiodic; +3. Lim(B, C, γ) is only defined for stationary distributions γ with γ = γ · P if +Di is periodic for any 1 ⩽ i ⩽ k. +Proof. (1.) By Lemma 2, every cutset distribution with γ = γ · P is a convex +combination of the steady-state distributions for the BSCCs. Thus, for k = 1 +a unique distribution γ exists, whereas for k > 1, there are infinitely many +real-valued distributions in the convex hull. +(2.) A Markov chain is aperiodic if all its BSCCs are aperiodic. Aperiodicity +suffices for the limit limn→∞ γn with γn+1 = γn · P to exist for every γ0. Then +limn→∞ γ′ +n with γ′ +n+1 = Next(B, C, γ′ +n)|C exists as well by Lemma 2. +(3.) Assume some BSCC D is periodic with a period of p. Then, for any γ0 ∈ +Dist(Asg(C)), γn+1 = γn · P, and νn = γn|D, we have νp·n = νn. Now consider +γ0 and γ1 = γ0 · P. If γ0 = γ1, then γ0 = γn for all n ∈ N and γ0 = limn→∞ γn +holds. Otherwise, if γ0 ̸= γ1, the following non-convergent sequence exists: +ν0, ν1, . . . , νp, νp+1, . . . , ν2p, ν2p+1, . . . +Then limn→∞ γn cannot converge either, so Lim(B, C, γ0) is undefined. +⊓⊔ +Lemma 5. Let B be a smooth GBN and C a cutset of B. Then the graph of the +cutset Markov chain CMC(B, C) is a complete digraph. +Proof. The graph of CMC(B, C) = ⟨Asg(C), P⟩ is a complete digraph iff each +entry in P is positive. Thus, for each two assignments b, c ∈ Asg(C), we need to + +On the Foundations of Cycles in Bayesian Networks +25 +show P(b, c) > 0. Let Bb = Dissect(B, C, Dirac(b)). Then from Definition 8, we +have +P(b, c) = Next(Dirac(b), B, C)(c) += distBN(Bb)(c′). +The probability dist BN(Bb)(c′) is given by the sum over all full assignments +v ∈ Asg(V) that agree with c′ on the assignment of the cutset node copies C′. +Further, the sum can be partitioned into those v that agree with assignment b +on C and those that do not: +distBN(Bb)(c′) = +� +v∈Asg(V) +s.t. c′⊂v, b⊂v +distBN(Bb)(v) + +� +v∈Asg(V) +s.t. c⊂v, b̸⊂v +dist BN(Bb)(v). +By the definition of the standard BN-semantics, we have +distBN(Bb)(v) = ι +� +vInit(G) +� +· Dirac(b)(vC) · +� +X∈V\C +Pr +� +vX | vPre(X) +� +. +Now consider the second sum in the previous equation where b ̸⊂ v. For those +assignments, Dirac(b)(vC) = 0 and thus the whole sum equals zero. For the +first sum, we have vC = b, so Dirac(b)(vC) = 1 and we only need to consider +the product with X ∈ V \ C and the initial distribution over Init(G). By the +construction of Bb, the CPTs of all X ∈ V \C are the original CPTs from B, thus +their entries all fall within the open interval ]0, 1[ by the smoothness assumption +of B. The same holds for the value ι +� +vInit(G) +� +. Thus, the whole product resides +in ]0, 1[ as well. Finally, note that the sum is non-empty as C′ and C are disjoint, +so there exists at least one v ∈ Asg(V) with c ⊂ v and b ⊂ v. As a non-empty +sum over values in ]0, 1[ is necessarily positive, we have distBN(Bb)(c′) > 0 and +the claim follows. +⊓⊔ +Corollary 1. The limit semantics of a smooth GBN B is a singleton for every +cutset C of B and Lim(B, C, γ0) is defined for all γ0 ∈ Dist(Asg(C)). +Proof. Follows from Lemma 4 and Lemma 5 because every complete graph forms +a single bottom SCC and is necessarily aperiodic. +⊓⊔ +Lemma 6. Let B be a GBN over nodes V, C ⊆ V a cutset for B, and µ ∈ +�B�MC-C. Then µ is strongly CPT-consistent for all nodes in V\C and weakly +CPT-consistent for the nodes in C. +Proof. By definition, µ = Extend(B, C, γ) for some γ ∈ Dist(Asg(C)) with +γ = γ · P. As Extend(B, C, γ) is the standard BN semantics for the acyclic +BN Dissect(B, C, γ) without the copies of the cutset nodes, CPT-consistency for +the nodes in V \ C follows directly from the CPT-consistency of the standard +semantics for acyclic BNs. + +26 +C. Baier et al. +It remains to prove weak CPT-consistency for the cutset nodes. Let δ = +distBN(Dissect(B, C, γ)) ∈ Dist(Asg(V ∪ C′)). Thus, µ = δ|V and γ = δ|C. Then +for each assignment b ∈ Asg(C), we have +µ(b) = γ(b) = (γ · P)(b) = δ(b′) +where b′ ∈ Asg(C′) is given by b′(Y ′) = b(Y ) for all Y ∈ C. In particular, for each +Y ∈ C: +µ(Y=T) = δ(Y ′=T) +Let D = Asg(Pre(Y )) where Pre(·) refers to the original scGBN. For c ∈ +Asg(C), we write Dc for the set of all assignments d ∈ D that comply with c in +the sense that if Z ∈ C ∩ Pre(Y ) then c(Z) = d(Z). In this case, c and d can be +combined to an assignment for C ∪Pre(Y ). Similarly, if d ∈ D, then the notation +Asgd(C) is used for the set of assignments c ∈ Asg(C) that comply with d. Then: +δ(Y ′=T) = +� +c∈Asg(C) +δ(Y ′=T | c) · µ(c) += +� +c∈Asg(C) +� +d∈Dc +δ(Y ′=T | c, d) +� +�� +� +Pr(Y =T|d) +· δ(d | c) +� �� � +µ(d|c) +· δ(c) +���� +µ(c) += +� +d∈D +Pr(Y =T | d) · +� +c∈Asgd(C) +µ(d | c) · µ(c) += +� +d∈D +Pr(Y =T | d) · µ(d). +Putting everything together, we obtain: +µ(Y =T) = δ(Y ′=T) = +� +d∈D +Pr(Y =T | d) · µ(d). +Thus, µ is weakly CPT-consistent for Y ∈ C. +⊓⊔ +Lemma 7. Let B be a GBN over nodes V and C1, . . . , Ck cutsets of B s.t. for +each node X ∈ V there is an i ∈ {1, . . . , k} with X /∈ Ci. Then +� +0⩽i⩽k +�B�MC-Ci ⊆ �B�Cpt. +Proof. We need to show CPT-consistency for every node under µ ∈ � +i�B�MC-Ci. +Let X ∈ V. Then we choose a cutset Ci s.t. X /∈ Ci and CPT consistency follows +from Lemma 6. +⊓⊔ +Lemma 8. Let B be a GBN with cutset C and IC = d-sep +� +Close(Close(G)[C]) +� +. +Then we have +�B�Cpt-IC ⊆ �B�MC-C. + +On the Foundations of Cycles in Bayesian Networks +27 +Proof. Let µ ∈ �B�Cpt-IC and γ = µ|C. The task is to show that γ satisfies the +fixed point equation γ = γ · P. +The standard BN semantics δ = dist BN(Dissect(B, C, γ)) of the dissected BN +is the unique distribution over Asg(V ∪ C′) that +– is CPT-consistent w.r.t. the conditional probability tables in Dissect(B, C, γ), +– agrees with γ when restricted to the assignments for C, and +– satisfies the conditional independencies in IC. +Consider the distribution ˜µ ∈ Dist +� +Asg(V∪C′) +� +defined as follows for b ∈ Asg(V) +and c′ ∈ Asg(C′): +˜µ(b, c′) := µ(b) · +� +Y∈C +Pr +� +Y=c′(Y ′) | bPre(Y ) +� +. +Then, ˜µ satisfies the above three constraints. Hence, ˜µ = δ. +For c ∈ Asg(C), let c′ ∈ Asg(C′) denote the corresponding assignment with +c′(Y ′) = c(Y ) for Y ∈ C. +(γ · P)(c) = δ(c′) = ˜µ(c′) += +� +d∈Asg(Pre(C)) +µ(d) · +� +Y∈C +Pr(Y=c′(Y ′) | d) +� +�� +� +Pr(Y=c(Y )|d) += µ(c) = γ(c). +Hence, γ = γ · P and µ ∈ �B�MC-C. +⊓⊔ + diff --git a/1dFAT4oBgHgl3EQfkB03/content/tmp_files/load_file.txt b/1dFAT4oBgHgl3EQfkB03/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d9040859d73b8b1d10c97864a047bc39760312e4 --- /dev/null +++ b/1dFAT4oBgHgl3EQfkB03/content/tmp_files/load_file.txt @@ -0,0 +1,907 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf,len=906 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='08608v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='AI] 20 Jan 2023 On the Foundations of Cycles in Bayesian Networks⋆ Christel Baier1, Clemens Dubslaff1, Holger Hermanns2,3, and Nikolai K¨afer1 1 TU Dresden, Dresden, Germany 2 Saarland University, Saarbr¨ucken, Germany 3 Institute of Intelligent Software, Guangzhou, China Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Bayesian networks (BNs) are a probabilistic graphical model widely used for representing expert knowledge and reasoning under un- certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Traditionally, they are based on directed acyclic graphs that capture dependencies between random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' However, directed cy- cles can naturally arise when cross-dependencies between random vari- ables exist, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', for modeling feedback loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Existing methods to deal with such cross-dependencies usually rely on reductions to BNs without cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' These approaches are fragile to generalize, since their justifica- tions are intermingled with additional knowledge about the application context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In this paper, we present a foundational study regarding seman- tics for cyclic BNs that are generic and conservatively extend the cycle- free setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' First, we propose constraint-based semantics that specify requirements for full joint distributions over a BN to be consistent with the local conditional probabilities and independencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Second, two kinds of limit semantics that formalize infinite unfolding approaches are intro- duced and shown to be computable by a Markov chain construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 1 Introduction A Bayesian network (BN) is a probabilistic graphical model representing a set of random variables and their conditional dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' BNs are ubiquitous across many fields where reasoning under uncertainties is of interest [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Specifically, a BN is a directed acyclic graph with the random variables as nodes and edges manifesting conditional dependencies, quantified by conditional probability ta- bles (CPTs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The probability of any random variable can then be deduced by the CPT entries along all its predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Here, these probabilities are indepen- dent of all variables that are no (direct or transitive) predecessors in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Acyclicity is hence crucial and commonly assumed to be rooted in some sort of causality [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' A classical use of BNs is in expert systems [22] where BNs ag- gregate statistical data obtained by several independent studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In the medical ⋆ This work was partially supported by the DFG in projects TRR 248 (CPEC, see https://perspicuous-computing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='science, project ID 389792660) and EXC 2050/1 (CeTI, project ID 390696704, as part of Germany’s Excellence Strategy), and the Key-Area Research and Development Program Grant 2018B010107004 of Guang- dong Province.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 2 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Baier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' X Y X=T X=F F s1 1 − s1 T s2 1 − s2 Y X Y=T Y=F F t1 1 − t1 T t2 1 − t2 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 1: A cyclic GBN with CPTs for X and Y domain, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', they can capture the correlation of certain symptoms, diseases, and human factors [11,15,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Imagine for instance an expert system for supporting diagnosis of Covid- 19, harvesting multiple clinical studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' One study might have investigated the percentage of patients who have been diagnosed with fever also having Covid- 19, while another study in turn might have investigated among the Covid-19 patients whether they have fever, too.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Clearly, both studies investigate the de- pendency between fever and Covid-19, but under different conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Fever may weaken the immune system and could increase the risk of a Covid-19 infection, while Covid-19 itself has fever as a symptom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In case there is uniform knowledge about “which symptom was first” in each of the constituent studies, then dy- namic Bayesian networks (DBNs) [19] could be used as a model for the expert system, breaking the interdependence of fever and Covid-19 through a prece- dence relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' However, this implies either to rely only on studies where these temporal dependencies are clearly identified or to introduce an artificial notion of time that might lead to spurious results [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' A naive encoding into the BN framework always yields a graph structure that contains cycles, as is the case in our small example shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 1 where X and Y stand for the random variables of diagnosing Covid-19 and fever, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' That cycles might be unavoidable has already been observed in seminal pa- pers such as [22,15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' But acyclicity is crucial for computing the joint probability distribution of a BN, and thereby is a prerequisite for, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', routine inference tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Existing literature that considers cycles in BNs mainly recommends re- ducing questions on the probability values to properties in acyclic BNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For instance, in [11] nodes are collapsed towards removing cycles, while [22] suggests to condition on each value combination on a cycle, generating a decomposition into tree-like BNs and then averaging over the results to replace cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Some- times, application-specific methods that restructure the cyclic BN towards an acyclic BN by introducing additional nodes [26,8] or by unrolling cycles up to a bounded depth [17,2] have been reported to give satisfactory results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Other ap- proaches either remove edges that have less influence or reverse edges on cycles (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', [10]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' However, such approaches are highly application dependent and hinge on knowledge about the context of the statistical data used to construct the BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Furthermore, as already pointed out by [30], they usually reduce the solution space of families of joint distributions to a single one, or introduce so- lutions not consistent with the CPTs of the original cyclic BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' While obviously many practitioners have stumbled on the problem how to treat cycles in BNs On the Foundations of Cycles in Bayesian Networks 3 and on the foundational question “What is the meaning of a cyclic BN?”' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', there is very little work on the foundations of Bayesian reasoning with cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In this paper, we approach this question by presenting general semantics for BNs with cycles, together with algorithms to compute families of joint distri- butions for such BNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' First, we investigate how the two main constituents of classical BNs, namely consistency with the CPTs and independencies induced by the graph structure, influence the joint distributions in the presence of cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' This leads to constraints semantics for cyclic BNs that comprise all those joint distributions respecting the constraints, being either a single uniquely defined one, none, or infinitely many distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Second, we present semantics that formalize unfolding approaches and depend on the choice of a cutset, a set of random variables that break every cycle in a cyclic BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Intuitively, such cutsets form the seams along which feedback loops can be unraveled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' These semantics are defined in terms of the limit (or limit average) of a sequence of distribu- tions at descending levels in the infinite unfolding of the BN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We show that the same semantics can be defined using a Markov chain construction and sub- sequent long-run frequency analysis, which enables both precise computation of the semantics and deep insights in the semantics’ behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Among others, an immediate result is that the family of distributions induced with respect to the limit semantics is always non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As we will argue, the limit semantics have obvious relations to a manifold of approaches that have appeared in the literature, yet they have not been spelled out and studied explicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='1 Notation Let V be a set of Boolean random variables4 over the domain B = {F, T}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We usually denote elements of V by X, Y, or Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' An assignment over V is a function b: V → B which we may specify through set notation, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', b = {X=T, Y=F} for b(X) = T and b(Y ) = F, or even more succinctly as XY .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The set of all possible assignments over V is denoted by Asg(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We write bU for the restriction of b to a subset U ⊆ V, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', b{X} = {X=T}, and may omit set braces, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', bX,Y = b{X,Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' A distribution over a set S is a function µ: S → [0, 1] where � s∈S µ(s) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The set of all distributions over S is denoted by Dist(S).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For |S| = n, µ will occasionally be represented as a vector of size n for some fixed order on S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In the following, we are mainly concerned with distributions over assignments, that is distributions µ ∈ Dist(Asg(V)) for some set of random variables V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Each such distribution µ induces a probability measure (also called µ) on 2Asg(V).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Thus, for a set of assignments φ ⊆ Asg(V), we have µ(φ) = � b∈φ µ(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We are often interested in the probability of a partial assignment d ∈ Asg(U) on a subset U ⊊ V of variables, which is given as the probability of the set of all full 4 We use Boolean random variables for simplicity of representation, an extension of the proposed semantics over random variables with arbitrary finite state spaces is certainly possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 4 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Baier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' assignments b ∈ Asg(V) that agree with d on U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As a shorthand, we define µ(d) := µ � {b ∈ Asg(V) : bU = d} � = � b∈Asg(V) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' bU =d µ(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The special case µ(X=T) is called the marginal probability of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The restriction of µ ∈ Dist(Asg(V)) to U, denoted µ|U ∈ Dist(Asg(U)), is given by µ|U(d) := µ(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For a set W disjoint from V and ν ∈ Dist(Asg(W)), the product distribution of µ and ν is given by (µ ⊗ ν)(c) := µ(cV) · ν(cW) for every c ∈ Asg(V ∪ W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' µ is called a Dirac distribution if µ(b) = 1 for some assignment b ∈ Asg(V) and thus µ(c) = 0 for all other assignments c ̸= b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' A Dirac distribution derived from a given assignment b is denoted by Dirac(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Graph Notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For a graph G = ⟨V, E⟩ with nodes V and directed edges E ⊆ V × V, we may represent an edge (X, Y ) ∈ E as X → Y if E is clear from context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Pre(X) := {Y ∈ V : Y → X} denotes the set of parents of a node X ∈ V, and Post∗(X) := {Y ∈ V : X → · · · → Y } is the set of nodes reachable from X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' A node X is called initial if Pre(X) = ∅, and Init(G) is the set of all nodes initial in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' A graph G is strongly connected if each node in V is reachable from every other node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' A set of nodes D is a strongly connected component (SCC) of G if all nodes in D can reach each other and D is not contained in another SCC, and a bottom SCC (BSCC) if no node in V \\ D can be reached from D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Markov Chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' A discrete-time Markov chain (DTMC) is a tuple M = ⟨S, P⟩ where S is a finite set of states and P: S × S → [0, 1] a function such that P(s, ·) ∈ Dist(S) for all states s ∈ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The underlying graph GM = ⟨S, E⟩ is defined by E = {(s, t) ∈ S × S : P(s, t) > 0}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The transient distribution πι n ∈ Dist(S) at step n is defined through the probability πι n(s) to be in state s after n steps if starting with initial state distribution ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' It satisfies (in matrix-vector notation) πι n = ι · Pn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We are also interested in the long-run frequency of state occupancies when n tends to infinity, defined as the Ces`aro limit lrfι : S → [0, 1]: lrfι(s) := lim n→∞ 1 n + 1 n � i=0 πι n(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (LRF) This limit always exists and corresponds to the long-run fraction of time spent in each state [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The limit probability limn→∞ πι n is arguably more intuitive as a measure of the long-run behavior, but may not exist (due to periodicity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In case of existence, it agrees with the Ces`aro limit lrfι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' If GM forms an SCC, the limit is independent of the choice of ι and the superscript can be dropped.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We denote this limit by lrfM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 2 Generalized Bayesian Networks We introduce generalized Bayesian networks (GBNs) as a BN model that does not impose acyclicity and comes with a distribution over initial nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' On the Foundations of Cycles in Bayesian Networks 5 Definition 1 (Generalized BN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' A GBN B is a tuple ⟨G, P, ι⟩ where – G = ⟨V, E⟩ is a directed graph with nodes V and an edge relation E ⊆ V × V, – P is a function that maps all non-initial nodes X ∈ V\\Init(G) paired with each of their parent assignments b ∈ Asg(Pre(X)) to a distribution P(X, b): Asg � {X} � → [0, 1], – ι is a distribution over the assignments for the initial nodes Init(G), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', ι ∈ Dist � Asg(Init(G)) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The distributions P(X, b) have the same role as the entries in a conditional probability table (CPT) for X in classical BNs: they specify the probability for X=T or X=F depending on the assignments of the predecessors of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' To this end, for X ∈ V\\Init(G) and b ∈ Asg(Pre(X)), we also write Pr(X=T | b) for P(X, b)(X=T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In the literature, initial nodes are often assigned a marginal prob- ability via a CPT as well, assuming independence of all initial nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Differently, in our definition of GBNs, it is possible to specify an arbitrary distribution ι over all initial nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' If needed, P can be easily extended to initial nodes by setting P(X, ∅) := ι|{X} for all X ∈ Init(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Hence, classical BNs arise as a special instance of GBNs where the graph G is acyclic and initial nodes are pairwise independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In that case, the CPTs given by P are a compact representation of a single unique full joint distribution dist BN(B) over all random variables X ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For every assignment b ∈ Asg(V), we can compute dist BN(B)(b) by the so-called chain rule: dist BN(B)(b) := ι � bInit(G) � � X∈V\\Init(G) Pr � bX | bPre(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (CR) In light of the semantics introduced later on, we define the standard BN-semantics of an acyclic GBN B as the set �B�BN := {distBN(B)}, and �B�BN := ∅ if B con- tains cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The distribution dist BN(B) satisfies two crucial properties: First, it is con- sistent with the CPT entries given by P and the distribution ι, and second, it observes the independencies encoded in the graph G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In fact, those two properties are sufficient to uniquely characterize distBN(B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We briefly review the notion of independence and formally define CPT consistency later on in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Independence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Any full joint probability distribution µ ∈ Dist(Asg(V)) may in- duce a number of conditional independencies among the random variables in V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For X, Y, and Z disjoint subsets of V, the random variables in X and Y are independent under µ given Z if the conditional probability of each assignment over the nodes in X given an assignment for Z is unaffected by further condi- tioning on any assignment of Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Formally, the set Indep(µ) contains the triple (X, Y, Z) iff for all a ∈ Asg(X), b ∈ Asg(Y), and c ∈ Asg(Z), we have µ(a | b, c) = µ(a | c) or µ(b, c) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We also write (X ⊥ Y | Z) for (X, Y, Z) ∈ Indep(µ) and may omit the set brackets of X, Y, and Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 6 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Baier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' d-separation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For classical BNs, the graph topology encodes independencies that are necessarily satisfied by any full joint distribution regardless of the CPT entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Given two random variables X and Y as well as a set of observed variables Z, then X and Y are conditionally independent given Z if the corresponding nodes in the graph are d-separated given Z [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' To establish d-separation, all simple undirected paths5 between X and Y need to be blocked given Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let W denote such a simple path W0, W1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' , Wk with W0 = X, Wk = Y, and either Wi → Wi+1 or Wi ← Wi+1 for all i < k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then W is blocked given Z if and only if there exists an index i, 0 < i < k, such that one of the following two conditions holds: (1) Wi is in Z and is situated in a chain or a fork in W, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', – Wi−1 → Wi → Wi+1 (forward chain) – Wi−1 ← Wi ← Wi+1 (backward chain) and Wi ∈ Z, – Wi−1 ← Wi → Wi+1 (fork) (2) Wi is in a collider and neither Wi nor any descendant of Wi is in Z, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', – Wi−1 → Wi ← Wi+1 (collider) and Post∗(Wi) ∩ Z = ∅.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Two sets of nodes X and Y are d-separated given a third set Z if for each X ∈ X and Y ∈ Y, X and Y are d-separated given Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Notably, the d-separation criterion is applicable also in presence of cycles [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For a graph G = ⟨V, E⟩ of a GBN, we define the set d-sep(G) as d-sep(G) := � (X, Y, Z) ∈ (2V)3 : X and Y are d-separated given Z � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For acyclic Bayesian networks it is well known that the independencies ev- ident from the standard BN semantics’ distribution include the independencies derived from the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' That is, for acyclic GBNs B̸⟳ = ⟨G, P, ι⟩ where all initial nodes are pairwise independent under ι, we have d-sep(G) ⊆ Indep � dist BN(B̸⟳) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For an arbitrary initial distribution, the above relation does not necessarily hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' However, we can still find a set of independencies that are necessarily observed by the standard BN semantics and thus act as a similar lower bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We do so by assuming the worst case, namely that each initial node depends on every other initial node under ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Formally, given a graph G = ⟨V, E⟩, we define a closure operation Close(·) as follows and compute the set d-sep � Close(G) � : Close(G) := � V, E ∪ {(A, B) for A, B ∈ Init(G), A ̸= B} � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B̸⟳ = ⟨G, P, ι⟩ be an acyclic GBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then d-sep � Close(G) � ⊆ Indep � distBN(B̸⟳) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As intuitively expected, the presence of cycles in G generally reduces the number of graph independencies, though note that also in strongly connected graphs independencies may exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For example, if G is a four-node cycle with nodes W, X, Y, and Z, then d-sep(G) = � (W ⊥ Y | X, Z), (X ⊥ Z | W, Y ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 5 A path is simple if no node occurs twice in the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' “Undirected” in this context means that edges in either direction can occur along the path.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' On the Foundations of Cycles in Bayesian Networks 7 3 Constraints Semantics For classical acyclic BNs there is exactly one distribution that agrees with all CPTs and satisfies the independencies encoded in the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' This distribution can easily be constructed by means of the chain rule (CR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For cyclic GBNs, applying the chain rule towards a full joint distribution is not possible in general, as the result is usually not a valid probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Still, we can look for distributions consistent with a GBN’s CPTs and the independencies derived from its graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Depending on the GBN, we will see that there may be none, exactly one, or even infinitely many distributions fulfilling these constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='1 CPT-consistency We first provide a formal definition of CPT consistency in terms of linear con- straints on full joint distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Definition 2 (Strong and weak CPT-consistency).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN with nodes V and X ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then µ is called strongly CPT-consistent for X in B (or simply CPT-consistent) if for all c ∈ Asg(Pre(X)) µ(X=T, c) = µ(c) · Pr(X=T | c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (Cpt) We say that µ is weakly CPT-consistent for X in B if µ(X=T) = � c∈Asg(Pre(X)) µ(c) · Pr(X=T | c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (wCpt) Intuitively, the constraint (Cpt) is satisfied for µ if the conditional proba- bility µ(X=T | c) equals the entry in the CPT for X under assignment c, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', µ(X=T | c) = Pr(X=T | c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In the weak case (wCpt), only the resulting marginal probability of X needs to agree with the CPTs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Definition 3 (Cpt and wCpt semantics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For a GBN B = ⟨G, P, ι⟩, the CPT-semantics �B�Cpt is the set of all distributions µ ∈ Dist(Asg(V)) where µ|Init(G) = ι and µ is CPT-consistent for every node X ∈ V\\Init(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The weak CPT-semantics �B�wCpt is defined analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Clearly, we have �B�Cpt ⊆ �B�wCpt for all B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The next example shows that depending on the CPT values, the set �B�Cpt may be empty, a singleton, or of infinite cardinality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Example 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' To find CPT-consistent distributions for the GBN from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 1, we construct a system of linear equations whose solutions form distributions µ ∈ Dist � Asg({X, Y }) � , represented as vectors in the space [0, 1]4: \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed s1 0 s1−1 0 0 s2 0 s2−1 t1 t1−1 0 0 0 0 t2 t2−1 1 1 1 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed µXY µXY µXY µXY \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ec \uf8ed 0 0 0 0 1 \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f7 \uf8f8 8 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Baier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' where, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', µXY abbreviates µ(X=T, Y=F).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The first line of the matrix states the (Cpt) constraint for node X and the parent assignment c = {Y=F}: 0 = s1 · µXY + 0 · µXY + (s1−1) · µXY + 0 · µXY µXY = (µXY + µXY ) · s1 µXY = µY · Pr(X=T | Y=F) µ(X=T, c) = µ(c) · Pr(X=T | c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Analogously, the following three rows encode the CPT constraints for X, Y, and their remaining parent assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The last row ensures that solutions are indeed probability distributions satisfying � c µ(c) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The number of solutions for the system now depends on the CPT entries s1, s2, t1, and t2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For s1 = t2 = 0 and s2 = t1 = 1, no solution exists as the first four equations require µ(b) = 0 for all b ∈ Asg({X, Y }), while the last equation ensures µXY + µXY + µXY + µXY = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For s1 = t1 = 0 and s2 = t2 = 1, all distributions with µXY = 1 − µXY and µXY = µXY = 0 are solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Finally, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', for s1 = t1 = 3/4 and s2 = t2 = 1/2, there is exactly one solution with µXY = 1/10 and µ(b) = 3/10 for the other three assignments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='2 Independence-consistency We extend Cpt semantics with a set of independencies that need to be observed by all induced distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Definition 4 (Cpt-I semantics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For a GBN B = ⟨G, P, ι⟩ and a set of inde- pendencies I, the CPT-I semantics �B�Cpt-I is defined as the set of all CPT- consistent distributions µ for which I ⊆ Indep(µ) holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Technically, the distributions in �B�Cpt-I have to fulfill the following polynomial constraints in addition to the CPT-consistency constraints: µ(b) · µ(bW) = µ(b{X}∪W) · µ(bU∪W) (Cpt-I) for each independence (X ⊥ U | W) ∈ I with variable X∈V and sets of variables U, W ⊆ V, and for each assignment b ∈ Asg({X} ∪ U ∪ W).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Note that in case µ(bW) > 0, (Cpt-I) is equivalent to the constraint µ(bX | bU∪W) = µ(bX | bW).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We can now formally state the alternative characterization of the standard BN semantics as the unique CPT-consistent distribution that satisfies the d- separation independencies of the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For each classical BN B with acyclic graph G and I = d-sep(G), we have �B�BN = {distBN(B)} = �B�Cpt-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Thus, the Cpt-I semantics provides a conservative extension of the standard BN semantics to GBNs with cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' However, in practice, its use is limited since there might be no distribution that satisfies all constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In fact, the case where �B�Cpt-I = ∅ is to be expected for most cyclic GBNs, given that the resulting constraint systems tend to be heavily over-determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The next section introduces semantics that follow a more constructive ap- proach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We will see later on in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='1 that the families of distributions induced by these semantics are always non-empty and usually singletons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' On the Foundations of Cycles in Bayesian Networks 9 X Y Z Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 2: The graph of a strongly connected GBN X0 Y0 Z0 X1 Y1 Z1 X2 Y2 Z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (a) Unfolding along all nodes Z0 X1 Y1 Z1 X2 Y2 Z2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (b) Unfolding along the Z nodes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 3: Two infinite unfoldings of the graph in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 2 4 Limit and Limit Average Semantics We first develop the basic ideas underling the semantics by following an example, before giving a formal treatment in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='1 Intuition Consider the GBN B whose graph G is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' One way to get rid of the cycles is to construct an infinite unfolding of B as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In this new graph G∞, each level contains a full copy of the original nodes and corresponds to some n ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For any edge X → Y in the original graph, we add edges Xn → Yn+1 to G∞, such that each edge descends one level deeper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Clearly any graph constructed in this way is acyclic, but this fact alone does not aid in finding a matching distribution since we dearly bought it by giving up finiteness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' However, we can consider what happens when we plug in some initial distribution µ0 over the nodes X0, Y0, and Z0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Looking only at the first two levels, we then get a fully specified acyclic BN by using the CPTs given by P for X1, Y1, and Z1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For this sub-BN, the standard BN semantics yields a full joint distribution over the six nodes from X0 to Z1, which also induces a distribution µ1 over the three nodes at level 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' This procedure can then be repeated to construct a distribution µ2 over the nodes X2, Y2, and Z2, and, more generally, to get a distribution µn+1 given a distribution µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Recall that 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Baier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' each of those distributions can be viewed as vector of size 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Considering the sequence µ0, µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' , the question naturally arises whether a limit exists, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', a distribution/vector µ such that µ = lim n→∞ µn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Example 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Consider the GBN from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 1 with CPT entries s1 = t2 = 1 and s2 = t1 = 0, which intuitively describe the contradictory dependencies “X iff not Y ” and “Y iff X”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For any initial distribution µ0 = ⟨e f g h⟩, the construction informally described above yields the following sequence of distributions µn: µ0 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed e f g h \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8, µ1 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed f h e g \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8, µ2 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed h g f e \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8, µ3 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed g e h f \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8, µ4 = \uf8eb \uf8ec \uf8ec \uf8ec \uf8ed e f g h \uf8f6 \uf8f7 \uf8f7 \uf8f7 \uf8f8, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As µ4 = µ0, the sequence starts to cycle infinitely between the first four distribu- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The series converges for e = f = g = h = 1/4 (in which case the sequence is constant), but does not converge for any other initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The example shows that the existence of the limit depends on the given initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In case no limit exists because some distributions keep repeating without ever converging, it is possible to determine the limit average (or Ces`aro limit) of the sequence: ˜µ = lim n→∞ 1 n + 1 n � i=0 µi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The limit average has three nice properties: First, if the regular limit µ exists, then the limit average ˜µ exists as well and is identical to µ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Second, in our use case, ˜µ in fact always exists for any initial distribution µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' And third, as we will see in Section 5, the limit average corresponds to the long-run frequency of certain Markov chains, which allows us both to explicitly compute and to derive important properties of the limit distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Example 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Continuing Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 2, the limit average of the sequence µ0, µ1, µ2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' is the uniform distribution ˜µ = ⟨1/4 1/4 1/4 1/4⟩, regardless of the choice of µ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Before we formally define the infinite unfolding of GBNs and the resulting limit semantics, there is one more observation to be made.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' To ensure that the unfolded graph G∞ is acyclic, we redirected every edge of the GBN B to point one level deeper, resulting in the graph displayed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 3b, we also get an acyclic unfolded graph by only redirecting the edges originating in the Z nodes to the next level and keeping all other edges on the same level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The relevant property is to pick a set of nodes such that for each cycle in the original GBN B, at least one node in the cycle is contained in the set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We call such sets the cutsets of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' On the Foundations of Cycles in Bayesian Networks 11 Definition 5 (Cutset).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be an GBN with graph G = ⟨V, E⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' A subset C ⊆ V is a cutset for B if every cycle in G contains at least one node from C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Example 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The GBN in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 2 has the following cutsets: {Y }, {Z}, {X, Y }, {X, Z}, {Y, Z}, and {X, Y, Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Note that {X} does not form a cutset as no node from the cycle Y → Z → Y is contained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' So far we implicitly used the set V of all nodes for the unfolding, which always trivially forms a cutset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The following definitions will be parameterized with a cutset, as the choice of cutsets influences the resulting distributions as well as the time complexity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='2 Formal Definition Let Vn := {Xn : X ∈ V} denote the set of nodes on the nth level of the unfolding in G∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For C ⊆ V a cutset of the GBN, the subset of cutset nodes on that level is given by Cn := {Xn ∈ Vn : X ∈ C}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then a distribution γn ∈ Dist(Asg(Cn)) for the cutset nodes in Cn suffices to get a full distribution µn+1 ∈ Dist(Asg(Vn+1)) over all nodes on the next level, n + 1: We look at the graph fragment Gn+1 of G∞ given by the nodes Cn ∪ Vn+1 and their respective edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In this fragment, the cutset nodes are initial, so the cutset distribution γn can be combined with the initial distribution ι to act as new initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For the nodes in Vn+1, the corresponding CPTs as given by P can be used, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', Pn(Xn, ·) = P(X, ·) for Xn ∈ Vn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Putting everything together, we obtain an acyclic GBN Bn+1 = ⟨Gn+1, Pn+1, ι ⊗ γn⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' However, GBNs constructed in this way for each level n > 0 are all isomorphic and only differ in the given cutset distribution γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For simplicity and in light of later use, we thus define a single representative GBN Dissect(B, C, γ) that represents a dissection of B along a given cutset C, with ι ⊗ γ as initial distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Definition 6 (Dissected GBN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B = ⟨G, P, ι⟩ be a GBN with graph G = ⟨V, E⟩ and C ⊆ V a cutset for B with distribution γ ∈ Dist(Asg(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then, the C-dissected GBN Dissect(B, C, γ) is the acyclic GBN ⟨GC, PC, ι ⊗ γ⟩ with graph GC = ⟨V ∪ C′, EC⟩ where – C′ := {X′ : X ∈ C} extends V by fresh copies of all cutset nodes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' – incoming edges to nodes in C are redirected to their copies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', EC := � (X, Y ′) : (X, Y ) ∈ E, Y ∈ C � ∪ � (X, Y ) : (X, Y ) ∈ E, Y /∈ C � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' – the function PC uses the CPT entries given by P for the cutset nodes as entries for their copies and the original entries for all other nodes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', we have PC(Y ′, a) = P(Y, a) for each node Y ′ ∈ C′ and parent assignment a ∈ Asg(Pre(Y ′)), and PC(X, b) = P(X, b) for X ∈ V\\C and b ∈ Asg(Pre(X)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 4 shows two examples of dissections on the GBN of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As any dissected GBN is acyclic by construction, the standard BN semantics yields a full joint distribution over all nodes in V ∪ C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We restrict this distribution to the nodes 12 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Baier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' X Y Z X′ Y ′ Z′ (a) Cutset C = {X, Y, Z} Z X Y Z′ (b) Cutset C = {Z} Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 4: Dissections of the GBN in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 2 for two cutsets in (V \\ C) ∪ C′, as those are the ones on the “next level” of the unfolding, while re-identifying the cutset node copies with the original nodes to get a distribution over V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Formally, we define the distribution Next(B, C, γ) for each assignment b ∈ Asg(V) as Next(B, C, γ)(b) := dist BN � Dissect(B, C, γ) � (b′) where the assignment b′ ∈ Asg � (V\\C) ∪ C′� is given by b′(X) = b(X) for all X ∈ V\\C and b′(Y ′) = b(Y ) for all Y ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In the unfolded GBN, this allows us to get from a cutset distribution γn to the next level distribution µn+1 = Next(B, C, γn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The next cutset distribution γn+1 is then given by restricting the full distribution to the nodes in C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', γn+1 = Next(B, C, γn)|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='6 Vice versa, a cutset distribution γ suffices to recover the full joint distribution over all nodes V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Again using the standard BN semantics of the dissected GBN, we define the distribution Extend(B, C, γ) ∈ Dist(Asg(V)) as Extend(B, C, γ) := dist BN � Dissect(B, C, γ) ��� V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' With these definitions at hand, we can formally define the limit and limit average semantics described in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Definition 7 (Limit and limit average semantics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN over nodes V with cutset C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The limit semantics of B w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' C is the partial function Lim(B, C, ·) : Dist � Asg(C) � ⇀ Dist � Asg(V) � from initial cutset distributions γ0 to full distributions µ = Extend(B, C, γ) where γ = lim n→∞ γn and γn+1 = Next(B, C, γn)|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The set �B�Lim-C is given by the image of Lim(B, C, ·), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', �B�Lim-C := {Lim(B, C, γ0) : γ0 ∈ Dist(Asg(C)) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Lim(B, C, γ0) is defined}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The limit average semantics of B w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' C is the partial function LimAvg(B, C, ·) : Dist � Asg(C) � ⇀ Dist � Asg(V) � 6 Recall that we may view distributions as vectors which allows us to equate distribu- tions over different but isomorphic domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' On the Foundations of Cycles in Bayesian Networks 13 X=T Y=T X=T Y=F X=F Y=T X=F Y=F 3/8 3/8 1/8 1/8 1/2 1/2 3/4 1/4 1 X=T Y=T X=T Y=F X=F Y=T X=F Y=F Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 5: A cutset Markov chain for a cutset C = {X, Y } from γ0 to distributions µ = Extend(B, C, γ) where γ = lim n→∞ 1 n + 1 n � i=0 γn and γn+1 = Next(B, C, γn)|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The set �B�LimAvg-C is likewise given by the image of LimAvg(B, C, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We know that the limit average coincides with the regular limit if the lat- ter exists, so for every initial cutset distribution γ0, we have Lim(B, C, γ0) = LimAvg(B, C, γ0) if Lim(B, C, γ0) is defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Thus, �B�Lim-C ⊆ �B�LimAvg-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 5 Markov Chain Semantics While we gave some motivation for the limit and limit average semantics, their definitions do not reveal an explicit way to compute their member distributions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In this section we introduce the (cutset) Markov chain semantics which offers explicit construction of distributions and is shown to coincide with the limit average semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' It further paves the way for proving several properties of both limit semantics in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' At the core of the cutset Markov chain semantics lies the eponymous cut- set Markov chain which captures how probability mass flows from one cutset assignment to the others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' To this end, the Dirac distributions corresponding to each assignment are used as initial distributions in the dissected GBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' With the Next function we then get a new distribution over all cutset assignments, and the probabilities assigned by this distribution are used as transition probabilities for the Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Definition 8 (Cutset Markov chain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN with cutset C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The cutset Markov chain CMC(B, C) = ⟨Asg(C), P⟩ w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' B and C is a DTMC where the transition matrix P is given for all cutset assignments b, c ∈ Asg(C) by P(b, c) := Next � B, C, Dirac(b) � (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Example 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 5 shows the cutset Markov chain for the GBN from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 1 with CPT entries s1 = 1/4, s2 = 1, t1 = 1/2, t2 = 0, and cutset C = {X, Y }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Exem- plarily, the edge at the bottom from assignment b = {X=F, Y=F} to assignment 14 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Baier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' c = {X=F, Y=T} with label 3/8 is derived as follows: P(b, c) = Next � B, C, Dirac(b) � (c) = dist BN � Dissect(B, C, Dirac(b)) � (c′) = � a∈Asg(VC) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' c′⊆a dist BN � Dissect(B, C, Dirac(b)) � (a) = � a∈Asg(VC) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' c′⊆a Dirac(b)(aX,Y ) · Pr(X′=F | aY ) · Pr(Y ′=T | aX) = Pr(X′=F | Y=F) · Pr(Y ′=T | X=F) = (1 − s1) · t1 = 3/8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Note that in the second-to-last step, in the sum over all full assignments a which agree with the partial assignment c′, only the assignment which also agrees with b remains as for all other assignments we have Dirac(b)(aX,Y ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Given a cutset Markov chain with transition matrix P and an initial cutset distribution γ0, we can compute the uniquely defined long-run frequency distri- bution lrfγ0 (see Section 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then the Markov chain semantics is given by the extension of this distribution over the whole GBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Definition 9 (Markov chain semantics).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN over nodes V with a cutset C ⊆ V and cutset Markov chain CMC(B, C) = ⟨Asg(C), P⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then the Markov chain semantics of B w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' C is the function MCS(B, C, ·) : Dist � Asg(C) � → Dist � Asg(V) � from cutset distributions γ0 to full distributions µ = Extend(B, C, lrfγ0) where lrfγ0 = lim n→∞ 1 n+1 n � i=0 γi and γi+1 = γi · P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The set �B�MC-C is defined as the image of MCS(B, C, ·).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In the following lemma, we give four equivalent characterizations of the long- run frequency distributions of the cutset Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN with cutset C, cutset distribution γ ∈ Dist(Asg(C)), and M = ⟨Asg(C), P⟩ the cutset Markov chain CMC(B, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then the following statements are equivalent: (a) γ = γ · P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (b) There exists γ0 ∈ Dist(Asg(C)) such that for γi+1 = γi · P, we have γ = lim n→∞ 1 n+1 n � i=0 γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (c) γ belongs to the convex hull of the long-run frequency distributions lrfD of the bottom SCCs D of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (d) γ = Next(B, C, γ)|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' On the Foundations of Cycles in Bayesian Networks 15 Following Lemma 2, we can equivalently define the cutset Markov chain se- mantics as the set of extensions of all stationary distributions for P: �B�MC-C := � Extend(B, C, γ) : γ ∈ Dist � Asg(C) � s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' γ = γ · P � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Example 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Continuing Ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 5, there is a unique stationary distribution γ with γ = γ · P for the cutset Markov chain in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 5: γ = ⟨48/121 18/121 40/121 15/121⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As in this case the cutset C = {X, Y } equals the set of all nodes V, we have Extend(B, C, γ) = γ and thus �B�MC-{X,Y } = {γ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As shown by Lemma 2, the behavior of the Next function is captured by multiplication with the transition matrix P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Both the distributions in the limit average semantics and the long-run frequency distributions of the cutset Markov chain are defined in terms of a Ces`aro limit, the former over the sequence of distributions obtained by repeated application of Next, the latter by repeated multiplication with P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Thus, both semantics are equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then for any cutset C of B and initial distribution γ0 ∈ Dist(Asg(C)), we have MCS(B, C, γ0) = LimAvg(B, C, γ0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We know that Lim(B, C, γ0) is not defined for all initial distributions γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' However, the set of all limits that do exist contains exactly the distributions admitted by the Markov chain and limit average semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then for any cutset C of B, we have �B�MC-C = �B�LimAvg-C = �B�Lim-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='1 Properties By the equivalences established in Theorem 1 and Lemma 3, we gain profound insights about the limit and limit average distributions by Markov chain analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As every finite-state Markov chain has at least one stationary distribution, it immediately follows that �B�MC-C—and thus �B�LimAvg-C and �B�Lim-C—is always non-empty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Further, if the cutset Markov chain is irreducible, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', the graph is strongly connected, the stationary distribution is unique and �B�MC-C is a singleton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The existence of the limit semantics for a given initial distribution γ0 hinges on the periodicity of the cutset Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Example 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We return to Example 2 and construct the cutset Markov chain CMC(B, C) = ⟨Asg(C), P⟩ for the (implicitly used) cutset C = {X, Y }: X=T Y=T X=T Y=F X=F Y=T X=F Y=F 1 1 1 1 16 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Baier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The chain is strongly connected and has a period of length 4, which explains the observed behavior that for any initial distribution γ0, we got the sequence γ0, γ1, γ2, γ3, γ0, γ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' This sequence obviously converges only for initial distributions that are station- ary, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', if we have γ0 = γ0 · P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The following lemma summarizes the implications that can be drawn from close inspection of the cutset Markov chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Lemma 4 (Cardinality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN with cutset C and cutset Markov chain CMC(B, C) = ⟨Asg(C), P⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Further, let k > 0 denote the number of bottom SCCs D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' , Dk of CMC(B, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' the cardinality of the cutset Markov chain semantics is given by ���B�MC-C �� = � 1 if k = 1, ∞ if k > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Lim(B, C, γ0) is defined for all γ0 ∈ Dist(Asg(C)) if all Di are aperiodic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Lim(B, C, γ) is only defined for stationary distributions γ with γ = γ · P if Di is periodic for any 1 ⩽ i ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' A handy sufficient (albeit not necessary) criterion for both aperiodicity and the existence of a single bottom SCC in the cutset Markov chain is the absence of zero and one entries in the CPTs and the initial distribution of a GBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Definition 10 (Smooth GBNs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' A GBN B = ⟨G, P, ι⟩ is called smooth iff all CPT entries as given by P and all values in ι are in the open interval ]0, 1[.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a smooth GBN and C a cutset of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then the graph of the cutset Markov chain CMC(B, C) is a complete digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The limit semantics of a smooth GBN B is a singleton for every cutset C of B and Lim(B, C, γ0) is defined for all γ0 ∈ Dist(Asg(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As noted in [14], one rarely needs to assign a probability of zero (or, con- versely, of one) in real-world applications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' and doing so in cases where some event is extremely unlikely but not impossible is a common modeling error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' This observation gives reason to expect that most GBNs encountered in practice are smooth, and their semantics is thus, in a sense, well-behaved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='2 Relation to Constraints Semantics We take a closer look at how the cutset semantics relates to the CPT-consistency semantics defined in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' CPTs of nodes outside cutsets remain unaffected in the dissected BNs from which the Markov chain semantics is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Since there are cyclic GBNs for which no CPT-consistent distribution exists (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Ex- ample 1) while Markov chain semantics always yields at least one solution due to Lemma 4, it cannot be expected that cutset nodes are necessarily CPT- consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' However, they are always weakly CPT-consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' On the Foundations of Cycles in Bayesian Networks 17 Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN over nodes V, C ⊆ V a cutset for B, and µ ∈ �B�MC-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then µ is strongly CPT-consistent for all nodes in V\\C and weakly CPT-consistent for the nodes in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The lemma shows a way to find fully CPT consistent distributions: Consider there is a distribution µ ∈ �B�MC-C ∩ �B�MC-D for two disjoint cutsets C and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then by Lemma 6 the nodes in V \\ C and V \\ D are CPT consistent, so in fact µ is CPT consistent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In general, we get the following result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN over nodes V and C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' , Ck cutsets of B s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' for each node X ∈ V there is an i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' , k} with X /∈ Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then � 0⩽i⩽k �B�MC-Ci ⊆ �B�Cpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We take a look at which independencies are necessarily observed by the distributions in �B�MC-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let γ ∈ Dist(Asg(C)) be the cutset distribution and let G[C] denote the graph of Dissect(B, C, γ) restricted to the nodes in V such that the cutset nodes in C are initial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then by Lemma 1, the d-separation in- dependencies of the closure of G[C] hold in all distributions µ ∈ �B�MC-C, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', d-sep � Close(G[C]) � ⊆ Indep(µ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The next lemma states that any Cpt-consistent distribution that satisfies these independence constraints for some cutset C also belongs to �B�MC-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN with cutset C and IC = Close(G[C]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then we have �B�Cpt-IC ⊆ �B�MC-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Combining Lemma 7 and Lemma 8 yields the following equivalence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For a GBN B with cutsets C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' , Ck as in Lemma 7 and the independence set I = � 0⩽i⩽k Close(G[Ci]), we have � 0⩽i⩽k �B�MC-Ci = �B�Cpt-I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='3 Overview Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 6 gives an overview of the relations between all proposed semantics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Boxes represent the set of distributions induced by the respective semantics and arrows stand for set inclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For the non-trivial inclusions the arrows are annotated with the respective lemma or theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As an example, Cpt→wCpt states that �B�Cpt ⊆ �B�wCpt holds for all GBNs B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The three semantics in the top row parameterized with a cutset C and a distribution γ stand for the singleton set containing the respective function applied to γ, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', �B�Lim-C-γ = {Lim(B, C, γ)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' � C MC-C stands for the intersection of the Markov chain semantics for various cutsets as in Lemma 7, and the incoming arrow from Cpt-IC holds for the set of independencies IC as in Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 18 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Baier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Lim-C-γ LimAvg-C-γ MC-C-γ Lim-C LimAvg-C MC-C � C MC-C Cpt-IC Cpt wCpt-IC wCpt C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='2 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='6 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='7 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='8 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='3 L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='3 T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='1 Lim-C-γ LimAvg-C-γ MC-C-γ Lim-C LimAvg-C MC-C � C MC-C Cpt-IC Cpt wCpt-IC wCpt Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 6: Relations between different variations of limit, limit average, and Markov chain semantics (blue) as well as strong and weak CPT-consistency semantics (yellow resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' orange) 6 Related Work That cycles in a BN might be unavoidable when learning its structure is well known for more than 30 years [15,22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' During the learning process of BNs, cy- cles might even be favorable as demonstrated in the context of gene regulatory networks where cyclic structures induce monotonic scores [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' That work only discusses learning algorithms, but does not deal with evaluating the joint dis- tribution of the resulting cyclic BNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In most applications, however, cycles have been seen as a phenomenon to be avoided to ease the computation of the joint distribution in BNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' By an example BN comprising a single isolated cycle, [30] showed that reversing or removing edges to avoid cycles may reduce the solution space from infinitely many joint distributions that are (weakly) consistent with the CPTs to a single one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In this setting, our results on weak CPT-semantics also provide that wCpt cannot express conditions on the relation of variables like implications or mutual exclusion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' This is rooted in the fact that the solution space of weak CPT-semantics always contains at least one full joint distribu- tion with pairwise independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' An example where reversing edges led to satisfactory results has been considered in [3], investigating the impact of reinforced defects by steel corrosion in concrete structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Unfolding cycles up to a bounded depth has been applied in the setting of a robotic sensor system by [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In their use case, only cycles of length two may appear, and only the nodes appearing on the cycles are implicitly used as cutset for the unfolding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In [13], the set of all nodes is used for unfolding (correspond- ing to a cutset C = V in our setting) and subsequent limit construction, but restricted to cases where the limit exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' There have been numerous variants of BNs that explicitly or implicitly ad- dress cyclic dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Dynamic Bayesian networks (DBNs) [19] extend BNs by an explicit notion of discrete time steps that could break cycles through On the Foundations of Cycles in Bayesian Networks 19 timed ordering of random variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Cycles in BNs could be translated to the DBN formalism by introducing a notion of time, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', following [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Our cutset approach is orthogonal, choosing a time-abstract view on cycles and treating them as stabilizing feedback loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Learning DBNs requires “relatively large time-series data” [32] and thus, may be computationally demanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In [18] ac- tivator random variables break cycles in DBNs to circumvent spurious results in DBN reasoning when infinitesimal small time steps would be required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Causal BNs [23] are BNs that impose a meaning on the direction of an edge in terms of causal dependency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Several approaches have been proposed to extend causal BNs for modeling feedback loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In [25], an equilibrium se- mantics is sketched that is similar to our Markov chain semantics, albeit based on variable oderings rather than cutsets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Determining independence relations, Markov properties, and joint distributions are central problems addressed for cyclic causal BNs [2,5,20,24,29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Markov properties and joint distributions for extended versions of causal BNs have been considered recently, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', in directed graphs with hyperedges (HEDGes) [5] and cyclic structural causal models (SCMs) [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Besides others, they show that in presence of cycles, there might be multiple solutions for a joint distribution or even no solution at all [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' While we consider all random variables to be observable, the latter approaches focus on models with latent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Further, while our focus in this paper is not on causality, our approach is surely also applicable to causal BNs with cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Recursive relational Bayesian networks (RRBNs) [9] allow representing prob- abilistic relational models where the random variables are given by relations over varying domains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The resulting first-order dependencies can become quite com- plex and may contain cycles, though semantics are given only for the acyclic cases by the construction of corresponding standard BNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Bayesian attack graphs (BAGs) [16] are popular to model and reason about security vulnerabilities in computer networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Learned graphs and thus their BN semantics frequently contain cycles, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', when using the tool MulVAL [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In [27], “handling cycles correctly” is identified as “a key challenge” in security risk analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Resolution methods for cyclic patterns in BAGs [1,4,17,31] are mainly based on context-specific security considerations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', to break cycles by removing edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The semantic foundations for cyclic BNs laid in this paper do not require graph manipulations and decouple the probability theoretic basis from context-specific properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 7 Conclusion This paper has developed a foundational perspective on the semantics of cycles in Bayesian networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Constraint-based semantics provide a conservative extension of the standard BN semantics to the cyclic setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' While conceptually impor- tant, their practical use is limited by the fact that for many GBNs, the induced constraint system is unsatisfiable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' On the other hand, the two introduced limit semantics echo in an abstract and formal way what practitioners have been devis- ing across a manifold of domain-specific situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In this abstract perspective, 20 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Baier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' cutsets are the ingredients that enable a controlled decoupling of dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The appropriate choice of cutsets is where, in our view, domain-specific knowl- edge is confined to enter the picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Utilizing the constructively defined Markov chain semantics, we established key results relating and demarcating the differ- ent semantic notions and showed that for the ubiquitous class of smooth GBNs a unique full joint distribution always exists.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', Gayraud, G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=': Structure learning of bayesian networks involving cyclic structures (2020) 22 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Baier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' A Appendix The appendix contains the proofs omitted from the body of the submission “On the Foundations of Cycles in Bayesian Networks” due to space constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B̸⟳ = ⟨G, P, ι⟩ be an acyclic GBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then d-sep � Close(G) � ⊆ Indep � distBN(B̸⟳) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The idea is to show that the dependencies of every possible BN structure for the initial distribution ι are covered by the closure operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let the graph G⋆ ι = ⟨Init(G), E⋆⟩ be a DAG that is an I-map for ι, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=', d-sep(G⋆ ι ) ⊆ Indep(ι).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then ι factorizes according to G⋆ ι , that is for every assignment b ∈ Asg(Init(G)), we have ι(b) = � X∈Init(G) ι � bX | bPreG⋆ι (X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Now consider the BN B⋆ ̸⟳ with graph G⋆ = ⟨V, E ∪ E⋆⟩ where we add the edges of G⋆ ι to G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The CPTs for the nodes in V \\ Init(G) are given by P whereas the new CPTs (according to the structure in G⋆ ι ) for the nodes in Init(G) are derived from ι.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then for every assignment c ∈ Asg(V): distBN(B⋆ ̸⟳)(c) = � X∈V Pr � cX | cPre(X) � = � X∈Init(G) ι � cX | cPreG⋆ι (X) � � X∈V\\Init(G) Pr � cX | cPre(X) � = ι � cInit(G) � � X∈V\\Init(G) Pr � cX | cPre(X) � = distBN(B̸⟳)(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As B⋆ ̸⟳ is a regular BN without an initial distribution, we have d-sep(G⋆) ⊆ Indep(dist BN(B⋆ ̸⟳)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We proceed to show d-sep(Close(G)) ⊆ d-sep(G⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let (X ⊥ Y | Z) ∈ d-sep(Close(G)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then each path from X to Y in Close(G) is blocked by the nodes in Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As Close(G) contains all possible edges between the nodes Init(G) but G⋆ only a subset thereof, it is clear that each path in G⋆ also exists in Close(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Thus, there cannot be an unblocked path from X to Y given Z in G⋆ either, so (X ⊥ Y | Z) ∈ d-sep(G⋆).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Altogether, we have d-sep � Close(G) � ⊆ d-sep(G⋆) ⊆ Indep � dist BN(B⋆ ̸⟳) � = Indep � distBN(B̸⟳) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' ⊓⊔ Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN with cutset C, cutset distribution γ ∈ Dist(Asg(C)), and M = ⟨Asg(C), P⟩ the cutset Markov chain CMC(B, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then the following statements are equivalent: (a) γ = γ · P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' On the Foundations of Cycles in Bayesian Networks 23 (b) There exists γ0 ∈ Dist(Asg(C)) such that for γi+1 = γi · P, we have γ = lim n→∞ 1 n+1 n � i=0 γi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (c) γ belongs to the convex hull of the long-run frequency distributions lrfD of the bottom SCCs D of M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (d) γ = Next(B, C, γ)|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (a) =⇒ (b): If we have γ = γ · P, then statement (b) is obtained by considering γ0 = γ, as then γi = γ for all i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (b) =⇒ (c): The proof of the implication relies on the following standard facts about finite-state Markov chains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Given a BSCC D and an arbitrary distribution ν0 ∈ Dist(Asg(D)), the distribution lrfD agrees with the Ces`aro limit of the sequence (νi)i⩾0 where νi+1 = νi · PD and PD denotes the restriction of P to assignments on D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' That is, lrfD = lim n→∞ 1 n+1 n � i=0 νi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Vice versa, for γ0 ∈ Dist(Asg(C)) and γi+1 = γi · P, then the Ces`aro limit γ of the sequence (γi)i⩾0 has the form γ = � D λ(D) · lrfD where D ranges over all BSCCs of M, λ(D) is the probability for reaching D in M with the initial distribution γ0, and all vectors lrfD are padded with zero entries to range over the whole state space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In particular, γ is a convex combination of the distributions lrfD as 0 ⩽ λ(D) ⩽ 1 and � D λ(D) = 1 (because every finite-state Markov chain almost surely reaches a BSCC).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (c) =⇒ (a): Suppose γ = � D λ(D)·lrfD where 0 ⩽ λ(D) ⩽ 1, � D λ(D) = 1, and each lrfD is padded appropriately as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then: γ · P = � D λ(D) · lrfD · P = � D λ(D) · lrfD = γ where we use the fact that lrfD = lrfD · P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (a) ⇐⇒ (d): Because γ can be represented as convex combination of Dirac distributions as γ = � c∈Asg(C) γ(c) · Dirac(c), we know: Next(B, C, γ) = � c∈Asg(C) γ(c) · Next � B, C, Dirac(c) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As P(c, b) = Next � B, C, Dirac(c) � (b) for any assignment b ∈ Asg(C), and assum- ing γ = γ · P, we get Next(B, C, γ)(b) = � c∈Asg(C) γ(c) · P(c, b) = (γ · P)(b) = γ(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Conversely, assuming Next(B, C, γ)|C = γ, we yield γ = γ · P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' ⊓⊔ 24 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Baier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Lemma 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then for any cutset C of B, we have �B�MC-C = �B�LimAvg-C = �B�Lim-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We have �B�MC-C = �B�LimAvg-C by Theorem 1 and know �B�Lim-C ⊆ �B�LimAvg-C, so it remains to show �B�MC-C ⊆ �B�Lim-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let µ ∈ �B�MC-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then there exists a cutset distribution γ s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' µ = Extend(B, C, γ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We need to show there exists an initial distribution γ0 ∈ Dist(Asg(C)) such that γ = limn→∞ γi where γi+1 = Next(B, C, γi)|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let us choose γ0 = γ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then we know γ0 = Next(B, C, γ0)|C by Lemma 2, so γi = γ0 for all i ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Thus, γ = limi→∞ γi and therefore µ ∈ �B�Lim-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' ⊓⊔ Lemma 4 (Cardinality).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN with cutset C and cutset Markov chain CMC(B, C) = ⟨Asg(C), P⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Further, let k > 0 denote the number of bottom SCCs D1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' , Dk of CMC(B, C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' the cardinality of the cutset Markov chain semantics is given by ���B�MC-C �� = � 1 if k = 1, ∞ if k > 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Lim(B, C, γ0) is defined for all γ0 ∈ Dist(Asg(C)) if all Di are aperiodic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Lim(B, C, γ) is only defined for stationary distributions γ with γ = γ · P if Di is periodic for any 1 ⩽ i ⩽ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=') By Lemma 2, every cutset distribution with γ = γ · P is a convex combination of the steady-state distributions for the BSCCs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Thus, for k = 1 a unique distribution γ exists, whereas for k > 1, there are infinitely many real-valued distributions in the convex hull.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=') A Markov chain is aperiodic if all its BSCCs are aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Aperiodicity suffices for the limit limn→∞ γn with γn+1 = γn · P to exist for every γ0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then limn→∞ γ′ n with γ′ n+1 = Next(B, C, γ′ n)|C exists as well by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=') Assume some BSCC D is periodic with a period of p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then, for any γ0 ∈ Dist(Asg(C)), γn+1 = γn · P, and νn = γn|D, we have νp·n = νn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Now consider γ0 and γ1 = γ0 · P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' If γ0 = γ1, then γ0 = γn for all n ∈ N and γ0 = limn→∞ γn holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Otherwise, if γ0 ̸= γ1, the following non-convergent sequence exists: ν0, ν1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' , νp, νp+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' , ν2p, ν2p+1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then limn→∞ γn cannot converge either, so Lim(B, C, γ0) is undefined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' ⊓⊔ Lemma 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a smooth GBN and C a cutset of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then the graph of the cutset Markov chain CMC(B, C) is a complete digraph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The graph of CMC(B, C) = ⟨Asg(C), P⟩ is a complete digraph iff each entry in P is positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Thus, for each two assignments b, c ∈ Asg(C), we need to On the Foundations of Cycles in Bayesian Networks 25 show P(b, c) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let Bb = Dissect(B, C, Dirac(b)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then from Definition 8, we have P(b, c) = Next(Dirac(b), B, C)(c) = distBN(Bb)(c′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The probability dist BN(Bb)(c′) is given by the sum over all full assignments v ∈ Asg(V) that agree with c′ on the assignment of the cutset node copies C′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Further, the sum can be partitioned into those v that agree with assignment b on C and those that do not: distBN(Bb)(c′) = � v∈Asg(V) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' c′⊂v, b⊂v distBN(Bb)(v) + � v∈Asg(V) s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' c⊂v, b̸⊂v dist BN(Bb)(v).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' By the definition of the standard BN-semantics, we have distBN(Bb)(v) = ι � vInit(G) � Dirac(b)(vC) · � X∈V\\C Pr � vX | vPre(X) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Now consider the second sum in the previous equation where b ̸⊂ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For those assignments, Dirac(b)(vC) = 0 and thus the whole sum equals zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For the first sum, we have vC = b, so Dirac(b)(vC) = 1 and we only need to consider the product with X ∈ V \\ C and the initial distribution over Init(G).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' By the construction of Bb, the CPTs of all X ∈ V \\C are the original CPTs from B, thus their entries all fall within the open interval ]0, 1[ by the smoothness assumption of B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The same holds for the value ι � vInit(G) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Thus, the whole product resides in ]0, 1[ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Finally, note that the sum is non-empty as C′ and C are disjoint, so there exists at least one v ∈ Asg(V) with c ⊂ v and b ⊂ v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As a non-empty sum over values in ]0, 1[ is necessarily positive, we have distBN(Bb)(c′) > 0 and the claim follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' ⊓⊔ Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The limit semantics of a smooth GBN B is a singleton for every cutset C of B and Lim(B, C, γ0) is defined for all γ0 ∈ Dist(Asg(C)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Follows from Lemma 4 and Lemma 5 because every complete graph forms a single bottom SCC and is necessarily aperiodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' ⊓⊔ Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN over nodes V, C ⊆ V a cutset for B, and µ ∈ �B�MC-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then µ is strongly CPT-consistent for all nodes in V\\C and weakly CPT-consistent for the nodes in C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' By definition, µ = Extend(B, C, γ) for some γ ∈ Dist(Asg(C)) with γ = γ · P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' As Extend(B, C, γ) is the standard BN semantics for the acyclic BN Dissect(B, C, γ) without the copies of the cutset nodes, CPT-consistency for the nodes in V \\ C follows directly from the CPT-consistency of the standard semantics for acyclic BNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' 26 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Baier et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' It remains to prove weak CPT-consistency for the cutset nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let δ = distBN(Dissect(B, C, γ)) ∈ Dist(Asg(V ∪ C′)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Thus, µ = δ|V and γ = δ|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then for each assignment b ∈ Asg(C), we have µ(b) = γ(b) = (γ · P)(b) = δ(b′) where b′ ∈ Asg(C′) is given by b′(Y ′) = b(Y ) for all Y ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In particular, for each Y ∈ C: µ(Y=T) = δ(Y ′=T) Let D = Asg(Pre(Y )) where Pre(·) refers to the original scGBN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For c ∈ Asg(C), we write Dc for the set of all assignments d ∈ D that comply with c in the sense that if Z ∈ C ∩ Pre(Y ) then c(Z) = d(Z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' In this case, c and d can be combined to an assignment for C ∪Pre(Y ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Similarly, if d ∈ D, then the notation Asgd(C) is used for the set of assignments c ∈ Asg(C) that comply with d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then: δ(Y ′=T) = � c∈Asg(C) δ(Y ′=T | c) · µ(c) = � c∈Asg(C) � d∈Dc δ(Y ′=T | c, d) � �� � Pr(Y =T|d) δ(d | c) � �� � µ(d|c) δ(c) ���� µ(c) = � d∈D Pr(Y =T | d) · � c∈Asgd(C) µ(d | c) · µ(c) = � d∈D Pr(Y =T | d) · µ(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Putting everything together, we obtain: µ(Y =T) = δ(Y ′=T) = � d∈D Pr(Y =T | d) · µ(d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Thus, µ is weakly CPT-consistent for Y ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' ⊓⊔ Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN over nodes V and C1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' , Ck cutsets of B s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' for each node X ∈ V there is an i ∈ {1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' , k} with X /∈ Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then � 0⩽i⩽k �B�MC-Ci ⊆ �B�Cpt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' We need to show CPT-consistency for every node under µ ∈ � i�B�MC-Ci.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let X ∈ V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then we choose a cutset Ci s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' X /∈ Ci and CPT consistency follows from Lemma 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' ⊓⊔ Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let B be a GBN with cutset C and IC = d-sep � Close(Close(G)[C]) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then we have �B�Cpt-IC ⊆ �B�MC-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' On the Foundations of Cycles in Bayesian Networks 27 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Let µ ∈ �B�Cpt-IC and γ = µ|C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The task is to show that γ satisfies the fixed point equation γ = γ · P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' The standard BN semantics δ = dist BN(Dissect(B, C, γ)) of the dissected BN is the unique distribution over Asg(V ∪ C′) that – is CPT-consistent w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' the conditional probability tables in Dissect(B, C, γ), – agrees with γ when restricted to the assignments for C, and – satisfies the conditional independencies in IC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Consider the distribution ˜µ ∈ Dist � Asg(V∪C′) � defined as follows for b ∈ Asg(V) and c′ ∈ Asg(C′): ˜µ(b, c′) := µ(b) · � Y∈C Pr � Y=c′(Y ′) | bPre(Y ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Then, ˜µ satisfies the above three constraints.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Hence, ˜µ = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' For c ∈ Asg(C), let c′ ∈ Asg(C′) denote the corresponding assignment with c′(Y ′) = c(Y ) for Y ∈ C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' (γ · P)(c) = δ(c′) = ˜µ(c′) = � d∈Asg(Pre(C)) µ(d) · � Y∈C Pr(Y=c′(Y ′) | d) � �� � Pr(Y=c(Y )|d) = µ(c) = γ(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' Hence, γ = γ · P and µ ∈ �B�MC-C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} +page_content=' ⊓⊔' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1dFAT4oBgHgl3EQfkB03/content/2301.08608v1.pdf'} diff --git a/1tFST4oBgHgl3EQfWzjN/content/tmp_files/2301.13782v1.pdf.txt b/1tFST4oBgHgl3EQfWzjN/content/tmp_files/2301.13782v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..53548c047df1c80de6756f41eb775c2175e5681b --- /dev/null +++ b/1tFST4oBgHgl3EQfWzjN/content/tmp_files/2301.13782v1.pdf.txt @@ -0,0 +1,1803 @@ +Active Nematic Multipoles: Flow Responses and the Dynamics of Defects and Colloids +Alexander J.H. Houston1 and Gareth P. Alexander1, 2, ∗ +1Department of Physics, Gibbet Hill Road, University of Warwick, Coventry, CV4 7AL, United Kingdom. +2Centre for Complexity Science, Zeeman Building, +University of Warwick, Coventry, CV4 7AL, United Kingdom. +(Dated: Wednesday 1st February, 2023) +We introduce a general description of localised distortions in active nematics using the framework +of active nematic multipoles. +We give the Stokesian flows for arbitrary multipoles in terms of +differentiation of a fundamental flow response and describe them explicitly up to quadrupole order. +We also present the response in terms of the net active force and torque associated to the multipole. +This allows the identification of the dipolar and quadrupolar distortions that generate self-propulsion +and self-rotation respectively and serves as a guide for the design of arbitrary flow responses. Our +results can be applied to both defect loops in three-dimensional active nematics and to systems with +colloidal inclusions. They reveal the geometry-dependence of the self-dynamics of defect loops and +provide insights into how colloids might be designed to achieve propulsive or rotational dynamics, +and more generally for the extraction of work from active nematics. Finally, we extend our analysis +also to two dimensions and to systems with chiral active stresses. +I. +INTRODUCTION +Active liquid crystals model a wide range of materials, both biological and synthetic [1–3], including cell mono- +layers [4], tissues [5], bacteria in liquid crystalline environments [6] and bacterial suspensions [7], and synthetic +suspensions of microtubules [8]. +Nematic and polar phases have been the focus of attention but smectic [9, 10], +cholesteric [11, 12] and hexatic [13] phases have also been considered. Key features and motifs of the active nematic +state include self-propelled topological defects [14–16], spontaneous flows and vortices, and on how these may be +controlled through boundary conditions, confinement [17–19], external fields, geometry or topology. Active defects, +in particular, have been related to processes of apoptosis in epithelial sheets [5], tissue dynamics, bacterial spreading +and biofilm formation, and morphogenesis in Hydra [20]. +In three-dimensional active nematics the fundamental excitations are defect loops and system-spanning lines [21, 22]. +The defect loops actively self-propel [23], and self-orient [24], in addition to undergoing deformations in shape. Their +finite extent means that they represent localised distortions to the nematic director, on scales larger than their size, +and this facilitates a description through elastic multipoles [24]. It also invites comparison with colloidal inclusions in +passive liquid crystals, which create localised realignments of the director and act as elastic multipoles [25–27]. These +multipole distortions mediate interactions between colloids and allow for a means of controlling both the colloidal +inclusions and the host material. For instance, they facilitate self-assembly and the formation of metamaterials [28, 29], +and enable novel control of topological defects [27, 30, 31]. While there have been studies of active nematic droplets +in a host passive liquid crystal [32, 33], colloidal inclusions in host active nematics have not been looked at previously. +The multipole approach to describing colloidal inclusions and localised director distortions in general, offers an +equally fruitful paradigm in active nematics. Here, we present a generic analysis of the active flows generated by +multipole director distortions in an active nematic and predict that the presence of colloids transforms their behaviour +similarly to the passive case. These active multipole flows represent the responses of the active nematic both to +localised features, such as defect loops, and to colloidal inclusions. This allows us to identify those distortions which +produce directed or rotational flows and show that such distortions may be naturally induced by colloids. We also +characterise the response in terms of the active forces and torques that they induce. This general connection can +serve as a guide for using colloidal inclusions as a means to control active nematics, or how to design them to engineer +a desired response, or extract work. The properties of inclusions have been studied in scalar active matter [34], as +have active droplets in passive nematics [35], but while there have been specific demonstrations of propulsive colloids +[36, 37] the general responses of inclusions in active nematics have not previously been considered. Understanding +how such responses relate to local manipulations and molecular fields in active nematics will bring both fundamental +insights and the potential for control of active metamaterials. +The remainder of this paper is structured as follows. In Section 2 we briefly review the equations of active nemato- +hydrodynamics and describe the regime in which our linear multipole approach applies. In Section 3 we present these +∗ G.P.Alexander@warwick.ac.uk +arXiv:2301.13782v1 [cond-mat.soft] 31 Jan 2023 + +2 +multipoles as complex derivatives acting on 1/r, showing how this naturally elucidates their symmetries. In Section +4 we show that the linear active response to a harmonic distortion is generated by the same complex derivatives +acting on fundamental flow and pressure solutions and highlight certain examples that illustrate the self-propulsive +and rotational dynamics that can arise. We then show in Section 5 that these phenomenological responses can be +discerned from integrals of the active stress, allowing the identification of the distortion which produces propulsion +along or rotation about a given axis. Sections 6 and 7 contain extensions of our approach, first to two-dimensional +systems and then to those with chiral active stresses. Section 8 gives a discussion and summary. +II. +HYDRODYNAMICS OF ACTIVE NEMATICS +We summarise the hydrodynamics of active nematics as described by their director field n and fluid velocity u. The +fluid flow satisfies the continuity ∂iui = 0 and Stokes ∂jσij = 0 equations, with stress tensor [1–3] +σij = −pδij + 2µDij + ν +2 +� +nihj + hinj +� ++ 1 +2 +� +nihj − hinj +� ++ σE +ij − ζninj. +(1) +Here, p is the pressure, µ is the viscosity, Dij = 1 +2(∂iuj + ∂jui) is the symmetric part of the velocity gradients, ν is +the flow alignment parameter, hi = −δF/δni is the molecular field associated with the Frank free energy F, σE +ij is the +Ericksen stress, and ζ is the magnitude of the activity. The active nematic is extensile when ζ > 0 and contractile +when ζ < 0. The director field satisfies the relaxational equation +∂tni + uj∂jni + Ωijnj = 1 +γ hi − ν +� +Dijnj − ni(njDjknk) +� +, +(2) +where γ is a rotational viscosity and Ωij = 1 +2(∂iuj − ∂jui) is the antisymmetric part of the velocity gradients. We +adopt a one-elastic-constant approximation for the Frank free energy [38] +F = +� K +2 +� +∂inj +�� +∂inj +� +dV, +(3) +for which the molecular field is hi = K +� +∇2ni − ninj∇2nj +� +and the Ericksen stress is σE +ij = −K∂ink ∂jnk. +An often-used analytical approximation is to consider the active flows generated by an equilibrium director field. +This approximation has been used previously in the theoretical description of the active flows generated by defects +in both two [16, 39] and three dimensions [23], including on curved surfaces [40], and in active turbulence [41]. It +may be thought of in terms of a limit of weak activity, however, even when the activity is strong enough to generate +defects, their structure may still be close to that of equilibrium defects and the approximation remain good and the +comparison of active defect motion and flows described in this way with full numerical simulations suggests that this +is at least qualitatively the case. The equations can then be reduced to h = 0 for the director field and the Stokes +equation +− ∇p + µ∇2u = ζ∇ · +� +nn +� +, +(4) +for the active flow. Here we have neglected the Ericksen stress since for an equilibrium director field it can be balanced +by a contribution to the pressure (representing nematic hydrostatic equilibrium). +We limit our analysis to director fields that can be linearised around a (locally) uniformly aligned state, n = ez +δn, +with δn · ez = 0, for which the equations reduce to +∇2δn = 0, +(5) +∇ · u = 0, +(6) +−∇p + µ∇2u = ζ +� +ez +� +∇ · δn +� ++ ∂zδn +� +. +(7) +These correspond to elastic multipole states in the director field, which are often thought of as an asymptotic de- +scription, however, they provide a close approximation even at only moderate distances outside a ‘core’ region that +is the source of the multipole. To illustrate this we show in Fig. 1 a comparison between the exact director field (red +streamlines) and linear multipole approximation (blue rods) for the most slowly varying monopole distortion created +by uniformly rotating the director by an angle θ0 within a sphere of radius a. The agreement is close anywhere outside +the sphere and only deviates significantly in the near-field region inside it. This is relevant to the active system as it +is well-known that the uniformly aligned active nematic state is fundamentally unstable [42] and active nematics are +turbulent on large enough scales. Our solutions should be interpreted as describing the behaviour on intermediate +scales, larger than the core structure of the source but smaller than the scale on which turbulence takes over. + +3 +FIG. 1. Comparison of the exact director field (red streamlines) and linearised multipole approximation (blue rods) for the +most slowly decaying monopole distortion. This is produced by uniformly rotating the director by an angle θ0 within a spherical +volume of radius a, indicated by the grey disc; the alignment inside the sphere is indicated by the thick red line. The figure +shows only the xz-plane in which the director rotates and in which the comparison is most strict. +III. +MULTIPOLE DIRECTOR DISTORTIONS +In this section, we describe the multipole director fields satisfying (5). The far-field orientation ez gives a splitting +of directions in space into those parallel and perpendicular to it. We complexify the perpendicular plane to give the +decomposition as R3 ∼= C ⊕ R and convert the director deformation δn to the complex form δn = δnx + iδny. The +real and imaginary parts of δn are harmonic, meaning that at order l they may be expressed as spherical harmonics +1/rl+1Y l +m or, as we shall do, as l derivatives of 1/r [43–45]. These order l multipole solutions form a 2(2l + 1)-real- +dimensional vector space. Associated to the C ⊕ R splitting is a local symmetry group isomorphic to U(1), preserving +ez, whose irreducible representations provide a natural basis for the vector space of multipoles at each order. We write +the complex derivatives on C as ∂w = 1 +2(∂x − i∂y) and ∂ ¯ +w = 1 +2(∂x + i∂y) in terms of which the director deformation +can be written +δn = +∞ +� +l=0 +l +� +m=−l +qlm al+1 ∂m +¯ +w ∂l−m +z +1 +r , +(8) +where qlm are complex coefficients and a is a characteristic length scale of the multipole, as might be set by the radius +of a colloid. For compactness of notation it is to be understood that when m is negative ∂m +¯ +w represents ∂|m| +w . The +index m denotes the topological charge of the phase winding of the spherical harmonic. This gives the spin of the +corresponding vector field as 1 − m, where the 1 is due to a vector (δn or δn) being a spin-1 object. The multipoles +at order l therefore have spins that range from 1 − l to 1 + l. They are illustrated up to quadrupole order in Fig. 2, +along with a representation in terms of topological defects which we shall elaborate upon shortly. The structure of +Fig. 2 is such that differentiation maps the distortions of one order to the next, with ∂z leaving the distortion in the +same spin class, ∂ ¯ +w moving it one column to the left and ∂w moving it one column to the right. The operators ∂w and +∂ ¯ +w play the same role as the raising and lowering operators in quantum mechanics and the shift by one in the spin +values simply results from the object on which they act being a spin-1 director deformation as opposed to a spin-0 +wavefunction. +The monopole distortions, with l = 0, result from a rotation of the director by an angle θ0 in a sphere of radius +a [46]. They form a two-real-dimensional vector space for which a basis may be taken to be the distortions 1 +r and i 1 +r. +These are shown at the top of Fig. 2 and can be controllably created in passive nematics using platelet inclusions [47]. +The director distortions of dipole type, with l = 1, form a six-real-dimensional vector space that splits into two- + +24 +FIG. 2. The multipolar director distortions up to quadrupole order. The director is shown on a planar cross-section as blue +rods, along with a topological skeleton corresponding to the spherical harmonic, where appropriate. Defect loops are coloured +according to wedge (blue) or twist (red-green) type and the charge of point defects is indicated through the use of opposing +colour pairs: red (+1) and cyan (−1), yellow (+2) and blue (−2), and green (+3) and magenta (−3). Their charge is further +indicated by a local decoration of the director with an orientation, indicated by black arrows. Each multipole order is classified +into vertical pairs according to the spin of the distortion. For the chiral multipoles, the visualisation instead shows the director +along some of its integral curves (orange). + +-1 +0 +2 +3 +Monopoles +Dipoles +Quadrupoles5 +real-dimensional subspaces for each value of the spin (0, 1, or 2) as +p0 = +� +∂ ¯ +w +1 +r , i ∂ ¯ +w +1 +r +� +∼ − 1 +2r3 +� +x ex + y ey, −y ex + x ey +� +∼ 1 +r2 +� +Y 1 +1 , i Y 1 +1 +� +, +(9) +p1 = +� +∂z +1 +r , i ∂z +1 +r +� +∼ − 1 +r3 +� +z ex, z ey +� +∼ 1 +r2 +� +Y 0 +1 , i Y 0 +1 +� +, +(10) +p2 = +� +∂w +1 +r , i ∂w +1 +r +� +∼ − 1 +2r3 +� +x ex − y ey, y ex + x ey +� +∼ 1 +r2 +� +Y −1 +1 +, i Y −1 +1 +� +. +(11) +For comparison, we have presented three representations for the distortions of each spin class: in terms of complex +derivatives of 1/r, two-component vectors whose coefficients are homogenous polynomials of degree 1 and complex +spherical harmonics. In the interest of space we have suppressed certain prefactors in the last of these, but note +the difference in sign, and in some cases normalisation, between our representation as complex derivatives and the +standard form of the harmonic distortions as two-component vectors [48]. The two basis functions of any spin class +are related by a factor of i, which corresponds to a local rotation of the transverse director distortion by π +2 . For a +spin-s distortion this is equivalent to a global rotation by +π +2s, with the pair of distortions having the same character +and simply providing a basis for all possible orientations. The exception is when s = 0, such distortions lack an +orientation and the local rotation produces two distinct states that transform independently under rotations as a +scalar and pseudoscalar. In the dipole case the first is the isotropic distortion recognisable as the UPenn dipole [25] +and the second is an axisymmetric chiral distortion with the far-field character of left-handed double twist. Separating +p0 into its isotropic and chiral components allows a decomposition of the dipole director deformations into the basis +p = pI ⊕ pC ⊕ p1 ⊕ p2, +(12) +a decomposition which was presented in [49]. +Similarly, the quadrupolar distortions (l = 2) form a ten-real-dimensional vector space that splits into a sum of +two-real-dimensional subspaces for each value of the spin +Q−1 = +� +∂2 +¯ +w +1 +r , i ∂2 +¯ +w +1 +r +� +∼ +3 +4r5 +� +(x2 − y2) ex + 2xy ey, −2xy ex + (x2 − y2) ey +� +∼ 1 +r3 +� +Y 2 +2 , i Y 2 +2 +� +, +(13) +Q0 = +� +∂2 +¯ +wz +1 +r , i ∂2 +¯ +wz +1 +r +� +∼ +3 +2r5 +� +xz ex + yz ey, −yz ex + xz ey +� +∼ 1 +r3 +� +Y 1 +2 , i Y 1 +2 +� +, +(14) +Q1 = +� +∂2 +z +1 +r , i ∂2 +z +1 +r +� +∼ 1 +r5 +� +(2z2 − x2 − y2) ex, (2z2 − x2 − y2) ey +� +∼ 1 +r3 +� +Y 0 +2 , i Y 0 +2 +� +, +(15) +Q2 = +� +∂2 +wz +1 +r , i ∂2 +wz +1 +r +� +∼ +3 +2r5 +� +xz ex − yz ey, yz ex + xz ey +� +∼ 1 +r3 +� +Y −1 +2 +, i Y −1 +2 +� +, +(16) +Q3 = +� +∂2 +w +1 +r , i ∂2 +w +1 +r +� +∼ +3 +4r5 +� +(x2 − y2) ex − 2xy ey, 2xy ex + (x2 − y2) ey +� +∼ 1 +r3 +� +Y −2 +2 +, i Y −2 +2 +� +. +(17) +Once again the spin-0 distortions can be further partitioned into those that transform as a scalar and pseudoscalar, +these being the Saturn’s ring distortion [50] and a chiral quadrupole with opposing chirality in the two hemispheres, +respectively. This yields the basis for the quadrupolar director deformations +Q = Q−1 ⊕ QI ⊕ QC ⊕ Q1 ⊕ Q2 ⊕ Q3. +(18) +The well-known multipoles, the UPenn dipole and Saturn ring quadrupole, are associated to a configuration of +topological defects in the core region and we describe now an extension of this association to all of the multipoles. In +general, such an association is not unique, for instance, the colloidal ‘bubblegum’ configuration [51] represents the same +far field quadrupole as the Saturn ring, however, for each multipole we can construct a representative arrangement of +topological defects which produce it in the far field on the basis of commensurate symmetries and defects of a type and +location corresponding to the nodal set of the harmonic. This correspondence allow us to condense the visualisation +of complicated three-dimensional fields into a few discrete elements, suggests means by which such distortions might +be induced and enables us to build an intuition for their behaviour in active systems through established results for +defects [23]. +We first describe some examples, shown in Fig. 3. On the left is the spherical harmonic that describes the UPenn +dipole, with the form ∂ ¯ +w 1 +r ∼ eiφ sin θ, visualised on a spherical surface. This has nodes at the two poles about which +the phase has −1 winding and so we can infer similar winding of the director in the transverse plane. Supplementing + +6 +FIG. 3. +The connection between spherical harmonics and nematic topological defects. +The coloured spheres indicate the +phase of the complex spherical harmonics with the nodal set shown in white for simplicity. A representative skeleton of the +corresponding nematic distortion is shown in black and the red arrows indicate the winding vector of the director. +with the far-field alignment along ez yields the familiar picture of a pair of oppositely charged hedgehog defects. +Similarly, the Saturn ring quadrupole, described by ∂ ¯ +wz 1 +r ∼ eiφ sin 2θ, has zeros at the poles and around the equator. +The winding about the poles is still +1, but the sign change in the lower hemisphere means that in the transverse +plane around the south pole the vector points inwards, resulting in both point defects having topological charge +1. +With regards to the equatorial line, since the director is everywhere radial the winding vector must be tangential to +the defect loop, shown by the red arrows in Fig. 3. As the phase changes by π on passing from one hemisphere to the +other the winding must be ±1 and the far-field alignment allows us to determine it to be −1. For a general multipole +distortion of the form ∂m +¯ +w ∂l−m +z +(1/r) the nodal set is the poles along with l − m lines of latitude. The phase winding +of the spherical harmonic dictates the transverse winding of the director and, when supplemented with the far-field +alignment, allows us to associate topological point defects with the poles. Similarly, nodal lines may be connected +with defect loops with integer winding and a winding vector that rotates according to eimφ. In Fig. 3 we illustrate +this for the case ∂2 +¯ +w∂3 +z(1/r) ∼ −Y 5 +2 /r6. +We now describe briefly the correspondence for our basis of dipolar and quadrupolar distortions. As already stated, +the isotropic scalar in p0 is the UPenn dipole, its pseudoscalar counterpart a chiral splay-free twist-bend distortion +whose integral curves are shown in orange in Fig. 2. As a twist-bend mode it may be of particular relevance to +extensional systems given their instability to bend distortions. The two dipoles of p1 are transverse to the far-field +alignment, they are related to those resulting from a defect loop of wedge-twist type [21]. The distortions of p2 have +a hyperbolic character; they describe the far field of a pair of point defects both of which have a hyperbolic structure. +Such hyperbolic defect pairs arise in toron configurations in frustrated chiral nematics [52, 53]. +Similarly, Q0 contains the Saturn ring quadurpole as the scalar, with the pseudoscalar a pure bend chiral distortion. +For the latter, the integral curves of the director possess opposing chirality in the two hemispheres, which could be +generated by an appropriately coated Janus particle. The director distortion exhibits a helical perversion in the z = 0 +plane and, being a local rotation of the Saturn ring distortion, may be viewed as resulting from a pair of vortex point +defects along with a pure twist defect loop with integer winding. This is similar to the bubblegum defect lines [51, 54] +that appear between a colloid diad with normal anchoring, suggesting that this chiral quadrupole could be formed by +two colloids with opposing chiral tangential anchoring. +The spin-1 quadrupoles consist of pairs of wedge-twist defect loops. The distortions of Q2 may be associated with +a pair of hyperbolic defects along with a defect ring with the appropriate symmetry. The harmonics of spin −1 and +3 contain no z-derivatives and so are associated with pairs of point defects only. +IV. +FLOWS FROM MULTIPOLE DISTORTIONS +In this section we calculate the active flow generated by an arbitrary director multipole. We present this initially in +vectorial form, converting to the complex representation subsequently. As (7) is linear the responses due to the two +components of δn are independent and so to simplify the derivation we consider only distortions in the x-component +for now and extend to the general case afterwards. Within this restriction a generic multipole distortion at order l + +22 +75 +r3 +r6 +r +Y7 +may be written as +δnx = al∇v1 · · · ∇vl +a +r , +(19) +where v1, . . . , vl are l directions for the differentiation. Substituting this into (7) gives the Stokes equation in the +form +− ∇p(x) + µ∇2u(x) = al+1ζ∇v1 · · · ∇vl +� +ex ∂z + ez ∂x +�1 +r , +(20) +where the use of the superscript (x) is to emphasise that we are only treating the response to distortions in the +x-component of the director. Taking the divergence of both sides we have +− ∇2p(x) + µ∇2∇ · u(x) = al+1ζ∇v1 · · · ∇vl∂2 +xz +2 +r . +(21) +Making use of the continuity equation ∇ · u(x) = 0 in conjunction with the identity ∇2r = 2 +r we arrive at the solution +for the pressure +p(x) = −al+1ζ∇v1 · · · ∇vl ∂x∂zr = al+1ζ∇v1 · · · ∇vl +xz +r3 . +(22) +Substituting this back into the Stokes equation (20) we obtain +µ∇2u(x) = al+1ζ∇v1 · · · ∇vl +� +ex ∂z +�1 +r − ∂x∂xr +� +− ey ∂x∂y∂zr + ez ∂x +�1 +r − ∂z∂zr +�� +, +(23) +which can be integrated using the identity ∇2r3 = 12r to find +u(x) = al+1 ζ +4µ∇v1 · · · ∇vl +� +ex +�z +r + x2z +r3 +� ++ ey +xyz +r3 + ez +�x +r + xz2 +r3 +�� +. +(24) +Both the pressure and flow solutions for a generic multipole distortion are given in terms of derivatives of a +fundamental response to a monopole deformation, namely +p(x) = aζ xz +r3 , +(25) +u(x) = aζ +4µ +� +ex +�z +r + x2z +r3 +� ++ ey +xyz +r3 + ez +�x +r + xz2 +r3 +�� +. +(26) +This flow response, shown as the top panel in Fig. 4, is primarily extensional in the xz-plane. Interestingly, the flow +solution (26) does not decay with distance; this reflects the generic hydrodynamic instability of active nematics [42] +providing a real-space local response counterpart to the usual Fourier mode analysis. +However, the active flow +produced by any higher multipole does decay and vanishes at large distances. +The pressure and flow solutions in (25) and (26) are complemented by analogous ones resulting from distortions +in the y-component of the director, obtained by simply interchanging x and y. The linearity of (7) makes these +fundamental responses sufficient to obtain the active flow induced by an arbitrary multipole distortion through taking +derivatives appropriate to describe the x and y components of the director, respectively. +We now convert this description to the complex notation used in § III. This is achieved by taking the combinations +p = p(x) − ip(y) and u = u(x) − iu(y). To see this consider the multipole distortion δn = (Lx + iLy)1/r, where the Li +are generic real differential operators which generate the i-component of the director by acting on 1/r. This distortion +has a conjugate partner given by i(Lx + iLy)1/r = (−Ly + iLx)1/r. Acting with this same operator on u(x) − iu(y) +we have +(Lx + iLy)(u(x) − iu(y)) = (Lxu(x) + Lyu(y)) − i(−Lyu(x) + Lxu(y)), +(27) +and can see that the flow response for our original distortion forms the real part and that for its conjugate partner +the coefficient of −i and the same holds for the pressure response. This leads us to a complex fundamental pressure +response +˜p = aζ ¯wz +r3 , +(28) + +8 +FIG. 4. The active flows due to three-dimensional nematic multipole distortions up to quadrupole order. The flows are grouped +according to their spin, in correspondence with the distortions in Fig. 2. Green and red arrows indicate the net active force +and torque for the relevant dipoles and quadrupoles respectively, see §V. +and, introducing complex basis vectors ew = ex + iey and e ¯ +w = ex − iey, a complex-valued fundamental flow vector +˜u = aζ +4µ +� +ew +¯w2z +2r3 + e ¯ +w +�z +r + w ¯wz +r3 +� ++ ez +¯w +r +� +1 + z2 +r2 +�� +. +(29) +We use a tilde to distinguish these fundamental responses from those that result due to a generic distortion and which +may be found by appropriate differentiation. This provides a unified framework in which the active response to a +generic nematic multipole can be calculated through the application of the same complex derivatives that we have +used to describe the director distortion. The resulting active flows for distortions up to quadrupole order are shown + +-1 +0 +2 +3 +Monopoles +Dipoles +Quadrupoles9 +in Fig. 4, with their layout corresponding to that of the nematic distortions in Fig. 2 which induce them. We now +describe some examples in more detail. +A. +UPenn and chiral dipole +Typically the active responses induced by the two distortions in a spin class will, like the distortions themselves, be +related by a global rotation such that while both are needed to form a sufficient basis, the real part essentially serves +as a proxy for the pair. This is not true for the spin-0 distortions, due to their rotational symmetry, and so we use +them in providing an explicit illustration of the active flow calculation. We begin with the UPenn dipole [25] and its +partner the chiral dipole, for which the far-field transverse director is +δn ≈ αa ∂ ¯ +w +a +r , +(30) +where α is a dimensionless coefficient, and the corresponding derivative of the fundamental flow solution in (29) gives +αa∂ ¯ +w˜u = ζαa2 +4µr5 +� +ew z ¯w(4z2 + w ¯w) − e ¯ +w 3zw2 ¯w + ez 2 +� +3z4 + (z2 + w ¯w)2�� +. +(31) +Taking the real part gives, after some manipulation, the flow induced by the UPenn dipole as +u = αa R ∂ ¯ +w˜u = ζαa2 +8µ +� +ez +�1 +r + z2 +r3 +� ++ er +z +r2 +�3z2 +r2 − 1 +�� +, +(32) +where er is the unit vector in the radial direction. The flow response to the conjugate distortion, the isotropic chiral +dipole is given by +u = −αa I ∂ ¯ +w˜u = −ζαa2 +4µ +z +r2 eφ, +(33) +with eφ the azimuthal unit vector. Both flows decay at large distances like 1/r and are highlighted in the top row of +Fig. 5. The UPenn dipole flow has a striking net flow directed along the z-axis, reminiscent of that of the Stokeslet +flow [55, 56] associated with a point force along ez. The chiral dipole generates an axisymmetric flow composed +of two counter-rotating vortices aligned along ez, mirroring the circulating flows produced by spiral defects in two +dimensions [57]. The 1/r decay of these active vortex flows is unusually slow, slower than the decay of a point torque +in Stokesian hydrodynamics [56]. +Despite the similarity between the active flow induced by the UPenn dipole and a Stokeslet, there is a key difference +in their angular dependence. +In a Stokeslet, and all related squirming swimmer flows [58, 59] that result from +derivatives of it, the terms with higher angular dependence decay more quickly such that the lowest order terms +dominate the far field. By contrast, distortions in active nematics produce asymptotic flow fields in which all terms +decay at the same rate regardless of their angular dependence as they all result from the same derivative of the +fundamental flow. Thus, even if the same angular terms are present in both systems, the lowest order ones will +dominate in the squirming case while the far field will bear the signature of the highest order in the active nematics. +A closer point of comparison comes from the flows induced by active colloids within a passive nematic [35, 60]. +Calculation of the relevant Green’s functions [61] has shown that the anisotropy of the medium leads to a difference +in effective viscosities such that a Stokeslet aligned along the director pumps more fluid in this direction. This fits +with the anisotropy displayed in (32), reaffirming the similarity between the flow induced by the UPenn dipole and +the Stokeslet. +Considering the pressure response for these distortions in the same way we have +αa∂ ¯ +w ˜p = ζαa2 +2r5 z(2z2 − w ¯w) = ζαa2z +2r3 +�3z2 +r2 − 1 +� +. +(34) +As this expression is purely real it comprises the response due to the UPenn dipole in its entirety; the vanishing +of the imaginary part shows that the chiral dipole is compatible with a zero pressure solution. Our complexified +construction allows this property to be read off immediately, since ∂ ¯ +w( ¯wzm/rn) will be real for any m and n, with +this also resulting in the vanishing z-component of flow for the chiral dipole. Indeed, this property of pure realness is +unchanged by the action of ∂z, it being real itself, and so extends to higher order distortions. + +10 +FIG. 5. The active flows induced by spin 0 dipole (top row) and quadrupole (bottom row) distortions. The flow is indicated +by blue arrows and superposed upon integral curves of the director, shown in orange. On the left are the UPenn dipole and +Saturn ring quadrupole and on the right their chiral counterparts. +B. +Saturn ring and chiral quadrupole +Proceeding in the same fashion for the spin-0 quadrupoles, for which δn ≈ αa2∂2 +¯ +wza/r, we find that the complexified +flow is +αa2∂2 +¯ +wz˜u = −ζαa3 +4µr7 +� +−ew ¯w(w2 ¯w2 + 8w ¯wz2 − 8z4) + e ¯ +w3w2 ¯w(w ¯w − 4z2) ++ez2z(w2 ¯w2 − 10w ¯wz2 + 4z2) +� +. +(35) +Taking the real part gives the flow induced by the Saturn ring quadrupole as +u = αa2R∂2 +¯ +wz˜u = −ζαa3 +2µr6 (r4 − 12z2r2 + 15z4)er, +(36) +that is a purely radial flow reminiscent of a stresslet along ez, shown in the bottom left of Fig. 5. The purely radial +nature is a result of the divergencelessness of the flow, combined with the 1/r2 decay and rotational invariance about +ez. Working in spherical coordinates we have +∇ · u = 1 +r2 ∂r(r2ur) + +1 +r sin θ [∂θ(uθ sin θ) + ∂φuφ] = 0 +(37) +All active flows induced by quadrupole distortions decay as 1/r2 and so ∂r(r2ur) = 0. The distortion is rotationally +symmetric and achiral, meaning uφ = 0 and the condition of zero divergence reduces to +1 +r sin θ∂θ(uθ sin θ) = 0. +(38) +The only non-singular solution is uθ = 0, resulting in ur being the only non-zero flow component. The corresponding +pressure is given by +αa2∂2 +¯ +wz ˜p = −3αa3 +2r7 (r4 − 12z2r2 + 15z4). +(39) + +wz11 +Taking the imaginary part of (35) reveals the flow response of the chiral quadrupole to be +u = −αa2I∂2 +¯ +wz˜u = ζαa3 +µr2 (3 cos2 θ − 1) sin θeφ. +(40) +As illustrated in Fig. 5 this is a purely azimuthal flow corresponding to rotation about the z axis and, as for the +chiral dipole, is compatible with a zero pressure solution. The 1/r2 decay of this rotational flow is the same as that +which results from the rotlet [55, 56], but unlike the rotlet the flow direction is not uniform. Rather, as can be seen +in Fig. 5, there is an equatorial band of high-velocity flow accompanied by two slowly counter-rotating polar regions. +The distribution of flow speeds is such that the net flow is along −eφ, consistent with a rotlet along −ez. +C. +Other multipoles +For the remaining multipoles up to quadrupole order we do not provide the same explicit calculation but instead +highlight the key features of the active flows they induce. In full we find that half of the dipole distortions contain +directed components in their active flow responses. Along with the isotropic UPenn dipole which produces flow along +ez the two spin-1 dipoles produce directed flows transverse to it. These directed flows indicated that were the source +of the distortion free to move it would exhibit active self-propulsion. The net transverse flows for the dipoles of p1 is +in accordance with the previously established motile nature of such defect loops [23]. A more complete description of +the active dynamics of defect loops via their multipole distortions is presented in Section IV D and [24]. +Along with the chiral dipole, the two additional dipoles which do not generate directed flows are those with spin 2. +These produce active flows which are extensional with the expected two-fold rotational symmetry about the z-axis. +Direct calculation shows that the flows resulting from spin-2 distortions have zero azimuthal component. Once again, +this observation is unaffected by z-derivatives and so holds true for the higher-order multipoles of the form ∂n +z ∂w(1/r). +Similarly, there are ten linearly independent quadrupoles, five of which can be seen from Fig. 4 to generate rotational +flows. As expected, it is the four modes of Q±1 that generate rotations about transverse directions and QC that +produces rotation around ez. For two of these, namely those in Q1, the director distortions are planar, suggesting +a two-dimensional analogue and the potential to generate them with cogs or gears [62]. These distortions may be +associated with a pair of opposingly oriented charge-neutral defect loops and so the rotational flow generated by these +distortions is in accordance with their antiparallel self-propulsion. +The quadrupoles of Q−1 are composed of pairs of point defects with topological charge +2. Using ∂2 +¯ +w +1 +r as an +example, the rotation can be understood by considering the splay distortions in the xz plane. The splay changes sign +for positive and negative x, leading to antiparallel forces. The active forces are greatest in this plane, as this is where +the transverse distortion is radial resulting in splay and bend distortions. Along ey the distortions are of twist type +and so do not contribute to the active force. This results in the rotational flow shown in Fig. 4. The stretching of the +flow along ez is as observed for a rotlet in a nematic environment [61]. +Although they lack the rotational symmetry of a stresslet, the flows produced by the quadrupoles of Q2 are also +purely radial. The argument is largely the same as for the Saturn ring distortion, except that the vanishing of uφ is +not due to rotational invariance but a property inherited from the spin-2 dipoles. +The quadrupoles of Q3 produce extensional flows whose spin-3 behaviour under rotations about ez is commensurate +with that of the distortions. Although they visually resemble the similarly extensional flows produced by the dipoles +of p2, they do not share the property of a vanishing azimuthal flow component. +D. +Defect loops +Of particular relevance to the dynamics of three-dimensional active nematics are charge-neutral defect loops [21, +23, 24]. For such defect loops the director field has the planar form +n = cos Υ +4 ez + sin Υ +4 ex, +(41) +where Υ is the solid angle function for the loop [43, 63], and is a critical point of the Frank free energy in the one- +elastic-constant approximation [64]. This allows a multipole expansion for the director at distances larger than the +loop size in which the multipole coefficients are determined explicitly by the loop geometry [24] +Υ(x) = 1 +2 +� +K +ϵijk yj dyk ∂i +1 +r − 1 +3 +� +K +ϵikl ylyk dyl ∂i∂j +1 +r + . . . , +(42) + +12 +FIG. 6. Additonal flow solutions induced by spin-1 nematic multipoles. The nematic multipoles which induce the flows are +shown below them as complex derivatives of 1/r. The red arrows indicate the net active torque. +where y labels the points of the loop K and r = |x| with the ‘centre of mass’ of the loop defined to be at x = 0. The +dipole moment vector is the projected area of the loop, while the quadrupole moment is a traceless and symmetric +tensor with an interpretation via the first moment of area or, in the case of loops weakly perturbed from circular, the +torsion of the curve. +The planar form of the director field (41) corresponds to a restricted class of director deformations in which δn is +purely real. This disrupts the complex basis we have adopted for the representation of multipoles, so that another +choice is to be preferred. +We may say that the planar director selects a real structure for the orthogonal plane +C, breaking the U(1) symmetry, and the restricted multipoles should then be decomposed with respect to this real +structure. Accordingly, the pressure and flow responses may be generated by derivatives of the fundamental responses +for distortions in ex, (25) and (26), with these derivatives corresponding to the multipole expansion of the solid +angle shown in (42). The details of this approach along with the consequences it has for both the self-propulsive and +self-rotational dynamics of active nematic defect loops are given in [24]. +E. +Technical note +We conclude this section with a technical note on the flow solutions that we have presented. The construction +for calculating active flow responses that we have developed in this section requires knowledge of the multipole +as a specified set of derivatives of 1/r. +The harmonic director components satisfy ∇2ni ∝ δ(r) and while this +delta function does not affect the far-field director it impacts the flow solutions. Consequently, at quadrupole order +and higher, distinct derivatives of 1 +r can produce the same multipole distortion in the director but have different +associated active flows. As an explicit example we take the spin-1 quadrupole shown in Fig. 2, which may be written +as n = a2∂2 +z +a +r ex +ez and therefore induces an active flow given by the action of a2∂2 +z on 29, as is illustrated in Fig. 4. +However the same director distortion is captured by n = −4a2∂2 +w ¯ +w +a +r ex + ez, for which the corresponding active flow +is shown in Fig. 6. A partial resolution to this ambiguity is that any non-equilibrium phenomenological features such +as propulsion or rotation will be invariant to this choice of derivatives since, as we shall show in the following section, +they can be expressed directly in terms of the director components. As a more complete resolution we reiterate that +whenever an exact solution for the director is known the appropriate derivatives can be determined, as demonstrated +earlier for defect loops [24], and so the apparent ambiguity disappears. +V. +ACTIVE FORCES AND TORQUES +The directed and rotational active flow components highlighted above result in viscous stresses whose net effect +must be balanced by their active counterparts, since the net force and torque must be zero. Consequently, these +generic aspects of the response of an active nematic can be identified by considering the contribution that the active + +ww13 +stresses make to the force and torque +f a = +� +ζnn · dA ≈ +� +ζ +� +ex +z δnx +r ++ ey +z δny +r ++ ez +x δnx + y δny +r +� +dA, +(43) +τ a = +� +x × ζnn · dA ≈ +� +ζ +� +ex +�xy δnx +r ++ (y2 − z2)δny +r +� ++ ey +�(z2 − x2)δnx +r +− xy δny +r +� ++ ez +z(−y δnx + x δny) +r +� +dA, +(44) +integrating over a large sphere of radius r. These integrals depend on the surface of integration, as the active stresses +are neither divergenceless nor compactly supported. However, a spherical surface is concordant with the multipole +approach we are taking and the results are then independent of the radius, as a direct consequence of the orthogonality +of spherical harmonics. From these expressions we can read off the multipole that will generate any desired active +force or torque; dipoles generate forces and quadrupoles generate torques. When the active torque is non-zero, the +compensating viscous torque will drive a persistent rotation of the multipole, creating an active ratchet; similarly, a +non-zero active force will generate directed fluid flow. The above integrals therefore provide a solution to the inverse +problem: given a particular non-equilibrium response, which distortion induces it? Hence they serve as a design guide +for generating out of equilibrium responses in active nematics. +If the multipole is free to move it will self-propel and rotate. The translational and rotational velocities are related +to the viscous forces and torques by a general mobility matrix [65]. +In passive nematics, experiments [66] and +simulations [67, 68] have found that it is sufficient to take a diagonal form for the mobility (no translation-rotation +coupling) with separate viscosities for motion parallel, µ∥, and perpendicular, µ⊥, to the director, with typical ratio +of viscosities µ⊥/µ∥ ∼ 1.6 [66–68]. This has the consequence that in general the force and velocity are not colinear +U = −1 +6πa +� 1 +µ∥ +f a +∥ ez + 1 +µ⊥ +f a +⊥ +� +. +(45) +We again use the UPenn dipole as an example. Integrating the active stresses over a spherical surface of radius R we +find an active force +� +ζnn · dA ≈ −ζαa2 +2 +� � +ex +xz +R4 + ey +yz +R4 + ez +� z +R + x2 + y2 +R4 +�� +dA = −4πζαa2 +3 +ez. +(46) +Balancing this against Stokes drag predicts a ‘self-propulsion’ velocity for the active dipole of +U = 2ζαa +9µ∥ +ez. +(47) +For extensile activity (ζ > 0) the dipole moves ‘hyperbolic hedgehog first’ and with a speed that increases linearly +with the core size a. This self-propulsion is in accordance with the directed component of the active flow, as can be +seen in Fig. 5. The same self-propulsion speed along ex and ey is found for the transverse dipoles of p1, except that +the parallel viscosity µ∥ should be replaced with µ⊥. Again, this self-propulsion agrees with the directed flow induced +by these distortions, as calculated through the multipole approach, shown in Fig. 4 [24] and also with the results of +both a local flow analysis and simulations [23]. The same directed motion has been observed in a related system of an +active droplet within a passive nematic [35], with the droplet inducing a UPenn dipole in the nematic and moving in +the direction of the hedgehog defect at a speed that grew with the droplet radius. The mechanism at play is different +however; the motion results from directional differences in viscosity resulting from the anisotropic environment. +To illustrate the rotational behaviour we use a member of Q1, ∂2 +z(1/r), as an example. We find an active torque +� +ζx × nn · dA ≈ ζαa3 +� +1 +r6 (2z2 − x2 − y2) +� +xyex + (z2 − x2)ey − yzez +� +dA +(48) += 8πζαa3 +5 +ey. +(49) +Balancing against Stokes drag as was done in the dipole case gives an angular velocity +Ω = −ζα +5µey. +(50) + +14 +We note that for this and all other distortions which result in net torques the angular velocity is independent of the +colloid size. In accordance with the relation ∂2 +z +4∂2 +w ¯ +w(1/r) = 0, the torque resulting from ∂2 +w ¯ +w(1/r) is of the opposite +sign and a quarter the strength. The net active torques due to harmonics of Q0 and Q−1 have the directions indicated +in Fig. 4 and half the magnitude of (49). +Let us consider the approximate magnitude of the effects we have described. Beginning with the self-propulsion +speed, the fluid viscosity is roughly 10−2 Pa s [17], although effects due to the elongated form of the nematogens +could increase this by a factor of 30 or so [69, 70]. Both the activity [16] and the dipole moment constant [48] are +of order unity, meaning the colloid would approximately cover its radius in a second. Similar approximations for the +quadrupole give an angular velocity of about 2/3 rad s−1. For a colloid of radius 10 µm this has an associated power +of the order of femtowatts, the same as predicted for bacterial ratchets [71]. +VI. +TWO-DIMENSIONAL SYSTEMS AND RATCHETS +As noted above, the planar nature of the rotational distortions in Q1 suggests the existence of two-dimensional +analogues. In part motivated by this we now discuss the active response of multipolar distortions in two dimensions, +again beginning with the connection between these multipoles and topological defect configurations. +A. +Multipoles and topological defects +The categorisation of the harmonic distortions in two dimensions is much simpler, but we provide it here for +completeness. Taking the asymptotic alignment to be along ey the symmetry of the far-field director is now described +by the order 2 group {1, Ry}, with Ry reflection with axis ey, under which the monopole distortion nx ∼ A log(r/a) +is antisymmetric. The higher-order distortions are once again generated via differentiation of the monopole, with ∂y +leaving the symmetry under Ry unchanged and ∂x inverting it. +It should be noted that the potential multiplicity of differential representations of harmonics that arose in three +dimensions does not occur in two dimensions. This is because, under the assumption of a single elastic constant, the +director angle φ may be written as the imaginary part of a meromorphic function of a single complex variable and +this naturally defines the appropriate set of derivatives. Making z = x+iy our complex variable we write φ = I {f(z)} +which upon performing a Laurent expansion of f(z) around z = 0 and assuming the existence of a uniform far-field +alignment gives +φ = I +� +0 +� +n=−∞ +anzn +� += I +� +a0 + +∞ +� +n=1 +(−1)n−1 +an +(n − 1)!∂n +z (ln z) +� +. +(51) +Hence at every order there is a one parameter family of distortions, corresponding to the phase of the an. A natural +basis at order n is provided by {R {∂n +z (ln z)} , I {∂n +z (ln z)}}. This basis consists of a symmetric and anti-symmetric +distortion under the action of Ry, the roles alternating with order, and of course correspond to the two harmonic +functions cos nθ/rn and sin nθ/rn. +In two dimensions the connection between defect configurations and far-field multipole distortions can be made +concrete, and also serves as an illustration of how a particular set of derivatives is determined. For defects with +topological charges sj at locations zj the angle that the director makes to ex is given by +φ = φ0 + +� +j +sjI +� +ln +�z − zj +a +�� +, +(52) +which, upon performing a series expansion, gives +φ = φ0 + +� +j +sjI {ln(z/a)} − +∞ +� +n=1 +I +�� +j sjzn +j ¯zn� +n|z|2n +, +(53) += φ0 + +� +j +sjI {ln(z/a)} + +∞ +� +n=1 +(−1)nI +�� +j sjzn +j ∂n +z ln z +� +n! +, +(54) +Provided the total topological charge is zero the winding term proportional to ln w vanishes and φ0 is the far-field +alignment. The distortions are given as a series of harmonics in which the coefficient of the nth harmonic is determined +by a sum of zn +j weighted by the defect charges. + +15 +We would like to have a basis of representative defect configurations for each harmonic distortion. However, it +can be seen from (54) that the correspondence between arrangements of topological defects and the leading order +nematic multipole is not one-to-one. Two defect-based representations of harmonic will prove particularly useful to +us. The first, which we develop in this chapter, provides a representation in terms of half-integer defects on the disc +and allows an intuition for the response to multipole distortions in active nematics through known results for such +defects [15, 16]. The second uses the method of images to construct defect arrangements corresponding to a specific +anchoring condition on the disc, with the same multipoles dominating the nematic distortion in the far field. This +representation naturally lends itself to the control of induced multipoles through colloidal geometry and is explored +fully in [62]. Nonetheless, both of these representations will be of use to us in the remainder of this chapter and as +they are equally valid near-field representations for the asymptotic distortions that we are considering we will pass +fairly freely between them. +With this aforementioned half-integer representation in mind, let us consider sets of 2m defects sitting on the unit +circle, with −1/2 defects at the mth roots of unity and +1/2 defects at the intermediate points. A useful formula here +is the following for the sum of a given power of these roots of unity, after first rotating them all by a given angle θ +m−1 +� +k=0 +� +eiθei 2π +m k�n += +� +meinθ, +if m|n +0, +otherwise . +(55) +The vanishing of this sum for values of n that are not multiples of m comes directly from the expression for the +geometric sum and is a consequence of the cyclic group structure of the roots of unity. It means that the lowest order +multipole distortion induced by such an arrangement of defects is order m and so allows a desired multipole distortion +to be selected as the dominant far-field contribution. Explicitly, the director angle is given by +φ = φ0 + +� +k odd +I +� +¯zmk� +k|z|2mk = φ0 + I {¯zm} +|z|2m + O +� 1 +z3m +� +, +(56) +with the approximation becoming rapidly better for higher-order multipoles due to the condition that n must be an +odd multiple of the number of defects. Rotating the entire set of defects rigidly by an angle −π/(2m) generates the +conjugate multipole as the dominant far-field contribution +φ = φ0 + +� +k odd +I +� +(−i)k¯zmk� +k|z|2mk += φ0 − R {¯zm} +|z|2m ++ O +� 1 +z3m +� +, +(57) +with the natural interpolation between these two harmonics as the defect configuration is rigidly rotated. +Hence we can interchange between a given harmonic distortion and a defect arrangement which has this harmonic +as its dominant far-field contribution, with the correspondence becoming rapidly more accurate for higher orders, +allowing us to relate the existing results for the behaviour of active defects [15, 16] to ours and vice versa. This +correspondence is illustrated in Fig. 7. The locations of +1/2 and −1/2 defects are indicated with red and cyan dots +respectively and the background colouring denotes the phase of the complex function � sj ln(z−zj), whose imaginary +part provides the director angle for the given defect arrangement. The integral curves of this director field are shown +in black and are remarkably well matched by those of the leading multipole, shown in white, despite the asymptotic +nature of the approximation. In this context we are able to make precise the notion of a core region of a singular +distortion, outside of which our multipole approach applies. The series in (54) is attained through a Taylor series of +terms of the form ln(1 − 1/z), which are convergent for |z| > 1. More generally the greatest radial displacement of a +defect defines a core radius, outside of which the multipole series converges onto the exact director angle. +B. +Flows from multipole distortions +We can proceed analogously to our three-dimensional calculation in generating the active flows from a fundamental +response in two dimensions, provided we are mindful of the logarithmic form that the monopole now has. A director +rotation by θ0 inside a disc of radius a results in an equilibrium texture given by +n = cos +�θ0 log(r/R) +log(a/R) +� +ey + sin +�θ0 log(r/R) +log(a/R) +� +ex, +(58) +which in the far field tends to a monopole distortion n ≈ ey + θ0 log(r/R) +log(a/R) ex. Due to the logarithmic divergence of the +fundamental harmonic in two dimensions it is necessary to normalise through a large length R such that a uniformly +aligned far-field director is recovered. + +16 +FIG. 7. Representative defect configurations for nematic multipoles in two dimensions. The red and cyan dots indicate the +locations of +1/2 and −1/2 defects respectively. The black curves are the integral curves of the corresponding director field +and the background colour shows the phase of the complex function whose imaginary part gives the exact director angle, as +in (52). The white lines are the integral curves of the dominant multipole, that is the leading term of (54). The multipole +series converges onto the exact director angle outside a core region, shown as a white disc, and the leading multipole provides +a remarkably good approximation in this region. +Following our three-dimensional analysis we solve Stokes’ equations to linear order in nematic deformations for a +monopole distortion. We write Stokes’ equations in terms of complex derivatives as +2∂¯z(−p + iµω) = f, +(59) +where we have used that 2∂zu = ∇ · u + iω, with ω the vorticity. Hence we seek f as a ¯z-derivative, implicitly +performing a Helmholtz derivative with the real and imaginary parts of the differentiated term corresponding to the +scalar and vector potentials respectively. Expressing the active force in this way we have +2∂¯z(−p + iµω) = +ζθ0 +log(a/R)∂¯z +�i¯z +z +� +(60) +and so +− p + iµω = +ζθ0 +2 log(a/R) +i¯z +z . +(61) +Reading off the pressure and vorticity, solving for the flow and converting back to Cartesians the fundamental flow +response is now found to be +˜u = +ζθ0 +8µ log(a/R) +�x2 − y2 +r2 +(−yex + xey) + 2 log +� r +R +� +(yex + xey) +� +, +(62) +˜p = − +ζθ0 +log(a/R) +xy +r2 . +(63) +There is a clear similarity between these solutions and their three-dimensional counterparts, but while the fundamental +flow response is still extensional it now grows linearly with distance from the distortion, with this change in scaling +inherited by the subsequent harmonics. + +b) +a +) +(p17 +As in the three-dimensional case we can gain general insight into the active response of a nematic by considering +the net contribution of the active stresses to the force and torque when integrated over a large circle of radius r +� +ζnn · erdr ≈ +� +ζ +�yδnx +r +ex + xδnx +r +ey +� +dr, +(64) +� +x × ζnn · erdr ≈ +� +ζ (y2 − x2)δnx +r +dr. +(65) +We see that in two dimensions both dipoles will self-propel if free to move and there is a single chiral quadrupole +which produces rotations. +The far-field flow solutions for distortions up to dipole order are illustrated in Fig. 8, superposed over the nematic +director. +Both dipoles are now motile and as in the three-dimensional case they set up flows reminiscent of the +Stokeslet. +Vertical and horizontal self-propulsive modes may be viewed as resulting from normal and tangential +anchoring respectively of the nematic on a disc. Interpolating between these orthogonal modes the angle of motility +changes commensurately with the anchoring angle, such that sufficient control of the boundary conditions would allow +for self-propulsion at an arbitrary angle with respect to the far-field alignment. This change in the dipole character +can be represented by rigidly rotating the defect pair around the unit circle and the resulting motility is as would be +expected from the position and orientation of the +1/2 defect [16, 72, 73]. Determining the motility induced by these +dipolar modes is complicated by the Stokes paradox and although this can be circumvented by various means we do +not pursue this here. If such dipolar colloids were fixed within the material they would pump the ambient fluid and +so it should be possible to use them to produce the concentration, filtering and corralling effects observed previously +by funneling motile bacteria [74]. +In line with our discussion at the beginning of this section, the basis quadrupoles are given by the real and +imaginary parts of ∂2 +z, these being an achiral and chiral mode respectively, which are shown along with their flows in +Fig. 8. The flow generated by the achiral quadrupole in Fig. 8(d) is purely radial and resembles the stresslet flow, +unsurprising as it results from differentiating the vertical dipole in the same way as the stresslet is related to the +Stokeslet. It is produced by a quadrupole distortion which may be associated with normal anchoring on the disc – +its counterpart with tangential anchoring has all the charges in its representative defect configuration inverted and +a reversed flow response. Just as for the dipole distortions, the character of the quadrupole can be smoothly varied +through adapting the boundary condition and the topological defects which represent the harmonic rotate rigidly in +step with the changing anchoring angle. A generic anchoring angle will produce a net active torque, maximised for an +angle of π/4 as illustrated for the chiral quadrupole shown in Fig. 8(e). For extensile activity this distortion generates +clockwise rotation, as can easily be justified via our representation of the far-field director structure as arising from a +square arrangement of two +1/2 and two −1/2 defects – the dual mode with the defect charges interchanged rotates +anticlockwise. By choosing boundary conditions such that the defects are positioned closer to the mid-line of the +colloid the strength of the active torque can be tuned. +VII. +CHIRAL ACTIVE STRESSES +Chirality is a ubiquitous trait, in living systems and liquid crystals alike. In active matter it opens a wealth of new +phenomena, including odd viscous [75] and elastic responses [76, 77], surface waves, rotating crystals [78] and non- +reciprocal interactions [79]. Chiral active stresses induce vortex arrays in active cholesterics [12] and have also been +shown to be important in nematic cell monolayers where they modify collective motion, the motility of topological +defects and generate edge currents [80, 81]. We now consider the effects of such chiral active stresses on nematic +multipoles, both in two and three dimensions. +A. +Two dimensions +For chiral stresses in two dimensions, the active stress tensor has the form σc = χJ(nn − n⊥n⊥)/2, where J is the +complex structure defined by Jn = n⊥ and Jn⊥ = −n. The chiral active force is +∇ · σc = χJ +� +∇ · (nn) +� +, +(66) +and is simply a π/2 rotation of the achiral active force. Accordingly we can modify (61) to give +− p + iµω = − +ζθ0 +2 log(a/R) +¯z +z , +(67) + +18 +FIG. 8. Distortions up to quadrupole order in two-dimensional active nematics. The active flow in white is superposed on the +pressure field, with the integral curves of the director shown in black. (a) The fundamental monopole response is extensional +and grows linearly with distance from the distortion. (b) and (c) show the flows induced by dipole distortions, labelled by the +appropriate derivative of the nematic monopole, with the green arrows indicating the direction of self-propulsion that would +result from net active forces in extensile systems. The vertical and horizontal dipoles are the far-field director responses to +normal and tangential anchoring respectively and may also be interpreted as arising from a pair of +1/2 (cyan) and −1/2 (red) +defects. The self-propulsion matches that expected for the +1/2 defect. +and solve as before to find +˜u = +χθ0 +8µ log(a/R) +�2xy +r2 (−yex + xey) + 2 log +� r +R +� +(−xex + yey) +� +, +(68) +˜p = +χθ0 +log(a/R) +x2 − y2 +2r2 +. +(69) +Another way to understand the relation between achiral and chiral stresses is that, since the monopole active force +field is spin-2, the π/2 local rotation of the active force results in a global rotation by π/4 of the force field and hence +the fundamental flow responses. The action of this global rotation, denoted Rπ/4, may be seen by comparing the +monopole flow responses for achiral and chiral stresses, shown in Fig. 8(a) and Fig. 9(a) respectively. For distortions +of order n there are two basis flows, ur and ui, corresponding to the real and imaginary parts of ∂n +z respectively. +The rotation of the monopole response has the consequence that for achiral and chiral active stresses these flows are +related by +uc +r = Rπ/4 +� +cos +�nπ +4 +� +ua +r − sin +�nπ +4 +� +ua +i +� +, +(70) +uc +i = Rπ/4 +� +sin +�nπ +4 +� +ua +r + cos +�nπ +4 +� +ua +i +� +, +(71) + +a) +b) +-% +(p +e) +02 +hc19 +FIG. 9. Distortions up to quadrupole order in two-dimensional active nematics with purely chiral stresses. The active flow +in white is superposed on the pressure field, with the integral curves of the director shown in black. (a) The fundamental +monopole response is extensional and grows linearly with distance from the distortion. (b) and (c) show the flows induced +by dipole distortions, labelled by the appropriate derivative of the nematic monopole, with the green arrows indicating the +direction of self-propulsion that would result from net active forces in extensile systems. +where the superscripts denote the nature of the stresses as achiral or chiral. Hence flow solutions for chiral and achiral +stresses are related by a clockwise rotation by nπ/4 in the space of solutions followed by a rigid spatial rotation +anticlockwise by π/4, as can be seen in Fig 9. At dipole order the chiral flow fields are rotated superpositions of +the achiral ones, with the overall effect of chirality being to rotate the self-propulsion direction anticlockwise by π/2, +interchanging the roles of horizontal and vertical propulsion. For a generic mixture of achiral and chiral stresses +the direction of self-propulsion is rotated from the achiral case by an angle arctan(χ/ζ), mirroring the effect such +stresses have on the flow profile of a +1/2 defect [80]. For the quadrupole distortions we have uc +i = Rπ/4ua +r and +uc +i = Rπ/4(−ua +i ) = ua +i , again swapping which distortion produces a chiral or achiral flow response. +It is worth +emphasising that the sign of the macroscopic rotation is not necessarily the same as the sign of the chiral stresses, +rather it is the product of the signs of the activity and the distortion, just as for achiral stresses. + +a) +b) +(p +22 +e) +hc +C20 +FIG. 10. The active flows induced by spin 0 dipole distortions with chiral active stresses. The flow is superposed upon the +integral curves of the director, shown in orange, for the UPenn dipole (left) and chiral dipole (right). +B. +Three dimensions +In three dimensions the chiral active force is χ∇×[∇ · (nn)] [12] and so, by linearity, the fundamental flow responses +are given by the curl of those derived earlier, namely +u(x) = +aχ +2µr3 +� +−exxy + ey(x2 − z2) + ezyz +� +, +(72) +u(y) = +aχ +2µr3 +� +−ex(y2 − z2) + eyxy + −ezxz +� +, +(73) +for monopole distortions in the x- and y-components respectively. Just as for achiral active stresses, we can combine +these into a single complex fundamental flow response as u(x) − iu(y), giving +˜u = i +r3 +� +− ¯w2ew + (w ¯w − 2z2)e ¯ +w + 2 ¯wzez +� +. +(74) +Since the active chiral force is a pure curl the corresponding pressure is constant. +Owing to the additional derivative the functional behaviour of the flow responses is shifted up one order of distortion +compared to achiral stresses, meaning dipole distortions induce rotations, although it should be noted that monopoles +do not produce propulsive flows. The monopole flow responses are still spin-1, but since the flow response for a +monopole distortion in nx for achiral stresses is primarily in the x − z plane, the action of curl produces a flow that is +dominantly in the y-direction and similarly the response to a monopole distortion in ny is mainly along ex. Together +these ingredients mean that heuristically the flow response of a given distortion with chiral active stresses will resemble +the achiral active stress flow response of the conjugate distortion at one higher order and with the same spin, that +is the distortion reached by the action of i∂z. This is illustrated in Fig. 10 for the spin-0 dipoles. The UPenn dipole +induces rotation about ez while the chiral dipole produces a purely radial flow, resembling the achiral flow responses +of the chiral quadrupole and Saturn’s ring quadurpole respectively. +The phenomenological response can again be captured through integration of the stress tensor over a large sphere +of radius r, just as was done for achiral active stresses. To enable us to reduce the active torque to a single boundary +integral we use the symmetric form of the chiral active stress tensor [12], σc +ij = [∇ × (nn)]ij + [∇ × (nn)]ji, such that + +d21 +to linear order in director distortions we have +f a = +� +χσc · dA ≈ 0, +(75) +τ a = +� +x × χσc · dA ≈ +� +χ +� +ex +�xz ∂xδnx − 2yz∂yδnx + (y2 − z2)∂zδnx +r +� ++ ey +�yz∂yδny − 2xz∂xδny + (x2 − z2)∂zδny +r +� ++ ez +−2xy(∂yδnx + ∂xδny) − (x2 − y2)(∂xδnx − ∂yδny) + z(x∂zδnx + y∂zδny) +r +� +dA. +(76) +From the first of these equations we see that, to linear order, there are no harmonic distortions which produce net +forces in a nematic with chiral active stresses. With regard to the net active torques, the x− and y− components +involve only δnx and δny respectively and each term yields a non-zero integral only for δni ∼ ∂z1/r, hence the two +spin-1 dipoles produce transverse torques. Turning to the z-component, each term gives a non-zero integral only for +δni ∼ ∂i1/r, and as the expression is symmetric under interchange of x and y we see that only the UPenn dipole +produces torques around ez. In other words, a dipolar director distortion which produces a net active force along +a given direction in an achiral active nematic produces a net torque around the same direction in a chiral active +nematic. +These results of course accord with our earlier statements regarding the spins of distortions which are +capable of producing torques about given axes. Performing the integrals we find that in each case the net active +torque has magnitude −12πχαa2/5. Balancing this against Stokes drag gives, using the UPenn dipole as an example, +an angular velocity +Ω = 3χα +10µaez. +(77) +While the angular velocity in achiral active nematics is independent of the distortion size, in chiral active nematics +it is inversely proportional to the radius, a direct consequence of the additional derivative in the active stress tensor. +Accordingly, in chiral active nematics the rotational velocity is largest for smaller colloids. +VIII. +DISCUSSION +We have introduced active nematic multipoles as a novel framework for understanding the dynamics of active +nematics. Although only formally valid on mesoscopic lengthscales, this approach produces results for the propulsive +dynamics of defect loops that agree with those of a local analysis [23, 24]. It also provides various testable predictions, +for example for the axis of self-propulsion or rotation induced by a distortion or how the corresponding velocities +would scale with the size of a colloid. +More broadly, our results reveal self-propulsion and rotation as generic non-equilibrium responses that naturally +arise due to colloidal inclusions in active nematics but also provide a template for the tailored design of particular +dynamics. This provides insight into the issue of harnessing the energy of active systems to perform useful work, +something which has been demonstrated in bacterial suspensions [71, 82] and is now receiving greater attention in +the nematic context [36, 37, 83, 84]. Specific anchoring conditions on colloids have been investigated as a means of +generating directed motion [36]. Our results suggest that sufficient control of the anchoring conditions would allow for +steerable and targeted colloidal delivery [85], although there may be routes to a similar degree of dynamical control +through colloidal geometry alone [62]. +The transformative power of colloids in passive nematics was revealed in their collective behaviour, forming crys- +talline structures [28, 86–89] which can serve as photonic metamaterials [90]. While our predictions for the dynamics +of individual colloids have utility in their own right, there is again considerable interest in the collective dynamics +which might emerge [91]. Although our results are insufficient to fully address these questions, some basic points +can nonetheless be extracted from the flow solutions. The long-range nature of the active flows suggests that the hy- +drodynamic interactions will be dominant over elastic ones. The leading contribution to the pair-wise hydrodynamic +interactions will be the advection of each colloid by the flow field generated by the other, and the even inversion +symmetry of dipole flows implies that this provides a mechanism for pair-wise propulsion, even for colloids which are +not self-propulsive themselves. +To conclude, it has been long-established that the distinct symmetries of ±1/2 nematic defects can be directly +related to the qualitatively different dynamics they display in active systems [15, 16]. The aim of this paper is to +bring the insights of this symmetry-based approach to generic nematic distortions. + +22 +ACKNOWLEDGMENTS +This work was supported by the UK EPSRC through Grant No. EP/N509796/1. +[1] S. Ramaswamy. The mechanics and statistics of active matter. Annu. Rev. Condens. Matter Phys., 1(1):323–345, 2010. +[2] M.C. Marchetti, J.-F. Joanny, S. Ramaswamy, T.B. Liverpool, J. Prost, M. Rao, and R.A. Simha. Hydrodynamics of soft +active matter. Rev. 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Nature, 503(7474):95–98, 2013. + diff --git a/1tFST4oBgHgl3EQfWzjN/content/tmp_files/load_file.txt b/1tFST4oBgHgl3EQfWzjN/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..ee9992613acb527dd375558520dbaa8eb3891aa3 --- /dev/null +++ b/1tFST4oBgHgl3EQfWzjN/content/tmp_files/load_file.txt @@ -0,0 +1,1437 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf,len=1436 +page_content='Active Nematic Multipoles: Flow Responses and the Dynamics of Defects and Colloids Alexander J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Houston1 and Gareth P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Alexander1, 2, ∗ 1Department of Physics, Gibbet Hill Road, University of Warwick, Coventry, CV4 7AL, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 2Centre for Complexity Science, Zeeman Building, University of Warwick, Coventry, CV4 7AL, United Kingdom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (Dated: Wednesday 1st February, 2023) We introduce a general description of localised distortions in active nematics using the framework of active nematic multipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We give the Stokesian flows for arbitrary multipoles in terms of differentiation of a fundamental flow response and describe them explicitly up to quadrupole order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We also present the response in terms of the net active force and torque associated to the multipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This allows the identification of the dipolar and quadrupolar distortions that generate self-propulsion and self-rotation respectively and serves as a guide for the design of arbitrary flow responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Our results can be applied to both defect loops in three-dimensional active nematics and to systems with colloidal inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' They reveal the geometry-dependence of the self-dynamics of defect loops and provide insights into how colloids might be designed to achieve propulsive or rotational dynamics, and more generally for the extraction of work from active nematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Finally, we extend our analysis also to two dimensions and to systems with chiral active stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' INTRODUCTION Active liquid crystals model a wide range of materials, both biological and synthetic [1–3], including cell mono- layers [4], tissues [5], bacteria in liquid crystalline environments [6] and bacterial suspensions [7], and synthetic suspensions of microtubules [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Nematic and polar phases have been the focus of attention but smectic [9, 10], cholesteric [11, 12] and hexatic [13] phases have also been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Key features and motifs of the active nematic state include self-propelled topological defects [14–16], spontaneous flows and vortices, and on how these may be controlled through boundary conditions, confinement [17–19], external fields, geometry or topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Active defects, in particular, have been related to processes of apoptosis in epithelial sheets [5], tissue dynamics, bacterial spreading and biofilm formation, and morphogenesis in Hydra [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In three-dimensional active nematics the fundamental excitations are defect loops and system-spanning lines [21, 22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The defect loops actively self-propel [23], and self-orient [24], in addition to undergoing deformations in shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Their finite extent means that they represent localised distortions to the nematic director, on scales larger than their size, and this facilitates a description through elastic multipoles [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' It also invites comparison with colloidal inclusions in passive liquid crystals, which create localised realignments of the director and act as elastic multipoles [25–27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' These multipole distortions mediate interactions between colloids and allow for a means of controlling both the colloidal inclusions and the host material.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' For instance, they facilitate self-assembly and the formation of metamaterials [28, 29], and enable novel control of topological defects [27, 30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' While there have been studies of active nematic droplets in a host passive liquid crystal [32, 33], colloidal inclusions in host active nematics have not been looked at previously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The multipole approach to describing colloidal inclusions and localised director distortions in general, offers an equally fruitful paradigm in active nematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Here, we present a generic analysis of the active flows generated by multipole director distortions in an active nematic and predict that the presence of colloids transforms their behaviour similarly to the passive case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' These active multipole flows represent the responses of the active nematic both to localised features, such as defect loops, and to colloidal inclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This allows us to identify those distortions which produce directed or rotational flows and show that such distortions may be naturally induced by colloids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We also characterise the response in terms of the active forces and torques that they induce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This general connection can serve as a guide for using colloidal inclusions as a means to control active nematics, or how to design them to engineer a desired response, or extract work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The properties of inclusions have been studied in scalar active matter [34], as have active droplets in passive nematics [35], but while there have been specific demonstrations of propulsive colloids [36, 37] the general responses of inclusions in active nematics have not previously been considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Understanding how such responses relate to local manipulations and molecular fields in active nematics will bring both fundamental insights and the potential for control of active metamaterials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The remainder of this paper is structured as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In Section 2 we briefly review the equations of active nemato- hydrodynamics and describe the regime in which our linear multipole approach applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In Section 3 we present these ∗ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content='P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content='Alexander@warwick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content='uk arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content='13782v1 [cond-mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content='soft] 31 Jan 2023 2 multipoles as complex derivatives acting on 1/r, showing how this naturally elucidates their symmetries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In Section 4 we show that the linear active response to a harmonic distortion is generated by the same complex derivatives acting on fundamental flow and pressure solutions and highlight certain examples that illustrate the self-propulsive and rotational dynamics that can arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We then show in Section 5 that these phenomenological responses can be discerned from integrals of the active stress, allowing the identification of the distortion which produces propulsion along or rotation about a given axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Sections 6 and 7 contain extensions of our approach, first to two-dimensional systems and then to those with chiral active stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Section 8 gives a discussion and summary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' HYDRODYNAMICS OF ACTIVE NEMATICS We summarise the hydrodynamics of active nematics as described by their director field n and fluid velocity u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The fluid flow satisfies the continuity ∂iui = 0 and Stokes ∂jσij = 0 equations, with stress tensor [1–3] σij = −pδij + 2µDij + ν 2 � nihj + hinj � + 1 2 � nihj − hinj � + σE ij − ζninj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (1) Here, p is the pressure, µ is the viscosity, Dij = 1 2(∂iuj + ∂jui) is the symmetric part of the velocity gradients, ν is the flow alignment parameter, hi = −δF/δni is the molecular field associated with the Frank free energy F, σE ij is the Ericksen stress, and ζ is the magnitude of the activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The active nematic is extensile when ζ > 0 and contractile when ζ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The director field satisfies the relaxational equation ∂tni + uj∂jni + Ωijnj = 1 γ hi − ν � Dijnj − ni(njDjknk) � , (2) where γ is a rotational viscosity and Ωij = 1 2(∂iuj − ∂jui) is the antisymmetric part of the velocity gradients.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We adopt a one-elastic-constant approximation for the Frank free energy [38] F = � K 2 � ∂inj �� ∂inj � dV, (3) for which the molecular field is hi = K � ∇2ni − ninj∇2nj � and the Ericksen stress is σE ij = −K∂ink ∂jnk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' An often-used analytical approximation is to consider the active flows generated by an equilibrium director field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This approximation has been used previously in the theoretical description of the active flows generated by defects in both two [16, 39] and three dimensions [23], including on curved surfaces [40], and in active turbulence [41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' It may be thought of in terms of a limit of weak activity, however, even when the activity is strong enough to generate defects, their structure may still be close to that of equilibrium defects and the approximation remain good and the comparison of active defect motion and flows described in this way with full numerical simulations suggests that this is at least qualitatively the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The equations can then be reduced to h = 0 for the director field and the Stokes equation − ∇p + µ∇2u = ζ∇ · � nn � , (4) for the active flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Here we have neglected the Ericksen stress since for an equilibrium director field it can be balanced by a contribution to the pressure (representing nematic hydrostatic equilibrium).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We limit our analysis to director fields that can be linearised around a (locally) uniformly aligned state, n = ez +δn, with δn · ez = 0, for which the equations reduce to ∇2δn = 0, (5) ∇ · u = 0, (6) −∇p + µ∇2u = ζ � ez � ∇ · δn � + ∂zδn � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (7) These correspond to elastic multipole states in the director field, which are often thought of as an asymptotic de- scription, however, they provide a close approximation even at only moderate distances outside a ‘core’ region that is the source of the multipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' To illustrate this we show in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 1 a comparison between the exact director field (red streamlines) and linear multipole approximation (blue rods) for the most slowly varying monopole distortion created by uniformly rotating the director by an angle θ0 within a sphere of radius a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The agreement is close anywhere outside the sphere and only deviates significantly in the near-field region inside it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This is relevant to the active system as it is well-known that the uniformly aligned active nematic state is fundamentally unstable [42] and active nematics are turbulent on large enough scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Our solutions should be interpreted as describing the behaviour on intermediate scales, larger than the core structure of the source but smaller than the scale on which turbulence takes over.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Comparison of the exact director field (red streamlines) and linearised multipole approximation (blue rods) for the most slowly decaying monopole distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This is produced by uniformly rotating the director by an angle θ0 within a spherical volume of radius a, indicated by the grey disc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' the alignment inside the sphere is indicated by the thick red line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The figure shows only the xz-plane in which the director rotates and in which the comparison is most strict.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' MULTIPOLE DIRECTOR DISTORTIONS In this section, we describe the multipole director fields satisfying (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The far-field orientation ez gives a splitting of directions in space into those parallel and perpendicular to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We complexify the perpendicular plane to give the decomposition as R3 ∼= C ⊕ R and convert the director deformation δn to the complex form δn = δnx + iδny.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The real and imaginary parts of δn are harmonic, meaning that at order l they may be expressed as spherical harmonics 1/rl+1Y l m or, as we shall do, as l derivatives of 1/r [43–45].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' These order l multipole solutions form a 2(2l + 1)-real- dimensional vector space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Associated to the C ⊕ R splitting is a local symmetry group isomorphic to U(1), preserving ez, whose irreducible representations provide a natural basis for the vector space of multipoles at each order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We write the complex derivatives on C as ∂w = 1 2(∂x − i∂y) and ∂ ¯ w = 1 2(∂x + i∂y) in terms of which the director deformation can be written δn = ∞ � l=0 l � m=−l qlm al+1 ∂m ¯ w ∂l−m z 1 r , (8) where qlm are complex coefficients and a is a characteristic length scale of the multipole, as might be set by the radius of a colloid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' For compactness of notation it is to be understood that when m is negative ∂m ¯ w represents ∂|m| w .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The index m denotes the topological charge of the phase winding of the spherical harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This gives the spin of the corresponding vector field as 1 − m, where the 1 is due to a vector (δn or δn) being a spin-1 object.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The multipoles at order l therefore have spins that range from 1 − l to 1 + l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' They are illustrated up to quadrupole order in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 2, along with a representation in terms of topological defects which we shall elaborate upon shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The structure of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 2 is such that differentiation maps the distortions of one order to the next, with ∂z leaving the distortion in the same spin class, ∂ ¯ w moving it one column to the left and ∂w moving it one column to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The operators ∂w and ∂ ¯ w play the same role as the raising and lowering operators in quantum mechanics and the shift by one in the spin values simply results from the object on which they act being a spin-1 director deformation as opposed to a spin-0 wavefunction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The monopole distortions, with l = 0, result from a rotation of the director by an angle θ0 in a sphere of radius a [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' They form a two-real-dimensional vector space for which a basis may be taken to be the distortions 1 r and i 1 r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' These are shown at the top of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 2 and can be controllably created in passive nematics using platelet inclusions [47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The director distortions of dipole type, with l = 1, form a six-real-dimensional vector space that splits into two- 24 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The multipolar director distortions up to quadrupole order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The director is shown on a planar cross-section as blue rods, along with a topological skeleton corresponding to the spherical harmonic, where appropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Defect loops are coloured according to wedge (blue) or twist (red-green) type and the charge of point defects is indicated through the use of opposing colour pairs: red (+1) and cyan (−1), yellow (+2) and blue (−2), and green (+3) and magenta (−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Their charge is further indicated by a local decoration of the director with an orientation, indicated by black arrows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Each multipole order is classified into vertical pairs according to the spin of the distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' For the chiral multipoles, the visualisation instead shows the director along some of its integral curves (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 1 0 2 3 Monopoles Dipoles Quadrupoles5 real-dimensional subspaces for each value of the spin (0, 1, or 2) as p0 = � ∂ ¯ w 1 r , i ∂ ¯ w 1 r � ∼ − 1 2r3 � x ex + y ey, −y ex + x ey � ∼ 1 r2 � Y 1 1 , i Y 1 1 � , (9) p1 = � ∂z 1 r , i ∂z 1 r � ∼ − 1 r3 � z ex, z ey � ∼ 1 r2 � Y 0 1 , i Y 0 1 � , (10) p2 = � ∂w 1 r , i ∂w 1 r � ∼ − 1 2r3 � x ex − y ey, y ex + x ey � ∼ 1 r2 � Y −1 1 , i Y −1 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (11) For comparison, we have presented three representations for the distortions of each spin class: in terms of complex derivatives of 1/r, two-component vectors whose coefficients are homogenous polynomials of degree 1 and complex spherical harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In the interest of space we have suppressed certain prefactors in the last of these, but note the difference in sign, and in some cases normalisation, between our representation as complex derivatives and the standard form of the harmonic distortions as two-component vectors [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The two basis functions of any spin class are related by a factor of i, which corresponds to a local rotation of the transverse director distortion by π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' For a spin-s distortion this is equivalent to a global rotation by π 2s, with the pair of distortions having the same character and simply providing a basis for all possible orientations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The exception is when s = 0, such distortions lack an orientation and the local rotation produces two distinct states that transform independently under rotations as a scalar and pseudoscalar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In the dipole case the first is the isotropic distortion recognisable as the UPenn dipole [25] and the second is an axisymmetric chiral distortion with the far-field character of left-handed double twist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Separating p0 into its isotropic and chiral components allows a decomposition of the dipole director deformations into the basis p = pI ⊕ pC ⊕ p1 ⊕ p2, (12) a decomposition which was presented in [49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Similarly,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' the quadrupolar distortions (l = 2) form a ten-real-dimensional vector space that splits into a sum of two-real-dimensional subspaces for each value of the spin Q−1 = � ∂2 ¯ w 1 r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' i ∂2 ¯ w 1 r � ∼ 3 4r5 � (x2 − y2) ex + 2xy ey,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' −2xy ex + (x2 − y2) ey � ∼ 1 r3 � Y 2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' i Y 2 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (13) Q0 = � ∂2 ¯ wz 1 r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' i ∂2 ¯ wz 1 r � ∼ 3 2r5 � xz ex + yz ey,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' −yz ex + xz ey � ∼ 1 r3 � Y 1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' i Y 1 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (14) Q1 = � ∂2 z 1 r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' i ∂2 z 1 r � ∼ 1 r5 � (2z2 − x2 − y2) ex,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (2z2 − x2 − y2) ey � ∼ 1 r3 � Y 0 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' i Y 0 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (15) Q2 = � ∂2 wz 1 r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' i ∂2 wz 1 r � ∼ 3 2r5 � xz ex − yz ey,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' yz ex + xz ey � ∼ 1 r3 � Y −1 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' i Y −1 2 � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (16) Q3 = � ∂2 w 1 r ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' i ∂2 w 1 r � ∼ 3 4r5 � (x2 − y2) ex − 2xy ey,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 2xy ex + (x2 − y2) ey � ∼ 1 r3 � Y −2 2 ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' i Y −2 2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (17) Once again the spin-0 distortions can be further partitioned into those that transform as a scalar and pseudoscalar, these being the Saturn’s ring distortion [50] and a chiral quadrupole with opposing chirality in the two hemispheres, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This yields the basis for the quadrupolar director deformations Q = Q−1 ⊕ QI ⊕ QC ⊕ Q1 ⊕ Q2 ⊕ Q3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (18) The well-known multipoles, the UPenn dipole and Saturn ring quadrupole, are associated to a configuration of topological defects in the core region and we describe now an extension of this association to all of the multipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In general, such an association is not unique, for instance, the colloidal ‘bubblegum’ configuration [51] represents the same far field quadrupole as the Saturn ring, however, for each multipole we can construct a representative arrangement of topological defects which produce it in the far field on the basis of commensurate symmetries and defects of a type and location corresponding to the nodal set of the harmonic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This correspondence allow us to condense the visualisation of complicated three-dimensional fields into a few discrete elements, suggests means by which such distortions might be induced and enables us to build an intuition for their behaviour in active systems through established results for defects [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We first describe some examples, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' On the left is the spherical harmonic that describes the UPenn dipole, with the form ∂ ¯ w 1 r ∼ eiφ sin θ, visualised on a spherical surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This has nodes at the two poles about which the phase has −1 winding and so we can infer similar winding of the director in the transverse plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Supplementing 6 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The connection between spherical harmonics and nematic topological defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The coloured spheres indicate the phase of the complex spherical harmonics with the nodal set shown in white for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' A representative skeleton of the corresponding nematic distortion is shown in black and the red arrows indicate the winding vector of the director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' with the far-field alignment along ez yields the familiar picture of a pair of oppositely charged hedgehog defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Similarly, the Saturn ring quadrupole, described by ∂ ¯ wz 1 r ∼ eiφ sin 2θ, has zeros at the poles and around the equator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The winding about the poles is still +1, but the sign change in the lower hemisphere means that in the transverse plane around the south pole the vector points inwards, resulting in both point defects having topological charge +1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' With regards to the equatorial line, since the director is everywhere radial the winding vector must be tangential to the defect loop, shown by the red arrows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' As the phase changes by π on passing from one hemisphere to the other the winding must be ±1 and the far-field alignment allows us to determine it to be −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' For a general multipole distortion of the form ∂m ¯ w ∂l−m z (1/r) the nodal set is the poles along with l − m lines of latitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The phase winding of the spherical harmonic dictates the transverse winding of the director and, when supplemented with the far-field alignment, allows us to associate topological point defects with the poles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Similarly, nodal lines may be connected with defect loops with integer winding and a winding vector that rotates according to eimφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 3 we illustrate this for the case ∂2 ¯ w∂3 z(1/r) ∼ −Y 5 2 /r6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We now describe briefly the correspondence for our basis of dipolar and quadrupolar distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' As already stated, the isotropic scalar in p0 is the UPenn dipole, its pseudoscalar counterpart a chiral splay-free twist-bend distortion whose integral curves are shown in orange in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' As a twist-bend mode it may be of particular relevance to extensional systems given their instability to bend distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The two dipoles of p1 are transverse to the far-field alignment, they are related to those resulting from a defect loop of wedge-twist type [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The distortions of p2 have a hyperbolic character;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' they describe the far field of a pair of point defects both of which have a hyperbolic structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Such hyperbolic defect pairs arise in toron configurations in frustrated chiral nematics [52, 53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Similarly, Q0 contains the Saturn ring quadurpole as the scalar, with the pseudoscalar a pure bend chiral distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' For the latter, the integral curves of the director possess opposing chirality in the two hemispheres, which could be generated by an appropriately coated Janus particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The director distortion exhibits a helical perversion in the z = 0 plane and, being a local rotation of the Saturn ring distortion, may be viewed as resulting from a pair of vortex point defects along with a pure twist defect loop with integer winding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This is similar to the bubblegum defect lines [51, 54] that appear between a colloid diad with normal anchoring, suggesting that this chiral quadrupole could be formed by two colloids with opposing chiral tangential anchoring.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The spin-1 quadrupoles consist of pairs of wedge-twist defect loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The distortions of Q2 may be associated with a pair of hyperbolic defects along with a defect ring with the appropriate symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The harmonics of spin −1 and 3 contain no z-derivatives and so are associated with pairs of point defects only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' FLOWS FROM MULTIPOLE DISTORTIONS In this section we calculate the active flow generated by an arbitrary director multipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We present this initially in vectorial form, converting to the complex representation subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' As (7) is linear the responses due to the two components of δn are independent and so to simplify the derivation we consider only distortions in the x-component for now and extend to the general case afterwards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Within this restriction a generic multipole distortion at order l 22 75 r3 r6 r Y7 may be written as δnx = al∇v1 · · · ∇vl a r , (19) where v1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' , vl are l directions for the differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Substituting this into (7) gives the Stokes equation in the form − ∇p(x) + µ∇2u(x) = al+1ζ∇v1 · · · ∇vl � ex ∂z + ez ∂x �1 r , (20) where the use of the superscript (x) is to emphasise that we are only treating the response to distortions in the x-component of the director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Taking the divergence of both sides we have − ∇2p(x) + µ∇2∇ · u(x) = al+1ζ∇v1 · · · ∇vl∂2 xz 2 r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (21) Making use of the continuity equation ∇ · u(x) = 0 in conjunction with the identity ∇2r = 2 r we arrive at the solution for the pressure p(x) = −al+1ζ∇v1 · · · ∇vl ∂x∂zr = al+1ζ∇v1 · · · ∇vl xz r3 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (22) Substituting this back into the Stokes equation (20) we obtain µ∇2u(x) = al+1ζ∇v1 · · · ∇vl � ex ∂z �1 r − ∂x∂xr � − ey ∂x∂y∂zr + ez ∂x �1 r − ∂z∂zr �� , (23) which can be integrated using the identity ∇2r3 = 12r to find u(x) = al+1 ζ 4µ∇v1 · · · ∇vl � ex �z r + x2z r3 � + ey xyz r3 + ez �x r + xz2 r3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (24) Both the pressure and flow solutions for a generic multipole distortion are given in terms of derivatives of a fundamental response to a monopole deformation, namely p(x) = aζ xz r3 , (25) u(x) = aζ 4µ � ex �z r + x2z r3 � + ey xyz r3 + ez �x r + xz2 r3 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (26) This flow response, shown as the top panel in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 4, is primarily extensional in the xz-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Interestingly, the flow solution (26) does not decay with distance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' this reflects the generic hydrodynamic instability of active nematics [42] providing a real-space local response counterpart to the usual Fourier mode analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' However, the active flow produced by any higher multipole does decay and vanishes at large distances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The pressure and flow solutions in (25) and (26) are complemented by analogous ones resulting from distortions in the y-component of the director, obtained by simply interchanging x and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The linearity of (7) makes these fundamental responses sufficient to obtain the active flow induced by an arbitrary multipole distortion through taking derivatives appropriate to describe the x and y components of the director, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We now convert this description to the complex notation used in § III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This is achieved by taking the combinations p = p(x) − ip(y) and u = u(x) − iu(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' To see this consider the multipole distortion δn = (Lx + iLy)1/r, where the Li are generic real differential operators which generate the i-component of the director by acting on 1/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This distortion has a conjugate partner given by i(Lx + iLy)1/r = (−Ly + iLx)1/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Acting with this same operator on u(x) − iu(y) we have (Lx + iLy)(u(x) − iu(y)) = (Lxu(x) + Lyu(y)) − i(−Lyu(x) + Lxu(y)), (27) and can see that the flow response for our original distortion forms the real part and that for its conjugate partner the coefficient of −i and the same holds for the pressure response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This leads us to a complex fundamental pressure response ˜p = aζ ¯wz r3 , (28) 8 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The active flows due to three-dimensional nematic multipole distortions up to quadrupole order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The flows are grouped according to their spin, in correspondence with the distortions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Green and red arrows indicate the net active force and torque for the relevant dipoles and quadrupoles respectively, see §V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' and, introducing complex basis vectors ew = ex + iey and e ¯ w = ex − iey, a complex-valued fundamental flow vector ˜u = aζ 4µ � ew ¯w2z 2r3 + e ¯ w �z r + w ¯wz r3 � + ez ¯w r � 1 + z2 r2 �� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (29) We use a tilde to distinguish these fundamental responses from those that result due to a generic distortion and which may be found by appropriate differentiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This provides a unified framework in which the active response to a generic nematic multipole can be calculated through the application of the same complex derivatives that we have used to describe the director distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The resulting active flows for distortions up to quadrupole order are shown 1 0 2 3 Monopoles Dipoles Quadrupoles9 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 4, with their layout corresponding to that of the nematic distortions in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 2 which induce them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We now describe some examples in more detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' UPenn and chiral dipole Typically the active responses induced by the two distortions in a spin class will, like the distortions themselves, be related by a global rotation such that while both are needed to form a sufficient basis, the real part essentially serves as a proxy for the pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This is not true for the spin-0 distortions, due to their rotational symmetry, and so we use them in providing an explicit illustration of the active flow calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We begin with the UPenn dipole [25] and its partner the chiral dipole, for which the far-field transverse director is δn ≈ αa ∂ ¯ w a r , (30) where α is a dimensionless coefficient, and the corresponding derivative of the fundamental flow solution in (29) gives αa∂ ¯ w˜u = ζαa2 4µr5 � ew z ¯w(4z2 + w ¯w) − e ¯ w 3zw2 ¯w + ez 2 � 3z4 + (z2 + w ¯w)2�� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (31) Taking the real part gives, after some manipulation, the flow induced by the UPenn dipole as u = αa R ∂ ¯ w˜u = ζαa2 8µ � ez �1 r + z2 r3 � + er z r2 �3z2 r2 − 1 �� , (32) where er is the unit vector in the radial direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The flow response to the conjugate distortion, the isotropic chiral dipole is given by u = −αa I ∂ ¯ w˜u = −ζαa2 4µ z r2 eφ, (33) with eφ the azimuthal unit vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Both flows decay at large distances like 1/r and are highlighted in the top row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The UPenn dipole flow has a striking net flow directed along the z-axis, reminiscent of that of the Stokeslet flow [55, 56] associated with a point force along ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The chiral dipole generates an axisymmetric flow composed of two counter-rotating vortices aligned along ez, mirroring the circulating flows produced by spiral defects in two dimensions [57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The 1/r decay of these active vortex flows is unusually slow, slower than the decay of a point torque in Stokesian hydrodynamics [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Despite the similarity between the active flow induced by the UPenn dipole and a Stokeslet, there is a key difference in their angular dependence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In a Stokeslet, and all related squirming swimmer flows [58, 59] that result from derivatives of it, the terms with higher angular dependence decay more quickly such that the lowest order terms dominate the far field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' By contrast, distortions in active nematics produce asymptotic flow fields in which all terms decay at the same rate regardless of their angular dependence as they all result from the same derivative of the fundamental flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Thus, even if the same angular terms are present in both systems, the lowest order ones will dominate in the squirming case while the far field will bear the signature of the highest order in the active nematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' A closer point of comparison comes from the flows induced by active colloids within a passive nematic [35, 60].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Calculation of the relevant Green’s functions [61] has shown that the anisotropy of the medium leads to a difference in effective viscosities such that a Stokeslet aligned along the director pumps more fluid in this direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This fits with the anisotropy displayed in (32), reaffirming the similarity between the flow induced by the UPenn dipole and the Stokeslet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Considering the pressure response for these distortions in the same way we have αa∂ ¯ w ˜p = ζαa2 2r5 z(2z2 − w ¯w) = ζαa2z 2r3 �3z2 r2 − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (34) As this expression is purely real it comprises the response due to the UPenn dipole in its entirety;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' the vanishing of the imaginary part shows that the chiral dipole is compatible with a zero pressure solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Our complexified construction allows this property to be read off immediately, since ∂ ¯ w( ¯wzm/rn) will be real for any m and n, with this also resulting in the vanishing z-component of flow for the chiral dipole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Indeed, this property of pure realness is unchanged by the action of ∂z, it being real itself, and so extends to higher order distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 10 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The active flows induced by spin 0 dipole (top row) and quadrupole (bottom row) distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The flow is indicated by blue arrows and superposed upon integral curves of the director, shown in orange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' On the left are the UPenn dipole and Saturn ring quadrupole and on the right their chiral counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Saturn ring and chiral quadrupole Proceeding in the same fashion for the spin-0 quadrupoles, for which δn ≈ αa2∂2 ¯ wza/r, we find that the complexified flow is αa2∂2 ¯ wz˜u = −ζαa3 4µr7 � −ew ¯w(w2 ¯w2 + 8w ¯wz2 − 8z4) + e ¯ w3w2 ¯w(w ¯w − 4z2) +ez2z(w2 ¯w2 − 10w ¯wz2 + 4z2) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (35) Taking the real part gives the flow induced by the Saturn ring quadrupole as u = αa2R∂2 ¯ wz˜u = −ζαa3 2µr6 (r4 − 12z2r2 + 15z4)er, (36) that is a purely radial flow reminiscent of a stresslet along ez, shown in the bottom left of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The purely radial nature is a result of the divergencelessness of the flow, combined with the 1/r2 decay and rotational invariance about ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Working in spherical coordinates we have ∇ · u = 1 r2 ∂r(r2ur) + 1 r sin θ [∂θ(uθ sin θ) + ∂φuφ] = 0 (37) All active flows induced by quadrupole distortions decay as 1/r2 and so ∂r(r2ur) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The distortion is rotationally symmetric and achiral, meaning uφ = 0 and the condition of zero divergence reduces to 1 r sin θ∂θ(uθ sin θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (38) The only non-singular solution is uθ = 0, resulting in ur being the only non-zero flow component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The corresponding pressure is given by αa2∂2 ¯ wz ˜p = −3αa3 2r7 (r4 − 12z2r2 + 15z4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (39) wz11 Taking the imaginary part of (35) reveals the flow response of the chiral quadrupole to be u = −αa2I∂2 ¯ wz˜u = ζαa3 µr2 (3 cos2 θ − 1) sin θeφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (40) As illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 5 this is a purely azimuthal flow corresponding to rotation about the z axis and, as for the chiral dipole, is compatible with a zero pressure solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The 1/r2 decay of this rotational flow is the same as that which results from the rotlet [55, 56], but unlike the rotlet the flow direction is not uniform.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Rather, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 5, there is an equatorial band of high-velocity flow accompanied by two slowly counter-rotating polar regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The distribution of flow speeds is such that the net flow is along −eφ, consistent with a rotlet along −ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Other multipoles For the remaining multipoles up to quadrupole order we do not provide the same explicit calculation but instead highlight the key features of the active flows they induce.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In full we find that half of the dipole distortions contain directed components in their active flow responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Along with the isotropic UPenn dipole which produces flow along ez the two spin-1 dipoles produce directed flows transverse to it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' These directed flows indicated that were the source of the distortion free to move it would exhibit active self-propulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The net transverse flows for the dipoles of p1 is in accordance with the previously established motile nature of such defect loops [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' A more complete description of the active dynamics of defect loops via their multipole distortions is presented in Section IV D and [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Along with the chiral dipole, the two additional dipoles which do not generate directed flows are those with spin 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' These produce active flows which are extensional with the expected two-fold rotational symmetry about the z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Direct calculation shows that the flows resulting from spin-2 distortions have zero azimuthal component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Once again, this observation is unaffected by z-derivatives and so holds true for the higher-order multipoles of the form ∂n z ∂w(1/r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Similarly, there are ten linearly independent quadrupoles, five of which can be seen from Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 4 to generate rotational flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' As expected, it is the four modes of Q±1 that generate rotations about transverse directions and QC that produces rotation around ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' For two of these, namely those in Q1, the director distortions are planar, suggesting a two-dimensional analogue and the potential to generate them with cogs or gears [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' These distortions may be associated with a pair of opposingly oriented charge-neutral defect loops and so the rotational flow generated by these distortions is in accordance with their antiparallel self-propulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The quadrupoles of Q−1 are composed of pairs of point defects with topological charge +2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Using ∂2 ¯ w 1 r as an example, the rotation can be understood by considering the splay distortions in the xz plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The splay changes sign for positive and negative x, leading to antiparallel forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The active forces are greatest in this plane, as this is where the transverse distortion is radial resulting in splay and bend distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Along ey the distortions are of twist type and so do not contribute to the active force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This results in the rotational flow shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The stretching of the flow along ez is as observed for a rotlet in a nematic environment [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Although they lack the rotational symmetry of a stresslet, the flows produced by the quadrupoles of Q2 are also purely radial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The argument is largely the same as for the Saturn ring distortion, except that the vanishing of uφ is not due to rotational invariance but a property inherited from the spin-2 dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The quadrupoles of Q3 produce extensional flows whose spin-3 behaviour under rotations about ez is commensurate with that of the distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Although they visually resemble the similarly extensional flows produced by the dipoles of p2, they do not share the property of a vanishing azimuthal flow component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Defect loops Of particular relevance to the dynamics of three-dimensional active nematics are charge-neutral defect loops [21, 23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' For such defect loops the director field has the planar form n = cos Υ 4 ez + sin Υ 4 ex, (41) where Υ is the solid angle function for the loop [43, 63], and is a critical point of the Frank free energy in the one- elastic-constant approximation [64].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This allows a multipole expansion for the director at distances larger than the loop size in which the multipole coefficients are determined explicitly by the loop geometry [24] Υ(x) = 1 2 � K ϵijk yj dyk ∂i 1 r − 1 3 � K ϵikl ylyk dyl ∂i∂j 1 r + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' , (42) 12 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Additonal flow solutions induced by spin-1 nematic multipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The nematic multipoles which induce the flows are shown below them as complex derivatives of 1/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The red arrows indicate the net active torque.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' where y labels the points of the loop K and r = |x| with the ‘centre of mass’ of the loop defined to be at x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The dipole moment vector is the projected area of the loop, while the quadrupole moment is a traceless and symmetric tensor with an interpretation via the first moment of area or, in the case of loops weakly perturbed from circular, the torsion of the curve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The planar form of the director field (41) corresponds to a restricted class of director deformations in which δn is purely real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This disrupts the complex basis we have adopted for the representation of multipoles, so that another choice is to be preferred.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We may say that the planar director selects a real structure for the orthogonal plane C, breaking the U(1) symmetry, and the restricted multipoles should then be decomposed with respect to this real structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Accordingly, the pressure and flow responses may be generated by derivatives of the fundamental responses for distortions in ex, (25) and (26), with these derivatives corresponding to the multipole expansion of the solid angle shown in (42).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The details of this approach along with the consequences it has for both the self-propulsive and self-rotational dynamics of active nematic defect loops are given in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Technical note We conclude this section with a technical note on the flow solutions that we have presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The construction for calculating active flow responses that we have developed in this section requires knowledge of the multipole as a specified set of derivatives of 1/r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The harmonic director components satisfy ∇2ni ∝ δ(r) and while this delta function does not affect the far-field director it impacts the flow solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Consequently, at quadrupole order and higher, distinct derivatives of 1 r can produce the same multipole distortion in the director but have different associated active flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' As an explicit example we take the spin-1 quadrupole shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 2, which may be written as n = a2∂2 z a r ex +ez and therefore induces an active flow given by the action of a2∂2 z on 29, as is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' However the same director distortion is captured by n = −4a2∂2 w ¯ w a r ex + ez, for which the corresponding active flow is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' A partial resolution to this ambiguity is that any non-equilibrium phenomenological features such as propulsion or rotation will be invariant to this choice of derivatives since, as we shall show in the following section, they can be expressed directly in terms of the director components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' As a more complete resolution we reiterate that whenever an exact solution for the director is known the appropriate derivatives can be determined, as demonstrated earlier for defect loops [24], and so the apparent ambiguity disappears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' ACTIVE FORCES AND TORQUES The directed and rotational active flow components highlighted above result in viscous stresses whose net effect must be balanced by their active counterparts, since the net force and torque must be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Consequently, these generic aspects of the response of an active nematic can be identified by considering the contribution that the active ww13 stresses make to the force and torque f a = � ζnn · dA ≈ � ζ � ex z δnx r + ey z δny r + ez x δnx + y δny r � dA, (43) τ a = � x × ζnn · dA ≈ � ζ � ex �xy δnx r + (y2 − z2)δny r � + ey �(z2 − x2)δnx r − xy δny r � + ez z(−y δnx + x δny) r � dA, (44) integrating over a large sphere of radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' These integrals depend on the surface of integration, as the active stresses are neither divergenceless nor compactly supported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' However, a spherical surface is concordant with the multipole approach we are taking and the results are then independent of the radius, as a direct consequence of the orthogonality of spherical harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' From these expressions we can read off the multipole that will generate any desired active force or torque;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' dipoles generate forces and quadrupoles generate torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' When the active torque is non-zero, the compensating viscous torque will drive a persistent rotation of the multipole, creating an active ratchet;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' similarly, a non-zero active force will generate directed fluid flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The above integrals therefore provide a solution to the inverse problem: given a particular non-equilibrium response, which distortion induces it?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Hence they serve as a design guide for generating out of equilibrium responses in active nematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' If the multipole is free to move it will self-propel and rotate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The translational and rotational velocities are related to the viscous forces and torques by a general mobility matrix [65].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In passive nematics, experiments [66] and simulations [67, 68] have found that it is sufficient to take a diagonal form for the mobility (no translation-rotation coupling) with separate viscosities for motion parallel, µ∥, and perpendicular, µ⊥, to the director, with typical ratio of viscosities µ⊥/µ∥ ∼ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content='6 [66–68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This has the consequence that in general the force and velocity are not colinear U = −1 6πa � 1 µ∥ f a ∥ ez + 1 µ⊥ f a ⊥ � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (45) We again use the UPenn dipole as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Integrating the active stresses over a spherical surface of radius R we find an active force � ζnn · dA ≈ −ζαa2 2 � � ex xz R4 + ey yz R4 + ez � z R + x2 + y2 R4 �� dA = −4πζαa2 3 ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (46) Balancing this against Stokes drag predicts a ‘self-propulsion’ velocity for the active dipole of U = 2ζαa 9µ∥ ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (47) For extensile activity (ζ > 0) the dipole moves ‘hyperbolic hedgehog first’ and with a speed that increases linearly with the core size a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This self-propulsion is in accordance with the directed component of the active flow, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The same self-propulsion speed along ex and ey is found for the transverse dipoles of p1, except that the parallel viscosity µ∥ should be replaced with µ⊥.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Again, this self-propulsion agrees with the directed flow induced by these distortions, as calculated through the multipole approach, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 4 [24] and also with the results of both a local flow analysis and simulations [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The same directed motion has been observed in a related system of an active droplet within a passive nematic [35], with the droplet inducing a UPenn dipole in the nematic and moving in the direction of the hedgehog defect at a speed that grew with the droplet radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The mechanism at play is different however;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' the motion results from directional differences in viscosity resulting from the anisotropic environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' To illustrate the rotational behaviour we use a member of Q1, ∂2 z(1/r), as an example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We find an active torque � ζx × nn · dA ≈ ζαa3 � 1 r6 (2z2 − x2 − y2) � xyex + (z2 − x2)ey − yzez � dA (48) = 8πζαa3 5 ey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (49) Balancing against Stokes drag as was done in the dipole case gives an angular velocity Ω = −ζα 5µey.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (50) 14 We note that for this and all other distortions which result in net torques the angular velocity is independent of the colloid size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In accordance with the relation ∂2 z +4∂2 w ¯ w(1/r) = 0, the torque resulting from ∂2 w ¯ w(1/r) is of the opposite sign and a quarter the strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The net active torques due to harmonics of Q0 and Q−1 have the directions indicated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 4 and half the magnitude of (49).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Let us consider the approximate magnitude of the effects we have described.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Beginning with the self-propulsion speed, the fluid viscosity is roughly 10−2 Pa s [17], although effects due to the elongated form of the nematogens could increase this by a factor of 30 or so [69, 70].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Both the activity [16] and the dipole moment constant [48] are of order unity, meaning the colloid would approximately cover its radius in a second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Similar approximations for the quadrupole give an angular velocity of about 2/3 rad s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' For a colloid of radius 10 µm this has an associated power of the order of femtowatts, the same as predicted for bacterial ratchets [71].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' TWO-DIMENSIONAL SYSTEMS AND RATCHETS As noted above, the planar nature of the rotational distortions in Q1 suggests the existence of two-dimensional analogues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In part motivated by this we now discuss the active response of multipolar distortions in two dimensions, again beginning with the connection between these multipoles and topological defect configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Multipoles and topological defects The categorisation of the harmonic distortions in two dimensions is much simpler, but we provide it here for completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Taking the asymptotic alignment to be along ey the symmetry of the far-field director is now described by the order 2 group {1, Ry}, with Ry reflection with axis ey, under which the monopole distortion nx ∼ A log(r/a) is antisymmetric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The higher-order distortions are once again generated via differentiation of the monopole, with ∂y leaving the symmetry under Ry unchanged and ∂x inverting it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' It should be noted that the potential multiplicity of differential representations of harmonics that arose in three dimensions does not occur in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This is because, under the assumption of a single elastic constant, the director angle φ may be written as the imaginary part of a meromorphic function of a single complex variable and this naturally defines the appropriate set of derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Making z = x+iy our complex variable we write φ = I {f(z)} which upon performing a Laurent expansion of f(z) around z = 0 and assuming the existence of a uniform far-field alignment gives φ = I � 0 � n=−∞ anzn � = I � a0 + ∞ � n=1 (−1)n−1 an (n − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content='∂n z (ln z) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (51) Hence at every order there is a one parameter family of distortions, corresponding to the phase of the an.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' A natural basis at order n is provided by {R {∂n z (ln z)} , I {∂n z (ln z)}}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This basis consists of a symmetric and anti-symmetric distortion under the action of Ry, the roles alternating with order, and of course correspond to the two harmonic functions cos nθ/rn and sin nθ/rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In two dimensions the connection between defect configurations and far-field multipole distortions can be made concrete, and also serves as an illustration of how a particular set of derivatives is determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' For defects with topological charges sj at locations zj the angle that the director makes to ex is given by φ = φ0 + � j sjI � ln �z − zj a �� , (52) which, upon performing a series expansion, gives φ = φ0 + � j sjI {ln(z/a)} − ∞ � n=1 I �� j sjzn j ¯zn� n|z|2n , (53) = φ0 + � j sjI {ln(z/a)} + ∞ � n=1 (−1)nI �� j sjzn j ∂n z ln z � n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' , (54) Provided the total topological charge is zero the winding term proportional to ln w vanishes and φ0 is the far-field alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The distortions are given as a series of harmonics in which the coefficient of the nth harmonic is determined by a sum of zn j weighted by the defect charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 15 We would like to have a basis of representative defect configurations for each harmonic distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' However, it can be seen from (54) that the correspondence between arrangements of topological defects and the leading order nematic multipole is not one-to-one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Two defect-based representations of harmonic will prove particularly useful to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The first, which we develop in this chapter, provides a representation in terms of half-integer defects on the disc and allows an intuition for the response to multipole distortions in active nematics through known results for such defects [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The second uses the method of images to construct defect arrangements corresponding to a specific anchoring condition on the disc, with the same multipoles dominating the nematic distortion in the far field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This representation naturally lends itself to the control of induced multipoles through colloidal geometry and is explored fully in [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Nonetheless, both of these representations will be of use to us in the remainder of this chapter and as they are equally valid near-field representations for the asymptotic distortions that we are considering we will pass fairly freely between them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' With this aforementioned half-integer representation in mind, let us consider sets of 2m defects sitting on the unit circle, with −1/2 defects at the mth roots of unity and +1/2 defects at the intermediate points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' A useful formula here is the following for the sum of a given power of these roots of unity, after first rotating them all by a given angle θ m−1 � k=0 � eiθei 2π m k�n = � meinθ, if m|n 0, otherwise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (55) The vanishing of this sum for values of n that are not multiples of m comes directly from the expression for the geometric sum and is a consequence of the cyclic group structure of the roots of unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' It means that the lowest order multipole distortion induced by such an arrangement of defects is order m and so allows a desired multipole distortion to be selected as the dominant far-field contribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Explicitly, the director angle is given by φ = φ0 + � k odd I � ¯zmk� k|z|2mk = φ0 + I {¯zm} |z|2m + O � 1 z3m � , (56) with the approximation becoming rapidly better for higher-order multipoles due to the condition that n must be an odd multiple of the number of defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Rotating the entire set of defects rigidly by an angle −π/(2m) generates the conjugate multipole as the dominant far-field contribution φ = φ0 + � k odd I � (−i)k¯zmk� k|z|2mk = φ0 − R {¯zm} |z|2m + O � 1 z3m � , (57) with the natural interpolation between these two harmonics as the defect configuration is rigidly rotated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Hence we can interchange between a given harmonic distortion and a defect arrangement which has this harmonic as its dominant far-field contribution, with the correspondence becoming rapidly more accurate for higher orders, allowing us to relate the existing results for the behaviour of active defects [15, 16] to ours and vice versa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This correspondence is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The locations of +1/2 and −1/2 defects are indicated with red and cyan dots respectively and the background colouring denotes the phase of the complex function � sj ln(z−zj), whose imaginary part provides the director angle for the given defect arrangement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The integral curves of this director field are shown in black and are remarkably well matched by those of the leading multipole, shown in white, despite the asymptotic nature of the approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In this context we are able to make precise the notion of a core region of a singular distortion, outside of which our multipole approach applies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The series in (54) is attained through a Taylor series of terms of the form ln(1 − 1/z), which are convergent for |z| > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' More generally the greatest radial displacement of a defect defines a core radius, outside of which the multipole series converges onto the exact director angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Flows from multipole distortions We can proceed analogously to our three-dimensional calculation in generating the active flows from a fundamental response in two dimensions, provided we are mindful of the logarithmic form that the monopole now has.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' A director rotation by θ0 inside a disc of radius a results in an equilibrium texture given by n = cos �θ0 log(r/R) log(a/R) � ey + sin �θ0 log(r/R) log(a/R) � ex, (58) which in the far field tends to a monopole distortion n ≈ ey + θ0 log(r/R) log(a/R) ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Due to the logarithmic divergence of the fundamental harmonic in two dimensions it is necessary to normalise through a large length R such that a uniformly aligned far-field director is recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 16 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Representative defect configurations for nematic multipoles in two dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The red and cyan dots indicate the locations of +1/2 and −1/2 defects respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The black curves are the integral curves of the corresponding director field and the background colour shows the phase of the complex function whose imaginary part gives the exact director angle, as in (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The white lines are the integral curves of the dominant multipole, that is the leading term of (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The multipole series converges onto the exact director angle outside a core region, shown as a white disc, and the leading multipole provides a remarkably good approximation in this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Following our three-dimensional analysis we solve Stokes’ equations to linear order in nematic deformations for a monopole distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We write Stokes’ equations in terms of complex derivatives as 2∂¯z(−p + iµω) = f, (59) where we have used that 2∂zu = ∇ · u + iω, with ω the vorticity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Hence we seek f as a ¯z-derivative, implicitly performing a Helmholtz derivative with the real and imaginary parts of the differentiated term corresponding to the scalar and vector potentials respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Expressing the active force in this way we have 2∂¯z(−p + iµω) = ζθ0 log(a/R)∂¯z �i¯z z � (60) and so − p + iµω = ζθ0 2 log(a/R) i¯z z .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (61) Reading off the pressure and vorticity, solving for the flow and converting back to Cartesians the fundamental flow response is now found to be ˜u = ζθ0 8µ log(a/R) �x2 − y2 r2 (−yex + xey) + 2 log � r R � (yex + xey) � , (62) ˜p = − ζθ0 log(a/R) xy r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (63) There is a clear similarity between these solutions and their three-dimensional counterparts, but while the fundamental flow response is still extensional it now grows linearly with distance from the distortion, with this change in scaling inherited by the subsequent harmonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' b) a ) (p17 As in the three-dimensional case we can gain general insight into the active response of a nematic by considering the net contribution of the active stresses to the force and torque when integrated over a large circle of radius r � ζnn · erdr ≈ � ζ �yδnx r ex + xδnx r ey � dr, (64) � x × ζnn · erdr ≈ � ζ (y2 − x2)δnx r dr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (65) We see that in two dimensions both dipoles will self-propel if free to move and there is a single chiral quadrupole which produces rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The far-field flow solutions for distortions up to dipole order are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 8, superposed over the nematic director.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Both dipoles are now motile and as in the three-dimensional case they set up flows reminiscent of the Stokeslet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Vertical and horizontal self-propulsive modes may be viewed as resulting from normal and tangential anchoring respectively of the nematic on a disc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Interpolating between these orthogonal modes the angle of motility changes commensurately with the anchoring angle, such that sufficient control of the boundary conditions would allow for self-propulsion at an arbitrary angle with respect to the far-field alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This change in the dipole character can be represented by rigidly rotating the defect pair around the unit circle and the resulting motility is as would be expected from the position and orientation of the +1/2 defect [16, 72, 73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Determining the motility induced by these dipolar modes is complicated by the Stokes paradox and although this can be circumvented by various means we do not pursue this here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' If such dipolar colloids were fixed within the material they would pump the ambient fluid and so it should be possible to use them to produce the concentration, filtering and corralling effects observed previously by funneling motile bacteria [74].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In line with our discussion at the beginning of this section, the basis quadrupoles are given by the real and imaginary parts of ∂2 z, these being an achiral and chiral mode respectively, which are shown along with their flows in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The flow generated by the achiral quadrupole in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 8(d) is purely radial and resembles the stresslet flow, unsurprising as it results from differentiating the vertical dipole in the same way as the stresslet is related to the Stokeslet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' It is produced by a quadrupole distortion which may be associated with normal anchoring on the disc – its counterpart with tangential anchoring has all the charges in its representative defect configuration inverted and a reversed flow response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Just as for the dipole distortions, the character of the quadrupole can be smoothly varied through adapting the boundary condition and the topological defects which represent the harmonic rotate rigidly in step with the changing anchoring angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' A generic anchoring angle will produce a net active torque, maximised for an angle of π/4 as illustrated for the chiral quadrupole shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 8(e).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' For extensile activity this distortion generates clockwise rotation, as can easily be justified via our representation of the far-field director structure as arising from a square arrangement of two +1/2 and two −1/2 defects – the dual mode with the defect charges interchanged rotates anticlockwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' By choosing boundary conditions such that the defects are positioned closer to the mid-line of the colloid the strength of the active torque can be tuned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' CHIRAL ACTIVE STRESSES Chirality is a ubiquitous trait, in living systems and liquid crystals alike.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In active matter it opens a wealth of new phenomena, including odd viscous [75] and elastic responses [76, 77], surface waves, rotating crystals [78] and non- reciprocal interactions [79].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Chiral active stresses induce vortex arrays in active cholesterics [12] and have also been shown to be important in nematic cell monolayers where they modify collective motion, the motility of topological defects and generate edge currents [80, 81].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' We now consider the effects of such chiral active stresses on nematic multipoles, both in two and three dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Two dimensions For chiral stresses in two dimensions, the active stress tensor has the form σc = χJ(nn − n⊥n⊥)/2, where J is the complex structure defined by Jn = n⊥ and Jn⊥ = −n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The chiral active force is ∇ · σc = χJ � ∇ · (nn) � , (66) and is simply a π/2 rotation of the achiral active force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Accordingly we can modify (61) to give − p + iµω = − ζθ0 2 log(a/R) ¯z z , (67) 18 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Distortions up to quadrupole order in two-dimensional active nematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The active flow in white is superposed on the pressure field, with the integral curves of the director shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (a) The fundamental monopole response is extensional and grows linearly with distance from the distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (b) and (c) show the flows induced by dipole distortions, labelled by the appropriate derivative of the nematic monopole, with the green arrows indicating the direction of self-propulsion that would result from net active forces in extensile systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The vertical and horizontal dipoles are the far-field director responses to normal and tangential anchoring respectively and may also be interpreted as arising from a pair of +1/2 (cyan) and −1/2 (red) defects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The self-propulsion matches that expected for the +1/2 defect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' and solve as before to find ˜u = χθ0 8µ log(a/R) �2xy r2 (−yex + xey) + 2 log � r R � (−xex + yey) � , (68) ˜p = χθ0 log(a/R) x2 − y2 2r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (69) Another way to understand the relation between achiral and chiral stresses is that, since the monopole active force field is spin-2, the π/2 local rotation of the active force results in a global rotation by π/4 of the force field and hence the fundamental flow responses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The action of this global rotation, denoted Rπ/4, may be seen by comparing the monopole flow responses for achiral and chiral stresses, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 8(a) and Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 9(a) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' For distortions of order n there are two basis flows, ur and ui, corresponding to the real and imaginary parts of ∂n z respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The rotation of the monopole response has the consequence that for achiral and chiral active stresses these flows are related by uc r = Rπ/4 � cos �nπ 4 � ua r − sin �nπ 4 � ua i � , (70) uc i = Rπ/4 � sin �nπ 4 � ua r + cos �nπ 4 � ua i � , (71) a) b) % (p e) 02 hc19 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Distortions up to quadrupole order in two-dimensional active nematics with purely chiral stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The active flow in white is superposed on the pressure field, with the integral curves of the director shown in black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (a) The fundamental monopole response is extensional and grows linearly with distance from the distortion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (b) and (c) show the flows induced by dipole distortions, labelled by the appropriate derivative of the nematic monopole, with the green arrows indicating the direction of self-propulsion that would result from net active forces in extensile systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' where the superscripts denote the nature of the stresses as achiral or chiral.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Hence flow solutions for chiral and achiral stresses are related by a clockwise rotation by nπ/4 in the space of solutions followed by a rigid spatial rotation anticlockwise by π/4, as can be seen in Fig 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' At dipole order the chiral flow fields are rotated superpositions of the achiral ones, with the overall effect of chirality being to rotate the self-propulsion direction anticlockwise by π/2, interchanging the roles of horizontal and vertical propulsion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' For a generic mixture of achiral and chiral stresses the direction of self-propulsion is rotated from the achiral case by an angle arctan(χ/ζ), mirroring the effect such stresses have on the flow profile of a +1/2 defect [80].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' For the quadrupole distortions we have uc i = Rπ/4ua r and uc i = Rπ/4(−ua i ) = ua i , again swapping which distortion produces a chiral or achiral flow response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' It is worth emphasising that the sign of the macroscopic rotation is not necessarily the same as the sign of the chiral stresses, rather it is the product of the signs of the activity and the distortion, just as for achiral stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' a) b) (p 22 e) hc C20 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The active flows induced by spin 0 dipole distortions with chiral active stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The flow is superposed upon the integral curves of the director, shown in orange, for the UPenn dipole (left) and chiral dipole (right).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Three dimensions In three dimensions the chiral active force is χ∇×[∇ · (nn)] [12] and so, by linearity, the fundamental flow responses are given by the curl of those derived earlier, namely u(x) = aχ 2µr3 � −exxy + ey(x2 − z2) + ezyz � , (72) u(y) = aχ 2µr3 � −ex(y2 − z2) + eyxy + −ezxz � , (73) for monopole distortions in the x- and y-components respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Just as for achiral active stresses, we can combine these into a single complex fundamental flow response as u(x) − iu(y), giving ˜u = i r3 � − ¯w2ew + (w ¯w − 2z2)e ¯ w + 2 ¯wzez � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (74) Since the active chiral force is a pure curl the corresponding pressure is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Owing to the additional derivative the functional behaviour of the flow responses is shifted up one order of distortion compared to achiral stresses, meaning dipole distortions induce rotations, although it should be noted that monopoles do not produce propulsive flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The monopole flow responses are still spin-1, but since the flow response for a monopole distortion in nx for achiral stresses is primarily in the x − z plane, the action of curl produces a flow that is dominantly in the y-direction and similarly the response to a monopole distortion in ny is mainly along ex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Together these ingredients mean that heuristically the flow response of a given distortion with chiral active stresses will resemble the achiral active stress flow response of the conjugate distortion at one higher order and with the same spin, that is the distortion reached by the action of i∂z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 10 for the spin-0 dipoles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The UPenn dipole induces rotation about ez while the chiral dipole produces a purely radial flow, resembling the achiral flow responses of the chiral quadrupole and Saturn’s ring quadurpole respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The phenomenological response can again be captured through integration of the stress tensor over a large sphere of radius r, just as was done for achiral active stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' To enable us to reduce the active torque to a single boundary integral we use the symmetric form of the chiral active stress tensor [12], σc ij = [∇ × (nn)]ij + [∇ × (nn)]ji, such that d21 to linear order in director distortions we have f a = � χσc · dA ≈ 0, (75) τ a = � x × χσc · dA ≈ � χ � ex �xz ∂xδnx − 2yz∂yδnx + (y2 − z2)∂zδnx r � + ey �yz∂yδny − 2xz∂xδny + (x2 − z2)∂zδny r � + ez −2xy(∂yδnx + ∂xδny) − (x2 − y2)(∂xδnx − ∂yδny) + z(x∂zδnx + y∂zδny) r � dA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (76) From the first of these equations we see that, to linear order, there are no harmonic distortions which produce net forces in a nematic with chiral active stresses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' With regard to the net active torques, the x− and y− components involve only δnx and δny respectively and each term yields a non-zero integral only for δni ∼ ∂z1/r, hence the two spin-1 dipoles produce transverse torques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Turning to the z-component, each term gives a non-zero integral only for δni ∼ ∂i1/r, and as the expression is symmetric under interchange of x and y we see that only the UPenn dipole produces torques around ez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' In other words, a dipolar director distortion which produces a net active force along a given direction in an achiral active nematic produces a net torque around the same direction in a chiral active nematic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' These results of course accord with our earlier statements regarding the spins of distortions which are capable of producing torques about given axes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Performing the integrals we find that in each case the net active torque has magnitude −12πχαa2/5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Balancing this against Stokes drag gives, using the UPenn dipole as an example, an angular velocity Ω = 3χα 10µaez.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' (77) While the angular velocity in achiral active nematics is independent of the distortion size, in chiral active nematics it is inversely proportional to the radius, a direct consequence of the additional derivative in the active stress tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Accordingly, in chiral active nematics the rotational velocity is largest for smaller colloids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' VIII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' DISCUSSION We have introduced active nematic multipoles as a novel framework for understanding the dynamics of active nematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Although only formally valid on mesoscopic lengthscales, this approach produces results for the propulsive dynamics of defect loops that agree with those of a local analysis [23, 24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' It also provides various testable predictions, for example for the axis of self-propulsion or rotation induced by a distortion or how the corresponding velocities would scale with the size of a colloid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' More broadly, our results reveal self-propulsion and rotation as generic non-equilibrium responses that naturally arise due to colloidal inclusions in active nematics but also provide a template for the tailored design of particular dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' This provides insight into the issue of harnessing the energy of active systems to perform useful work, something which has been demonstrated in bacterial suspensions [71, 82] and is now receiving greater attention in the nematic context [36, 37, 83, 84].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Specific anchoring conditions on colloids have been investigated as a means of generating directed motion [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Our results suggest that sufficient control of the anchoring conditions would allow for steerable and targeted colloidal delivery [85], although there may be routes to a similar degree of dynamical control through colloidal geometry alone [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The transformative power of colloids in passive nematics was revealed in their collective behaviour, forming crys- talline structures [28, 86–89] which can serve as photonic metamaterials [90].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' While our predictions for the dynamics of individual colloids have utility in their own right, there is again considerable interest in the collective dynamics which might emerge [91].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Although our results are insufficient to fully address these questions, some basic points can nonetheless be extracted from the flow solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The long-range nature of the active flows suggests that the hy- drodynamic interactions will be dominant over elastic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The leading contribution to the pair-wise hydrodynamic interactions will be the advection of each colloid by the flow field generated by the other, and the even inversion symmetry of dipole flows implies that this provides a mechanism for pair-wise propulsion, even for colloids which are not self-propulsive themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' To conclude, it has been long-established that the distinct symmetries of ±1/2 nematic defects can be directly related to the qualitatively different dynamics they display in active systems [15, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The aim of this paper is to bring the insights of this symmetry-based approach to generic nematic distortions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' 22 ACKNOWLEDGMENTS This work was supported by the UK EPSRC through Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' EP/N509796/1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Ramaswamy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' The mechanics and statistics of active matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Annu.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Emergence of macroscopic directed motion in populations of motile colloids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} +page_content=' Nature, 503(7474):95–98, 2013.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/1tFST4oBgHgl3EQfWzjN/content/2301.13782v1.pdf'} diff --git a/29FQT4oBgHgl3EQf2zan/content/tmp_files/2301.13425v1.pdf.txt b/29FQT4oBgHgl3EQf2zan/content/tmp_files/2301.13425v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e62405441f99f91f3357d7a807ed14cf1adc6385 --- /dev/null +++ b/29FQT4oBgHgl3EQf2zan/content/tmp_files/2301.13425v1.pdf.txt @@ -0,0 +1,986 @@ +Towards Mechatronics Approach of System Design, +Verification and Validation for Autonomous Vehicles +Chinmay Samak∗, Tanmay Samak∗, Venkat Krovi +Abstract—Modern-day autonomous vehicles are increasingly +becoming complex multidisciplinary systems composed of me- +chanical, electrical, electronic, computing and information sub- +systems. Furthermore, the individual constituent technologies em- +ployed for developing autonomous vehicles have started maturing +up to a point, where it seems beneficial to start looking at the +synergistic integration of these components into sub-systems, +systems, and potentially, system-of-systems. Hence, this work +applies the principles of mechatronics approach of system design, +verification and validation for the development of autonomous +vehicles. Particularly, we discuss leveraging multidisciplinary co- +design practices along with virtual, hybrid and physical proto- +typing and testing within a concurrent engineering framework +to develop and validate a scaled autonomous vehicle using +the AutoDRIVE ecosystem. We also describe a case-study of +autonomous parking application using a modular probabilistic +framework to illustrate the benefits of the proposed approach. +Index Terms—Autonomous vehicles, mechatronics approach, +multidisciplinary design, simulation and virtual prototyping, +rapid prototyping, verification and validation. +I. INTRODUCTION +A +UTOMOTIVE vehicles have evolved significantly over +the course of time [1]. The gradual transition from purely +mechanical automobiles to those with greater incorporation of +electrical, electronic and computer-controlled sub-systems oc- +curred in phases over the course of the past century; with each +phase improving performance, convenience and reliability of +these systems. Modern vehicles are increasingly adopting +electrical, electronic, computing and information sub-systems +along with software algorithms for low-level control as well +as high-level advanced driver assistance system (ADAS) or +autonomous driving (AD) features [2]. This naturally brings in +the interplay between different levels of mechanical, electrical, +electronic, networking and software sub-systems among a +single vehicle system, thereby transforming them from purely +mechanical systems, which they were in the past, to complex +multidisciplinary systems [3]. As such, while it may have +been justifiable for earlier ADAS/AD feature developers to +focus on core software development, the increasing complexity +and interdisciplinary nature of modern automotive systems can +benefit from synergistic hardware-software co-design comple- +mented with integrated verification and validation by following +the mechatronics principles. +∗These authors contributed equally. +C. V. Samak, T. V. Samak and V. N. Krovi are with the Automation, +Robotics and Mechatronics Laboratory (ARMLab), Department of Automo- +tive Engineering, Clemson University International Center for Automotive +Research (CU-ICAR), Greenville, SC 29607, USA. Email: {csamak, +tsamak, vkrovi}@clemson.edu +������������������ +������������������ +������������������ +������������������ +������������������ +������������ +������������������ +������������ +�������� +������������ +�������� +������������ +���������� +������������ +����������� +������������ +�������� +������������ +���� +������������ +����������� +������ +����������� +������� +���������� +��������� +������������� +���������� +���� +�������������� +���������� +����������������� +����������� +�������� +���������������������� +�������� +����������� +������������� +���������� +������������� +���������� +������������������� +������������������� +�������������������� +���������� +�������������������� +���������� +���������� +��������������� +���������� +��������������� +�������� +���������� +���� +���������� +�������� +���������� +����������� +����������� +�������������������� +����������� +���������������� +����������� +������������������ +������������� +������������������ +����������������� +������������������ +������� +������������������ +��������������������������� +����������� +����������� +����������� +���������� +����������� +����������� +�������������������������������� +��������������� +���������������������� +�������������� +���������������������� +���������������� +����������� +���������� +���������� +���������� +����������� +����������� +����� +��������� +����������������� +������ +���������� +��������� +������� +������ +����������� +����������� +������� +Fig. 1. Extended V-model fostering mechatronics approach of system design, +verification and validation for autonomous vehicles. The model depicts evolu- +tion of a concept into a product through decomposition, design, development, +integration and testing across component, sub-system, system and system-of- +systems levels in a unified concurrent interdisciplinary engineering framework. +Mechatronics engineering [4]–[6] focuses on concurrent and +synergistic integration of mechanical, electrical and electronics +engineering, computer science and information technology +for development and validation of complex interdisciplinary +systems. This ideology is derived from the fact that various +components of a “mechatronic” system, often belonging to +a multitude of disciplines, influence each other and hence +have a design impact at the component, sub-system, system +and system-of-systems levels. The resulting ”mechatronic” +realization now builds on capabilities endowed by the vari- +ous constituent layers. In such a milieu, the system devel- +opment approach has also evolved from relatively ad-hoc +to the more formal V-model [7], building on the modular +software development and validation roadmap [8]. This model +has evolved through several state-of-the-art progressions [9] +and our work seeks to further formalize the adoption of +mechatronics approach of system conceptualization, design, +development, integration and testing for autonomous vehicles +(refer Fig. 1). +A recent book [10] highlights best practices for industrial +design, development and validation of autonomous vehicles +and notes the significant adoption of model-based design +(MBD) for system integration and testing. However, similar +arXiv:2301.13425v1 [cs.RO] 31 Jan 2023 + +Freewheeling +Front Wheel Hub +Rear Wheel +Drive Actuator +Front Wheel +Drive Actuator +Front Monocular +Camera +Front Binocular +Camera +Rear Monocular +Camera +Reversed Inertial +Measurement Unit +Reversed LIDAR +and AprilTag Marker + +? +Power +Computation +Communication +Others +Lights +Actuators +Sensors +Software +Chassis +A +B +Fig. 2. +AutoDRIVE ecosystem fosters mechatronics design principles at two levels: [A] primitive reconfigurability allows permutations and combinations +of addition, removal or replacement of selective components and sub-assemblies of the vehicle to better suit the application; [B] advanced reconfigurability +allows complete modification of existing hardware and software architectures, and provides an opportunity for introducing new features and functionalities to +the ecosystem. +adoption of such streamlined workflows by academia has +lagged behind [10]. This gap could be explained by the virtue +of standardization (e.g., ISO 26262 [11], ISO/IEC 33061 [12], +VDI 2221 [13], VDI 2206 [14], AUTOSAR [15], etc.) in +industries versus the fact that majority of academic projects +are deployed using fragmented hardware-software ecosystems +(e.g. hobby platforms) with a key focus on developing low-cost +initial proof-of-concept implementations. Additionally, such an +opportunistic and potentially uninformed selection of hardware +[16]–[18] and software [19]–[21] toolchains hinders adoption +of co-design and concurrent engineering thinking to full extent. +In this paper, we discuss the design philosophy and one +of the key motivation factors behind AutoDRIVE ecosystem1 +[22], [23] – adopting and promoting mechatronics approach of +system design, verification and validation for autonomous ve- +hicles, with an emphasis on creating a streamlined pathway for +seamless transition to ultimate industrial practice. This paper +also describes a detailed case-study which demonstrates the +methodical adoption of mechatronics approach for designing, +developing and validating a scaled vehicle in the context of +autonomous parking2 application using a modular probabilistic +framework. +II. MULTIDISCIPLINARY DESIGN +AutoDRIVE offers an open-access, open-interface and flex- +ible ecosystem for scaled autonomous vehicle development +by permitting access to and alteration of hardware as well as +software aspects of the multidisciplinary autonomous vehicle +design, thereby making it an apt framework for demonstrat- +ing the claims and contributions of this work. Particularly, +AutoDRIVE ecosystem offers the following two levels of +reconfigurability, thereby promoting hardware-software co- +design (refer Fig. 2). +1Webpage: https://autodrive-ecosystem.github.io +2Video: https://youtu.be/piCyvTM2dek +• Primitive Reconfigurability: The native vehicle of Au- +toDRIVE ecosystem, called “Nigel”, is modular enough +to support out-of-the-box hardware reconfigurability in +terms of swapping and replacing selective components +and sub-assemblies of the vehicle, in addition to flexibly +updating the vehicle firmware and/or autonomous driving +software stack (ADSS) to better suit the target applica- +tion. +• Advanced Reconfigurability: The completely open- +hardware, open-software architecture of AutoDRIVE +ecosystem allows modification of vehicle chassis param- +eters (different form factors and aspect ratios), power- +train configuration (variable driving performance), com- +ponent mounting profiles (relocation/replacement of com- +ponents), as well as firmware and ADSS architecture +(software flexibility). +The fundamental step in system design is requirement spec- +ification, without which the design cannot be truly validated to +be right or wrong, it can only be surprising [24]. Since Auto- +DRIVE was intended to be a generic ecosystem for rapidly +prototyping autonomous driving solutions, the requirement +elicitation resulted in a superset of requirements demanded by +the application case study discussed in this paper. Furthermore, +with AutoDRIVE, there is always a scope for updating the +designs of various components, sub-systems and systems for +expanding the ecosystem. That being said, following is a +summary of functional requirement specifications for Nigel +as of this version of the ecosystem. +• General design guidelines: +– Open-source hardware and software +– Inexpensive and user-friendly architecture +– Manufacturing technology agnostic designs +– Modularly reconfigurable components/sub-systems +– Integrated and comprehensive resources and tools + +O +D +Q +C +0 +CddB +A +Chassis +Power Electronics +Computation +Communication +Software +Sensors +Actuators +Lights +PiCamera V2.1 +Robot Operating System +NVIDIA JetPack SDK +Throttle Feedback +Steering Feedback +AutoDRIVE Devkit +RPLIDAR A1 +PiCamera V2.1 +Ethernet +WiFi +Jetson Nano B01 +Arduino Nano V3.0 +Firmware +MPU-6050 IMU +Headlights +Taillights +3A +Master Switch +10A +Buck Converter +11.1V 5200mAh +LiPo Battery +LiPo Voltage +Checker Module +6V DC 160 RPM 120:1 Geared Motor +6V DC 160 RPM 120:1 Geared Motor +Incremental Encoder +Incremental Encoder +20A +Motor Driver +Rear Wheels +AprilTag Marker +MG996R Servo Motor +Steering Mechanism +Front Wheels +Turning Indicators +Reverse Indicators +Left Ticks +Right Ticks +INT +Filtering +Fusion +I2C +GPIO +PWM +USB +Lights +Arduino Nano V3.0 +Jetson Nano B01 +Encoders +IMU +Actuators +Intensity +Timing +Throttle +Steering +SLAM +x +y +1 m +STATIC MAP +ODOMETRY +LOCALIZATION +NAVIGATION +Global +Costmap +Local +Costmap +Global +Planner +Local +Planner +Controller +VEHICLE +Parking +Pose +Throttle/Brake +Steering +LIDAR Scan +Save Map +Load Map +TF +TF +Odometry +C +Fig. 3. +Conceptualization and design of scaled autonomous vehicle: [A] hardware-software architecture; [B] firmware design specifications; [C] modular +perception, planning and control architecture for autonomous parking application. +• Perception sub-system shall offer: +– Ranging measurements (preferably 360◦) +– RGB visual feed (preferably front as well as rear) +– Positional measurements/estimation +– Inertial measurements/estimation +– Actuation feedback measurements/estimation +• Computation and communication sub-systems shall offer: +– Hierarchical computation topology +– GPU-enabled high-level edge computation platform +– Embedded low-level computation platform +– Vehicle-to-everything communication interface +• Locomotion and signaling sub-systems shall offer: +– Kinodynamically constrained drivetrain and steering +– Standard automotive lighting and signaling +The functional system requirements were decomposed into +mechanical, electronics, firmware and ADSS design specifica- +tions and carefully studied to analyze any potential trade-offs +so as to finalize the components and ultimately come up with +a refined system architecture design (refer Fig. 3). +The proposed hardware-software architecture of the scaled +autonomous vehicle system is divided into eight sub-systems +viz. chassis, power, computation, communication, software, +sensors, actuators and lights, each with its own share of com- +ponents (refer Fig. 3-A). The embedded firmware architecture +for low-level data acquisition and control is depicted in Fig. +3-B, which links the data sources to the respective data sinks +after processing the information. +Finally, Fig. 3-C depicts high-level architecture of the +autonomous parking solution described in this paper. Particu- +larly, it is shown how this candidate autonomy solution uses +modular algorithms for simultaneous localization and mapping +(SLAM) [25], odometry estimation [26], localization [27], +global [28] and local [29] path planning, and motion control. +Implementation descriptions are necessarily brief due to the +space limitations; however, further details can be found in this +technical report [23]. +III. VIRTUAL PROTOTYPING AND TESTING +Virtual prototypes help expedite the design process by +validating the designs against system requirements through +simulation, and suggesting design revisions at an early stage. +The scaled autonomous vehicle system was virtually pro- +totyped and tested in three phases. First, the mechanical +specifications, motions and fit were carefully analyzed using +a parametric computer aided design (CAD) assembly of the +system in conjunction with the physical modeling approach +for multi-body dynamic systems (refer Fig. 4-A). Parallelly, +the electronic sub-systems were prototyped using the physical +modeling approach, and also by adopting electronic design au- +tomation (EDA) workflow (refer Fig. 4-B). Next, the firmware +for low-level control (front wheel steering angle and rear wheel +velocity) of the vehicle was verified to produce reliable results +(within a specified tolerance of 3e-2 rad for steering angle +and 3e-1 rad/s for wheel velocity) through model-in-the-loop +(MIL) and software-in-the-loop (SIL) testing (refer Fig. 4-C). +The knowledge gained through this process was used to +update the AutoDRIVE Simulator (refer Fig. 4-D) from its +initial version discussed in [30], [31] to the one described +in [22], [23]. The updated simulator was then employed for +verification and validation of individual ADSS modules and +finally, the integrated autonomous parking solution was also +verified using the same toolchain (refer Fig. 5-A). Particularly, +we tested the vehicle in multiple environments, which included +unit tests for validating the SLAM, odometry, localization, +planning and control algorithms, followed by verification +of the integrated pipeline with and without the addition of +dynamic obstacles, which were absent while mapping the en- +vironment. The autonomous navigation behavior was analyzed +for 5 sample trials and verified to fit within an acceptable +tolerance threshold of 2.5e-2 m; the acceptable parking pose +tolerance was set to be 5e-2 m for linear positions in X and +Y directions and 8.73e-2 rad for the angular orientation about +Z-axis. + +A +B +D +Firmware +Model +Vehicle +Model +MIL +SIL +Firmware +Code +Vehicle +Model +Vehicle Model +Embedded +Firmware +PIL +Embedded +Firmware +Real-Time +Vehicle Model +HIL +VIL +Embedded +Firmware +Physical +Vehicle +C +E +Fig. 4. Development and system integration of scaled autonomous vehicle: [A] mechanical assembly; [B] electronic schematic; [C] MBD workflow depicting +MIL, SIL, PIL, HIL and VIL testing of vehicle firmware; [D] virtual prototype in AutoDRIVE Simulator; [E] physical prototype in AutoDRIVE Testbed. +IV. HYBRID PROTOTYPING AND TESTING +All models or virtual prototypes involve certain degrees of +abstraction, ranging from model fidelity to simulation settings, +and as such, cannot be treated as perfect representations +of their real-world counterparts. Therefore, once the virtual +prototyping and preliminary testing of the system has been +accomplished, the next step is to prototype and validate it in +a hybrid fashion (partly virtual and partly physical), focusing +more on high-level system integration. This method of hybrid +prototyping and testing is extremely beneficial since it follows +a gradual transition from simulation to reality, thereby enabling +a more faithful system verification framework and providing +a room for potential design revisions even before complete +physical prototyping is accomplished. +The scaled vehicle system was subjected to hybrid testing +by running processor-in-the-loop (PIL), hardware-in-the-loop +(HIL) and vehicle-in-the-loop (VIL) tests on the embedded +firmware for confirming minimum deviation from MIL and +SIL results, specified by the same tolerance values of 3e-2 +rad for steering angle and 3e-1 rad/s for wheel velocity (refer +Fig. 4-C). The performance of integrated autonomous vehicle +system was then validated using hybrid testing in two phases. +First, we deployed the ADSS on the physical vehicle’s +on-board computer, which was interfaced with AutoDRIVE +Simulator to receive live sensor feed from the virtual vehicle, +process it and generate appropriate control commands, and +finally relay these commands back to the simulated vehicle. +Specifically, for the autonomous parking solution (refer Fig. +5-A), we deployed and tested each of the SLAM, odometry, +localization, planning and control algorithms for satisfactory +performance. This was naturally followed by deployment and +validation of the integrated pipeline for accomplishing reliable +(within a specified tolerance of 2.5e-2 m) source-to-goal +navigation (within a goal pose tolerance of 5e-2 m and 8.73e- +2 rad) in different environments, wherein a subset of cases +included dynamic obstacles as discussed earlier. +Next, we collected real-world sensor data using AutoDRIVE +Testbed and replayed it as a real-time stimulus to the ADSS +deployed on the physical vehicle’s on-board computer run- +ning in-the-loop with AutoDRIVE Simulator. This way, we +increased the “real-world” component of the hybrid test and +verified the autonomous parking solution for expected perfor- +mance (within same tolerance values as mentioned earlier). +Particularly, the real-world data being collected/replayed was +occupancy-grid map of the environment built by executing +the SLAM module on the physical vehicle, which inherently +resulted as a unit test of this module in real-world conditions. +The simulated vehicle had to then localize against this real- +world map while driving in the virtual scene and navigate +autonomously from source to goal (parking) pose, which +further tested the robustness of the integrated pipeline against +minor environmental variations and/or vehicle behavior. +V. PHYSICAL PROTOTYPING AND TESTING +Once the system confirms satisfactory performance un- +der hybrid testing conditions, the next and final stage in +mechatronic system development is physical prototyping and +testing (refer Fig. 4-E). In order to physically validate the +modular autonomy application (refer Fig. 5-B), we initially +carried out unit tests to confirm the performance of each + +MPU9250 +Right Indicators +LeftIndicators +个个 +ArduinoNano +JetsonNano +RESET +VIN +Switch +(Rev3.0) +GND +5V +D11/MOSI +D12/MISO +DrivePower +D13/SCK +3V3 +DO/F +D1/TX +EncoderPower +V +D10 +交 +8 +8 +LED GND +Drive GND +1111 +Encoder Signal +Signal +Taillights +(LowBeam) +Headlights (High Beam) +Reverse Indicators +Drive +Headlights +个个个 +SteerO +OSteering Angle (rad) +0.5 +MIL Test +SIL Test +C +PIL Test +HIL Test +VIL Test +-0.5 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +Time (s)Wheel Velocity (rad/s) +10 +MIL Test +SIL Test +C +PIL Test +HIL Test +VIL Test +10 +0 +1 +2 +3 +4 +5 +6 +7 +8 +9 +10 +11 +12 +13 +14 +15 +Time (s)24 s +16 s +08 s +A +00 s +09 s +18 s +00 s +12 s +24 s +08 s +16 s +24 s +24 s +08 s +16 s +24 s +16 s +08 s +ii +ii +iii +iv +v +00 s +09 s +18 s +00 s +12 s +24 s +08 s +16 s +24 s +24 s +08 s +16 s +Start +Finish +ii +ii +iii +iv +v +C +B +Fig. 5. Verification and validation of scaled autonomous vehicle performance: [A] virtual/hybrid and [B] physical validation of (i) integrated system, unit +testing of (ii) SLAM, (iii) odometry, (iv) localization, (v) planning and control modules in AutoDRIVE Simulator/Testbed; [C] repeatability/reliability analysis +represented as mean and standard deviation of 5 trials for each deployment type with acceptable trajectory tolerance in green and parking tolerance in purple. +of the SLAM, odometry, localization, planning and control +algorithms followed by deployment of the integrated stack +for autonomous parking application (refer Fig. 5-C). The +vehicle was confirmed to exhibit a reliable (within a specified +tolerance of 2.5e-2 m) source-to-goal navigation (within a goal +pose tolerance of 5e-2 m and 8.73e-2 rad). Again, for testing +the robustness of ADSS we introduced dynamic obstacles that +were not existent while environment mapping was performed. +VI. CONCLUSION +In this work, we presented an extended V-model fostering +mechatronics approach of system design, verification and +validation for autonomous vehicles. Further, we discussed +the design philosophy of AutoDRIVE ecosystem, which is +to exploit and promote the mechatronics approach for au- +tonomous vehicle development across scales and inculcate a +habit of following it from academic education and research to +industrial deployments. We also demonstrated the methodical +adoption of mechatronics approach for designing, developing +and validating a scaled autonomous vehicle in the context of +a detailed case study pertaining to autonomous parking using +a modular probabilistic framework; including both qualitative +and quantitative remarks. 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Available: https://arxiv.org/abs/2211.07022 + diff --git a/29FQT4oBgHgl3EQf2zan/content/tmp_files/load_file.txt b/29FQT4oBgHgl3EQf2zan/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..bec9a7108191edc3a8fb1d3accd24be68f3c4fe3 --- /dev/null +++ b/29FQT4oBgHgl3EQf2zan/content/tmp_files/load_file.txt @@ -0,0 +1,584 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf,len=583 +page_content='Towards Mechatronics Approach of System Design, Verification and Validation for Autonomous Vehicles Chinmay Samak∗, Tanmay Samak∗, Venkat Krovi Abstract—Modern-day autonomous vehicles are increasingly becoming complex multidisciplinary systems composed of me- chanical, electrical, electronic, computing and information sub- systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Furthermore, the individual constituent technologies em- ployed for developing autonomous vehicles have started maturing up to a point, where it seems beneficial to start looking at the synergistic integration of these components into sub-systems, systems, and potentially, system-of-systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Hence, this work applies the principles of mechatronics approach of system design, verification and validation for the development of autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Particularly, we discuss leveraging multidisciplinary co- design practices along with virtual, hybrid and physical proto- typing and testing within a concurrent engineering framework to develop and validate a scaled autonomous vehicle using the AutoDRIVE ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' We also describe a case-study of autonomous parking application using a modular probabilistic framework to illustrate the benefits of the proposed approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Index Terms—Autonomous vehicles, mechatronics approach, multidisciplinary design, simulation and virtual prototyping, rapid prototyping, verification and validation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' INTRODUCTION A UTOMOTIVE vehicles have evolved significantly over the course of time [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' The gradual transition from purely mechanical automobiles to those with greater incorporation of electrical, electronic and computer-controlled sub-systems oc- curred in phases over the course of the past century;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' with each phase improving performance, convenience and reliability of these systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Modern vehicles are increasingly adopting electrical, electronic, computing and information sub-systems along with software algorithms for low-level control as well as high-level advanced driver assistance system (ADAS) or autonomous driving (AD) features [2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' This naturally brings in the interplay between different levels of mechanical, electrical, electronic, networking and software sub-systems among a single vehicle system, thereby transforming them from purely mechanical systems, which they were in the past, to complex multidisciplinary systems [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' As such, while it may have been justifiable for earlier ADAS/AD feature developers to focus on core software development, the increasing complexity and interdisciplinary nature of modern automotive systems can benefit from synergistic hardware-software co-design comple- mented with integrated verification and validation by following the mechatronics principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' ∗These authors contributed equally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Samak, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Samak and V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Krovi are with the Automation, Robotics and Mechatronics Laboratory (ARMLab), Department of Automo- tive Engineering, Clemson University International Center for Automotive Research (CU-ICAR), Greenville, SC 29607, USA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Email: {csamak, tsamak, vkrovi}@clemson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='edu ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='������������������ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='������������������ ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Extended V-model fostering mechatronics approach of system design, verification and validation for autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' The model depicts evolu- tion of a concept into a product through decomposition, design, development, integration and testing across component, sub-system, system and system-of- systems levels in a unified concurrent interdisciplinary engineering framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Mechatronics engineering [4]–[6] focuses on concurrent and synergistic integration of mechanical, electrical and electronics engineering, computer science and information technology for development and validation of complex interdisciplinary systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' This ideology is derived from the fact that various components of a “mechatronic” system, often belonging to a multitude of disciplines, influence each other and hence have a design impact at the component, sub-system, system and system-of-systems levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' The resulting ”mechatronic” realization now builds on capabilities endowed by the vari- ous constituent layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' In such a milieu, the system devel- opment approach has also evolved from relatively ad-hoc to the more formal V-model [7], building on the modular software development and validation roadmap [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' This model has evolved through several state-of-the-art progressions [9] and our work seeks to further formalize the adoption of mechatronics approach of system conceptualization, design, development, integration and testing for autonomous vehicles (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' A recent book [10] highlights best practices for industrial design, development and validation of autonomous vehicles and notes the significant adoption of model-based design (MBD) for system integration and testing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' However, similar arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='13425v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='RO] 31 Jan 2023 Freewheeling Front Wheel Hub Rear Wheel Drive Actuator Front Wheel Drive Actuator Front Monocular Camera Front Binocular Camera Rear Monocular Camera Reversed Inertial Measurement Unit Reversed LIDAR and AprilTag Marker ?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Power Computation Communication Others Lights Actuators Sensors Software Chassis A B Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' AutoDRIVE ecosystem fosters mechatronics design principles at two levels: [A] primitive reconfigurability allows permutations and combinations of addition, removal or replacement of selective components and sub-assemblies of the vehicle to better suit the application;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' [B] advanced reconfigurability allows complete modification of existing hardware and software architectures, and provides an opportunity for introducing new features and functionalities to the ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' adoption of such streamlined workflows by academia has lagged behind [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' This gap could be explained by the virtue of standardization (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=', ISO 26262 [11], ISO/IEC 33061 [12], VDI 2221 [13], VDI 2206 [14], AUTOSAR [15], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=') in industries versus the fact that majority of academic projects are deployed using fragmented hardware-software ecosystems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' hobby platforms) with a key focus on developing low-cost initial proof-of-concept implementations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Additionally, such an opportunistic and potentially uninformed selection of hardware [16]–[18] and software [19]–[21] toolchains hinders adoption of co-design and concurrent engineering thinking to full extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' In this paper, we discuss the design philosophy and one of the key motivation factors behind AutoDRIVE ecosystem1 [22], [23] – adopting and promoting mechatronics approach of system design, verification and validation for autonomous ve- hicles, with an emphasis on creating a streamlined pathway for seamless transition to ultimate industrial practice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' This paper also describes a detailed case-study which demonstrates the methodical adoption of mechatronics approach for designing, developing and validating a scaled vehicle in the context of autonomous parking2 application using a modular probabilistic framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' MULTIDISCIPLINARY DESIGN AutoDRIVE offers an open-access, open-interface and flex- ible ecosystem for scaled autonomous vehicle development by permitting access to and alteration of hardware as well as software aspects of the multidisciplinary autonomous vehicle design, thereby making it an apt framework for demonstrat- ing the claims and contributions of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Particularly, AutoDRIVE ecosystem offers the following two levels of reconfigurability, thereby promoting hardware-software co- design (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 1Webpage: https://autodrive-ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='io 2Video: https://youtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='be/piCyvTM2dek Primitive Reconfigurability: The native vehicle of Au- toDRIVE ecosystem, called “Nigel”, is modular enough to support out-of-the-box hardware reconfigurability in terms of swapping and replacing selective components and sub-assemblies of the vehicle, in addition to flexibly updating the vehicle firmware and/or autonomous driving software stack (ADSS) to better suit the target applica- tion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Advanced Reconfigurability: The completely open- hardware, open-software architecture of AutoDRIVE ecosystem allows modification of vehicle chassis param- eters (different form factors and aspect ratios), power- train configuration (variable driving performance), com- ponent mounting profiles (relocation/replacement of com- ponents), as well as firmware and ADSS architecture (software flexibility).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' The fundamental step in system design is requirement spec- ification, without which the design cannot be truly validated to be right or wrong, it can only be surprising [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Since Auto- DRIVE was intended to be a generic ecosystem for rapidly prototyping autonomous driving solutions, the requirement elicitation resulted in a superset of requirements demanded by the application case study discussed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Furthermore, with AutoDRIVE, there is always a scope for updating the designs of various components, sub-systems and systems for expanding the ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' That being said, following is a summary of functional requirement specifications for Nigel as of this version of the ecosystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' General design guidelines: – Open-source hardware and software – Inexpensive and user-friendly architecture – Manufacturing technology agnostic designs – Modularly reconfigurable components/sub-systems – Integrated and comprehensive resources and tools O D Q C 0 CddB A Chassis Power Electronics Computation Communication Software Sensors Actuators Lights PiCamera V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='1 Robot Operating System NVIDIA JetPack SDK Throttle Feedback Steering Feedback AutoDRIVE Devkit RPLIDAR A1 PiCamera V2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='1 Ethernet WiFi Jetson Nano B01 Arduino Nano V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='0 Firmware MPU-6050 IMU Headlights Taillights 3A Master Switch 10A Buck Converter 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='1V 5200mAh LiPo Battery LiPo Voltage Checker Module 6V DC 160 RPM 120:1 Geared Motor 6V DC 160 RPM 120:1 Geared Motor Incremental Encoder Incremental Encoder 20A Motor Driver Rear Wheels AprilTag Marker MG996R Servo Motor Steering Mechanism Front Wheels Turning Indicators Reverse Indicators Left Ticks Right Ticks INT Filtering Fusion I2C GPIO PWM USB Lights Arduino Nano V3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='0 Jetson Nano B01 Encoders IMU Actuators Intensity Timing Throttle Steering SLAM x y 1 m STATIC MAP ODOMETRY LOCALIZATION NAVIGATION Global Costmap Local Costmap Global Planner Local Planner Controller VEHICLE Parking Pose Throttle/Brake Steering LIDAR Scan Save Map Load Map TF TF Odometry C Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Conceptualization and design of scaled autonomous vehicle: [A] hardware-software architecture;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' [B] firmware design specifications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' [C] modular perception, planning and control architecture for autonomous parking application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='Perception sub-system shall offer: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='– Ranging measurements (preferably 360◦) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='– RGB visual feed (preferably front as well as rear) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='– Positional measurements/estimation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='– Inertial measurements/estimation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='– Actuation feedback measurements/estimation ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='Computation and communication sub-systems shall offer: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='– Hierarchical computation topology ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='– GPU-enabled high-level edge computation platform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='– Embedded low-level computation platform ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='– Vehicle-to-everything communication interface ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='Locomotion and signaling sub-systems shall offer: ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='– Kinodynamically constrained drivetrain and steering ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='– Standard automotive lighting and signaling ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='The functional system requirements were decomposed into ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='mechanical,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' electronics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' firmware and ADSS design specifica- tions and carefully studied to analyze any potential trade-offs so as to finalize the components and ultimately come up with a refined system architecture design (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' The proposed hardware-software architecture of the scaled autonomous vehicle system is divided into eight sub-systems viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' chassis, power, computation, communication, software, sensors, actuators and lights, each with its own share of com- ponents (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 3-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' The embedded firmware architecture for low-level data acquisition and control is depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 3-B, which links the data sources to the respective data sinks after processing the information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Finally, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 3-C depicts high-level architecture of the autonomous parking solution described in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Particu- larly, it is shown how this candidate autonomy solution uses modular algorithms for simultaneous localization and mapping (SLAM) [25], odometry estimation [26], localization [27], global [28] and local [29] path planning, and motion control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Implementation descriptions are necessarily brief due to the space limitations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' however, further details can be found in this technical report [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' VIRTUAL PROTOTYPING AND TESTING Virtual prototypes help expedite the design process by validating the designs against system requirements through simulation, and suggesting design revisions at an early stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' The scaled autonomous vehicle system was virtually pro- totyped and tested in three phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' First, the mechanical specifications, motions and fit were carefully analyzed using a parametric computer aided design (CAD) assembly of the system in conjunction with the physical modeling approach for multi-body dynamic systems (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 4-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Parallelly, the electronic sub-systems were prototyped using the physical modeling approach, and also by adopting electronic design au- tomation (EDA) workflow (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 4-B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Next, the firmware for low-level control (front wheel steering angle and rear wheel velocity) of the vehicle was verified to produce reliable results (within a specified tolerance of 3e-2 rad for steering angle and 3e-1 rad/s for wheel velocity) through model-in-the-loop (MIL) and software-in-the-loop (SIL) testing (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 4-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' The knowledge gained through this process was used to update the AutoDRIVE Simulator (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 4-D) from its initial version discussed in [30], [31] to the one described in [22], [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' The updated simulator was then employed for verification and validation of individual ADSS modules and finally, the integrated autonomous parking solution was also verified using the same toolchain (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 5-A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Particularly, we tested the vehicle in multiple environments, which included unit tests for validating the SLAM, odometry, localization, planning and control algorithms, followed by verification of the integrated pipeline with and without the addition of dynamic obstacles, which were absent while mapping the en- vironment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' The autonomous navigation behavior was analyzed for 5 sample trials and verified to fit within an acceptable tolerance threshold of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='5e-2 m;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' the acceptable parking pose tolerance was set to be 5e-2 m for linear positions in X and Y directions and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='73e-2 rad for the angular orientation about Z-axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' A B D Firmware Model Vehicle Model MIL SIL Firmware Code Vehicle Model Vehicle Model Embedded Firmware PIL Embedded Firmware Real-Time Vehicle Model HIL VIL Embedded Firmware Physical Vehicle C E Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Development and system integration of scaled autonomous vehicle: [A] mechanical assembly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' [B] electronic schematic;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' [C] MBD workflow depicting MIL, SIL, PIL, HIL and VIL testing of vehicle firmware;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' [D] virtual prototype in AutoDRIVE Simulator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' [E] physical prototype in AutoDRIVE Testbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' HYBRID PROTOTYPING AND TESTING All models or virtual prototypes involve certain degrees of abstraction, ranging from model fidelity to simulation settings, and as such, cannot be treated as perfect representations of their real-world counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Therefore, once the virtual prototyping and preliminary testing of the system has been accomplished, the next step is to prototype and validate it in a hybrid fashion (partly virtual and partly physical), focusing more on high-level system integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' This method of hybrid prototyping and testing is extremely beneficial since it follows a gradual transition from simulation to reality, thereby enabling a more faithful system verification framework and providing a room for potential design revisions even before complete physical prototyping is accomplished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' The scaled vehicle system was subjected to hybrid testing by running processor-in-the-loop (PIL), hardware-in-the-loop (HIL) and vehicle-in-the-loop (VIL) tests on the embedded firmware for confirming minimum deviation from MIL and SIL results, specified by the same tolerance values of 3e-2 rad for steering angle and 3e-1 rad/s for wheel velocity (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 4-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' The performance of integrated autonomous vehicle system was then validated using hybrid testing in two phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' First, we deployed the ADSS on the physical vehicle’s on-board computer, which was interfaced with AutoDRIVE Simulator to receive live sensor feed from the virtual vehicle, process it and generate appropriate control commands, and finally relay these commands back to the simulated vehicle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Specifically, for the autonomous parking solution (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 5-A), we deployed and tested each of the SLAM, odometry, localization, planning and control algorithms for satisfactory performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' This was naturally followed by deployment and validation of the integrated pipeline for accomplishing reliable (within a specified tolerance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='5e-2 m) source-to-goal navigation (within a goal pose tolerance of 5e-2 m and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='73e- 2 rad) in different environments, wherein a subset of cases included dynamic obstacles as discussed earlier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Next, we collected real-world sensor data using AutoDRIVE Testbed and replayed it as a real-time stimulus to the ADSS deployed on the physical vehicle’s on-board computer run- ning in-the-loop with AutoDRIVE Simulator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' This way, we increased the “real-world” component of the hybrid test and verified the autonomous parking solution for expected perfor- mance (within same tolerance values as mentioned earlier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Particularly, the real-world data being collected/replayed was occupancy-grid map of the environment built by executing the SLAM module on the physical vehicle, which inherently resulted as a unit test of this module in real-world conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' The simulated vehicle had to then localize against this real- world map while driving in the virtual scene and navigate autonomously from source to goal (parking) pose, which further tested the robustness of the integrated pipeline against minor environmental variations and/or vehicle behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' PHYSICAL PROTOTYPING AND TESTING Once the system confirms satisfactory performance un- der hybrid testing conditions, the next and final stage in mechatronic system development is physical prototyping and testing (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 4-E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' In order to physically validate the modular autonomy application (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 5-B), we initially carried out unit tests to confirm the performance of each MPU9250 Right Indicators LeftIndicators 个个 ArduinoNano JetsonNano RESET VIN Switch (Rev3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='0) GND 5V D11/MOSI D12/MISO DrivePower D13/SCK 3V3 DO/F D1/TX EncoderPower V D10 交 8 8 LED GND Drive GND 1111 Encoder Signal Signal Taillights (LowBeam) Headlights (High Beam) Reverse Indicators Drive Headlights 个个个 SteerO OSteering Angle (rad) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='5 MIL Test SIL Test C PIL Test HIL Test VIL Test 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='5 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time (s)Wheel Velocity (rad/s) 10 MIL Test SIL Test C PIL Test HIL Test VIL Test 10 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Time (s)24 s 16 s 08 s A 00 s 09 s 18 s 00 s 12 s 24 s 08 s 16 s 24 s 24 s 08 s 16 s 24 s 16 s 08 s ii ii iii iv v 00 s 09 s 18 s 00 s 12 s 24 s 08 s 16 s 24 s 24 s 08 s 16 s Start Finish ii ii iii iv v C B Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Verification and validation of scaled autonomous vehicle performance: [A] virtual/hybrid and [B] physical validation of (i) integrated system, unit testing of (ii) SLAM, (iii) odometry, (iv) localization, (v) planning and control modules in AutoDRIVE Simulator/Testbed;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' [C] repeatability/reliability analysis represented as mean and standard deviation of 5 trials for each deployment type with acceptable trajectory tolerance in green and parking tolerance in purple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' of the SLAM, odometry, localization, planning and control algorithms followed by deployment of the integrated stack for autonomous parking application (refer Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' 5-C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' The vehicle was confirmed to exhibit a reliable (within a specified tolerance of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='5e-2 m) source-to-goal navigation (within a goal pose tolerance of 5e-2 m and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='73e-2 rad).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Again, for testing the robustness of ADSS we introduced dynamic obstacles that were not existent while environment mapping was performed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' CONCLUSION In this work, we presented an extended V-model fostering mechatronics approach of system design, verification and validation for autonomous vehicles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Further, we discussed the design philosophy of AutoDRIVE ecosystem, which is to exploit and promote the mechatronics approach for au- tonomous vehicle development across scales and inculcate a habit of following it from academic education and research to industrial deployments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' We also demonstrated the methodical adoption of mechatronics approach for designing, developing and validating a scaled autonomous vehicle in the context of a detailed case study pertaining to autonomous parking using a modular probabilistic framework;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' including both qualitative and quantitative remarks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' We showed that the design, devel- opment as well as verification and validation of the scaled autonomous vehicle with regard to the aforementioned case study could be successfully accomplished within a stringent time-frame of about one month [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' It is to be noted that al- though the exact timeline of any multidisciplinary project may vary depending upon factors such as skill set, experience and number of individuals involved, lead time in the supply chain, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=', the mechatronics approach definitely proves to be efficient in terms of minimizing the design-development iterations by the virtue of synergistic integration in a concurrent engineering thinking framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' This provides a room for the rectification of any design issues early in the development cycle, thereby increasing the chances of successful verification and validation with minimal loss of time and resources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' REFERENCES [1] S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Beiker, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Hansson, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Suneson, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Uhl, “How the convergence of automotive and tech will create a new ecosystem,” McKinsey Company, 2016.' metadata={'source': 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“AutoDRIVE Simulator - Technical Report,” 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' [Online].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content=' Available: https://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='org/abs/2211.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} +page_content='07022' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/29FQT4oBgHgl3EQf2zan/content/2301.13425v1.pdf'} diff --git a/2dE1T4oBgHgl3EQf5gVO/vector_store/index.faiss b/2dE1T4oBgHgl3EQf5gVO/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..810ca982a357e07d132d9f8e3d6302323aed488d --- /dev/null +++ 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sha256:fc0c0f702bb117af2144f30d4a0925ca1fd7304bab1cfa5f6d1377f2f2e51ae6 +size 321733 diff --git a/4NE3T4oBgHgl3EQfogr8/content/tmp_files/2301.04635v1.pdf.txt b/4NE3T4oBgHgl3EQfogr8/content/tmp_files/2301.04635v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..95976b5c6274eec3d478d91a866407f29e2dbe44 --- /dev/null +++ b/4NE3T4oBgHgl3EQfogr8/content/tmp_files/2301.04635v1.pdf.txt @@ -0,0 +1,1918 @@ +ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS +FEDERICO GLAUDO AND ANDREA CIPRIETTI +Abstract. Let A be a multiset with elements in an abelian group. Let FS(A) +be the multiset containing the 2|A| sums of all subsets of A. +We study the reconstruction problem “Given FS(A), is it possible to identify +A?”, and we give a satisfactory answer for all abelian groups. We prove that, +up to identifying multisets through a natural equivalence relation, the function +A �→ FS(A) is injective (and thus the reconstruction problem is solvable) if and +only if every order n of a torsion element of the abelian group satisfies a certain +number-theoretical property linked to the multiplicative group (Z/nZ)∗. +The core of the proof relies on a delicate study of the structure of cyclotomic +units. Moreover, as a tool, we develop an inversion formula for a novel discrete +Radon transform on finite abelian groups that might be of independent interest. +1. Introduction +Let G be an abelian group and let A = {a1, a2, . . . , a|A|} be a finite multiset (i.e., a +set with repeated elements) with elements in G (see Section 2.1 for a formal definition +of multiset). Its subset sums multiset FS(A), that is, the multiset containing the +2|A| sums over all subsets of A (taking into account multiplicities), is defined as +FS(A) := +� � +i∈I +ai : I ⊆ {1, 2, . . . , |A|} +� +. +We study the following reconstruction question: +If one is given FS(A), is it possible to identify A? +As we will see, this strikingly simple question features a rich structure and its solution +spans a wide range of mathematics: from the theory of cyclotomic units, to an +inversion formula for a novel discrete Radon transform. Before going deeper into +the problem, let us give some background on related results in the literature. +If, instead of FS(A), one is given the sums over all the +�|A| +s +� +subsets with fixed +size equal to s (e.g., if s = 2, the sums over all pairs), the reconstruction problem +has been studied in the case of a free abelian group G = Zd [SS58; GFS62]. For +pairs (i.e. s = 2), the reconstruction is possible when the size of A is not a power of +2 [SS58, Theorem 1 and Theorem 2]. For s-subsets with s > 2, the reconstruction +is possible if the size of A does not belong to a finite subset of bad sizes [GFS62, +Section 4]. See the recent survey [Fom19] for a detailed presentation of the history +of this problem. +It might seem that if one is only provided with the sums of s-subsets (i.e., subsets +with size s) then the reconstruction is strictly harder than if one is provided the +sums of all subsets. This is not true because the information is not ordered and +1 +arXiv:2301.04635v1 [math.NT] 11 Jan 2023 + +2 +FEDERICO GLAUDO AND ANDREA CIPRIETTI +thus, even if we have more information, it is also harder to determine which value +corresponds to which subset. +Let us now go back to the reconstruction problem for FS. The first important +observation is the following one. Given a multiset A and a subset B ⊆ A whose +sum equals 0 (i.e. � +b∈B b = 0), if we flip the signs of elements of B then FS does +not change. So, if A′ := (A \ B) ∪ (−B), then FS(A) = FS(A′) (see Fig. 1 for an +explanation). +A +B +C +A′ +−(B \ C) +C \ B +Figure 1. +Proof by picture of A ∼0 A′ =⇒ FS(A) = FS(A′). +The set C in A (highlighted in gray) and the set (C\B)∪(−(B\C)) +in A′ (highlighted in gray) have the same sum because the sum of +the elements in B is assumed to be 0. Thus, we have a bijection +between the subsets of A and A′ which keeps the sum unchanged, +hence FS(A) = FS(A′). +Hence, if we only know FS(A), the best we can hope for is to identify the equiv- +alence class of A with respect to the following equivalence relation. +Definition 1.1. Given two multisets A, A′ with elements in G, we say that A ∼0 A′ +if and only if A′ can be obtained from A by flipping the signs of the elements of a +subset of A with null sum, i.e., if there exists B ⊆ A, with � +b∈B b = 0, such that +A′ = (A \ B) ∪ (−B). +We have already observed that if A ∼0 A′ then FS(A) = FS(A′). If the group +is G = Z, this turns out to be an “if and only if” (see Proposition 6.3), while if +G = Z/2Z it is not (indeed, in Z/2Z one has FS({0, 1}) = {0, 0, 1, 1} = FS({1, 1})). +It is natural to consider the class of abelian groups such that the double implication +holds, i.e. the fibers of FS coincide with the equivalence classes of ∼0. +Definition 1.2. A group G is FS-regular if, for any two multisets A, A′ with ele- +ments in G, it holds FS(A) = FS(A′) if and only if A ∼0 A′. +We have already observed that Z/2Z is not FS-regular; moreover, any group con- +taining a subgroup that is not FS-regular cannot be FS-regular. The next smallest +non-FS-regular group is elusive; in fact, it turns out that Z/nZ is FS-regular for +n = 3, 5, 7, 9, 11, 13, 15. But Z/17Z is not FS-regular, and then Z/nZ is FS-regular +for n = 19, 21, 23, 25, 27, 29 and not FS-regular for m = 31, 33. These small exam- +ples suggest that the FS-regularity of G may be related to the behavior of powers +of two in G (notice that 17, 31, 33 are adjacent to a power of two). +Our main result is the characterization of FS-regular groups. In order to state +our result, we need to introduce a subset of the natural numbers. +Definition 1.3. Let OFS be the set of odd natural numbers n ≥ 1 such that (Z/nZ)∗ +is covered by {±2j : j ≥ 0}; more precisely, for each x ∈ Z relatively prime with n +there exists j ≥ 0 such that either x − 2j or x + 2j is divisible by n. + +ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS +3 +Remark. The first few elements of OFS are +OFS = {1, 3, 5, 7, 9, 11, 13, 15, 19, 21, 23, 25, 27, 29, 35, 37, 39, 45, 47, 49, 53, 55, . . . }, +and the first few missing odd numbers are +(2N + 1) \ OFS = {17, 31, 33, 41, 43, 51, 57, 63, 65, 73, 85, 89, 91, 93, 97, 99, 105, . . . }. +Let us remark that if n ∈ OFS then also all divisors of n belong to OFS. Moreover, +one can show that if p, q, r are distinct odd primes, then pqr ̸∈ OFS, and therefore +if n ∈ OFS then n has at most two distinct prime factors. +We can now state our main theorem. +Theorem 1.1 (Characterization of FS-regular groups). An abelian group G is FS- +regular if and only if ord(g) ∈ OFS for all g ∈ G with finite order. +As a tool in the proof of Theorem 1.1 (see Section 1.1) we define a novel discrete +Radon transform for abelian groups and we prove an inversion formula for it. We +refer to Section 5 for some motivation on the definition and for an in-depth discussion +of the existing related literature. Since the invertibility of the Radon transform may +have other applications beyond the scope of this paper, we state it here for the +interested readers. +Theorem 1.2 (Invertibility of the discrete Radon transform). Let n, d ≥ 1 be pos- +itive integers. Given a function f : (Z/nZ)d → C, its discrete Radon transform +Rf = Rn,df : Hom((Z/nZ)d, Z/nZ) × Z/nZ → C is defined as +Rf(ψ, c) = +� +x: ψ(x)=c +f(x). +This discrete Radon transform is invertible and admits an inversion formula (see +Definition 5.2). +1.1. Sketch of the proof and structure of the paper. Let us briefly describe +the strategy that the proof follows, postponing a more detailed presentation to the +dedicated sections. +For the negative part of the statement, it is sufficient to show that Z/nZ is +not FS-regular if n ̸∈ OFS. For this, we construct an explicit counterexample in +Proposition 4.1. +Proving that if the orders belong to OFS then the group is FS-regular is more +complicated and relies on some nontrivial properties of the units of cyclotomic fields +and on the inversion formula for a novel discrete Radon transform on finite abelian +groups. The proof is divided into three steps. +Step 1: Proof for G = Z/nZ. Through the polynomial identity +� +s∈FS(A) +ts ≡ +� +a∈A +(1 + ta) +(mod tn − 1), +we reduce the FS-regularity of Z/nZ to the study of the kernel of the map +Zn ∋ x = (x0, x1, . . . , xn−1) �→ +� n−1 +� +j=0 +(1 + ωj +d)xj� +d|n, + +4 +FEDERICO GLAUDO AND ANDREA CIPRIETTI +where ωd ∈ C is a d-th primitive root of unity and the codomain of the map consists +of tuples indexed by the divisors of n. Thanks to a dimensional argument, identifying +the kernel of such map is equivalent to identifying its image, which is exactly what +we do in Lemma 4.4. This is the hardest and most technical proof of the whole +paper. Up to this point, we have used only that n is odd. The fact that n ∈ OFS +is needed in the computation of the rank of the image, which relies heavily on the +theory of cyclotomic units (see Lemma 4.2). +This step is carried out in Section 4. +Step 2: Z/nZ is FS-regular +=⇒ +(Z/nZ)d is FS-regular. Take A, A′ multisets +with elements in (Z/nZ)d such that FS(A) = FS(A′). +Given a homomorphism +ψ : (Z/nZ)d → Z/nZ, by linearity, it holds FS(ψ(A)) = FS(ψ(A′)), and since Z/nZ +is FS-regular this implies that ψ(A) ∼0 ψ(A′). So, we know that ψ(A) ∼0 ψ(A′) +for all homomorphisms ψ : (Z/nZ)d → Z/nZ. In order to deduce that A ∼0 A′, +we introduce a novel discrete Radon transform (see Definition 5.1) and we prove an +inversion formula (see Definition 5.2 and Theorem 1.2) which may be of indepen- +dent interest. This allows us to reconstruct a multiset B ∈ M((Z/nZ)d) from its +projections {φ(B) : φ ∈ Hom((Z/nZ)d, Z/nZ)}. +This step is performed in Section 5. +Step 3: G is FS-regular +=⇒ +G ⊕ Z is FS-regular. +In this step, we exploit +crucially that Z is totally ordered. The argument is short and purely combinatorial. +This is done in Section 6. +Once these three steps are established, Theorem 1.1 follows naturally, as shown +in Section 7. Let us remark here that our proof is not constructive, hence it does +not provide an efficient algorithm to find the ∼0-equivalence class of A if FS(A) is +known1. +To make the paper accessible to a broad audience, in Section 2 we recall basic +facts about multisets, abelian groups, and cyclotomic units. +Acknowledgements. The authors are thankful to Fabio Ferri for providing valu- +able suggestions and references about the theory of cyclotomic units, and also to +Michele D’Adderio and Elia Bru`e for their comments and feedback on an early ver- +sion of the manuscript. The second author is supported by the National Science +Foundation under Grant No. DMS-1926686. +2. Notation and Preliminaries +2.1. Multisets. A multiset with elements in a set X is an unordered collection +of elements of X which may contain a certain element more than once [Bli89]. For +example, {1, 1, 2, 2, 3} is a multiset. Rigorously, a multiset A is encoded by a function +µA : X → Z≥0 (Z≥0 denotes the set of nonnegative integers) such that µA(x) +represents the multiplicity of the element x in A. For example, if A = {1, 1, 2, 2, 3} +then µA(1) = 2, µA(2) = 2, µA(3) = 1. +1The nonconstructive part of the proof is contained Section 4. In fact, we show that a certain +map is injective by proving its surjectivity and then applying a standard dimension argument. +This kind of reasoning does not produce an efficient way to invert the map we have proven to be +injective. + +ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS +5 +A multiset A is finite if � +x∈X µA(x) < ∞. The cardinality of a finite multiset +A ∈ M(X) is given by |A| := � +x∈X µA(x). +Given a set X, let us denote with M(X) the family of finite multisets with +elements in X. +Let us define the usual set operations on multisets. Notice that all of them are +the natural generalization of the standard version when one takes into account the +multiplicity of elements. Fix two multisets A, B ∈ M(X). +Membership: We say that x ∈ X is an element of A, denoted by x ∈ A, if +µA(x) ≥ 1. +Inclusion: We say that A is a subset of B, denoted by A ⊆ B, if µA(x) ≤ µB(x) +for all x ∈ X. +Union: The union A∪B ∈ M(X) is defined as µA∪B(x) := µA(x)+µB(x). Hence, +{1} ∪ {1, 2} = {1, 1, 2}. +Cartesian product: The Cartesian product A × B ∈ M(X × X) is defined as +µA×B((x1, x2)) = µA(x1)µB(x2). +Difference: If A ⊆ B, the difference B \A is defined as µB\A(x) := µB(x)−µA(x). +Pushforward: Given a function f : X → Y , the pushforward f(A) ∈ M(Y ) of the +multiset A (denoted also by {f(a) : a ∈ A}) is defined as +µf(A)(y) = +� +x∈f −1(y) +µA(x). +Power set: The power set of A (the family of subsets of A), denoted by P(A) ∈ +M(M(X)), is a multiset defined recursively as follows. For the empty mul- +tiset, we have P(∅) := {∅}; otherwise let a ∈ A be an element of A and +define +P(A) := P(A \ {a}) ∪ +� +A′ ∪ {a} : A′ ∈ P(A \ {a}) +� +. +Notice that |P(A)| = 2|A|. Whenever we iterate over the subsets of A (e.g., +{f(A′) : A′ ⊆ A} or � +A′⊆A f(A′)), the iteration has to be understood over +P(A) (hence the subsets are counted with multiplicity). +Taking the complement is an involution of the power set, i.e., P(A) = +{A \ A′ : A′ ∈ P(A)}, and we have the following identity for the power set +of a union +P(A ∪ B) = {A′ ∪ B′ : (A′, B′) ∈ P(A) × P(B)}. +Sum (and product): If the set X is an additive abelian group, we can define the +sum � A ∈ X of the elements of A as +� +A := +� +x∈X +µA(x)x. +Analogously, if X is a multiplicative abelian group, one can define the prod- +uct � A of the elements of A. +2.2. Abelian Groups. Let us recall some basic facts about abelian groups that we +will use extensively later on. + +6 +FEDERICO GLAUDO AND ANDREA CIPRIETTI +Any finitely generated abelian group is isomorphic to a finite product of cyclic +groups [Lan02, Chapter I, Section 8]. We denote with Z/nZ the cyclic group with +n elements. +Given some elements g1, g2, . . . , gk ∈ G of an abelian group, we denote with +⟨g1, g2, . . . , gk⟩ the subgroup generated by such elements. Given an element g ∈ G, +its order (which may be equal to ∞) is denoted by ord(g). +For an abelian group G, its rank rk(G) is the cardinality of a maximal set of +Z-independent2 elements of G. Let us list some useful properties of the rank (see +[Lan02, Chapter I and XVI]). +• Any finitely generated abelian group G is isomorphic to Zrk(G) ⊕ G′ where +G′ is a finite abelian group. +• Given two abelian groups G, H, it holds rk(G ⊕ H) = rk(G) + rk(H). +• For a homomorphism φ : G → H of abelian groups, it holds rk(G) = +rk(ker φ) + rk(Im φ). +• An abelian group has null rank if and only if all elements have finite order. +• Let G1, G2, G3 be three abelian groups and φ1 : G1 → G2, φ2 : G2 → +G3 be two homomorphisms with full rank, i.e. rk(Im φ1) = rk(G2) and +rk(Im φ2) = rk(G3). Then φ2 ◦ φ1 : G1 → G3 has full rank as well, i.e. +rk(Im φ2 ◦ φ1) = rk(G3) +• Given an abelian group G, let us denote with G ⊗ Q its tensor product (as +a Z-module) with Q (see [Lan02, Chapter XVI]). The dimension of G ⊗ Q +as vector space over Q coincides with rk(G). +• For a homomorphism φ : G → H of abelian groups, let φ⊗Q : G⊗Q → H⊗Q +be its tensorization with Q. It holds rk(Im φ) = dimQ(Im (φ ⊗ Q)). +2.3. Units of cyclotomic fields. Given n ≥ 1, let ωn := exp(2πi/n) be the prim- +itive n-th root of unity with minimum positive argument. +The algebraic number field Q(ωn) is called cyclotomic field. It is well-known that +the ring of integers of Q(ωn) coincides with Z[ωn]. Our main focus is the group of +units of Q(ωn), that consists of the invertible elements of its ring of integers. +For 0 < r < n and s ≥ 1 coprime with n, the element ξ := 1−ωrs +n +1−ωrn is a unit of +Q(ωn). Indeed ξ = 1+ωr +n +· · ·+ω(s−1)r +n +∈ Z[ωn] and, if u ∈ N is such that n divides +us − 1, then +ξ−1 = 1 − ωrus +n +1 − ωrs +n += 1 + ωrs +n + · · · + ω(u−1)rs +n +∈ Z[ωn]. +It turns out that these units are sufficient to generate a subgroup of finite index +of the units of Q(ωn). The following statement follows from [Was97, Theorem 8.3 +and Theorem 4.12]. +Theorem 2.1. For any odd n ≥ 3, the multiplicative group Cn ⊆ C generated by +�1 − ωrs +n +1 − ωrn +: 0 < r < n, s ≥ 1 coprime with n +� +is a subgroup of finite index of the units of Q(ωn). +2Some elements g1, g2, . . . , gk ∈ G are Z-independent if, whenever � +i aigi = 0 for some +a1, a2, . . . , ak ∈ Z, it holds a1 = a2 = · · · = ak = 0. + +ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS +7 +Thus, applying Dirichlet’s unit Theorem (see [Mar77, Theorem 38]), we are able +to compute the rank of Cn (since it coincides with the rank of the group of units of +Q(ωn)). +Corollary 2.2. For any odd n ≥ 3, we have rk(Cn) = ϕ(n) +2 +− 1, where ϕ is Euler’s +totient function (and Cn is defined in Theorem 2.1). +The units of Q(ωn) satisfy a family of nontrivial relations known as distribution +relations (see [Was97, p. 151]). We recall here the relations in the form we will need. +Notice that 1+ωj +n is a unit for 1 ≤ j < n because of the identity 1+ωj +n = 1−ω2j +n +1−ωj +n ∈ Cn. +Proposition 2.3 (Distribution relations). Let n ≥ 1 be an odd integer and let p be +one of its prime divisors3. For any 0 ≤ j < n +p , the identity +p−1 +� +k=0 +(1 + ωj+kn/p +n +) = 1 + ωjp +n +holds. +Proof. The numbers {1 + ωj+kn/p +n +}0≤k

0. +We claim that, for any 0 ≤ j < n, it holds +(4.3) +uS,j − 1 +p +p−1 +� +k=0 +uS,j+km/p = uS′,j − 1 +puS′,jp. +We prove Eq. (4.3) by looking at the projections of both sides onto Qd/Dd and +considering various cases depending on the divisor d. +• If d ̸∈ S, then d ̸∈ S′ (since S′ ⊆ S) and thus we have +ud +S,j − 1 +p +p−1 +� +k=0 +ud +S,j+km/p = 0 = ud +S′,j − 1 +pud +S′,jp. +• If d ∈ S and υp(d) < υp(S), then d | +m +p +and therefore ud +S,j+km/p = +[ed +j+km/p]Dd = [ed +j]Dd = ud +S,j. +Since υp(d) < υp(S) implies that d ̸∈ S′, +we deduce +ud +S,j − 1 +p +p−1 +� +k=0 +ud +S,j+km/p = ud +S,j − 1 +p +p−1 +� +k=0 +ud +S,j = 0 = ud +S′,j − 1 +pud +S′,jp. +• If d ∈ S and υp(d) = υp(S), then it holds +(4.4) +� +0, m +p mod d, 2m +p mod d, . . . , (p−1)m +p mod d +� += +� +0, d +p, 2d +p, . . . , (p−1)d +p +� +. +To prove the latter identity, notice that for any 0 ≤ k < p, we have +� +k m +p mod d +� += +� +k m +d mod p +�d +p +and therefore the identity between sets follows from the fact that m/d is not +divisible by p. +6Here υp(x) denotes the p-adic valuation of a nonzero integer x, i.e. the maximum exponent +h ≥ 0 such that ph divides x. + +14 +FEDERICO GLAUDO AND ANDREA CIPRIETTI +Exploiting Eq. (4.4) and recalling that vd +p,j ∈ Dd, we obtain +ud +S,j − 1 +p +p−1 +� +k=0 +ud +S, j+km/p = +� +ed +j − 1 +p +p−1 +� +k=0 +ed +j+km/p +� +Dd += +� +ed +j − 1 +p +p−1 +� +k=0 +ed +j+kd/p +� +Dd += +� +ed +j − 1 +p(ed +jp − vd +p,j) +� +Dd += +� +ed +j − 1 +ped +jp +� +Dd += ud +S′,j − 1 +pud +S′,jp, +where in the last steps we used that d ∈ S′ (which is equivalent to the +assumptions d ∈ S and υp(d) = υp(S)). +Since we have covered all possible cases, Eq. (4.3) is proven. +The set S is solvable, therefore the left-hand side of Eq. (4.3) belongs to the image +of Ψn ⊗ Q, and thus also uS′,j − 1 +puS′,jp belongs to Im (Ψn ⊗ Q) for all 0 ≤ j < n. +Lemma 4.3, applied with vj := uS′,j, λ := 1/p, and σ(j) := (jp mod n), guarantees +that also uS′,j belongs to the image of Ψn ⊗ Q for all 0 ≤ j < n, which proves that +S′ is solvable as desired. +□ +As a simple consequence of Lemma 4.5, we claim that if S is solvable, then, for +any prime divisor p of n and for any 0 ≤ h ≤ υp(n), we have that {s ∈ S : υp(s) = h} +is also solvable. Let us prove it by induction on h, starting from h = υp(n) and going +backward to h = 0. +If {s ∈ S : υp(s) = υp(n)} is empty, then it is solvable; otherwise we can apply +Lemma 4.5 and obtain again that it is solvable. Now, we assume that {s ∈ S : +υp(s) = h′} is solvable for h′ > h. Then, since the difference of solvable sets is +solvable, we deduce that ˜S := {s ∈ S : υp(s) ≤ h} is solvable. If {s ∈ S : υp(s) = h} +is empty, then it is solvable; otherwise we can apply Lemma 4.5 on the set ˜S and +obtain again that {s ∈ S : υp(s) = h} is solvable as desired. +We can now conclude by showing that singletons {d} are solvable for each d | n. +This follows directly from the fact that {d ≥ 1 : d | n} is solvable and that if S +is solvable then {s ∈ S : υp(s) = h} is solvable for all prime divisors p | n and all +h ≥ 0. +□ +Proposition 4.6. For any n ∈ OFS, the group Z/nZ is FS-regular. +Proof. Let A, A′ ∈ M(Z/nZ) be two multisets such that FS(A) = FS(A′); we shall +prove that A ∼0 A′. +By definition of the map FS, it holds the polynomial identity in Z[t]/(tn − 1) +n−1 +� +j=0 +µFS(A)(j)tj ≡ +� +s∈FS(A) +ts ≡ +� +a∈A +(1 + ta) ≡ +n−1 +� +j=0 +(1 + tj)µA(j) +(mod tn − 1), +Thus the condition FS(A) = FS(A′) is equivalent to +n−1 +� +j=0 +(1 + tj)µA(j) ≡ +n−1 +� +j=0 +(1 + tj)µA′(j) +(mod tn − 1). + +ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS +15 +For any divisor d | n, ωd is a root of tn − 1 and therefore the latter identity implies +n−1 +� +j=0 +(1 + ωj +d)µA(j) = +n−1 +� +j=0 +(1 + ωj +d)µA′(j) +which, recalling Definition 4.2, is equivalent to +Fd +� +πn +d +� +(µA(j) − µA′(j))0≤j 1 (in particular, proving the invertibility +of the (d − 1)-planes transform seems to be considerably easier due to the +larger number of symmetries). +• The recent work [CHM18] defines a Radon transform which is almost equiva- +lent to our discrete Radon transform on (Z/pZ)d, where p is a prime number. +In that paper the Radon transform (which they call classical Radon trans- +form to distinguish it from the one of Diaconis and Graham) coincides with +the restriction of ours to the homomorphisms ψ ∈ Hom((Z/pZ)d, Z/pZ) +such that ψ(0, 0, . . . , 0, 1) ̸= 0. Due to this restriction, they cannot establish +a full inversion formula [CHM18, Theorem 1]. +• In the work [AI08], the authors define a discrete Radon transform on Zd +which is equivalent to the Radon transform on Zd with our notation (if one +allows the group to be non-finite in the definition). An inversion formula +[AI08, Theorem 4.1] is proven for such discrete Radon transform. Joining +the methods of [AI08] with ours, it might be possible to produce inversion +formulas for the discrete Radon transform on groups (Z/nZ × Z)d that are +neither finite nor torsion-free. We do not investigate this as it goes beyond +the scope of the paper. +• An alternative definition of discrete Radon transform for finite abelian groups +is provided in [Ilm14]. The maximal Radon transform defined in this ref- +erence [Ilm14, Section 7.3] computes the sum of the function f over all +translations of maximal cyclic subgroups of G. +It is not hard to check that, for p prime, the maximal Radon transform +on (Z/pZ)2 coincides with ours. +In this special case, the author proves +the invertibility of the Radon transform [Ilm14, Lemma 3.4]. In general his +definition does not coincide with ours and, in particular, the maximal Radon + +18 +FEDERICO GLAUDO AND ANDREA CIPRIETTI +transform is not invertible in many important cases [Ilm14, Propositions 7.2, +7.3]. +Let us introduce the concept of inversion formula for our discrete Radon transform +(cp. [Hel99, Theorem 3.1], [Str82]). The main goal of this section is to obtain an +inversion formula (see Theorem 1.2). +Definition 5.2. Let n, d ≥ 1 be positive integers. We say that the Radon transform +on (Z/nZ)d (see Definition 5.1) admits an inversion formula if there exists a function +λ = λn,d : Hom((Z/nZ)d, Z/nZ) → Q such that +f(x) = +� +ψ∈Hom((Z/nZ)d, Z/nZ) +λ(ψ)Rf(ψ, ψ(x)), +for all functions f : (Z/nZ)d → C and all x ∈ (Z/nZ)d. +Let us begin with a simple but useful criterion for the existence of an inversion +formula. +Lemma 5.1. Let n, d ≥ 1 be positive integers. A function λ : Hom((Z/nZ)d, Z/nZ) → +Q induces an inversion formula for the discrete Radon transform on (Z/nZ)d (see +Definition 5.2) if and only if it satisfies, for all x ∈ (Z/nZ)d, +(5.5) +� +ψ∈Hom((Z/nZ)d, Z/nZ) +ψ(x)=0 +λ(ψ) = +� +1 +if x = 0, +0 +otherwise. +Proof. For any f : (Z/nZ)d → C, any λ : Hom((Z/nZ)d, Z/nZ) → Q and any +x ∈ (Z/nZ)d, it holds +� +ψ∈Hom((Z/nZ)d, Z/nZ) +λ(ψ)Rf(ψ, ψ(x)) = +� +ψ∈Hom((Z/nZ)d, Z/nZ) +λ(ψ) +� +x′∈(Z/nZ)d +ψ(x′)=ψ(x) +f(x′) += +� +x′∈(Z/nZ)d +f(x′) +� +ψ∈Hom((Z/nZ)d, Z/nZ) +ψ(x′−x)=0 +λ(ψ). +Thanks to this identity, it is clear that λ induces an inversion formula if and only if +Eq. (5.5) holds. +□ +In the next technical lemma we show that inversion formulas behave nicely with +respect to products. +Lemma 5.2. Let n, m, d ≥ 1 be positive integers such that n and m are coprime. +If the discrete Radon transforms on (Z/nZ)d and on (Z/mZ)d admit inversion for- +mulas, then also the Radon transform on (Z/nmZ)d admits an inversion formula. +Proof. To simplify the notation, let G := Z/nZ and H := Z/mZ. +Let ι : Hom(Gd, G)×Hom(Hd, H) → Hom(Gd⊕Hd, G⊕H) be the map such that +ι(ψ1, ψ2)(x1, x2) = (ψ1(x1), ψ2(x2)) for all ψ1 ∈ Hom(Gd, G), ψ2 ∈ Hom(Hd, H), +x1 ∈ Gd, x2 ∈ Hd. Since n, m are coprime the map ι is bijective. +Since we assume that the Radon transforms on Gd and Hd admit inversion for- +mulas, thanks to Lemma 5.1, we deduce the existence of λ1 : Hom(Gd, G) → Q and +λ2 : Hom(Hd, H) → Q satisfying Eq. (5.5). + +ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS +19 +Let λ : Hom(Gd ⊕ Hd, G ⊕ H) → Q be the function such that λ(ι(ψ1, ψ2)) = +λ1(ψ1)λ2(ψ2) for all ψ1 ∈ Hom(Gd, G), ψ2 ∈ Hom(Hd, H). For any x1 ∈ Gd and +x2 ∈ Hd, we have +� +ψ∈Hom(Gd⊕Hd, G⊕H) +ψ(x1,x2)=(0G,0H) +λ(ψ) = +� +ψ1∈Hom(Gd, G), ψ2∈Hom(Hd, H) +ψ1(x1)=0G, ψ2(x2)=0H +λ1(ψ1)λ2(ψ2) += +� +� +ψ1∈Hom(Gd, G) +ψ1(x1)=0G +λ1(ψ1) +�� +� +ψ2∈Hom(Hd, H) +ψ1(x2)=0H +λ2(ψ2) +� += +� +1 +if x1 = 0G and x2 = 0H, +0 +otherwise, +which is equivalent to the fact that the discrete Radon tranform on Gd ⊕ Hd ad- +mits an inversion formula thanks to Lemma 5.1. This is equivalent to the desired +statement since G ⊕ H ∼= Z/nmZ as a consequence of the coprimality of m, n. +□ +Our next goal is to show that the Radon transform on (Z/pkZ)d (with p prime) +admits an inversion formula (see Lemma 5.8). +Let us begin with a sequence of technical lemmas (Lemmas 5.3 to 5.6) concerning +the structure of (Z/pkZ)d, its automorphisms and its canonical scalar product. The +first two statements, Lemmas 5.3 and 5.4, are special cases of known results (see +[HR07; SS99]). For completeness, and because the proofs are much simpler compared +to the proofs of the statements we cite, we provide a self-contained proof for both +facts. +Lemma 5.3. Fix a prime p and and two exponents k, d ≥ 1. Given a d × d matrix +M ∈ (Z/pkZ)d×d, let mulM : (Z/pkZ)d → (Z/pkZ)d be the group homomorphism +given by the multiplication with the matrix M, i.e., for all x = (x1, x2, . . . , xd) ∈ +(Z/pkZ)d, +mulM(x) := +� +d +� +j=1 +Mijxj +� +i=1,...,d. +The group of automorphisms of (Z/pkZ)d is given by +Aut((Z/pkZ)d) = {mulM : M ∈ (Z/pkZ)d×d so that p does not divide det(M)}. +Proof. A homomorphism φ : (Z/pkZ)d → (Z/pkZ)d is uniquely determined by +the images of the d generators of (Z/pkZ)d, that is by the values φ(1, 0, . . . , 0), +φ(0, 1, 0, . . . , 0), . . . , φ(0, . . . , 0, 1). Let M ∈ (Z/pkZ)d×d be the matrix such that +the j-th column is given by the image through φ of the j-th generator. It holds +φ = mulM. +It remains to prove that mulM is an automorphism (i.e., its inverse is a homo- +morphism) if and only if det(M) is not divisible by p. Notice that mulM ◦ mulN = +mulMN, therefore mulM is an automorphism if and only if M is invertible modulo +pk, or equivalently det(M) is not divisible by p. +□ + +20 +FEDERICO GLAUDO AND ANDREA CIPRIETTI +Lemma 5.4. Fix a prime p and two exponents k, d ≥ 1. For any 0 ≤ h ≤ k, let +Eh ⊆ (Z/pkZ)d be the subset +Eh := {x = (x1, . . . , xd) ∈ (Z/pkZ)d : ph divides xi for all i = 1, 2, . . . , d}. +Moreover, let E∗ +k := Ek = {(0, . . . , 0)} ∈ (Z/pkZ)d and, for 0 ≤ h < k, E∗ +h := +Eh \ Eh+1. +The orbits of the action of the automorphism group of (Z/pkZ)d are exactly +E∗ +0, E∗ +1, . . . , E∗ +k. +Proof. The subset Eh coincides with the elements of (Z/pkZ)d with order at most +pk−h, hence E∗ +h coincides with the elements of (Z/pkZ)d with order equal to pk−h. +In particular, the image of E∗ +h through an automorphism coincides with E∗ +h. +To prove that E∗ +h is an orbit for the automorphism group, we show that given +x = (x1, . . . , xd) ∈ E∗ +h there exists an automorphism φ ∈ Aut((Z/pkZ)d) such that +φ(ph, 0, 0, . . . , 0) = x. By definition of E∗ +h, it holds υp(xi) ≥ h for all i = 1, 2, . . . , d +(recall that υp denotes the p-adic valuation), and without loss of generality we may +assume that υp(x1) = h. Consider the matrix M ∈ (Z/pkZ)d×d with the following +entries +M = +� +� +� +� +� +� +� +� +� +x1/ph +0 +0 +· · · +0 +0 +x2/ph +1 +0 +· · · +0 +0 +x3/ph +0 +1 +· · · +0 +0 +... +... +... +... +... +... +xd−1/ph +0 +0 +· · · +1 +0 +xd/ph +0 +0 +· · · +0 +1 +� +� +� +� +� +� +� +� +� +. +Since det(M) = x1/ph, which is not divisible by p, the classification of automor- +phisms proven in Lemma 5.3 guarantees that φ = mulM is an automorphism which +satisfies φ(ph, 0, . . . , 0) = x, as desired. +□ +Lemma 5.5. Fix a prime p and two exponents k, d ≥ 1. Denote with · : (Z/pkZ)d × +(Z/pkZ)d → Z/pkZ the scalar product x · y := x1y1 + x2y2 + · · · + xdyd. +For any automorphism φ ∈ Aut((Z/pkZ)d), there exists an automorphism φt ∈ +Aut((Z/pkZ)d) such that φ(x) · y = x · φt(y) for all x, y ∈ (Z/pkZ)d. +Proof. Thanks to Lemma 5.3, we know that there exists M ∈ (Z/pkZ)d×d such that +φ = mulM. It can be checked that φt := mulM t, where M t is the transpose of M, +satisfies the requirements of the statement. +□ +Lemma 5.6. Fix a prime p and two exponents k, d ≥ 1. Recall the definitions of +Eh and E∗ +h given in Lemma 5.4. +Given 0 ≤ h, h′ ≤ k, for any x ∈ E∗ +h it holds +|{y ∈ Eh′ : x · y = 0}| = p(d−1)(k−h′)+min{h, k−h′} +and, in particular, this quantity does not depend on the specific choice of x ∈ E∗ +h. +Proof. Thanks to Lemma 5.4, there exists an automorphism φ ∈ Aut((Z/pkZ)d) +such that φ((ph, 0, 0, . . . , 0)) = x. Notice (recall Lemma 5.5) that x · y = 0 if and + +ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS +21 +only if (ph, 0, 0, . . . , 0) · φt(y) = 0. Moreover, as a consequence of Lemma 5.4, we +have that φt(Eh′) = Eh′. Thus, we deduce +φt� +{y ∈ Eh′ : x · y = 0} +� += {y ∈ Eh′ : (ph, 0, 0, . . . , 0) · y = 0}. +In particular, we have shown that the cardinality of such set does not depend on +the specific choice of x ∈ E∗ +h and we may assume x = (ph, 0, 0, . . . , 0). +The condition y ∈ Eh′ is equivalent to the fact that y may be expressed as y = +ph′(z1, z2, . . . zd) with z1, z2, . . . , zd ∈ Z/pk−h′Z. The condition (ph, 0, 0, . . . , 0)·y = 0 +is equivalent to υp(z1) ≥ max(0, k−(h+h′)). To conclude, we distinguish two cases. +• If k ≤ h+h′, then the constraint υp(z1) ≥ max(0, k −(h+h′)) is empty, and +thus any choice of z1, z2, . . . , zd ∈ Z/pk−h′Z yields an element y = ph′z of +{y ∈ Eh′ : (ph, 0, 0, . . . , 0) · y = 0}, thus such set has cardinality pd(k−h′) = +p(d−1)(k−h′)+min{h,k−h′}. +• If k ≥ h + h′, then z1 must be divisible by pk−(h+h′), while the other zi +can be arbitrary values in Z/pk−h′Z. +Therefore, the cardinality of the +set {y ∈ Eh′ : (ph, 0, 0, . . . , 0) · y = 0} is p(d−1)(k−h′)+k−h′−(k−(h+h′)) = +p(d−1)(k−h′)+min{h,k−h′}. +□ +The only missing ingredient necessary to prove that the discrete Radon transform +on (Z/pkZ)d admits an inversion formula is the invertibility of a certain matrix, +which is promptly established in the following lemma. +Lemma 5.7. Fix a prime p and two exponents k, d ≥ 1. The (k + 1) × (k + 1) +matrix U (k) ∈ Q(k+1)×(k+1) with entries, for 0 ≤ i, j ≤ k, given by +U (k) +ij +:= p(d−1)(k−j)+min{i, k−j} +is invertible. +Proof. It is easier to work with ˜U (k) +ij +:= U (k) +i(k−j) (which is invertible if and only if U (k) +is invertible). Indeed, defining q := pd−1, one has ˜U (k) +ij += pmin{i,j}qj and therefore +˜U (k) = +� +� +� +� +� +� +� +� +� +1 +q +q2 +· · · +qk−1 +qk +1 +pq +pq2 +· · · +pqk−1 +pqk +1 +pq +p2q2 +· · · +p2qk−1 +p2qk +... +... +... +... +... +... +1 +pq +p2q2 +· · · +pk−1qk−1 +pk−1qk +1 +pq +p2q2 +· · · +pk−1qk−1 +pkqk +� +� +� +� +� +� +� +� +� +. +We prove the statement by induction on k. We have ˜U (0) = +�1� +, which is in- +vertible. For the inductive step, subtracting the second-to-last row of ˜U (k) from the +last, all the entries of the last row become zero, except for the last one, which turns +into qk(pk − pk−1) ̸= 0. Note further that the top-left k × k submatrix of ˜U (k) is +˜U (k−1). Therefore, det ˜U (k) = qk(pk − pk−1) det ˜U (k−1), which concludes. +□ +Lemma 5.8. Fix a prime p and two exponents k, d ≥ 1. +The discrete Radon +transform on (Z/pkZ)d admits an inversion formula. + +22 +FEDERICO GLAUDO AND ANDREA CIPRIETTI +Proof. Notice that (Z/pkZ)d ∼= Hom((Z/pkZ)d, Z/pkZ) and the isomorphism is +given by the map that takes y ∈ (Z/pkZ)d and produces the homomorphism (Z/pkZ)d ∋ +x → x · y (where the scalar product is defined in Lemma 5.5). Therefore, applying +Lemma 5.1, we have that the validity of an inversion formula for the discrete Radon +transform on (Z/pkZ)d is equivalent to the existence of a function λ : (Z/pkZ)d → Q +such that, for all x ∈ (Z/pkZ)d, +� +y∈(Z/pkZ)d +x·y=0 +λ(y) = +� +1 +if x = (0, 0, . . . , 0), +0 +otherwise. +We are going to construct a function λ with this property. +Let U (k) ∈ Q(k+1)×(k+1) be the matrix considered in Lemma 5.7. Let V (k) ∈ +Q(k+1)×(k+1) be the matrix given by +V (k) +ij += +� +U (k) +i,j +if j = k, +U (k) +i,j − U (k) +i,j+1 +if j < k. +Since V (k) can be obtained by U (k) through Gauss moves, Lemma 5.7 implies that +V (k) is invertible as well. Notice that, by definition of V (k), for any 0 ≤ i, j ≤ k, +Lemma 5.6 implies that V (k) +ij += |{y ∈ E∗ +j : x · y = 0}| for any x ∈ E∗ +i (recall that +E∗ +j = Ej \ Ej+1). +Let Λ ∈ Qk+1 be the solution of V (k)Λ = (1, 0, . . . , 0). +Let us define λ : +(Z/pkZ)d → Q as the function such that λ(y) := Λj when y ∈ E∗ +j . +We show +that this function satisfies the sought identity. +Given x ∈ E∗ +i , we have +� +y∈(Z/pkZ)d +x·y=0 +λ(y) = +k +� +j=0 +|{y ∈ E∗ +j : x · y = 0}|Λj = +k +� +j=0 +V (k) +ij Λj = (V (k)Λ)i, +which is the desired formula since the right-hand side is 1 if i = 0 (which is equivalent +to x = (0, 0, . . . , 0) ∈ (Z/pkZ)d) and 0 otherwise. +□ +We are ready to show the validity of an inversion formula for all instances of our +discrete Radon transform. +Proof of Theorem 1.2. By the classification of finite abelian groups (see Section 2.2), +the statement follows from Lemmas 5.2 and 5.8. +□ +Let us apply the inversion formula obtained in Theorem 1.2 to establish the FS- +regularity of the group (Z/nZ)d when n ∈ OFS. The idea is to project through an +homomorphism onto Z/nZ, use the FS-regularity of Z/nZ proven in Proposition 4.6, +and then recover the FS-regularity of (Z/nZ)d thanks to the invertibility of the +Radon transform on (Z/nZ)d. +Proposition 5.9. For any n ∈ OFS and any d ≥ 1, the group (Z/nZ)d is FS-regular. +Proof. For a multiset B ∈ M((Z/nZ)d), by definition of the Radon transform +on ((Z/nZ)d (see Definition 5.1), one has RµB(ψ, c) = µψ(B)(c) (recall that µB +denotes the multiplicity of elements in the multiset B, see Section 2.1) for any + +ON THE DETERMINATION OF SETS BY THEIR SUBSET SUMS +23 +ψ ∈ Hom((Z/nZ)d, Z/nZ) and any c ∈ Z/nZ. Therefore, the inversion formula of +Theorem 1.2 implies +(5.6) +µB(x) = +� +ψ∈Hom((Z/nZ)d, Z/nZ) +λ(ψ)µψ(B)(ψ(x)), +for all x ∈ (Z/nZ)d. Notice that this formula allows us to reconstruct B given all +its projections ψ(B) onto Z/nZ. +Take two multisets A, A′ ∈ M((Z/nZ)d) such that FS(A) = FS(A′); our goal is +to prove that A ∼0 A′. +For any ψ ∈ Hom((Z/nZ)d, Z/nZ), it holds FS(ψ(A)) = FS(ψ(A′)) and there- +fore, since we have shown that Z/nZ is FS-regular in Proposition 4.6, we have +ψ(A) ∼0 ψ(A′). +Thus (we use only ψ(A) ∼ ψ(A′)), we deduce that for any +ψ ∈ Hom((Z/nZ)d, Z/nZ), +(5.7) +µψ(A)(x) + µψ(A)(−x) = µψ(A′)(x) + µψ(A′)(−x) +for all x ∈ (Z/nZ)d. +Joining Eqs. (5.6) and (5.7), we obtain +µA(x) + µA(−x) = +� +ψ∈Hom((Z/nZ)d, Z/nZ) +λ(ψ) +� +µψ(A)(ψ(x)) + µψ(A)(−ψ(x)) +� += +� +ψ∈Hom((Z/nZ)d, Z/nZ) +λ(ψ) +� +µψ(A′)(ψ(x)) + µψ(A′)(−ψ(x)) +� += µA′(x) + µA′(−x) +for all x ∈ (Z/nZ)d. The latter identity is equivalent to A ∼ A′, which implies +A ∼0 A′ thanks to Lemma 3.1-(3). +□ +6. FS-regularity of products with Z +In this section we show that multiplying by Z does not break the FS-regularity of +a group (see Proposition 6.3). In order to do it, we will need two technical lemmas. +The second one, Lemma 6.2, gives a condition equivalent to FS-regularity which +comes handy in the proof of the main result of this section. +Lemma 6.1. Let G be an abelian group without elements of order 2. Given three +multisets A, A′, B ∈ M(G), if A + FS(B) = A′ + FS(B), then A = A′. +Proof. Let us first prove the result when B = {b} is a singleton. We prove the result +by induction on the cardinality of A. +If |A| = 0, then ∅ = A + FS(B) = A′ + FS(B) and thus A′ = ∅. +To handle the case |A| > 0, we begin by showing that A and A′ have a common +element. We argue by contradiction, hence we assume that A and A′ are disjoint. +Take any a ∈ A. We have a + b ∈ A + FS(B) = A′ + {0, b}. Since a ̸∈ A′, it +must hold a + b ∈ A′. By repeating this argument (swapping the role of A and A′ +and replacing a with a + b) we obtain that a + 2b ∈ A. Repeating such argument k +times, we obtain that a + kb ∈ A if k is even, and a + kb ∈ A′ if k is odd. Since A +and A′ are finite, b must have finite order, otherwise the elements (a+kb)k∈N would +be all distinct. Let ord(b) be the order of b; by assumption ord(b) is odd. We have + +24 +FEDERICO GLAUDO AND ANDREA CIPRIETTI +the contradiction A ∋ a = a + ord(b)b ∈ A′; therefore we have proven that A and A′ +have a common element. +Now pick ¯a ∈ A ∩ A′. It holds +(A \ {¯a}) + FS(B) = (A + FS(B)) \ {¯a, ¯a + b} += (A′ + FS(B)) \ {¯a, ¯a + b} = (A′ \ {¯a}) + FS(B). +Therefore, by the induction hypothesis, A \ {¯a} = A′ \ {¯a}, which is equivalent to +A = A′. +Let us now treat general multisets B. We proceed by induction on the cardinality +of B; the case |B| = 0 is trivial and the case |B| = 1 is already established, so we +may assume |B| > 1. +Pick an element ¯b ∈ B. We have +A + FS(B) = (A + FS(B \ {¯b})) + FS({¯b}), +and likewise for A′. Applying the induction hypothesis for the three multiset A + +FS(B\{¯b}), A′+FS(B\{¯b}), {¯b}, yields the relation A+FS(B\{¯b}) = A′+FS(B\{¯b}), +and one more application yields the sought A = A′. +□ +Lemma 6.2. An abelian group G is FS-regular if and only if, for all A, A′ ∈ M(G) +such that FS(A) = FS(A′) + g for some g ∈ G, it holds A ∼ A′. +Proof. Assume that G is FS-regular and take A, A′ ∈ M(G) such that FS(A) = +FS(A′) + g for some g ∈ G. Applying Lemma 3.1-(4), we produce a multiset A′′ ∈ +M(G) such that A′′ ∼ A′ and FS(A) = FS(A′′); then we deduce A ∼0 A′′ because +G is FS-regular. So, we get A ∼0 A′′ ∼ A′ which implies A ∼ A′ by transitivity. +Let us now show the converse. Given A, A′ ∈ M(G) such that FS(A) = FS(A′), +the condition described in the statement implies A ∼ A′ which implies A ∼0 A′ +thanks to Lemma 3.1-(3). Therefore we have proven the FS-regularity of G. +□ +Proposition 6.3. If G is a FS-regular abelian group, then also G⊕Z is FS-regular. +Proof. We begin by setting up some notation. For B ∈ M(G⊕Z) and z ∈ Z, define +B 0 and its position fixed at the origin. The corresponding +Schr¨odinger operator is +(1.1) +H = −∆ + V +where ∆ = �N +k=1 ∆xk is the Laplacian in R3N, i.e. ∆xk refers to the Laplacian applied +to the variable xk, and V is the Coulomb potential given by +(1.2) +V (x) = − +N +� +k=1 +Z +|xk| + +� +1≤j 0 define +hb(t) = +� +tmin{0,5−b} +if b ̸= 5 +log(t−1 + 2) +if b = 5. +We denote N0 = N ∪ {0}. For all R > 0 and α, β ∈ N3 +0 with |α|, |β| ≥ 1 there exists C +such that +(1.7) +|∂α +x ∂β +y γ(x, y)| ≤ C +� +1 + |x|2−|α|−|β| + |y|2−|α|−|β| ++ h|α|+|β|(|x − y|) +� +∥ρ∥1/2 +L1(B(x,R)) ∥ρ∥1/2 +L1(B(y,R)) +and for all |α| ≥ 1 there exists C such that +(1.8) +|∂α +x γ(x, y)| + |∂α +y γ(x, y)| ≤ C +� +1 + |x|1−|α| + |y|1−|α| ++ h|α|(|x − y|) +� +∥ρ∥1/2 +L1(B(x,R)) ∥ρ∥1/2 +L1(B(y,R)) + +ONE-PARTICLE DENSITY MATRIX +3 +for all x, y ∈ R3 with x ̸= 0, y ̸= 0 and x ̸= y. The notation ∂α +x refers to the α-partial +derivative in the x variable. The constant C depends on α, β, R, N and Z. The right- +hand side is finite because ψ is normalised and hence ρ ∈ L1(R3). The bounded first +derivative at the nucleus reflects local Lipschitz continuity of γ on R6. In addition, there +is local boundedness of up to four derivatives at the diagonal with at worst a logarithmic +singularity for the fifth derivative. The purpose of this current paper is to show local +boundedness of the fifth derivative at the diagonal. +The existence of the fifth-order cusp at the diagonal was previously demonstrated in +[6] (see also [7]). In this paper, quantum chemistry calculations show that for x, r ∈ R3, +x ̸= 0 and small r we have +Re[γ(x + r, x − r)] = γ(x, x) + C(x)|r|5 + R(x, r) +(1.9) +for some functions C(x) and R(x, r), the latter having no contribution from |r|k for +k = 0, 1, 3, 5 in the small |r| expansion at r = 0. +Our main result is as follows. +Theorem 1.1. Let ψ be an eigenfunction of (1.3). Define m(x, y) = min{1, |x|, |y|}. +Then for all |α| + |β| = 5 and R > 0 we have C, depending on R and also on Z, N and +E, such that +(1.10) +|∂α +x ∂β +y γ(x, y)| ≤ Cm(x, y)−4 ∥ρ∥1/2 +L1(B(x,R)) ∥ρ∥1/2 +L1(B(y,R)) +for all x, y ∈ R3 obeying 0 < |x − y| ≤ (2N)−1m(x, y). +Remark 1.2. +(1) Theorem 1.1 naturally extends to the case of a molecule with +several nuclei whose positions are fixed. The modifications are straightforward. +(2) The bound (1.10) naturally complements (1.7) and (1.8) for |α| + |β| ̸= 5. +(3) As a consequence of [5, Proposition 7.1], the inequality (1.10) shows that γ ∈ +C4,1 +loc +� +(R3\{0}) × (R3\{0}) +� +. +Notation. As mentioned earlier, we use standard notation whereby x = (x1, . . . , xN) ∈ +R3N, xj ∈ R3, j=1, . . . , N, and where N is the number of electrons. In addition, define +for 1 ≤ j, k ≤ N, j ̸= k, +ˆxj = (x1, . . . , xj−1, xj+1, . . . , xN) +(1.11) +ˆxjk = (x1, . . . , xj−1, xj+1, . . . , xk−1, xk+1, . . . , xN) +(1.12) +with obvious modifications if either j, k equals 1 or N, and if k < j. We define ˆx = ˆx1, +which will be used throughout. Variables placed before ˆxj and ˆxjk will be placed in the +removed slots as follows, for any x, y ∈ R3 we have +(x, ˆxj) = (x1, . . . , xj−1, x, xj+1, . . . , xN), +(1.13) +(x, y, ˆxjk) = (x1, . . . , xj−1, x, xj+1, . . . , xk−1, y, xk+1, . . . , xN). +(1.14) +In this way, x = (xj, ˆxj) = (xj, xk, ˆxjk). + +4 +PETER HEARNSHAW +We define a cluster to be any subset P ⊂ {1, . . . , N}. Denote P c = {1, . . . , N}\P, +P ∗ = P\{1}. We will also need cluster sets, P = (P1, . . . , PM), where M ≥ 1 and +P1, . . . , PM are clusters. +First-order cluster derivatives are defined, for a non-empty cluster P, by +(1.15) +Dα +P = +� +j∈P +∂α +xj +for α ∈ N3 +0, |α| = 1. +For P = ∅, Dα +P is defined as the identity. +Higher order cluster derivatives, for α = +(α′, α′′, α′′′) ∈ N3 +0 with |α| ≥ 2, are defined by successive application of first-order cluster +derivatives as follows, +(1.16) +Dα +P = (De1 +P )α′(De2 +P )α′′(De3 +P )α′′′ +where e1, e2, e3 are the standard unit basis vectors of R3. Let P = (P1, . . . , PM) and +α = (α1, . . . , αM), αj ∈ N3 +0, 1 ≤ j ≤ M, then we define the multicluster derivative (often +simply referred to as cluster derivative) by +(1.17) +Dα +P = Dα1 +P1 . . . DαM +PM . +It can readily be seen that cluster derivatives obey the Leibniz rule. +Throughout, the letter C refers to a positive constant whose value is unimportant but +may depend on Z, N and the eigenvalue E. +Distance function notation and elementary results. For non-empty cluster P, +define +ΣP = +� +x ∈ R3N : +� +j∈P +|xj| +� +l∈P +m∈P c +|xl − xm| = 0 +� +. +(1.18) +For P = ∅ we set ΣP := ∅. Denote Σc +P = R3N\ΣP. For each P we have ΣP ⊂ Σ where +(1.19) +Σ = +� +x ∈ R3N : +� +1≤j≤N +|xj| +� +1≤l 0, +f∞(x; r; u) := ∥∇u∥L∞(B(x,r)) + ∥u∥L∞(B(x,r)) +(1.28) +for x ∈ R3N. The ball B(x, r) is considered in R3N. Largely, this notation will be used +for u = ψ and in this case we have the notation, +f∞(x; r) := f∞(x; r; ψ). +(1.29) +A pointwise cluster derivative bound. To prove Theorem 1.1 we will state and +prove a new pointwise bound to cluster derivatives of eigenfunctions ψ, which may itself +be of independent interest. It will be shown by elliptic regularity that for all α the weak +cluster derivatives Dα +Pψ exist in the set Σc +α. It is therefore interesting to consider how +such cluster derivatives behave as the set Σα is approached. +Previously, S. Fournais and T. Ø. Sørensen have given bounds to local Lp-norms of +cluster derivatives of ψ for a single cluster P. Indeed, in [8, Proposition 1.10] it is shown +that for any multiindex 0 ̸= α ∈ N3 +0, p ∈ (1, ∞] and any 0 < r < R < 1 there exists C, +depending on r, R, p and α, such that +(1.30) +∥Dα +Pψ∥Lp(B(x,rλP (x))) ≤ CλP(x)1−|α|� +∥∇ψ∥Lp(B(x,RλP (x))) + ∥ψ∥Lp(B(x,RλP (x))) +� + +6 +PETER HEARNSHAW +for all x ∈ Σc +P. Notice that for every x ∈ Σc +P, we have B(x, rλP(x)) ⊂ Σc +P by the +definition of λP(x). Therefore, the set ΣP is avoided when evaluating Dα +Pψ in the Lp- +norms. +The objective of the following theorem is to extend the bounds (1.30) in the case of +p = ∞ and for cluster derivatives for cluster sets P. In particular, estimates are obtained +which depend on the order of derivative for each of the respective clusters in P. In the +following, ∇ denotes the gradient operator in R3N. +Theorem 1.3. For every cluster set P = (P1, . . . , PM), multiindex α ∈ N3M +0 +and any +0 < r < R < 1 there exists C, depending on α, r and R, such that for k = 0, 1, +(1.31) +��Dα +P∇kψ +�� +L∞(B(x,rλα(x))) ≤ Cλα(x)1−kλP1(x)−|α1| . . . λPM(x)−|αM|f∞(x; R) +for all x ∈ Σc +α. +Furthermore, for each |α| ≥ 1 there exists a function Gα +P : Σc +α → C3N such that +(1.32) +Dα +P∇ψ = Gα +P + ψDα +P∇Fc +and for every b ∈ [0, 1) there exists C, depending on α, r, R and b, such that +(1.33) +∥Gα +P∥L∞(B(x,rλα(x))) ≤ Cµα(x)bλP1(x)−|α1| . . . λPM(x)−|αM|f∞(x; R) +for all x ∈ Σc +α. +Remark 1.4. +(1) In the case of a single cluster and k = 0, the bound (1.31) reestab- +lishes (1.30) in the case of p = ∞, albeit with a slightly larger radius in the +L∞-norms on the right-hand side. +(2) When k = 0 and α ̸= 0, the presence of a single power of λα(x) in the bound +will cancel a single negative power of λPj(x) for j such that λPj(x) ≤ λPi(x) for +all i = 1, . . . , M with αi ̸= 0. Notice that the appropriate j will depend on x. +(3) The bound in (1.33) is stronger than that of (1.31) with k = 1. This is because +a positive power of µα(x) will partially cancel a single negative power of λPj(x) +for j such that λPj(x) ≥ λPi(x) for all i = 1, . . . , M with αi ̸= 0. +The proof of Theorem 1.3 will follow a similar strategy to that of [8, Proposition 1.10]. +An additional result will be required to prove (1.33). This result, [9], shows that ψ can +be made C1,1(R3N) upon multiplication by a factor, universal in the sense that the factor +depends only on N and Z. +We will require an elliptic regularity result, stated below, which will be used in the +proofs. Beforehand, we clarify the precise form of definitions which we will be using. +Let Ω be open, θ ∈ (0, 1] and k = N0. We formally define the θ-H¨older seminorms for a +function f by +[f]θ,Ω = sup +x,y∈Ω +x̸=y +|f(x) − f(y)| +|x − y|θ +, +[∇kf]θ,Ω = sup +|α|=k +[∂αf]θ,Ω. + +ONE-PARTICLE DENSITY MATRIX +7 +The space Ck,θ(Ω) is defined as all f ∈ Ck(Ω) where [∇kf]θ,Ω′ is finite for each Ω′ +compactly contained in Ω. In addition, the space Ck,θ(Ω) is defined as all f ∈ Ck(Ω) +where [∇kf]θ,Ω is finite. This space has a norm given by +∥f∥Ck,θ(Ω) = ∥f∥Ck(Ω) + [∇kf]θ,Ω. +For open Ω ⊂ Rn we can consider the following elliptic equation, +(1.34) +Lu := −∆u + c · ∇u + du = g +for some c : Ω → Cn and d, g : Ω → C. The corresponding bilinear form for operator L +is defined formally as +L(u, χ) = +� +Ω +� +∇u · ∇χ + (c · ∇u)χ + duχ +� +dx +for all u ∈ H1 +loc(Ω) and χ ∈ C∞ +c (Ω). We say that a function u ∈ H1 +loc(Ω) is a weak +solution to the equation (1.34) in Ω if L(u, χ) = +� +Ω gχ dx for every χ ∈ C∞ +c (Ω). +The following theorem is a restatement of [5, Proposition 3.1] ([8, Proposition A.2] is +similar), with additional H¨older regularity which follows from the proof. +Theorem 1.5. Let x0 ∈ Rn, R > 0 and c, d, g ∈ L∞(B(x0, R)) and u ∈ H1(B(x0, R)) +be a weak solution to (1.34) then for each θ ∈ [0, 1) we have u ∈ C1,θ(B(x0, R)) ∩ +H2 +loc(B(x0, R)), and for any r ∈ (0, R) we have +(1.35) +∥u∥C1,θ(B(x0,r)) ≤ C(∥u∥L2(B(x0,R)) + ∥g∥L∞(B(x0,R))) +for C = C(n, K, r, R, θ) where +∥c∥L∞(B(x0,R)) + ∥d∥L∞(B(x0,R)) ≤ K. +2. Proof of Theorem 1.3 +Our strategy for the proof will be to choose a suitable function F = F(x), dependent +only on N and Z, such that the function e−Fψ has greater regularity than ψ itself. Such +a multiplicative factor is frequently called a Jastrow factor in mathematical literature, +and this strategy has been used successfully in, for example, [10], [11] to elucidate reg- +ularity properties of ψ. The function e−Fψ will solve an elliptic equation with bounded +coefficients which behave suitably well under the action of cluster derivatives. Elliptic +regularity will then produce bounds to the cluster derivatives of e−Fψ. Such bounds can +then be used to obtain bounds to the cluster derivatives of ψ itself. +2.1. Jastrow factors. We begin by defining the function +(2.1) +F(x) = Fc(x) − Fs(x), + +8 +PETER HEARNSHAW +for x ∈ R3N, where +Fc(x) = −Z +2 +� +1≤j≤N +|xj| + 1 +4 +� +1≤l 0), such that +(2.10) +C ∥φ∥L∞(B(x,R)) ≤ ∥ψ∥L∞(B(x,R)) ≤ C′ ∥φ∥L∞(B(x,R)) +for all x ∈ R3N. +2.2. Derivatives of F. Informally, our objective is to take cluster derivatives of the +elliptic equation (2.8) and apply elliptic regularity. To do so, we require bounds to the +cluster derivatives of the coefficients present in this equation. This is the objective of the +current section. To begin, we state and prove the following preparatory lemma involving +the distances introduced in (1.21), (1.25) and (1.27). + +ONE-PARTICLE DENSITY MATRIX +9 +Lemma 2.1. For any σ = (σ1, . . . , σM) ∈ N3M +0 +we have for k = 0, 1, +(2.11) +µσ(y)k−|σ| ≤ λσ(y)kλP1(y)−|σ1| . . . λPM(y)−|σM| +for all y ∈ R3N. Furthermore, let β(1), . . . , β(n) be an arbitrary collection of multiindices +in N3M +0 +such that β(1) + · · · + β(n) = σ then +(2.12) +n +� +j=1 +λβ(j)(y) ≤ λσ(y) +for all y ∈ R3N. +Proof. The results are trivial in the case of σ = 0 therefore we assume in the following +that σ is non-zero. First, observe for all j = 1, . . . , M, +µσ(y)−|σj| ≤ λPj(y)−|σj| +µσ(y)1−|σj| ≤ λPj(y)1−|σj| +if σj ̸= 0 +by the definition of µσ. Now perform the following trivial expansion of the product, +µσ(y)−|σ| = µσ(y)−|σ1| . . . µσ(y)−|σM| +≤ λP1(y)−|σ1| . . . λPM(y)−|σM| +which proves (2.11) for k = 0. For k = 1, consider that for each y we can find l = +1, . . . , M such that λσ(y) = λPl(y) and σl ̸= 0. Then, +µσ(y)1−|σ| = µσ(y)−|σ1| . . . µσ(y)1−|σl| . . . µσ(y)−|σM| +≤ λP1(y)−|σ1| . . . λPl(y)1−|σl| . . . λPM(y)−|σM| += λσ(y)λP1(y)−|σ1| . . . λPM(y)−|σM| +as required. +. Finally we prove (2.12). As above, take arbitrary y and a corresponding l such that +λσ(y) = λPl(y) with σl ̸= 0. For each 1 ≤ j ≤ n we denote the N3 +0-components of β(j) as +β(j) = (β(j) +1 , . . . , β(j) +M ). We know β(1)+· · ·+β(n) = σ so in particular, β(1) +l ++· · ·+β(n) +l += σl. +Since σl ̸= 0 there exists at least one 1 ≤ r ≤ n such that β(r) +l +̸= 0. Hence by the definition +of λβ(r), +λβ(r)(y) = min{λPs(y) : β(r) +s +̸= 0, s = 1, . . . , M} ≤ λPl(y) = λσ(y). +The remaining factors λβ(j)(y), for j ̸= r, can each be bounded above by one. +□ +The following lemma will be useful in proving results about taking cluster derivatives +of F, as defined in (2.1). Later, we will apply it using f as the function |x| for x ∈ R3, +or derivatives thereof. + +10 +PETER HEARNSHAW +Lemma 2.2. Let f ∈ C∞(R3\{0}) and k ∈ N0 be such that for each σ ∈ N3 +0 there exists +C such that +(2.13) +|∂σf(x)| ≤ C|x|k−|σ| for all x ̸= 0. +Then for any α ̸= 0 with |α| ≥ k there exists some new C such that for any l, m = +1, . . . , N, the weak derivatives Dα +P(f(xl)) and Dα +P(f(xl −xm)) are both smooth in Σc +α and +obey +|Dα +P(f(xl))|, |Dα +P(f(xl − xm))| ≤ Cqα(x)k−|α| +for all x ∈ Σc +α. +Proof. Take any j = 1, . . . , M with αj ̸= 0, then we have D +αj +Pj(f(xl)) ≡ 0 for each l ∈ P c +j . +Therefore, for Dα +P(f(xl)) to not be identically zero we require that l ∈ Pj for each j with +αj ̸= 0. For such l we have xl ̸= 0 since x ∈ Σc +α, and +|xl| ≥ dPj(x) +for each j with αj ̸= 0 by (1.20). Therefore, for constant C in (2.13), we have +|Dα +P(f(xl))| = |∂α1+···+αMf(xl)| ≤ C|xl|k−|α| ≤ Cqα(x)k−|α| +because |α| ≥ k. Similarly, for each j = 1, . . . , M, with αj ̸= 0 we have D +αj +Pj(f(xl−xm)) ≡ +0 if either l, m ∈ Pj or l, m ∈ P c +j . Therefore, for Dα +P(f(xl − xm)) to not be identically +zero we require that +(l, m) ∈ +� +j : αj̸=0 +� +(Pj × P c +j ) ∪ (P c +j × Pj) +� +. +For such (l, m) we have xl ̸= xm since x ∈ Σc +α and +|xl − xm| ≥ +√ +2 dPj(x) +for each j with αj ̸= 0 by (1.20). Therefore, for some constant C′, +|Dα +P(f(xl − xm))| = |∂α1+···+αMf(xl − xm)| ≤ C|xl − xm|k−|α| ≤ C′qα(x)k−|α| +because |α| ≥ k. +□ +The following lemma provides pointwise bounds to cluster derivatives of functions +involving F. +Lemma 2.3. For any cluster set P and any |σ| ≥ 1 there exists C, which depends on +σ, such that for k = 0, 1, +(2.14) +��Dσ +P∇kF(y) +��, +��Dσ +P∇k(eF)(y) +�� ≤ Cλσ(y)1−kλP1(y)−|σ1| . . . λPM(y)−|σM| +(2.15) +��Dσ +P|∇F(y)|2�� ≤ CλP1(y)−|σ1| . . . λPM(y)−|σM| +for all y ∈ Σc +σ. The bound to the first object in (2.14) also holds when F is replaced by +Fc. + +ONE-PARTICLE DENSITY MATRIX +11 +Proof. Let τ be the function defined as τ(x) = |x| for x ∈ R3. Then, by definition (2.2) +we can write +(2.16) +Fc(y) = −Z +2 +� +1≤j≤N +τ(yj) + 1 +4 +� +1≤l 0 we can define the following two cutoff factors ζt = ζt(z) and +θt = θt(z) by +(3.2) +ζt(z) = χ +�4N|z| +t +� +, +θt(z) = 1 − ζt(z) +for z ∈ R3. We have the following support criteria for cutoff factors. For any z ∈ R3 and +t > 0, + +ONE-PARTICLE DENSITY MATRIX +19 +• If ζt(z) ̸= 0 then |z| < (2N)−1t, +• If θt(z) ̸= 0 then |z| > (4N)−1t. +Let 0 < δ < (4N)−1ǫ (the use of (4N)−1 is explained in the following lemma). We +define a biscaled cutoff, which depends on δ and ǫ as parameters, as a function Φ = +Φδ,ǫ(x, y, ˆx) defined by +(3.3) +Φδ,ǫ(x, y, ˆx) = +� +2≤j≤N +g(1) +j (x − xj) +� +2≤j≤N +g(2) +j (y − xj) +� +2≤k 0, +1t(z) = 1{(4N)−1t<|z|<(2N)−1t}(z) +(3.4) +1′ +t(z) = 1{(4N)−1t<|z|<1}(z) +(3.5) +for z ∈ R3. And for each t > 0 we define the function Mt = Mt(x, y, ˆx) by +Mt(x, y, ˆx) = +� +2≤j≤N +1t(x − xj) + +� +2≤j≤N +1t(y − xj) + +� +2≤k (4N)−1ǫ ≥ δ. For such z we therefore have θδ(z) = 1, by the definition +of θδ, which proves (3.7). For (3.8), if we also have ζδ(z) ̸= 0, then |z| < (2N)−1δ by the +support criteria for ζδ, giving a contradiction. For (3.9) we need only consider z such +that ζδ(z) ̸= 0, in which case |z| < (2N)−1δ. This gives 4N|z|ǫ−1 < (2N)−1 and hence, +by definition, ζǫ(z) = 1. +. For any A, B ⊂ {2, . . . , N} with A ∩ B = ∅ we define +τA,B(x) = +� +j∈A +ζδ(x1 − xj) +� +j∈B +(θδζǫ)(x1 − xj) +� +j∈{2,...,N}\(A∪B) +θǫ(x1 − xj) +and therefore +� +A⊂{2,...,N} +B⊂{2,...,N}\A +τA,B(x) = +� +2≤j≤N +� +ζδ(x1 − xj) + (θδζǫ)(x1 − xj) + θǫ(x1 − xj) +� += 1 +for all x ∈ R3N. Let Ξ = {(j, k) : 2 ≤ k < l ≤ N}. For each subset Y, Z ⊂ Ξ with +Y ∩ Z = ∅ we define +TY,Z(ˆx) = +� +(j,k)∈Y +ζδ(xj − xk) +� +(j,k)∈Z +(θδζǫ)(xj − xk) +� +(j,k) ∈ Ξ\(Y ∪Z) +θǫ(xj − xk) +and therefore +� +Y ⊂ Ξ +Z ⊂ Ξ\Y +TY,Z(ˆx) = +� +2≤j δ/2 by the +reverse triangle inequality. The case of k ∈ S∗ is analogous. + +22 +PETER HEARNSHAW +. Now let k ∈ Q∗. First, we consider the case where either g(1) +k +̸= θǫ or g(2) +k +̸= θǫ, or both. +Without loss, assume g(1) +k +̸= θǫ. Then either g(1) +k += ζδ or g(1) +k += θδζǫ. Hence by support +criteria we have the inequalities, +|x − xk| < (2N)−1ǫ, +|y − xk| ≤ |x − y| + |x − xk| ≤ 2δ + (2N)−1ǫ ≤ ǫ/N, +which gives the required inequality since N ≥ 2. Now, suppose that g(1) +k += g(2) +k += θǫ. +Then there exist pairwise distinct j1, . . . , js ∈ {2, . . . , N} with 1 ≤ s ≤ N − 2 such that +fj1,j2, fj2,j3, . . . , fjs,k ̸= θǫ and either g(1) +j1 ̸= θǫ or g(2) +j1 ̸= θǫ. As before, we see that +|x − xj1|, |y − xj1| ≤ ǫ/N +regardless of which (or both) of g(1) +j1 and g(2) +j1 are not θǫ. It’s also clear that by support +criteria, |xj1 − xj2|, . . . , |xjs − xk| ≤ (2N)−1ǫ. Therefore, by the triangle inequality, +|x − xk| ≤ |x − xj1| + |xj1 − xj2| + · · · + |xjs − xk| ≤ ǫ +N + ǫ(N − 2) +2N += ǫ +2, +and similarly we can show |y − xk| ≤ ǫ/2, completing the proof. +□ +3.4. Factorisation of biscaled cutoffs. Let Φ be given by (3.3). We can define a +partial product of Φ as a function of the form +(3.14) +Φ′(x, y, ˆx) = +� +j∈T1 +g(1) +j (x − xj) +� +j∈T2 +g(2) +j (y − xj) +� +(k,l)∈R1 +fkl(xk − xl) +where T1, T2 ⊂ {2, . . . , N}, R1 ⊂ {(k, l) : 2 ≤ k < l ≤ N}. +We now define classes of partial products of Φ which corresponding to a cluster. Let +T be an arbitrary cluster with 1 ∈ T. +Φ(x, y, ˆx; T) = +� +j∈T ∗ +g(1) +j (x − xj) +� +j∈T ∗ +g(2) +j (y − xj) +� +k,l∈T ∗ +k 0 +such that for any z0 ∈ R3 we get ∂σ +z |z + z0|s ≤ C|z + z0|s−|σ| for all z ∈ R3, z ̸= −z0. +Recall the function 1t was defined in (3.4). +Lemma 3.5. For any σ ∈ N3 +0 with |σ| ≥ 1 and any t > 0 there exists C, depending on +σ but independent of t, such that +(3.22) +|∂σζt(z)|, |∂σθt(z)| ≤ Ct−|σ|1t(z) +for all z ∈ R3. + +24 +PETER HEARNSHAW +Proof. Without loss we consider the case of θt, the case of ζt being similar. +In the +following, χ(j) refers to the j-th (univariate) derivative of the function χ defined in (3.1). +Now, since |σ| ≥ 1 the chain rule shows that ∂σθt(z) can be written as a sum of terms +of the form +(3.23) +�4N +t +�m +χ(m)�4N|z| +t +� +∂σ1 +z |z| . . . ∂σm +z |z| +where 1 ≤ m ≤ |σ|, and σ1, . . . , σm ∈ N3 +0 are non-zero multiindices obeying +σ1 + · · · + σm = σ. +Since m ≥ 1 we have that if χ(m)(s) ̸= 0 then s ∈ (1, 2), and therefore for any term +(3.23) to be non-zero we require that +(3.24) +(4N)−1t < |z| < (2N)−1t. +By the remark preceeding the current lemma, there exists C, dependent on σ1, . . . , σm, +such that +∂σ1 +z |z| . . . ∂σm +z |z| ≤ C|z|m−|σ| ≤ C(4N)|σ|−mtm−|σ|, +using (3.24). Therefore, the terms (3.23) can readily be bounded to give the desired +result. +□ +We now give bounds for the cluster derivatives (3.20)-(3.21) acting on cutoffs. +Lemma 3.6. Let Φ be any biscaled cutoff and let Q = Q(Φ). Then +Dα +x,y,QΦ( · ; Q) ≡ 0 +(3.25) +for all α ∈ N3 +0 with |α| ≥ 1. +Proof. By the chain rule, each function in the product (3.15) for Φ( · ; Q) has zero deriv- +ative upon action of Dα +x,y,Q. +□ +Recall that the notion of partial products was defined in (3.14) and the function Mt +for t > 0 was defined in (3.6). +Lemma 3.7. Let δ ≤ (4N)−1ǫ and Φ = Φδ,ǫ be a biscaled cutoff. Let Q = Q(Φ). Then +for any multiindex α ∈ N3N+3 +0 +there exists C, dependent on α but independent of δ and +ǫ, such that for any partial products Φ′ of Φ we have +|∂αΦ′(x, y, ˆx)| ≤ +� +Cǫ−|α| +if Φ′ = Φ( · ; Q, Qc) +C +� +ǫ−|α| + δ−|α|Mδ(x, y, ˆx) +� +otherwise. +(3.26) +Proof. Lemma 3.5 gives bounds for the partial derivatives of the function ζδ, θδ, ζǫ, θǫ. +Considering θδζǫ, we apply the Leibniz rule with σ ∈ N3 +0, |σ| ≥ 1, to obtain +∂σ(θδζǫ)(z) = +� +µ≤σ +�σ +µ +� +∂µθδ(z)∂σ−µζǫ(z) += ∂σθδ(z) + ∂σζǫ(z), +(3.27) + +ONE-PARTICLE DENSITY MATRIX +25 +since, for each µ ≤ σ with µ ̸= 0 and µ ̸= σ we have +∂µθδ(z) ∂σ−µζǫ(z) ≡ 0 +by Lemma 3.5 and the definition (3.4). +. Now, to evaluate ∂αΦ′ we apply the Leibniz rule to the product (3.14) using (3.27) +where appropriate. Differentiated cutoff factors are bounded by (3.22) and any remaining +undifferentiated cutoff factors are bounded above by 1. If Φ′ = Φ( · ; Q, Qc), all cutoff +factors are of the form θǫ by Lemma 3.4 and therefore we need only use the bounds in +(3.22) with t = ǫ. +□ +The derivative Dx,y,Q acting on Φ is special in that it contributes only powers of ǫ (and +not δ) to the bounds. This is shown in the next lemma. +Lemma 3.8. Let δ ≤ (4N)−1ǫ and Φ = Φδ,ǫ be a biscaled cutoff. Let Q = Q(Φ). For +any multiindices α ∈ N3 +0 and σ ∈ N3N+3 +0 +there exists C, independent of ǫ and δ, such +that +|∂σDα +x,y,QΦ(x, y, ˆx)| ≤ Cǫ−|α|� +ǫ−|σ| + δ−|σ|Mδ(x, y, ˆx) +� +for all x, y, ˆx. +Proof. First, set Φ′ = Φ( · ; Q) Φ( · ; Qc) and Φ′′ = Φ( · ; Q, Qc). We then have +Dα +x,y,QΦ = Φ′ Dα +x,y,QΦ′′ +which follows from Lemmas 3.3 and 3.6 and that Φ( · ; Qc) is not dependent on variables +involved in the Dα +x,y,Q-derivative. By the definition (3.21), the derivative Dα +x,y,QΦ′′ can +be written as a sum of partial derivatives of the form ∂αΦ′′ where α ∈ N3N+3 +0 +obeys +|α| = |α|. Now, by the Leibniz rule and Lemma 3.7 there exists some constants C and +C′, independent of δ and ǫ, such that +|∂σ(Φ′ ∂αΦ′′)| ≤ +� +τ≤σ +�σ +τ +� +|∂τΦ′| |∂σ−τ+αΦ′′| +≤ C +� +τ≤σ +(ǫ−|τ| + δ−|τ|Mδ)ǫ−|σ|+|τ|−|α| +≤ C′ǫ−|α|(ǫ−|σ| + δ−|σ|Mδ), +completing the proof. +□ +4. Proof of Theorem 1.1 +The idea of the proof is to turn partial derivatives of an integral, such as the density +matrix γ(x, y) weighted with a suitable cutoff, into cluster derivatives under the inte- +gral. For the density matrix we then estimate the resulting integrals involving cluster +derivatives of ψ using the pointwise bounds of Theorem 1.3. + +26 +PETER HEARNSHAW +Although Theorem 1.1 is stated using partial derivatives of the density matrix γ(x, y) in +the x- and y-variables, it is more appropriate to consider directional derivatives. Indeed, +we define new variables +u = (x + y)/2, +(4.1) +v = (x − y)/2, +(4.2) +and consider the ∂α +u ∂β +v -derivatives of the density matrix. The ∂u-derivatives act along +the direction parallel to the diagonal and it is found that they do not affect the non- +smoothness we get at the diagonal, regardless of how many of these derivatives are taken. +The ∂v-derivatives act in the direction perpendicular to the diagonal and these derivatives +are found to contribute to worsening the non-smoothness at the diagonal. +Consider derivatives of the form ∂α +x ∂β +y γ(x, y) where |α|+|β| = 2. As discussed in [5], the +case where |α| = |β| = 1 (the mixed derivatives) is particularly well-behaved in contrast +to the other cases. The reason behind this is that when |α| = |β| = 1, differentiation +under the integral leads to both ψ factors being differentiated exactly once. The greater +regularity in this case follows from the well known fact that ψ, ∇ψ ∈ L∞ +loc(R3N), first +proven in [12], whereas higher order derivatives of ψ do not have this locally boundedness. +This is used in the proof in the following way. In (4.34) below, it will be shown that +derivatives of the form ∂σ +v for |σ| = 2 can be written in terms of ∂σ +u and the mixed +derivatives ∂α +x ∂β +y γ(x, y) for some |α| = |β| = 1. The benefit of this identity is that two +v-derivatives (which act to worsen the singularity at the diagonal) have been transformed +into two u-derivatives (which do not worsen the singularity) along with mixed derivatives +which have good regularity. +This method only works for two v-derivatives, and the +strategy of using cluster derivatives, as described above, must be used in conjunction. +The difficulties encountered in the fifth derivative of the density matrix are described in +a later section. +4.1. Density matrix notation. The proof will require auxiliary functions related to +the density matrix which we introduce now. For l, m ∈ N3 +0 with |l|, |m| ≤ 1 define, +(4.3) +γl,m(x, y) = +� +R3N−3 ∂l +xψ(x, ˆx)∂m +y ψ(y, ˆx) dˆx. +In this notation, it is clear that γ = γ0,0. For any biscaled cutoff Φ, defined in (3.3), we +set +γl,m(x, y; Φ) = +� +R3N−3 ∂l +xψ(x, ˆx)∂m +y ψ(y, ˆx)Φ(x, y, ˆx) dˆx, +and define γ( · ; Φ) = γ0,0( · ; Φ). We will consider the above functions in the variables +(4.1) and (4.2). It is then natural to define for all u, v ∈ R3, +˜γl,m(u, v) = γl,m(u + v, u − v), +(4.4) +˜γl,m(u, v; Φ) = γl,m(u + v, u − v; Φ). +(4.5) + +ONE-PARTICLE DENSITY MATRIX +27 +4.2. Integrals involving f∞. The following proposition is a restatement of [5, Lemma +5.1] and is proven in that paper. Notice that the function Mt was defined in (3.6) and +has a slightly different form to the corresponding function used in the paper. +Proposition 4.1. Given R > 0, there exists C such that +(4.6) +� +R3N−3 f∞(x, ˆx; R)f∞(y, ˆx; R) dˆx ≤ C ∥ρ∥1/2 +L1(B(x,2R)) ∥ρ∥1/2 +L1(B(y,2R)) +for all x, y ∈ R3. In addition, given G ∈ L1(R3) there exists C, independent of G, such +that +(4.7) +� +R3N−3 +� +|G(xj − xk)| + |G(z − xk)| + |G(xj)| +� +f∞(x, ˆx; R)f∞(y, ˆx; R) dˆx +≤ C ∥G∥L1(R3) ∥ρ∥1/2 +L1(B(x,2R)) ∥ρ∥1/2 +L1(B(y,2R)) +for all x, y, z ∈ R3, and j, k = 2, . . . , N, j ̸= k. In particular, for any t > 0 there exists +C, independent of t, such that +(4.8) +� +R3N−3 Mt(x, y, ˆx)f∞(x, ˆx; R)f∞(y, ˆx; R) dˆx ≤ Ct3 ∥ρ∥1/2 +L1(B(x,2R)) ∥ρ∥1/2 +L1(B(y,2R)) +for all x, y ∈ R3. +We introduce the following quantities based on the bounded distances (1.21). For any +x, y ∈ R3 and ˆx ∈ R3N−3 define +λ(x, y, ˆx) = min{λP(x, ˆx), λS∗(x, ˆx), λP ∗(y, ˆx), λS(y, ˆx)}, +(4.9) +π(x, y, ˆx) = min{λQ(x, ˆx), λQ(y, ˆx)}. +(4.10) +Later, we will see that these quantities appear to negative powers when we apply Theorem +1.3 to clusters derivatives of ψ involving the clusters P, S and Q. The next lemma will +give conditions for when these quantities can be bounded away from zero on the support +of Φ. +Beforehand, we consider an alternative formulae for (4.10) and (4.9). +Recall that +1 ∈ P, S, Q by definition. Using (1.20) we find that +(4.11) +π(x, y, ˆx) = min{1, |x|, |y|, |xj| : j ∈ Q∗, 2−1/2|x − xk| : k ∈ Qc, +2−1/2|y − xk| : k ∈ Qc, 2−1/2|xj − xk| : j ∈ Q∗, k ∈ Qc}. +In the case of P ∗ ∩ S∗ = ∅ we similarly find that +(4.12) +λ(x, y, ˆx) = min{1, |x|, |y|, |xj| : j ∈ P ∗ ∪ S∗, 2−1/2|x − xk| : k ∈ P c, +2−1/2|y − xk| : k ∈ Sc, 2−1/2|xj − xk| : (j, k) ∈ (P ∗ × P c) ∪ (S∗ × Sc)}. + +28 +PETER HEARNSHAW +Lemma 4.2. Let δ ≤ (4N)−1ǫ, and ǫ ≤ 1 and let Φ = Φδ,ǫ be an arbitrary biscaled +cutoff. Let x, y ∈ R3 be such that δ ≤ |x − y| ≤ 2δ, and |x|, |y| ≥ ǫ. Then there exists a +constant C, dependent only on N, such that +π(x, y, ˆx) ≥ Cǫ +(4.13) +λ(x, y, ˆx) ≥ Cδ +(4.14) +whenever Φ(x, y, ˆx) ̸= 0. In addition, for all b ≥ 0 with b ̸= 3 and R > 0 there exists C, +depending on b and R but independent of δ, ǫ, x and y such that +(4.15) +� +supp Φ(x,y, · ) +λ(x, y, ˆx)−bf∞(x, ˆx; R)f∞(y, ˆx; R) dˆx +≤ C(ǫ−b + hb(δ)) ∥ρ∥1/2 +L1(B(x,2R)) ∥ρ∥1/2 +L1(B(y,2R)) +where, for all t > 0, we define +hb(t) = +� +0 +if b < 3 +t3−b +if b > 3. +The following corollary will be useful later. +Corollary 4.3. There exists C, depending on R but independent of δ and ǫ, such that +(4.16) +2 +� +r=1 +� +R3N−3 λ(x, y, ˆx)−2f∞(x, ˆx; R)f∞(y, ˆx; R)|∇rΦ(x, y, ˆx)| dˆx +≤ Cǫ−3 ∥ρ∥1/2 +L1(B(x,2R)) ∥ρ∥1/2 +L1(B(y,2R)) +for all x, y ∈ R3 with δ ≤ |x − y| ≤ 2δ and |x|, |y| ≥ ǫ. +Proof of Corollary 4.3. When r = 1 the bound for the integral is immediate. +Using +Lemma 3.8 and (4.14), the integral in (4.16) for r = 2 can be bounded by some constant +multiplying +� +supp Φ(x,y, · ) +� +ǫ−1λ(x, y, ˆx)−2 + δ−3Mδ(x, y, ˆx) +� +f∞(x, ˆx; R)f∞(y, ˆx; R) dˆx +The required bound follows from (4.15) of Lemma 4.2, (4.8) of Proposition 4.1 and that +ǫ < 1. +□ +Proof of Lemma 4.2. By Lemma 3.2 we need only consider Φ with P ∗ ∩ S∗ = ∅. By the +definition of the cluster Q, if j ∈ Q∗ and k ∈ Qc then fjk = θǫ in the formula (3.3) +defining Φ. Similarly, if k ∈ Qc then g(1) +k += g(2) +k += θǫ. Using the support criteria of θǫ we +therefore get +|x − xk|, |y − xk|, |xj − xk| ≥ (4N)−1ǫ +j ∈ Q∗, k ∈ Qc. +(4.17) + +ONE-PARTICLE DENSITY MATRIX +29 +In addition, if j ∈ Q∗ we get +(4.18) +|xj| ≥ |x| − |x − xj| ≥ ǫ/2 +j ∈ Q∗ +since for such j we have |x − xj| ≤ ǫ/2 by Lemma 3.2. The lower bound (4.13) then +follows from the formula (4.11) for π(x, y, ˆx). +. In a similar way we consider the clusters P and S. Indeed, let j ∈ P ∗, k ∈ P c or j ∈ S∗, +k ∈ Sc, then fjk ̸= ζδ. Similarly, if k ∈ P c then g(1) +k +̸= ζδ, and if k ∈ Sc then g(2) +k +̸= ζδ. +Notice that if a cutoff factor is not ζδ then it must either be θδζǫ or θǫ, both of which +are only supported away from zero. Therefore by the support criteria of these factors we +obtain the following inequalities, +|x − xk|, |y − xl| ≥ (4N)−1δ +k ∈ P c, l ∈ Sc +(4.19) +|xj − xk| ≥ (4N)−1δ +j ∈ P ∗, k ∈ P c or j ∈ S∗, k ∈ Sc. +(4.20) +We also have (4.18) for j ∈ P ∗ ∪S∗ since P, S ⊂ Q. The lower bound (4.14) then follows +from the formula (4.12) for λ(x, y, ˆx). +. We recall the function 1′ +δ was defined in (3.5). By the formula (4.12) we can use (4.18) +to obtain some C, depending on b, such that +(4.21) +λ(x, y, ˆx)−b ≤ C +� +ǫ−b + +� +k∈P c +1′ +δ(x − xk) |x − xk|−b + +� +k∈Sc +1′ +δ(y − xk) |y − xk|−b ++ +� +(j,k)∈(P ∗×P c) +∪(S∗×Sc) +1′ +δ(xj − xk) |xj − xk|−b� +, +where the upper bounds in the indicator functions (3.5) can be included because 1 lies in +the minimum (4.12) and the lower bounds follow from (4.18)-(4.20). The bound (4.15) +then follows from the above inequality along with both (4.6) and (4.7) of Proposition +4.1, where we choose G to be the function G(z) = 1′ +δ(z)|z|−b for z ∈ R3. +□ +4.3. Differentiating the density matrix - some required bounds. We collect cer- +tain results which will be used throughout this section. Let δ ≤ (4N)−1ǫ and ǫ ≤ 1 and +suppose Φ = Φδ,ǫ is a biscaled cutoff as defined in (3.3). As usual, we denote Q = Q(Φ), +P = P(Φ) and S = S(Φ). Let η ∈ N3 +0 and ν = (ν1, ν2) ∈ N6 +0 be arbitrary. Firstly, we +define +(4.22) +Φ(η,ν)(x, y, ˆx) = Dη +x,y,QDν1 +x,PDν2 +y,SΦ(x, y, ˆx) +where in the notation it is implicit the clusters used are Q, P and S corresponding to Φ. +By Lemma 3.8 and the definition of cluster derivatives (1.16) it can be shown that for +each η and ν there exists C, independent of δ and ǫ, such that +(4.23) +|Φ(η,ν)(x, y, ˆx)| ≤ Cǫ−|η|� +ǫ−|ν| + δ−|ν|Mδ(x, y, ˆx) +� +for all x, y ∈ R3 and ˆx ∈ R3N−3. + +30 +PETER HEARNSHAW +Now take any x, y ∈ R3 with δ ≤ |x−y| ≤ 2δ and |x|, |y| ≥ ǫ, and suppose Φ(x, y, ˆx) ̸= +0. +Then, as a consequence of Lemma 4.2 we have both π(x, y, ˆx) and λ(x, y, ˆx) are +positive. Therefore, by the definitions (4.9), (4.10) and (1.20), (1.21), +(x, ˆx) ∈ Σc +Q ∩ Σc +P ∩ Σc +S∗ +and +(y, ˆx) ∈ Σc +Q ∩ Σc +P ∗ ∩ Σc +S. +(4.24) +This allows us to apply Theorem 1.3. Indeed, for every η ∈ N3 +0, ν = (ν1, ν2) ∈ N6 +0 and +all R > 0 there exists C, independent of our choice of x, y and ˆx, such that for k = 0, 1, +��Dη +QDν +{P,S∗}∇kψ(x, ˆx) +�� ≤ Cπ(x, y, ˆx)−|η|λ(x, y, ˆx)−|ν|f∞(x, ˆx; R), +(4.25) +��Dη +QDν +{P ∗,S}∇kψ(y, ˆx) +�� ≤ Cπ(x, y, ˆx)−|η|λ(x, y, ˆx)−|ν|f∞(y, ˆx; R). +(4.26) +For convenience, we choose a bound which holds for both values of k. The functions λ +and π are defined in (4.9) and (4.10) respectively. +Now, let |ν| ≥ 1. By Theorem 1.3 with b = 1/2 we can write +Dν +{P,S∗}∇ψ(x, ˆx) = Gν +{P,S∗}(x, ˆx) + ψ(x, ˆx) +� +Dν +{P,S∗}∇Fc(x, ˆx) +� +(4.27) +Dν +{P ∗,S}∇ψ(y, ˆx) = Gν +{P ∗,S}(y, ˆx) + ψ(y, ˆx) +� +Dν +{P ∗,S}∇Fc(y, ˆx) +� +(4.28) +for functions Gν +{P,S∗} and Gν +{P ∗,S} which obey +��Gν +{P,S∗}(x, ˆx) +�� ≤ Cλ(x, y, ˆx)1/2−|ν|f∞(x, ˆx; R), +(4.29) +��Gν +{P ∗,S}(y, ˆx) +�� ≤ Cλ(x, y, ˆx)1/2−|ν|f∞(y, ˆx; R), +(4.30) +for some C, dependent on ν but independent of the choice of x, y and ˆx. +By Lemma 2.3 and (2.2), for each ν ∈ N6 +0 there exists C, independent of x, y and ˆx, +such that +��Dν +{P,S∗}∇Fc(x, ˆx) +�� + +��Dν +{P ∗,S}∇Fc(y, ˆx) +�� ≤ Cλ(x, y, ˆx)−|ν|. +(4.31) +4.4. Differentiating the density matrix - first and second derivatives. In the +following, let l, m ∈ N3 +0 obey |l| = |m| = 1. In standard notation, by ∂l +x1ψ we mean the +l-partial derivative in the first R3 component of ψ. Then by differentiation under the +integral, +∂l +u˜γ(u, v) = +� +R3N−3 ∂l +x1ψ(u + v, ˆx)ψ(u − v, ˆx) dˆx + +� +R3N−3 ψ(u + v, ˆx)∂lx1ψ(u − v, ˆx) dˆx +(4.32) += ˜γl,0(u, v) + ˜γ0,l(u, v), +and similarly, +∂l +v˜γ(u, v) = ˜γl,0(u, v) − ˜γ0,l(u, v) +(4.33) + +ONE-PARTICLE DENSITY MATRIX +31 +for all u, v ∈ R3. The above equalities are used to obtain a formula relating second order +u-derivatives to second order v-derivatives of ˜γ. Omitting the argument (u, v) we get +� +∂l+m +u +− ∂l+m +v +� +˜γ = ∂l +u(˜γm,0 + ˜γ0,m) − ∂l +v(˜γm,0 − ˜γ0,m) += +� +∂l +u + ∂l +v +� +˜γ0,m + +� +∂l +u − ∂l +v +� +˜γm,0 += 2(˜γl,m + ˜γm,l) +(4.34) +where the final equality is obtained by differentiation under the integral, as in (4.32). +4.5. Differentiating the density matrix - general derivatives. Partial derivatives +of γ are written as linear combinations of integrals involving cluster derivatives of ψ. +One such integral is bounded in the following lemma. Since it is more involved than the +other such integrals, the proof is postponed until later in the section. +Lemma 4.4. Let δ ≤ (4N)−1ǫ and ǫ ≤ 1 and let Φ = Φδ,ǫ be an arbitrary biscaled cutoff. +Let P = P(Φ) and S = S(Φ) obey P ∗ ∩ S∗ = ∅. For any α, β ∈ N6 +0 with |α| + |β| = 3, +any l, m ∈ N3 +0 with |l| = |m| = 1, and any R > 0 there exists C such that +(4.35) +��� +� +R3N−3 +� +Dα +{P,S∗}∂l +xψ(x, ˆx) +�� +Dβ +{P ∗,S}∂m +y ψ(y, ˆx) +� +Φ(x, y, ˆx) dˆx +��� +≤ Cǫ−3 ∥ρ∥1/2 +L1(B(x,R)) ∥ρ∥1/2 +L1(B(y,R)) +for all δ ≤ |x−y| ≤ 2δ and |x|, |y| ≥ ǫ. The constant C depends on R but is independent +of δ, ǫ. +In part two of the following lemma the conditions on η, µ, l and m are not the most +general, but for simplicity we restrict ourselves to these assumptions. We use notation +of cluster derivatives of Φ from (4.22). +Lemma 4.5. Let δ ≤ (4N)−1ǫ and ǫ ≤ 1 and Φ = Φδ,ǫ be an arbitrary biscaled cutoff. +Let η, µ ∈ N3 +0 be arbitrary and l, m ∈ N3 +0 be such that |l|, |m| ≤ 1. +(i) On the set of u, v such that δ/2 ≤ |v| ≤ δ and |u + v|, |u − v| ≥ ǫ, the derivative +∂η +u∂µ +v ˜γl,m(u, v; Φ) is equal to a linear combination of integrals of the form +(4.36) +� +R3N−3 +� +Dχ1 +Q Dα +{P,S∗}∂l +x1ψ(u + v, ˆx) +�� +Dχ2 +Q Dβ +{P ∗,S}∂m +x1ψ(u − v, ˆx) +� +· +Φ(χ3,σ)(u + v, u − v, ˆx) dˆx +where α, β, σ ∈ N6 +0 and χ = (χ1, χ2, χ3) ∈ N9 +0 obey |χ| = |η| and |α|+|β|+|σ| = +|µ|. +(ii) Furthermore, suppose either +• |µ| ≤ 2, or +• |µ| = 3, η = 0 and |l| = |m| = 1. + +32 +PETER HEARNSHAW +Then for all R > 0 we have some C0 such that +(4.37) +|∂η +u∂µ +v ˜γl,m(u, v; Φ)| ≤ C0ǫ−|η|−|µ| ∥ρ∥1/2 +L1(B(u+v,R)) ∥ρ∥1/2 +L1(B(u−v,R)) +for all δ/2 ≤ |v| ≤ δ and |u + v|, |u − v| ≥ ǫ. The constant C0 depends on R but +is independent of δ, ǫ. +Proof of i). By Lemma 3.2, we need only consider Φ such that P ∗ ∩ S∗ = ∅. For each +choice of u and v we define a u- and v-dependent change of variables for the integral +˜γlm(u, v; Φ) defined in (4.5). +To start, we define two vectors ˆa = (a2, . . . , aN), ˆb = +(b2, . . . , bN) ∈ R3N−3 by +ak = +� +u +if k ∈ Q∗ +0 +if k ∈ Qc +bk = + + + + + +v +if k ∈ P ∗ +−v +if k ∈ S∗ +0 +if k ∈ (P ∪ S)c, +(4.38) +and define ˆωu,v = ˆa + ˆb. We then apply a translational change of variables which allows +us to write +(4.39) +˜γlm(u, v; Φ) = +� +R3N−3 ∂l +x1ψ(u+v,ˆz+ ˆωu,v)∂m +x1ψ(u − v,ˆz + ˆωu,v)Φ(u+v, u−v,ˆz+ ˆωu,v) dˆz. +We will then apply differentiation under the integral. Beforehand, we show how such +derivatives will act on each function within the integrand. For a function f and any +r ∈ N3 +0 with |r| = 1 we see that by the chain rule +∂r +u[f(u ± v,ˆz + ˆωu,v)] = Dr +Qf(u ± v,ˆz + ˆωu,v) +∂r +v[f(u + v,ˆz + ˆωu,v)] = Dr +Pf(u + v,ˆz + ˆωu,v) − Dr +S∗f(u + v,ˆz + ˆωu,v) +∂r +v[f(u − v,ˆz + ˆωu,v)] = Dr +P ∗f(u − v,ˆz + ˆωu,v) − Dr +Sf(u − v,ˆz + ˆωu,v). +Applying repeatedly, we obtain for arbitrary σ, ν ∈ N3 +0, +∂σ +u∂ν +v[f(u + v,ˆz + ˆωu,v)] = +� +τ≤ν +cτ,ν +� +Dσ +QDτ +PDν−τ +S∗ f(u + v,ˆz + ˆωu,v) +� +∂σ +u∂ν +v [f(u − v,ˆz + ˆωu,v)] = +� +τ≤ν +cτ,ν +� +Dσ +QDτ +P ∗Dν−τ +S +f(u − v,ˆz + ˆωu,v) +� +. +where cτ,ν = (−1)|ν|−|τ|�ν +τ +� +. In a similar manner, for the cutoff we use the definitions +(3.20)-(3.21) and (4.22) to write +∂σ +u∂ν +v[Φ(u + v, u − v,ˆz + ˆωu,v)] = +� +τ≤ν +cτ,νΦ(σ,τ,ν−τ)(u + v, u − v,ˆz + ˆωu,v). +By differentiating (4.39) under the integral, applying the Leibniz rule and reversing the +change of variables, we find that ∂η +u∂µ +v ˜γlm(u, v; Φ) is a linear combination of terms of the +required form. + +ONE-PARTICLE DENSITY MATRIX +33 +Proof of ii). By part (i) it suffices to prove the required bound for integrals of the form +(4.36) with |χ| = |η| and |α| + |β| + |σ| = |µ|. Rewriting such integrals in the variables +x = u + v and y = u − v we get +(4.40) +� +R3N−3 +� +Dχ1 +Q Dα +{P,S∗}∂l +x1ψ(x, ˆx) +�� +Dχ2 +Q Dβ +{P ∗,S}∂m +x1ψ(y, ˆx) +� +Φ(χ3,σ)(x, y, ˆx) dˆx. +Notice that the variables x and y must obey δ ≤ |x − y| ≤ 2δ and |x|, |y| ≥ ǫ. Firstly, +we bound the above integral in the case where |α| + |β| ≤ 2 and |α| + |β| + |σ| ≤ 3 for +any |l|, |m| ≤ 1. By (4.23), (4.25), (4.26) followed by (4.13) of Lemma 4.2 we can bound +this integral in absolute value by some constant multiplied by +ǫ−|χ| +� +supp Φ(x,y,·) +� +ǫ−|σ| + δ−|σ|Mδ(x, y, ˆx) +� +λ(x, y, ˆx)−|α|−|β|f∞(x, ˆx; R)f∞(y, ˆx; R) dˆx. +Selective use of (4.14) of Lemma 4.2 allows us to bound this quantity by some constant +multiplied by +ǫ−|χ| +� +supp Φ(x,y,·) +� +ǫ−|σ|λ(x, y, ˆx)−|α|−|β| + δ−|α|−|β|−|σ|Mδ(x, y, ˆx) +� +f∞(x, ˆx; R)f∞(y, ˆx; R) dˆx. +We can use (4.15) of Lemma 4.2, using that |α| + |β| ≤ 2 by assumption, and (4.8) of +Proposition 4.1 to bound this quantity by some constant multiplied by +ǫ−|α|−|β|−|σ|−|χ| ∥ρ∥1/2 +L1(B(x,2R)) ∥ρ∥1/2 +L1(B(y,2R)) +where it was used that δ ≤ 1 ≤ ǫ−1 and |α| + |β| + |σ| ≤ 3. After a return to u, v- +variables, this proves the bound for integrals (4.36) in the case where |α| + |β| ≤ 2 +and |α| + |β| + |σ| ≤ 3. It remains to bound (4.40) in the case where |α| + |β| = 3, +|σ| = |χ| = 0 and |l| = |m| = 1. This follows directly from Lemma 4.4. +□ +Lemma 4.6. Take any η, µ, l, m ∈ N3 +0 as in Lemma 4.5(ii). Then for all R > 0 we have +C such that +(4.41) |∂η +u∂µ +v ˜γl,m(u, v)| ≤ C min{1, |u + v|, |u − v|}−|η|−|µ| ∥ρ∥1/2 +L1(B(u+v,R)) ∥ρ∥1/2 +L1(B(u−v,R)) +for all u, v ∈ R3 obeying 0 < |v| ≤ (4N)−1 min{1, |u + v|, |u − v|}. +Proof. Firstly, by Lemma 3.1 there exists a finite collection of biscaled cutoffs, Φ(j), +j = 1, . . . , J, such that +(4.42) +˜γl,m = +J +� +j=1 +˜γl,m +� +· ; Φ(j) +δ,ǫ +� +holds for all choices of 0 < δ ≤ (4N)−1ǫ. + +34 +PETER HEARNSHAW +Let C0 be the constant from Lemma 4.5 such that (4.40) holds for each Φ(j), j = +1, . . . , J. Fix any v0 ̸= 0 and u0 such that |v0| ≤ (4N)−1 min{1, |u0 + v0|, |u0 − v0|} and +set δ0 = |v0| and ǫ0 = min{1, |u0 + v0|, |u0 − v0|}. Then by (4.42), +|∂η +u∂µ +v ˜γl,m(u0, v0)| ≤ +J +� +j=1 +��∂η +u∂µ +v ˜γl,m +� +u0, v0; Φ(j) +δ0,ǫ0 +��� +≤ JC0 min{1, |u0 + v0|, |u0 − v0|}−|η|−|µ| ∥ρ∥1/2 +L1(B(u0+v0,R)) ∥ρ∥1/2 +L1(B(u0−v0,R)) . +Since C0 does not depend on the choice of δ0 and ǫ0, the constant JC0 does not depend +on the choice of u0 and v0. +□ +Proposition 4.7. For all α, β ∈ N3 +0 with |α| + |β| = 5 and all R > 0 there exists C, +depending on R, such that +(4.43) +|∂α +u∂β +v ˜γ(u, v)| ≤ C min{1, |u + v|, |u − v|}−4 ∥ρ∥1/2 +L1(B(u+v,R)) ∥ρ∥1/2 +L1(B(u−v,R)) +for all u, v ∈ R3 obeying 0 < |v| ≤ (4N)−1 min{1, |u + v|, |u − v|}. +Proof. First, we consider the case where |β| ≤ 3. We use (4.32) and (4.33) for when |β| ≤ +2 and |β| = 3 respectively. The bound then follows from Lemma 4.6 with |η| + |µ| = 4 +and |µ| ≤ 2. Now, consider 4 ≤ |β| ≤ 5. Take l, m ∈ N3 +0, |l| = |m| = 1 be such that +l + m ≤ β. Then by (4.34), +∂α +u∂β +v ˜γ(u, v) = ∂α+l+m +u +∂β−l−m +v +˜γ(u, v) − 2 +� +∂α +u∂β−l−m +v +˜γl,m(u, v) + ∂α +u∂β−l−m +v +˜γm,l(u, v) +� +. +The first term on the right-hand side has |β| − |l| − |m| ≤ 3 hence the required bound +follows from the previous step. The remaining terms can be bounded using Lemma 4.6 +□ +The proof of our main theorem is an immediate consequence of this proposition. +Proof of Theorem 1.1. The proof follows from Proposition 4.7 with u = (x + y)/2 and +v = (x − y)/2, along with the definition (4.4). +□ +4.6. Proof of Lemma 4.4. To prove Lemma 4.4, we will examine the cluster derivatives +of ψ present in (4.35) more closely. In particular, Theorem 1.3 allows us to write such +derivatives in terms of derivatives of Fc and a function, Gα +P, of higher regularity near +certain singularities. Sign cancellation allows uniform boundedness of the integral (4.35) +as x and y approach each other, and is more easily handled via derivatives of Fc rather +than those of ψ itself. Indeed, due to the simple formula definining Fc, it is possible +to characterise all its cluster derivatives explicitly. A series of steps, mostly involving +integration by parts, will complete the proof. +To begin, we use definition (2.2) to write +∇xFc(x, ˆx) = −Z +2 ∇x|x| + 1 +4 +� +2≤j≤N +∇x|x − xj| +(4.44) + +ONE-PARTICLE DENSITY MATRIX +35 +with the formula also holding when x is replaced by y. Let P and S be arbitrary clusters +with 1 ∈ P, S and P ∗ ∩ S∗ = ∅. Let α = (α1, α2) ∈ N6 +0, β = (β1, β2) ∈ N6 +0 and let +l, m ∈ N3 +0 obey |l| = |m| = 1. Then, +Dα +{P,S∗}∂l +x|x| = +� +∂α1+l +x +|x| +if α2 = 0 +0 +if |α2| ≥ 1, +(4.45) +Dβ +{P ∗,S}∂m +y |y| = +� +∂β2+m +y +|y| +if β1 = 0 +0 +if |β1| ≥ 1. +(4.46) +In the following, the cluster derivatives in (4.47) are understood to act with respect to +the ordered variables (x, x2, . . . , xN) and the cluster derivatives in (4.48) are understood +to act with respect to the ordered variables (y, x2, . . . , xN). For later convenience, on the +right-hand side of both formulae, all derivatives in the x- or y-variable are rewritten to +act on xj. Now assume |α|, |β| ≥ 1, then +Dα +{P,S∗}∂l +x|x − xj| = + + + + + + + + + +(−1)|α1|+1∂α1+α2+l +xj +|x − xj| +if |α2| ≥ 1 and j ∈ S∗ +0 +if |α2| ≥ 1 and j ∈ Sc +(−1)|α1|+1∂α1+α2+l +xj +|x − xj| +if α2 = 0 and j ∈ P c +0 +if α2 = 0 and j ∈ P ∗ +(4.47) +Dβ +{P ∗,S}∂m +y |y − xj| = + + + + + + + + + +(−1)|β2|+1∂β1+β2+m +xj +|y − xj| +if |β1| ≥ 1 and j ∈ P ∗ +0 +if |β1| ≥ 1 and j ∈ P c +(−1)|β2|+1∂β1+β2+m +xj +|y − xj| +if β1 = 0 and j ∈ Sc +0 +if β1 = 0 and j ∈ S∗. +(4.48) +In particular, since P ∗ ∩ S∗ = ∅, we have Dα +{P,S∗}∂l +x|x − xj| ≡ 0 unless j ∈ P c and +Dβ +{P ∗,S}∂m +y |y − xj| ≡ 0 unless j ∈ Sc. +We will often use the following elementary fact. For each z0 ∈ R3 and η ∈ N3 +0 there +exists C, dependent on η but independent of z0, such that +(4.49) +��∂η +z |z0 − z| +�� ≤ C|z0 − z|1−|η| +for all z ̸= z0. +Therefore, by (4.12) we have for each |η| ≥ 1 some C and C′ such that +��∂η +xj|x − xj| +�� + +��∂η +xk|y − xk| +�� ≤ C +� +|x − xj|1−|η| + |y − xk|1−|η|� +≤ C′λ(x, y, ˆx)1−|η| +(4.50) +for all 2 ≤ j, k ≤ N if |η| = 1, and all j ∈ P c, k ∈ Sc if |η| ≥ 2. Here, λ(x, y, ˆx) is defined +in (4.9) using the clusters P and S. +Lemma 4.8. Let P = P(Φ) and S = S(Φ) obey P ∗ ∩ S∗ = ∅, and take any R > 0. +Consider |α| + |β| = 3 and |l| = |m| = 1. For |α|, |β| ≥ 1, the integral +(4.51) +� +R3N−3 +� +Dα +{P,S∗}∂l +xFc(x, ˆx) +� +ψ(x, ˆx) +� +Dβ +{P ∗,S}∂m +y Fc(y, ˆx) +� +ψ(y, ˆx)Φ(x, y, ˆx) dˆx, + +36 +PETER HEARNSHAW +can be bounded as in (4.35). For |α| = 3, the integrals +� +R3N−3 +� +Dα +{P,S∗}∂l +xFc(x, ˆx) +� +ψ(x, ˆx)∂m +y F(y, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx, +(4.52) +� +R3N−3 +� +Dα +{P,S∗}∂l +xFc(x, ˆx) +� +ψ(x, ˆx)eF(y, ˆx)∂m +y φ(y, ˆx)Φ(x, y, ˆx) dˆx, +(4.53) +and, for |β| = 3, the integrals +� +R3N−3 ∂l +xF(x, ˆx)ψ(x, ˆx) +� +Dβ +{P ∗,S}∂m +y Fc(y, ˆx) +� +ψ(y, ˆx)Φ(x, y, ˆx) dˆx, +(4.54) +� +R3N−3 eF(x, ˆx)∂l +xφ(x, ˆx) +� +Dβ +{P ∗,S}∂m +y Fc(y, ˆx) +� +ψ(y, ˆx)Φ(x, y, ˆx) dˆx +(4.55) +can each be bounded as in (4.35). +Proof. We first prove the bound for (4.51). Use of (4.44) and (4.45)-(4.48) to expand +the integral will produce a linear combination of the following terms, where j ∈ P c and +k ∈ Sc, +∂α1+l +x +|x|∂β2+m +y +|y| +� +R3N−3 ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +if α2 = β1 = 0, +∂α1+l +x +|x| +� +R3N−3 ∂β1+β2+m +xk +|y − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +if α2 = 0, +∂β2+m +y +|y| +� +R3N−3 ∂α1+α2+l +xj +|x − xj|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +if β1 = 0, +� +R3N−3 ∂α1+α2+l +xj +|x − xj|∂β1+β2+m +xk +|y − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx. +Derivatives of |x| and |y| are bounded using (4.49) and |x|, |y| ≥ ǫ. Now we bound each +integral in absolute value. The first, second and third integrals are readily bounded by +(4.50) and Lemma 4.2. Finally, for the fourth integral we use (4.59) of Lemma 4.9. +Now suppose |α| = 3. +First, we prove the bound for the integral (4.52). +Recall +F = Fc − Fs. Using this, along with (4.44) and (4.45)-(4.48) we can expand the integral + +ONE-PARTICLE DENSITY MATRIX +37 +as linear combination of the following terms, where j ∈ P c and k ∈ {2, . . . , N}, +∂α1+l +x +|x|∂m +y |y| +� +R3N−3 ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +if α2 = 0, +∂α1+l +x +|x| +� +R3N−3 ∂m +xk|y − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +if α2 = 0, +∂α1+l +x +|x| +� +R3N−3 ∂m +xkFs(y, ˆx)ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +if α2 = 0, +∂m +y |y| +� +R3N−3 ∂α1+α2+l +xj +|x − xj|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx, +� +R3N−3 ∂α1+α2+l +xj +|x − xj|∂m +xk|y − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +� +R3N−3 ∂α1+α2+l +xj +|x − xj|∂m +xkFs(y, ˆx)ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +Derivatives of |x| and |y| are bounded using (4.49) and |x|, |y| ≥ ǫ. +The first three +integrals above are then readily bounded using (4.6) of Proposition 4.1 and that ∇Fs ∈ +L∞(R3N). For the fourth and sixth integral we use (4.57) of Lemma 4.9 with χ ≡ 1 +and χ(x, y, ˆx) = ∂m +xkFs(y, ˆx) respectively. Finally, for the fifth integral we use the same +lemma, specifically (4.59). This proves (4.52). The proof of (4.54) is similar in the case +where |β| = 3. +Next, using (4.44), (4.45) and (4.47) we can rewrite the integral (4.53) as a linear +combination of the following two terms, where k ∈ P c, +∂α1+l +x +|x| +� +R3N−3 ψ(x, ˆx)eF(y, ˆx)∂m +y φ(y, ˆx)Φ(x, y, ˆx) dˆx +if α2 = 0, +� +R3N−3 ∂α1+α2+l +xk +|x − xk|ψ(x, ˆx)eF(y, ˆx)∂m +y φ(y, ˆx)Φ(x, y, ˆx) dˆx. +Derivatives of |x| are bounded using (4.49) and |x| ≥ ǫ. The first integral is then bounded +by (2.6), (2.9), (2.10) and finally (4.6) of Proposition 4.1. The second integral is bounded +immediately from (4.69) of Lemma 4.10. This proves (4.53). The proof of (4.55) is similar +in the case where |β| = 3. +□ + +38 +PETER HEARNSHAW +Proof of Lemma 4.4. To prove the required inequality, first consider the case where +|α|, |β| ≥ 1. We can then use both (4.27) and (4.28) to write +� +R3N−3 +� +Dα +{P,S∗}∂l +xψ(x, ˆx) +�� +Dβ +{P ∗,S}∂m +y ψ(y, ˆx) +� +Φ(x, y, ˆx) dˆx += +� +R3N−3 +� +Gα,l +{P,S∗}(x, ˆx) +�� +Dβ +{P ∗,S}∂m +y ψ(y, ˆx) +� +Φ(x, y, ˆx) dˆx ++ +� +R3N−3 +� +Dα +{P,S∗}∂l +xFc(x, ˆx) +� +ψ(x, ˆx) +� +Gβ,m +{P ∗,S}(y, ˆx) +� +Φ(x, y, ˆx) dˆx ++ +� +R3N−3 +� +Dα +{P,S∗}∂l +xFc(x, ˆx) +� +ψ(x, ˆx) +� +Dβ +{P ∗,S}∂m +y Fc(y, ˆx) +� +ψ(y, ˆx)Φ(x, y, ˆx) dˆx +We bound each of these integrals. We start with the final integral on the right-hand side +which is just (4.51) of Lemma 4.8. Next, by using (4.25)-(4.26) and (4.29)-(4.31), the +first and second integrals on the right-hand side can be bounded in absolute value by +some constant multiplied by +(4.56) +� +R3N−3 λ(x, y, ˆx)−5/2f∞(x, ˆx; R)f∞(y, ˆx; R)Φ(x, y, ˆx) dˆx +≤ Cǫ−5/2 ∥ρ∥1/2 +L1(B(x,2R)) ∥ρ∥1/2 +L1(B(y,2R)) +where the inequality above holds for some C by (4.15) of Lemma 4.2. This proves the +required bound when |α|, |β| ≥ 1. +Now we consider the case where |α| = 3, and hence β = 0. We use that ∇ψ = +ψ∇F + eF∇φ and (4.27) to give +� +R3N−3 +� +Dα +{P,S∗}∂l +xψ(x, ˆx) +� +∂m +y ψ(y, ˆx)Φ(x, y, ˆx) dˆx += +� +R3N−3 +� +Gα,l +{P,S∗}(y, ˆx) +� +∂m +y ψ(y, ˆx)Φ(x, y, ˆx) dˆx ++ +� +R3N−3 +� +Dα +{P,S∗}∂l +xFc(x, ˆx) +� +ψ(x, ˆx)∂m +y F(y, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx ++ +� +R3N−3 +� +Dα +{P,S∗}∂l +xFc(x, ˆx) +� +ψ(x, ˆx)eF(y, ˆx)∂m +y φ(y, ˆx)Φ(x, y, ˆx) dˆx. +As before, the first integral on the right-hand side can be bounded in absolute value by +some constant multiplying (4.56). The second and third integrals on the right-hand side +are just (4.52) and (4.53) respectively. The proof of the |β| = 3 case is similar to the +|α| = 3 case. +□ +We now prove two lemmas which were used to prove Lemma 4.8. +Lemma 4.9. Let P = P(Φ) and S = S(Φ) obey P ∗ ∩ S∗ = ∅, and take any R > 0. + +ONE-PARTICLE DENSITY MATRIX +39 +(i) Let |η| = 4. Let χ = χ(x, y, ˆx) ∈ C∞(R3N+3) such that χ, ∇χ ∈ L∞(R3N+3) (for +example χ ≡ 1 may be chosen). Then, for all j ∈ P c, the integral +(4.57) +� +R3N−3 ∂η +xj|x − xj|χ(x, y, ˆx)ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx, +and, for all k ∈ Sc, the integral +(4.58) +� +R3N−3 ∂η +xk|y − xk|χ(x, y, ˆx)ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +can be bounded as in (4.35). +(ii) Let |α| + |β| = 3 and |l| = |m| = 1. Then for any pair 2 ≤ j, k ≤ N such that +j ∈ P c if |α| ≥ 1 and k ∈ Sc if |β| ≥ 1 we have that the integral +Ij,k = +� +R3N−3 ∂α+l +xj |x − xj|∂β+m +xk +|y − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx. +(4.59) +can be bounded as in (4.35). +Proof of i). Take any j ∈ P c. We bound (4.57). By a product rule for weak derivatives +we know that +∇xj +� +ψ(x, ˆx)ψ(y, ˆx) +� += +� +∇xjψ(x, ˆx) +� +ψ(y, ˆx) + ψ(x, ˆx) +� +∇xjψ(y, ˆx) +� +. +The functions χ and Φ are both smooth so are readily included in such a product rule. +Take any multiindex τ ≤ η with |τ| = 1. Using integration by parts we get that (4.57) +equals +− +� +R3N−3 ∂η−τ +xj |x − xj|∂τ +xj +� +χ(x, y, ˆx)ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) +� +dˆx +(4.60) +which, using (4.50), can be bounded in absolute value by some constant, depending on χ, +multipling the expression (4.16) which is bounded in Corollary 4.3. The proof of (4.58) +is similar. +Proof of ii). We first prove the case where j ̸= k, where integration by parts is particularly +simple. Suppose |α| ≥ 1. Using a strategy similar to the proof of i), we use integration +by parts to obtain +Ij,k = − +� +R3N−3 ∂α +xj|x − xj|∂β+m +xk +|y − xk|∂l +xj +� +ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) +� +dˆx +(4.61) +which can then be bounded using Corollary 4.3. The case of |β| ≥ 1 is similar except we +apply integration by parts on the ∂m +xk-derivative. This completes the proof where j ̸= k. +For the remainder of the proof we consider the case where k = j. Suppose, first, that +k ∈ (S ∪ P)c. To simplify calculations we write η = α + l and µ = β + m. Notice that + +40 +PETER HEARNSHAW +|η|, |µ| ≥ 1. Therefore we can find some multiindex µ1 ≤ µ with |µ1| = 1. Integration by +parts then gives +Ik,k = − +� +R3N−3 ∂η+µ1 +xk +|x − xk|∂µ−µ1 +xk +|y − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx +− +� +R3N−3 ∂η +xk|x − xk|∂µ−µ1 +xk +|y − xk|∂µ1 +xk +� +ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) +� +dˆx. +(4.62) +We leave untouched the second integral above. For the first, if |µ|−|µ1| ≥ 1 we can remove +another first-order derivative from |y − xk| by the same procedure - using integration by +parts to give two new terms as in (4.62). We retain the term where the derivative falls +on ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx). Whereas on the term where the derivative falls on |x − xk| +we repeat the procedure, so long as there remains a non-trivial derivative on |y − xk|. +Through this process, we obtain the following formula for Ik,k. Let T = |µ|. Then write +µ = �T +i=1 µi for some collection |µi| = 1, where 1 ≤ i ≤ T. Furthermore, define +µj = + + + + + +0 +if j = T +µT +if j = T − 1 +µj+1 + · · · + µT +if j ≤ T − 2. +Then, +(4.63) +Ik,k = (−1)T +� +R3N−3 ∂η+µ +xk |x − xk||y − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx + +T +� +j=1 +(−1)jI(j) +k,k +where +I(j) +k,k = +� +R3N−3 ∂η+µj +xk |y − xk|∂µj +xk +� +ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) +� +dˆx +Using (4.50), for each 1 ≤ j ≤ T − 1 we can bound |I(j) +k,k| by some constant multiplying +the expression (4.16), which is bounded in Corollary 4.3. +It remains to bound I(T) +k,k , +along with the first integral in the formula (4.63). Starting with the latter, we begin by +expanding |y − xk| = |x − xk| + +� +|y − xk| − |x − xk| +� +and noticing that +(4.64) +��|y − xk| − |x − xk| +�� ≤ |x − y| ≤ 2δ. +To bound the first integral in (4.63), it then suffices to bound the two integrals: +δ +� +R3N−3 +��∂η+µ +xk |x − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) +�� dˆx, +(4.65) +� +R3N−3 ∂η+µ +xk |x − xk||x − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx. +(4.66) + +ONE-PARTICLE DENSITY MATRIX +41 +We start with the first of these two integrals. Since k ∈ (P ∪ S)c we can use (4.50) to +show that (4.65) is bounded by some constant multiplied by +(4.67) +δ +� +R3N−3 λ(x, y, ˆx)−4f∞(x, ˆx; R)f∞(y, ˆx; R)Φ(x, y, ˆx) dˆx +≤ Cδ(ǫ−4 + δ−1) ∥ρ∥1/2 +L1(B(x,2R)) ∥ρ∥1/2 +L1(B(y,2R)) +where the bound holds by Lemma 4.2. We can then use the simplification δ(ǫ−4 +δ−1) ≤ +Cǫ−3 for some new constant C. Before looking at (4.66), we next bound I(T) +k,k . By (4.64) +we get +��I(T) +k,k +�� ≤ 2δ +� +R3N−3 +��∂η+µj = + + + + + +0 +if j = 5 +σ5 +if j = 4 +σj+1 + · · · + σ5 +if 1 ≤ j ≤ 3. +We now apply the same method used above, that is, we transfer successive first order +derivatives via integration by parts. To begin, we apply integration by parts to transfer +∂σ1 +xk from ∂σ +xk|x − xk|. We leave as a remainder the term where the derivative falls on +ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx). However, for the term where the derivative falls on |x − xk| we +continue the procedure to now remove ∂σ2 +xk from ∂σ−σ1 +xk +|x − xk| using integration by parts +again. Since |σ| = 5 is odd, the result after this procedure has occured five times is +that (4.66) is equal to minus the same integral plus remainder terms. This explains the + +42 +PETER HEARNSHAW +(1/2)-factor in the following formula, +� +R3N−3 ∂σ +xk|x − xk||x − xk|ψ(x, ˆx)ψ(y, ˆx)Φ(x, y, ˆx) dˆx += 1 +2 +5 +� +j=1 +(−1)j +� +R3N−3 ∂σ>j +xk |x − xk|∂σ 0. Let +η, l, m ∈ N3 +0 obey |η| = 4 and |l| = |m| = 1. Then, for each j ∈ P c, the integral +� +R3N−3 ∂η +xj|x − xj|ψ(x, ˆx)eF(y, ˆx)∂m +y φ(y, ˆx)Φ(x, y, ˆx) dˆx +(4.69) +and, for each k ∈ Sc, the integral +� +R3N−3 eF(x, ˆx)∂l +xφ(x, ˆx)∂η +xk|y − xk|ψ(y, ˆx)Φ(x, y, ˆx) dˆx +(4.70) +can be bounded as in (4.35). + +ONE-PARTICLE DENSITY MATRIX +43 +Proof. We prove the bound for (4.69). The case of (4.70) is similar. First, take some +function χ ∈ C∞ +c (R), 0 ≤ χ ≤ 1, with +χ(t) = +� +1 +if |t| ≤ 1 +0 +if |t| ≥ 2. +Furthermore, define χR(t) = χ(t/R) for all t ∈ R. +It suffices to bound the following two integrals +� +R3N−3 +� +1 − χR(|x − xj|) +� +∂η +xj|x − xj|ψ(x, ˆx)eF(y, ˆx)∂m +y φ(y, ˆx)Φ(x, y, ˆx) dˆx +(4.71) +� +R3N−3 χR(|x − xj|)∂η +xk|x − xj|ψ(x, ˆx)eF(y, ˆx)∂m +y φ(y, ˆx)Φ(x, y, ˆx) dˆx. +(4.72) +First, we see that when χR(|x − xj|) ̸= 1 we have |x − xj| > R. This, along with (2.6) +and (4.49) gives some constant C, depending on R, such that (4.71) can be bounded in +absolute value by +� +R3N−6 +� +{xj:|x−xj|>R} +��∂η +xj|x − xj|ψ(x, ˆx)eF(y, ˆx)∇φ(y, ˆx)Φ(x, y, ˆx) +�� dxjdˆx1j +≤ C +� +R3N−3 +��ψ(x, ˆx)∇φ(y, ˆx) +�� dˆx, +which itself can be bounded using (2.9)-(2.10) by some constant multiplying (4.6) with, +for example, the same R. The relevant bound then follows from Proposition 4.1, and +that ǫ < 1. +Recall the notation introduced in (1.11)-(1.14), namely we can write +(y, x, ˆx1j) = (y, x2, . . . , xj−1, x, xj+1, . . . , xN). +Using ∂m +y φ(y, ˆx) = ∂m +y φ(y, x, ˆx1j) + +� +∂m +y φ(y, ˆx) − ∂m +y φ(y, x, ˆx1j) +� +it follows that in order +to bound (4.72) it suffices to bound the following two integrals, +� +R3N−3 χR(|x − xj|)∂η +xj|x − xj|ψ(x, ˆx)eF(y, ˆx)∂m +y φ(y, x, ˆx1j)Φ(x, y, ˆx) dˆx, +(4.73) +� +R3N−3 χR(|x − xj|)∂η +xj|x − xj|ψ(x, ˆx)eF(y, ˆx) +� +∂m +y φ(y, ˆx) − ∂m +y φ(y, x, ˆx1j) +� +Φ(x, y, ˆx) dˆx. +(4.74) +Notice that when χR(|x − xj|) ̸= 0 we have |x − xj| < 2R. Take any θ ∈ (0, 1). Then +since φ ∈ C1,θ(R3N), we have local boundedness and local θ-H¨older continuity of ∇φ. +Therefore, by (2.9) and (2.10) there exists a constant C such that, when |x − xj| < 2R, +(4.75) +|∂m +y φ(y, x, ˆx1j)| ≤ ∥∇φ∥L∞(B((y,ˆx),2R)) ≤ Cf∞(y, ˆx; 4R) +(4.76) +��∂m +y φ(y, ˆx) − ∂m +y φ(y, x, ˆx1j) +�� ≤ |x − xj|θ[∇φ]θ,B((y,ˆx),2R) ≤ C|x − xj|θf∞(y, ˆx; 4R). + +44 +PETER HEARNSHAW +The constant C depends on R and θ but is independent of x, y and ˆx. +Integration by parts in the variable xj is used in (4.73) to remove a single derivative +from ∂η +xj|x−xj|. Since ∂m +y φ(y, x, ˆx1j) has no dependence on xj, this process avoids taking +a second derivative of φ. Take any τ ≤ η with |τ| = 1. Integral (4.73) can therefore be +rewritten as +− +� +R3N−3 ∂τ +xj +� +χR(|x − xj|) +� +∂η−τ +xj |x − xj| ψ(x, ˆx) eF(y, ˆx) ∂m +y φ(y, x, ˆx1j) Φ(x, y, ˆx) dˆx +− +� +R3N−3 χR(|x − xj|) ∂η−τ +xj |x − xj| ∂m +y φ(y, x, ˆx1j) ∂τ +xj +� +eF(y, ˆx)ψ(x, ˆx)Φ(x, y, ˆx) +� +dˆx +which, by (2.6), (4.50) and (4.75), can be bounded in absolute value by +C +� +R3N−3 λ(x, y, ˆx)−2f∞(x, ˆx; 4R)f∞(y, ˆx; 4R) +� +Φ(x, y, ˆx) + |∇Φ(x, y, ˆx)| +� +dˆx. +for some C depending on R and our choice of χ. The relevant bound then follows by +Corollary 4.3. +Using (2.6), (4.49), (4.50) and (4.76), the integral (4.74) can then be bounded in +absolute value by some constant multiplied by +� +R3N−3 λ(x, y, ˆx)−3+θf∞(x, ˆx; 4R)f∞(y, ˆx; 4R)Φ(x, y, ˆx) dˆx. +The relevant bound then follows by Lemma 4.2 with b = −3 + θ. +□ +Appendix A. Second derivatives of φ +Fix some function χ ∈ C∞ +c (R), 0 ≤ χ ≤ 1, with +χ(t) = +� +1 +if |t| ≤ 1 +0 +if |t| ≥ 2. +We also set +(A.1) +g(x, y) = (x · y) ln +� +|x|2 + |y|2� +for x, y ∈ R3. For each x ∈ R3N we can then define the function +(A.2) +G(x) = K0 +� +1≤j 0 there exists C, depending on r, R +and b but independent of ψ, such that for any non-empty cluster P and any η ∈ N3 +0 with +|η| = 1, +∥Dη +P∇φ∥L∞(B(x,rλP (x))) ≤ CλP(x)−b ∥φ∥L∞(B(x,R)) +for all x ∈ Σc +P. +Proof. To begin, we obtain bounds for derivatives of g(x, y) and G(x). It can be seen +that g ∈ C1,θ(R6) for all θ ∈ [0, 1). For the second derivatives, let α, β ∈ N3 +0 obey +|α| + |β| = 2, then there exists C such that +|∂α +x ∂β +y g(x, y)| ≤ +� +C + +�� ln +� +|x|2 + |y|2��� +if |α| = |β| = 1 +C +otherwise +(A.5) +for all x, y ∈ R3. It follows that +(A.6) +G, ∇G ∈ L∞(R3N), +and given b > 0 there exist constants C and C′, only the latter depending on b, such +that for any η ∈ N3 +0 with |η| = 1, and k = 1, . . . , N, we have +|∂η +xk∇G(y)| ≤ C +� +1 + +N +� +l=1 +l̸=k +χ(|yk|)χ(|yl|) +�� ln +� +|yk|2 + |yl|2��� +� +≤ C′� +1 + |yk|−b� +for all y = (y1, . . . , yN) ∈ R3N with yk ̸= 0. Using the above inequality for every k ∈ P +and the definition of cluster derivatives, (1.15), we can obtain some C, depending on b, + +46 +PETER HEARNSHAW +such that +|Dη +P∇G(y)| ≤ CλP(y)−b +(A.7) +for all y ∈ Σc +P. Here, we also used that λP ≤ 1, the definition of λP in (1.21), and the +formula (1.20). Now, take some x ∈ Σc +P. As in Lemma 2.4, we use (1.22) to show that +(1 − r)λP(x) ≤ λP(y) for each y ∈ B(x, rλP(x)). Therefore, for C as in (A.7), we have +∥Dη +P∇G∥L∞(B(x,rλP (x))) ≤ C(1 − r)−bλP(x)−b +(A.8) +for all x ∈ Σc +P. +We now in a position to consider derivatives of φ = eGφ′. Firstly, we have ∇φ = +eGφ′∇G + eG∇φ′. And therefore the following formula holds for each η ∈ N3 +0, |η| = 1, +Dη +P∇φ = +� +Dη +P∇G + Dη +PG ∇G +� +φ + eGDη +Pφ′ ∇G + eGDη +PG ∇φ′ + eGDη +P∇φ′. +Taking the norm and using (A.6) and (A.8) we can then obtain C such that +∥Dη +P∇φ∥L∞(B(x,rλP (x))) ≤ C +� +λP(x)−b ∥φ∥L∞(B(x,rλP (x))) + ∥φ′∥W 2,∞(B(x,rλP (x))) +� +. +(A.9) +for all x ∈ Σc +P. To the second term in the above bound we may then use Theorem +A.1 with constant C(r, R), followed by use of (A.6) to obtain another constant C′, also +dependent on r, R but independent of x, such that +∥φ′∥W 2,∞(B(x,rλP (x))) ≤ C(r, R) ∥φ′∥L∞(B(x,R)) ≤ C′ ∥φ∥L∞(B(x,R)) . +(A.10) +Together, (A.9) and (A.10) complete the proof. +□ +Acknowledgments. The author would like to thank A. V. Sobolev for helpful dis- +cussions in all matters of the current work. +References +[1] M. Reed and B. Simon. II: Fourier Analysis, Self-Adjointness. Elsevier, 1975. +[2] P. Hearnshaw and A.V. Sobolev. Analyticity of the one-particle density matrix. Ann. Henri. +Poincare, 23:707–738, 2022. +[3] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and T.Ø. Sørensen. Analyticity of the +density of electronic wavefunctions. Ark. Mat., 42:87–106, 2004. +[4] T. Jecko. A New Proof of the Analyticity of the Electronic Density of Molecules. Lett Math Phys, +(93):73–83, 2010. +[5] P. Hearnshaw and A.V. Sobolev. The diagonal behaviour of the one-particle coulombic density +matrix. 2022. arXiv, 2207.03963. +[6] J. Cioslowski. Off-diagonal derivative discontinuities in the reduced density matrices of electronic +systems. J. Chem. Phys, 153(154108), 2020. +[7] J. Cioslowski. Reverse engineering in quantum chemistry: How to reveal the fifth-order off-diagonal +cusp in the one-electron reduced density matrix without actually calculating it. Int J Quantum +Chem, 122(8)(e26651), 2022. + +ONE-PARTICLE DENSITY MATRIX +47 +[8] S. Fournais and T.Ø. Sørensen. Estimates on derivatives of coulombic wave functions and their elec- +tron densities. Journal f¨ur die reine und angewandte Mathematik (Crelles Journal), 2021(775):1–38, +2021. +[9] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and T.Ø. Sørensen. Sharp regularity +results for coulombic many-electron wave functions. Comm. Math. Phys., 255:183–227, 2005. +[10] S. Fournais, M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and T.Ø. Sørensen. The electron den- +sity is smooth away from the nuclei. Comm. Math. Phys., 228:401–415, 2002. +[11] M. Hoffmann-Ostenhof, T. Hoffmann-Ostenhof, and T.Ø. Sørensen. Electron wavefunctions and +densities for atoms. Ann. Henri. Poincare, 2:77–100, 2001. +[12] T. Kato. On the eigenfunctions of many-particle systems in quantum mechanics. Comm. Pure Appl. +Math., 10:151–177, 1957. +[13] L.C. Evans and R.F. Gariepy. Measure Theory and Fine Properties of Functions. CRC Press, 2015. +Department of Mathematics, University College London, Gower Street, London, +WC1E 6BT UK +Email address: peter.hearnshaw.18@ucl.ac.uk + diff --git a/59E1T4oBgHgl3EQfBQIH/content/tmp_files/load_file.txt b/59E1T4oBgHgl3EQfBQIH/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..3273cabac1c330f5241f75f4a61274c8d8670d32 --- /dev/null +++ b/59E1T4oBgHgl3EQfBQIH/content/tmp_files/load_file.txt @@ -0,0 +1,2107 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf,len=2106 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content='02848v1 [math-ph] 7 Jan 2023 BOUNDEDNESS OF THE FIFTH OFF-DIAGONAL DERIVATIVE FOR THE ONE-PARTICLE COULOMBIC DENSITY MATRIX PETER HEARNSHAW Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content=' Boundedness is demonstrated for the fifth derivative of the one-particle reduced density matrix for non-relativistic Coulombic wavefunctions in the vicinity of the diagonal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content=' Introduction and results We consider the non-relativistic quantum systems of N ≥ 2 electrons among N0 nuclei which represents the system of an atom or molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content=' For simplicity we restrict ourselves to the case of an atom (N0 = 1), although all results readily generalise to the molecular case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content=' The electrons have coordinates x = (x1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content=' , xN), xk ∈ R3, k = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content=' , N, and the nucleus has charge Z > 0 and its position fixed at the origin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content=' The corresponding Schr¨odinger operator is (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content='1) H = −∆ + V where ∆ = �N k=1 ∆xk is the Laplacian in R3N, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content=' ∆xk refers to the Laplacian applied to the variable xk, and V is the Coulomb potential given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/59E1T4oBgHgl3EQfBQIH/content/2301.02848v1.pdf'} +page_content='2) V (x) = − N � k=1 Z |xk| + � 1≤j |kt| belong to the fast- +wave regime in the dispersion diagram and can propagate in +free space. Other Floquet harmonics are surface waves. The +direction of the reflection can be calculated by the following +formula: +sin θr = kt/k0 = (k0 sin θi + 2πn/D)/k0. +(4) +For the chosen design parameters (θi = 0◦, θr = 70◦, f = +8 GHz), the period is equal to D = 1.0642λ, and the Floquet +expansion has three propagating harmonics (k0 > |kt|): zero +Floquet mode (0◦), −1 Floquet mode (−70◦), and +1 Floquet +mode (+70◦), as follows from Eq. (4). The field amplitudes +in these modes define the efficiency of power channeling from +one mode to another. +1) Performance of the surface-impedance models +Initially, the ideal impedance profile is considered when +the period is discretized to six elements. In other words, it +is assumed that the impedance boundary condition is applied +straight on the surface without considering actual realization +(Fig. 2). It is noteworthy to notice that the discretization of +the impedance profile deteriorates performance, however, it is +spoiled in the same way for all methods. Using such discretiza- +tion, reasonable results can be achieved rather fast. All the +methods except the phase gradient method use optimization, +therefore analytical closed-form formulas for the impedance +profiles do not exist. The list of optimized impedance values +of each unit cell in a period is presented in Table I for +all designed methods. For both designs based on the input +impedance model (phase-gradient and input impedance opti- +mization), we convert the obtained input impedance profile +to the grid impedance by using the equivalent transmission- +line model presented in Fig. 2. The input impedance can be +considered as that of a shunt connection of the grid impedance +to the transmission line modeling the grounded dielectric +substrate. For the non-local method, a pre-final optimization +is applied here similarly to what was done in [10]. As it +was discussed in Sec. II, for the non-local approach we can +assume an impedance profile using repeated receiving and re- +radiating leaky-wave sections to mimic the ideal active-lossy +profile. Therefore, the optimization at the level of the grid +impedance is an initial step before the final optimization for +the whole supercell in the real structure. Eventually, the same +configuration for all the methods enables us to complete a fair +comparison. +TABLE I: The impedance profile list for each unit cell (jΩ) +Cell1 +Cell2 +Cell3 +Cell4 +Cell5 +Cell6 +Phase +gradient +-97.6 +-82.1 +-51.4 ++2673.9 +-145.4 +-113.4 +Input +impedance +-291.7 +-141.3 +-114.0 +-98.0 +-83.8 +-85.3 +Grid +impedance +-73.8 +-1334.0 +-112.9 +-172.6 +-91.4 +-96.7 +Non- +local +-81.82 +-74.74 +-49.16 +-73.50 +-75.07 +-73.06 +Performance comparison of the discretized impedance pro- +files after optimization is made using full-wave simulators, +CST STUDIO [31] and ANSYS HFSS [32]. As it was dis- +cussed, there are three propagating Floquet harmonics (open +channels) in our specific example. Therefore, we can consider + +4 +(a) +(b) +(c) +(d) +(e) +(f) +(g) +(h) +No diffracted modes +No diffracted modes +No diffracted modes +No diffracted modes +Fig. 3: Power distribution between three propagating modes and scattered field distribution (bottom); (a,e) for the phase gradient, +(b,f) input impedance, (c,g) grid impedance, (d,h) non-local design method. The horizontal black lines in the scattered field +distribution figures illustrate the location of the metasurfaces where the IBC is applied. +these reflectors as three-port networks. Scattering parameters +(Sn1) can be determined numerically when the input wave +comes from Floquet port 1 and the output wave is observed +in the port number n. Consequently, the power efficiency is +found as squared scattering parameters (ηn = |Sn1|2) in the +full-wave simulators. The power efficiency for each mode +measures the fraction of power rerouted from the incident +wave (assuming that the incident port is 1) to the propagating +mode n. +Figures 3 (a-d) show ratios of power rerouted to propagating +channels n = 0, 1, and −1. In all cases, below 7.5 GHz +diffraction modes are not allowed, therefore all the energy +is reflected back to the normal direction. Designs based +on the phase-gradient, input impedance, and grid impedance +methods show broadband behavior as compared to the non- +local approach. The phase-gradient method does not take +evanescent modes into account, which results in the lowest +efficiency at the operational frequency. Power distribution and +the corresponding field amplitudes for all methods can be +found in Table II. +TABLE II: Amplitude/power ratio of propagating Floquet +modes and power efficiency level at 8 GHz +θi +θr +−θr +Phase gradient +0.33/0.11 +1.45/0.72 +0.71/0.17 +Input impedance +0.10/0.01 +1.67/0.95 +0.34/0.04 +Grid impedance +0.00/0.00 +1.71/1 +0.01/0.00 +Non-local +0.1/0.01 +1.70/0.99 +0.12/0.00 +It is noteworthy to sketch the scattered electric field distri- +butions (Fig. 3(e-h)). Efficiency for the phase-gradient method +is only 71.8%, and, correspondingly, Fig. 3(e) shows a field +distribution that is distorted by fields scattered into two par- +asitic propagating channels. For the other methods, efficiency +is nearly perfect, however, the near-field distributions are +different due to different methods used to optimize evanescent +modes. It is important to note that for perfect anomalous +reflection with ideal power efficiency, the power reflected to +the desired direction must be equal to the power of the incident +plane wave, and, as a result, the ratio between the amplitudes +of the reflected and incident fields for these angles should be +larger than one |Er| = |Ei| +� +cos(θi)/ cos(θr) (1.71 for our +example case) [10]. +2) Realizations as patch arrays +The next step is to compare actual structures designed using +the previously obtained and discussed impedance profiles. Fol- +lowing the procedures described in the corresponding papers, +we design supercells formed by six unit cells based on the +rectangular shape metal patches above the grounded dielectric +substrate (see Fig. 4 and Table III). +The corresponding efficiencies for all the considered meth- +ods are shown in Fig. 5. The frequency for the best per- +formance becomes shifted for all methods, except for the +non-local design, where optimization of the whole supercell +is implemented. In addition to that, dispersion and losses +deteriorate the efficiency in different ways. The absorption +levels as well as efficiency at the design frequency (8 GHz) +are reported in Table IV. The remained power is scattered to +other propagating Floquet modes that are not shown in Fig. 5. + +Re (E/E) +1.5 +0.8 +1 +0.5 +0.6 +0 +2 +0.4 +-0.5 +0.2 +-1 +-1.5 +0 +-0.5 +0 +0.5 +c/ DxRe (E/E) +1.5 +0.8 +1 +0.5 +0.6 +0 +2 +0.4 +-0.5 +0.2 +-1 +1.5 +0 +-0.5 +0 +0.5 +α/DxRe (E/E.) +1 +1.5 +0.8 +1 +0.5 +0.6 +0 +2 +0.4 +-0.5 +0.2 +-1 +1.5 +0 +-0.5 +0 +0.5 +α/DxRe (E/E) +1.5 +0.8 +1 +0.5 +0.6 +0 +2 +0.4 +-0.5 +0.2 +-1 +1.5 +0 +-0.5 +0 +0.5 +α/Dxn=0n=-1n=+1 +0.8 +Eficiency, In +0.6 +0.4 +0.2 +0 +7 +7.5 +8 +8.5 +9 +Frequency (GHz)n=0n=-1n=十1 +0.8 +0.6 +0.4 +0.2 +0 +7 +7.5 +8 +8.5 +9 +Frequency (GHz)n=0n=-1n=十1 +0.8 +Efficiency, n +0.6 +0.4 +0.2 +0 +7 +7.5 +8 +8.5 +9 +Frequency (GHz)n=0n=-1n=+1 +0.8 +Eficiency, Nn +0.6 +0.4 +0.2 +0 +7 +7.5 +8 +8.5 +9 +Frequency (GHz)5 +Cell #1 Cell #2 Cell #3 Cell #4 Cell #5 Cell #6 +𝐷 +𝑑 = +ൗ +𝐷 6 +Fig. 4: The configuration of supercells utilized for the designs +consisting of six unit cells. All the parameters of the dielectric +substrate are given in Sec. II. The period of the array (the +supercell size) is fixed to D = 39.9 mm, and the width of +a single unit cell is d = D/6. The width of metal strips +is 3.5 mm, while the strip lengths are different for different +design methods. +TABLE III: Lengths of metal strips for each unit cell (mm) +Strip1 +Strip2 +Strip3 +Strip4 +Strip5 +Strip6 +Phase +gradient +10.8 +11.41 +13.23 +0 +9.47 +10.29 +Input +impedance +7.3 +9.57 +10.28 +10.78 +11.3 +11.25 +Grid +impedance +3.71 +10.32 +8.89 +11.03 +10.84 +11.78 +Non- +local +10.47 +10.91 +11.26 +12.22 +11.30 +8.88 +TABLE IV: The best-performance frequency and the corre- +sponding efficiency versus the absorption rate and efficiency +for the design frequency. +Best performance +8 GHz +Frequency +Efficiency +Absorption +Efficiency +Phase +gradient +8.5 GHz +78.2(%) +3.9(%) +62.7(%) +Input +impedance +8.27 GHz +86.3(%) +3.2(%) +56.6(%) +Grid +impedance +8.2 GHz +95.0(%) +3.3(%) +73.6(%) +Non-local +8 GHz +96.6(%) +3.5(%) +96.6(%) +IV. ANGULAR RESPONSE +A very interesting property of anomalous reflectors which is +often left unstudied is the angular response, i.e., performance +of the structure for various incident angles θi, which can +be different from the design angle of incidence. Here, we +consider angular response for periodical arrays formed by +repeated supercells consisting of 6 unit cells with patches +printed on a grounded substrate and assume the periodic +boundary condition for this analysis. To distinguish between +the illumination angle and the incidence angle for which the +surface was designed, we denote this design incidence angle by +No diffracted modes +Fig. 5: Frequency dependence of efficiency for structures +realized with metallic rectangular patches. +θid. Worth to note that θid together with the required reflection +angle defines the period of the structure operating as an +anomalous reflector for these angles. The angular response is +studied by sweeping the incident angle θi for a fixed structure, +designed for the angle θid. The number of propagating Floquet +modes existing in the system is defined by the incident angle +θi, the period of the structure D, and the frequency f (see +Eq. (3)). The condition for the mode propagation can be +written as follows: +ktn < k0 → 2π +D |n| < 2π +λ → |n| < D +λ , +(5) +and their propagation directions can be calculated as [33], [34]: +θtn = arctan(ktn/knn), +(6) +where knn is the normal component of the wavenumber for +the nth mode, and knn = +� +k2 +0 − k2 +tn. If k0 > |ktn|, the +normal component of the nth wavenumber is purely real, +which corresponds to a propagating mode. Otherwise, the +wavenumber is imaginary, which corresponds to a surface +mode that propagates along the interface. Figure 6 shows that +for this fixed operational frequency and period of the structure, +only five propagating Floquet modes with n ∈ [−2, 2] are +allowed in the system when the angle of incidence is changing. +All other modes (|n| > 3) are surface modes. +Figure 7 depicts the spatial power distribution for prop- +agating modes versus the illumination angle at 8 GHz. At +the angle θi = 0◦ the incident angle is equal to the design +angle θid, therefore most of the power goes to mode +1, with +different efficiency for each method (see Table IV). Based on +the discussion in Refs. [33], [34], for the phase-gradient case, +there is a retro-reflection angle (at which all the energy is +reflected back at the angle of incidence), that can be calculated +as θretro = arcsin[(sin θi −sin θr)/2] and is equal to −28◦ for +the considered case. At this angle only two channels are open +(see Fig. 6), and the angle for the other channel is −θretro. +Ideally, 100% of the power should be scattered in the retro- +reflection direction, however, discretization and the presence +of losses decrease it down to 96%. Therefore, the rest of the +power goes to the remaining channel or gets absorbed. Due + +Phase grad Input imp Grid impNon-local +Efficiency, Mn +0.5 +0 +7 +7.5 +8 +8.5 +9 +Frequency +GHz6 +Fig. 6: Propagation angle for different Floquet modes with +respect to the incident angle. This figure is made using Eq. 6 +when the incidence angle is swept. +to reciprocity, the structure behaves in the same way when +illuminated from direction −θretro. It is important to notice +that for other design methods, retro-reflection occurs at the +angle +70◦, when three propagating channels are open. When +the structure is illuminated from the normal direction, most of +the energy couples to mode n = +1, where the reflection angle +is θr = +70◦. Therefore, channel n = −1 becomes decoupled +from the other two, and when the structure is illuminated from +the angle θi = −70◦, all the energy is reflected back to the +source. +Finally, a sweep of the incident angle reveals that the design +method based on grid impedance is the solution that has the +least sensitive response (see Fig. 7(c)). It means that when the +incident angle changes between −70◦ and +70◦, the power +couples primarily to the same modes, unlike for other methods. +Figure 8 illustrates and reports the results of the study for +the best performance frequency for each method. The result +is the same for the non-local optimization approach since in +this case, the best performance frequency matches the design +frequency. +V. FAR-FIELD SCATTERING FROM FINITE-SIZE +STRUCTURES +In the previous analysis, we considered infinite periodical +structures excited by plane waves. Here, we study far-field +scattering properties of finite-size structures. It is possible to +study metasurfaces on the grid or sheet impedance levels using +the mode-matching method for calculation of induced currents +[7], [35] and the far-field approximation for the calculation of +scattered fields [33], [36]. To do that, the following conditions +have to be met: +|r| ≫ λ, +(7a) +|r| ≫ L, +(7b) +L2/|r| ≪ λ, +(7c) +where |r| is the distance from the observation point to the +center of the structure, and L is max(2a, 2b), in which a +and b denote distances between the center of the metasur- +face and the edges of the structure along the x and y- +axes, respectively. Considering TE polarized incident waves +(Ei = E0e−jk(sin θix+cos θiz)ˆy) and selecting the observation +point in the plane of incidence (xy-plane), the normalized +scattering pattern in spherical coordinates can be determined +by the following expressions: +Fr(θ) = +1 +2 cos θi +� +n +rn(θi)(cos(θrn) + cos(θ))sinc(kaefn), +(8) +Fsh(θ) = +1 +2 cos θi +(cos(θ) − cos(θi))sinc(kaef), +(9) +where rn(θi) are the amplitudes of excited harmonics, de- +termined by the mode matching technique. Angle θrn shows +the reflected angle for each harmonic, and sinc(x) is a sinc +function. In addition, aefn and aef can be represented by +aefn = (sin θ − sin θrn)a and aefn = (sin θ − sin θi)a, re- +spectively. Finally, the total scattering pattern can be found as +the sum Fsc = Fr + Fsh. Worth to mention that normalization +is performed with respect to the maximum of the reflected +field. +Alternatively to the analytical approach, one can study +finite-size structures numerically using full-wave simulators. +The result shown in Fig. 9(a) corresponds to the analytical +solution, and in Fig. 9(b) to the full-wave simulations. In +both cases, the radiation pattern is calculated at 8 GHz +for structures with the size 11.7λ × 7λ in the xy-plane. +The discrepancy between the two radiation patterns, which +becomes more significant for side lobes (SL), is caused by +the neglected current distortions near the edges. Nevertheless, +the general behavior is similar. Parameters of the patterns +related to each method are shown in Table V. +The beam- +TABLE V: The normalized main lobe amplitude, the first side +lobe amplitude in linear scale, and their directions. +Main lobe +Amp/angle +corresponds to +n=+1. +SL Amp/angle +corresponds to +n=0 +SL Amp/angle +corresponds to +n=-1 +Phase gradient +0.94/68◦ +0.52/-1◦ +0.49/-67◦ +Input +impedance +0.95/68◦ +0.45/0◦ +0.35/-67◦ +Grid +impedance +0.99/69◦ +0.13/-1◦ +0.33/-67◦ +Non-local +1.00/69◦ +0.24/-2◦ +0.39/-68◦ +widths for all cases are similar and close to 9◦ (see Fig. 9). +Moreover, as it is shown in the inset, the maximum of the +scattered field is higher for the design methods based on grid +impedance and non-local solution because the power efficiency +is higher in these methods compared to the phase gradient +design and optimization based on the input impedance. The +most important difference corresponds to the side-lobe level +(SLL). As it is expected, the highest side lobes occur along +the θ = −70 and θ = 0, because there are two propagating +Floquet harmonics along these directions. +Figure 10 shows the scattering patterns of all the methods +at different frequencies, where all the patterns are normalized + +—n=0n=-1n=+1n=-2n=+2 +50 +0 +-50 +-50 +0 +50 +Angle of incidence, :7 +(a) +(b) +(c) +(d) +Fig. 7: Power distribution among different propagating modes depending on the incident angle at frequency 8 GHz for (a) +phase gradient, (b) input impedance optimization, (c) grid impedance optimization, and (d) non-local optimization method. +(a) +(b) +(c) +(d) +Fig. 8: Power distribution among different propagating modes depending on the incident angle for the best-performance +frequencies which are reported in Table IV, for (a) phase gradient, (b) input impedance optimization, (c) grid impedance +optimization, and (d) non-local optimization method. + +n=0 +n=-1 +n=+1n=-2 +2n=+2 +Eficiency, Mn +0.5 +-50 +0 +50 +Angle of incidence, :n=0 +-n=-1 +n=+1n=-2n=+2 +Efficiency, Mn +0.5 +-50 +0 +50 +Angle of incidence, :n=0 +-n=-1 +-n=+1-n=-2 +一n=+2 +Efficiency, Mn +0.5 +-50 +0 +50 +Angle of incidence, :n=0 +-n=-1 +n=+1n=-2 +n=+2 +Eficiency, Mn +0.5 +-50 +0 +50 +Angle of incidence, :n=0 +-n=-1 +-n=+1 +n=-2 +n=+2 +Efficiency, Mn +0.5 +-50 +0 +50 +Angle of incidence, :n=0n=-1n=+1n=-2n=+2 +Eficiency, Mn +0.5 +0 +-50 +0 +50 +Angle of incidence, O:n=0 +-n=-1 +n=+1n=-2 +n=+2 +Eficiency, Nn +0.5 +-50 +0 +50 +Angle of incidence, :n=0 +-n=-1 +-n=+1- +-n=-2 +一n=+2 +Efficiency, Mn +0.5 +0 +-50 +0 +50 +Angle of incidence, :8 +Fig. 9: Normalized radiation pattern in linear scale. All pat- +terns are normalized with respect to the main lobe amplitude +of the non-local method which has the highest gain compared +to the other approaches. (a) analytical pattern based on the +Huygens principle, (b) full-wave simulation in CST STUDIO. +to the main lobe. It is important to notice that due to the +fixed period of the structure (D = λ/(sin θr − sin θi)), by +sweeping the frequency, the angle of reflection changes. This +can be observed in Fig. 10. By changing the frequency, the +scattered n = +1 Floquet harmonic scans the space from the +desired reflection angle (+70◦) at 8 GHz to smaller angles +at higher frequencies and larger angles at lower frequencies. +Below 7.5 GHz there are no diffraction modes, therefore we +plot the scattering patterns starting from 7.75 GHz, where most +of the energy is reflected into the normal direction (see the +blue line in Fig. 10). The red line in the figure corresponds to +the scattering pattern at the design frequency. Eventually, the +radiation patterns for 8.25 and 8.5 GHz are shown by yellow +and purple lines, respectively. +VI. CONCLUSION +We have presented a comprehensive analysis of four main +design methods for anomalous reflectors. In order to provide a +meaningful comparison we chose design methods that can be +realized within the same topology. At first, we performed an +analysis of periodical infinite structures on the level of input +and grid impedances. Then we proceeded to design actual +implementations as supercells formed by six metal patches +placed on top of a grounded dielectric substrate. Further, we +analyzed the angular response of the designed metasurfaces +(a) +(b) +(c) +(d) +Fig. 10: Frequency bandwidth patterns. At 7.5 GHz, which is +not plotted here, there is no diffracted mode, and all the energy +goes back to the specular (normal) direction. The patterns are +plotted between 7.75 GHz to 8.5 GHz with a 0.25 GHz step. +Designed based on (a) phase gradient, (b) input impedance +optimization, (c) grid impedance optimization, (d) non-local +optimization. All patterns are normalized to the main lobe +amplitude for each case. +and finally presented far-field radiation patterns of finite-size +structures. +In this work, we provide a comparative summary of the +main features of previously introduced design methods as well +as present an original study of a property that is frequently left +unstudied: the angular response. This study can be considered +referential for engineers working on reconfigurable intelligent +surfaces, where similar design methods are utilized. +ACKNOWLEDGMENT +This work was supported by the European Union’s Hori- +zon 2020 research and innovation programme under the +Marie Skłodowska-Curie grant agreement No 956256 (project +METAWIRELESS), and the Academy of Finland (grant +345178). +REFERENCES +[1] V. S. 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Tretyakov, “On the integration +of reconfigurable intelligent surfaces in real-world environments: A +convenient approach for estimation reflection and transmission,” IEEE +Antennas and Propagation Magazine, vol. 64, no. 4, pp. 85–95, 2022. + diff --git a/6tE1T4oBgHgl3EQfBgIF/content/tmp_files/load_file.txt b/6tE1T4oBgHgl3EQfBgIF/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..dc32ed61815b7010bdd741c6d4af09cd0c3bdfe7 --- /dev/null +++ b/6tE1T4oBgHgl3EQfBgIF/content/tmp_files/load_file.txt @@ -0,0 +1,798 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf,len=797 +page_content='1 Comparison Between Different Designs and Realizations of Anomalous Reflectors Mostafa Movahediqomi1, Grigorii Ptitcyn1, and Sergei Tretyakov1, Fellow, IEEE 1Department of Electronics and Nanoengineering, School of Electrical Engineering, Aalto University, 02150 Espoo, Finland Metasurfaces enable efficient manipulation of electromagnetic radiation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' In particular, control over plane-wave reflection is one of the most useful features in many applications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Extensive research has been done in the field of anomalous reflectors over the past years, resulting in numerous introduced geometries and several distinct design approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Anomalously reflecting metasurfaces designed using different methods show different performances in terms of reflection efficiency, angular response, frequency bandwidth, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Without a comprehensive comparison between known design approaches, it is difficult to properly select the most appropriate design method and the most suitable metasurface geometry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Here, we consider four main approaches that can be used to design anomalous reflectors within the same basic topology of the structure and study the designed metasurfaces first on the level of the input impedance and then consider and compare the performance of the realized structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' We cover a wide range of performance aspects, such as the power efficiency and losses, angular response, and the scattering pattern of finite-size structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' We anticipate that this study will prove useful for developing new engineering methods and designing more sophisticated structures that include reconfigurable elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Furthermore, we believe that this study can be considered referential since it provides comparative physical insight into anomalous reflectors in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Index Terms—Anomalous reflectors, diffraction grating, phase gradient, surface wave, angular response, scattering parameters, far-field pattern.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' INTRODUCTION Wireless communication technologies constantly progress towards higher operational frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' This progress comes with smaller antenna sizes and, alas, at the expense of the need to use highly-directive and scanning antennas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Improve- ment of transmitters and receivers is limited, therefore com- munication engineers proposed to optimize the propagation environment using metasurfaces and metagratings [1]–[14], and reconfigurable intelligent metasurfaces (RIS) [15]–[24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The latter approach has gained increasing attention recently in communication communities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Often, reconfigurable structures are designed based on conventional fixed structures with the addition of tunable elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Therefore, a comparison of known approaches to design anomalous reflectors is timely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' There are two fundamentally different methods to realize a flat surface that reflects plane waves into plane waves along any desired direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' One is the use of periodical structures (diffraction gratings) whose period is chosen accordingly to the required angles of incidence and reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The other one is using aperiodically loaded antenna arrays whose geometrical period is fixed to usually λ/2 [25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The majority of works on anomalous reflectors use the first approach, and here we con- sider various designs of periodically modulated anomalously reflected boundaries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Perhaps, the most classical approach to manipulate the direction of reflection from a surface is the use of phased-array (reflectarray) antennas [26]–[28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Here, the phase distribution at the antenna aperture is tuned so that reflections from all antenna array elements interfere constructively along the desired direction of reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Generalizing this principle, a similar approach can be realized in a planar subwavelength- structured metasurface if the local reflection phase is made nonuniform over the surface, realizing a phase-gradient re- flector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Using this approach one can direct the reflected wave at will, beating the conventional law of reflection and realizing so-called anomalous reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The main drawback of this method is the low efficiency at large deviations from the usual law of reflection [2], [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Impedance mismatch between the incident and the reflected waves becomes significant, and it causes more scattering into parasitic propagating modes (See Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Theoretically, the problem of reduced efficiency at large deflection angles can be completely solved with the use of active and lossy inclusions in the metasurface [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Worth mentioning, that the average power produced by the surface would be zero, however, some parts of it must produce energy, and the other parts should absorb it, which is quite impractical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Another possibility is to use completely passive structures, where auxiliary surface waves in the near-field region are properly tuned [1], [3] as it is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Optimization of the evanescent modes can be performed in several differ- ent ways: based on the optimization of the input (surface) impedance [5], [6], grid (sheet) impedance [7]–[9], by direct optimization of the whole structure [10], [11], by finding an analytical solution [12], [13], and finally, introducing non- planar (power flow-conformal) structures [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' In this paper, we overview and compare some of these approaches in detail and discuss their differences and advantages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' We repeated the selected design methods and compared the most important characteristics of these works, including power efficiency, angular stability, far-field radiation patterns, and frequency bandwidth for the infinite and finite-size structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The paper is organized as follows: In Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' II the selected methods for comparison will be briefly introduced and the pros and cons for each of them will be highlighted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Then Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' III is devoted to the investigation of scattering parameters for both ideal and realized structures and the comparison of the power arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='02851v1 [physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='app-ph] 3 Jan 2023 2 efficiency of each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The angular response of an anoma- lous reflector is another important aspect that is discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Here, we show the behavior of reflectors when they are illuminated by waves at different angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Furthermore, reflection and scattering by a finite-size structure in the far zone is important for applications, and recently it has been considered in several studies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' We cover this issue in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Finally, conclusions are formulated in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' VI to finalize this comparison and make the advantages and drawbacks of each approach clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' CONSIDERED DESIGN METHODS To provide a fair comparison, we choose methods that can be realized using arrays of metallic patches or strips printed on a grounded dielectric substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' We select an example required performance: an anomalous reflection of normally incident plane waves with TE polarization to the 70◦-direction, at 8 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' All designs are based on the same basic platform: a metal patch array on a grounded dielectric substrate (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The chosen example substrate is Rogers 5880 with 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='2 per- mittivity, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='575 mm thickness, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='0002 loss tangent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' For all designs, we split the period into 6 sub-cells that are either impedance strips or shaped metal strips.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The use of the same parameters for all designs allows a meaningful comparison of performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The initial reference design is a phase-gradient metasur- face, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' [26]–[28], [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The unit cells are designed in the conventional locally periodical approximation so that at every point of the reflector the reflection phase (at normal incidence) from an infinite array of identical cells is as required by the linear phase gradient rule for the desired reflection angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' It means that the reflection properties of a metasurface can be defined by the “local reflection coefficient” which is assumed to be controlled by adjusting the geometrical parameters of the unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Strong coupling between the inclusions in an inhomogeneous array makes this approximation rather rough when the deflection angle is not small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' More advanced methods that aim at overcoming the inherent parasitic scattering of phase-gradient reflectors we classify based on the degree of use of homogenized boundary con- ditions: Method 1 (input impedance method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Here, the metasurface is designed at the level of the equivalent input impedance, also known as the impenetrable impedance boundary condition (IBC), see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The input impedance (Zinput) relates the tangential components of the electric field (Et) and magnetic field at the interface between the metasurface structure and free space: (Ht) Et = Zinput · ˆz × Ht |z=0+ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' (1) In this method, the input impedance distribution over the reflector surface is optimized with the goal to channel most of the reflected power into the desired direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Optimization algorithms vary the input impedance, ensuring zero normal component of the Poynting vector at every point of the surface so that the input impedance is purely reactive [5], [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' When the desired input impedance values at every point are found, 𝜃𝑟 −𝜃𝑟 Normal incident wave (Ei) Retro-reflection (Er0) Symmetry reflection (Er−1) Anomalous reflection (Er+1) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 1: Concept of periodical arrays acting as anomalous reflectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The case with three propagating Floquet harmonics is illustrated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The design goal is to suppress reflections into all propagating modes except the desired anomalous reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' the actual geometry of the structure is determined using the lo- cally periodic approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' That is, the continuous reactance profile is discretized, and the dimensions of each unit cell are optimized using periodical boundary conditions, ensuring that the plane-wave reflection phase (at normal incidence) from an infinite periodical array of this cell is the same as from a uniform boundary with the required input reactance at this point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Method 2 (grid impedance method).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' This design approach is based on the grid (or sheet) impedance model of a patch array that is also known as penetrable IBC, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 2(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' In this method, the impedance boundary condition is used to model only the array of metal patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The grid impedance (Zgrid) relates the surface-averaged electric field with the difference between the averaged tangential magnetic fields at both sides of the metasheet: Et = Zgrid · ˆz × (Ht |z=0+ −Ht |z=0−).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' (2) In this method, spatial dispersion of the grounded dielectric layer is taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The optimization process in this case considers a more practical structure, that treats waves inside the substrate in a more complete way compared to the first method [7]–[9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' In method 2 the locally periodical approximation is used to design reactive sheets in contrast with method 1 in which it is utilized to model the whole metasurface volume.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Method 3 (non-local design) accounts for all specific geo- metrical and electromagnetic features of the layer, not relying on homogenization methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The optimization usually starts from some initial settings in terms of the input impedance (for example, in [10] it was required that the reflector if formed by periodically arranged regions of receiving and re-radiating leaky-wave antennas), but the final steps optimize the whole supercell of the periodical lattice instead of individual patches in periodical arrays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Importantly, the normal component of the Poynting vector along the surface is not set to zero, cor- responding to the effective active-lossy behavior, although the overall structure remains completely passive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The drawback of this approach lies in the need for direct optimization, which 3 𝑍input 𝜂0 𝜂𝑑 𝜂0 𝑍grid PEC Substrate 𝜀𝑟 𝐄𝐭 = 𝑍grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' ො𝒛 × (𝐇𝐭ห𝑧 = 0+ − 𝐇𝐭ȁ𝑧 = 0−) Z= 0 d X Z X Z Z= 0 𝐄𝐭 = 𝑍input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' ො𝒛 × 𝐇𝐭ห𝑧 = 0+ (a) (b) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 2: Two types of IBCs: (a) input impedance, also known as impenetrable IBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The left side illustrates the conceptual struc- ture, and the right side shows the corresponding transmission- line model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' (b) Grid or sheet impedance is also known as penetrable IBC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The conceptual structure on the left consists of an impedance sheet placed on top of a grounded dielectric substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The equivalent transmission-line model is shown on the right side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' usually requires heavy computational facilities and might also become time-consuming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Other approaches realize perfect anomalous reflection using arrays of loaded wires [12], [13] or non-planar structures [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' However, to provide an insightful comparison, we chose only methods that are suitable for planar structures that can be realized as printed circuit boards with metallic patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Specif- ically, the phase-gradient sample is designed based on the required tangent-profile of the input impedance, for method 1 (impenetrable IBCs) we follow [6], paper [7] for method 2 (penetrable IBCs), and [10] for method 3 (non-local design).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' In the following sections, the scattering properties, angular response, as well as far-field characteristics of test finite-size structures for all aforementioned methods will be investigated and compared in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' SCATTERING PROPERTIES At first, the scattering properties of all the anomalous reflec- tors under study will be investigated assuming infinite period- ical structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Upon plane-wave illuminations, the structures support surface currents that are also periodic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The Floquet theory defines the tangential wavenumbers of modes supported by the surface: kt = kt0 + ktn = k0 sin θi + 2πn/D, (3) Where k0 is the wavenumber in free space, θi is the angle of the upcoming incident wave, n is an integer number that denotes the index of the Floquet mode, and D is the period of the surface pattern, determined by D = λ/(sin θr − sin θi).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' θr is the desired reflection angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' By choosing this period, the tangential component of the wavenumber is fixed so that one of the harmonics is reflected to the desired angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Floquet harmonics that satisfy criterion k0 > |kt| belong to the fast- wave regime in the dispersion diagram and can propagate in free space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Other Floquet harmonics are surface waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The direction of the reflection can be calculated by the following formula: sin θr = kt/k0 = (k0 sin θi + 2πn/D)/k0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' (4) For the chosen design parameters (θi = 0◦, θr = 70◦, f = 8 GHz), the period is equal to D = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='0642λ, and the Floquet expansion has three propagating harmonics (k0 > |kt|): zero Floquet mode (0◦), −1 Floquet mode (−70◦), and +1 Floquet mode (+70◦), as follows from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' (4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The field amplitudes in these modes define the efficiency of power channeling from one mode to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 1) Performance of the surface-impedance models Initially, the ideal impedance profile is considered when the period is discretized to six elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' In other words, it is assumed that the impedance boundary condition is applied straight on the surface without considering actual realization (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' It is noteworthy to notice that the discretization of the impedance profile deteriorates performance, however, it is spoiled in the same way for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Using such discretiza- tion, reasonable results can be achieved rather fast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' All the methods except the phase gradient method use optimization, therefore analytical closed-form formulas for the impedance profiles do not exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The list of optimized impedance values of each unit cell in a period is presented in Table I for all designed methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' For both designs based on the input impedance model (phase-gradient and input impedance opti- mization), we convert the obtained input impedance profile to the grid impedance by using the equivalent transmission- line model presented in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The input impedance can be considered as that of a shunt connection of the grid impedance to the transmission line modeling the grounded dielectric substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' For the non-local method, a pre-final optimization is applied here similarly to what was done in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' As it was discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' II, for the non-local approach we can assume an impedance profile using repeated receiving and re- radiating leaky-wave sections to mimic the ideal active-lossy profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Therefore, the optimization at the level of the grid impedance is an initial step before the final optimization for the whole supercell in the real structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Eventually, the same configuration for all the methods enables us to complete a fair comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' TABLE I: The impedance profile list for each unit cell (jΩ) Cell1 Cell2 Cell3 Cell4 Cell5 Cell6 Phase gradient 97.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6 82.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='1 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 +2673.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='9 145.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 113.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 Input impedance 291.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='7 141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='3 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='0 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='0 83.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 85.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='3 Grid impedance 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 1334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='0 112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='9 172.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6 91.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='7 Non- local 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='82 74.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='74 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='16 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='50 75.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='07 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='06 Performance comparison of the discretized impedance pro- files after optimization is made using full-wave simulators, CST STUDIO [31] and ANSYS HFSS [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' As it was dis- cussed, there are three propagating Floquet harmonics (open channels) in our specific example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Therefore, we can consider 4 (a) (b) (c) (d) (e) (f) (g) (h) No diffracted modes No diffracted modes No diffracted modes No diffracted modes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 3: Power distribution between three propagating modes and scattered field distribution (bottom);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' (a,e) for the phase gradient, (b,f) input impedance, (c,g) grid impedance, (d,h) non-local design method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The horizontal black lines in the scattered field distribution figures illustrate the location of the metasurfaces where the IBC is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' these reflectors as three-port networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Scattering parameters (Sn1) can be determined numerically when the input wave comes from Floquet port 1 and the output wave is observed in the port number n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Consequently, the power efficiency is found as squared scattering parameters (ηn = |Sn1|2) in the full-wave simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The power efficiency for each mode measures the fraction of power rerouted from the incident wave (assuming that the incident port is 1) to the propagating mode n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Figures 3 (a-d) show ratios of power rerouted to propagating channels n = 0, 1, and −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' In all cases, below 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 GHz diffraction modes are not allowed, therefore all the energy is reflected back to the normal direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Designs based on the phase-gradient, input impedance, and grid impedance methods show broadband behavior as compared to the non- local approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The phase-gradient method does not take evanescent modes into account, which results in the lowest efficiency at the operational frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Power distribution and the corresponding field amplitudes for all methods can be found in Table II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' TABLE II: Amplitude/power ratio of propagating Floquet modes and power efficiency level at 8 GHz θi θr −θr Phase gradient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='33/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='11 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='45/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='71/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='17 Input impedance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='10/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='67/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='34/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='04 Grid impedance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='00/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='71/1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='01/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='00 Non-local 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='1/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='01 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='70/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='12/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='00 It is noteworthy to sketch the scattered electric field distri- butions (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 3(e-h)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Efficiency for the phase-gradient method is only 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8%, and, correspondingly, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 3(e) shows a field distribution that is distorted by fields scattered into two par- asitic propagating channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' For the other methods, efficiency is nearly perfect, however, the near-field distributions are different due to different methods used to optimize evanescent modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' It is important to note that for perfect anomalous reflection with ideal power efficiency, the power reflected to the desired direction must be equal to the power of the incident plane wave, and, as a result, the ratio between the amplitudes of the reflected and incident fields for these angles should be larger than one |Er| = |Ei| � cos(θi)/ cos(θr) (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='71 for our example case) [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 2) Realizations as patch arrays The next step is to compare actual structures designed using the previously obtained and discussed impedance profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Fol- lowing the procedures described in the corresponding papers, we design supercells formed by six unit cells based on the rectangular shape metal patches above the grounded dielectric substrate (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 4 and Table III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The corresponding efficiencies for all the considered meth- ods are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The frequency for the best per- formance becomes shifted for all methods, except for the non-local design, where optimization of the whole supercell is implemented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' In addition to that, dispersion and losses deteriorate the efficiency in different ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The absorption levels as well as efficiency at the design frequency (8 GHz) are reported in Table IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The remained power is scattered to other propagating Floquet modes that are not shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Re (E/E) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='2 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 c/ DxRe (E/E) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='2 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 α/DxRe (E/E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=') 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='2 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 α/DxRe (E/E) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6 0 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='2 1 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 α/Dxn=0n=-1n=+1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 Eficiency, In 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='2 0 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 9 Frequency (GHz)n=0n=-1n=十1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='2 0 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 9 Frequency (GHz)n=0n=-1n=十1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 Efficiency, n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='2 0 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 9 Frequency (GHz)n=0n=-1n=+1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 Eficiency, Nn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='2 0 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 9 Frequency (GHz)5 Cell #1 Cell #2 Cell #3 Cell #4 Cell #5 Cell #6 𝐷 𝑑 = ൗ 𝐷 6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 4: The configuration of supercells utilized for the designs consisting of six unit cells.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' All the parameters of the dielectric substrate are given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The period of the array (the supercell size) is fixed to D = 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='9 mm, and the width of a single unit cell is d = D/6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The width of metal strips is 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 mm, while the strip lengths are different for different design methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' TABLE III: Lengths of metal strips for each unit cell (mm) Strip1 Strip2 Strip3 Strip4 Strip5 Strip6 Phase gradient 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='41 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='23 0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='47 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='29 Input impedance 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='3 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='57 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='28 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='78 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='3 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='25 Grid impedance 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='71 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='32 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='89 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='03 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='84 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='78 Non- local 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='47 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='91 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='26 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='22 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='30 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='88 TABLE IV: The best-performance frequency and the corre- sponding efficiency versus the absorption rate and efficiency for the design frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Best performance 8 GHz Frequency Efficiency Absorption Efficiency Phase gradient 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 GHz 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='2(%) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='9(%) 62.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='7(%) Input impedance 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='27 GHz 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='3(%) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='2(%) 56.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6(%) Grid impedance 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='2 GHz 95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='0(%) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='3(%) 73.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6(%) Non-local 8 GHz 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6(%) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5(%) 96.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6(%) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' ANGULAR RESPONSE A very interesting property of anomalous reflectors which is often left unstudied is the angular response, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=', performance of the structure for various incident angles θi, which can be different from the design angle of incidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Here, we consider angular response for periodical arrays formed by repeated supercells consisting of 6 unit cells with patches printed on a grounded substrate and assume the periodic boundary condition for this analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' To distinguish between the illumination angle and the incidence angle for which the surface was designed, we denote this design incidence angle by No diffracted modes Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 5: Frequency dependence of efficiency for structures realized with metallic rectangular patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' θid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Worth to note that θid together with the required reflection angle defines the period of the structure operating as an anomalous reflector for these angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The angular response is studied by sweeping the incident angle θi for a fixed structure, designed for the angle θid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The number of propagating Floquet modes existing in the system is defined by the incident angle θi, the period of the structure D, and the frequency f (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' (3)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The condition for the mode propagation can be written as follows: ktn < k0 → 2π D |n| < 2π λ → |n| < D λ , (5) and their propagation directions can be calculated as [33], [34]: θtn = arctan(ktn/knn), (6) where knn is the normal component of the wavenumber for the nth mode, and knn = � k2 0 − k2 tn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' If k0 > |ktn|, the normal component of the nth wavenumber is purely real, which corresponds to a propagating mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Otherwise, the wavenumber is imaginary, which corresponds to a surface mode that propagates along the interface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Figure 6 shows that for this fixed operational frequency and period of the structure, only five propagating Floquet modes with n ∈ [−2, 2] are allowed in the system when the angle of incidence is changing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' All other modes (|n| > 3) are surface modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Figure 7 depicts the spatial power distribution for prop- agating modes versus the illumination angle at 8 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' At the angle θi = 0◦ the incident angle is equal to the design angle θid, therefore most of the power goes to mode +1, with different efficiency for each method (see Table IV).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Based on the discussion in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' [33], [34], for the phase-gradient case, there is a retro-reflection angle (at which all the energy is reflected back at the angle of incidence), that can be calculated as θretro = arcsin[(sin θi −sin θr)/2] and is equal to −28◦ for the considered case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' At this angle only two channels are open (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 6), and the angle for the other channel is −θretro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Ideally, 100% of the power should be scattered in the retro- reflection direction, however, discretization and the presence of losses decrease it down to 96%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Therefore, the rest of the power goes to the remaining channel or gets absorbed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Due Phase grad Input imp Grid impNon-local Efficiency, Mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0 7 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 8 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 9 Frequency GHz6 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 6: Propagation angle for different Floquet modes with respect to the incident angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' This figure is made using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 6 when the incidence angle is swept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' to reciprocity, the structure behaves in the same way when illuminated from direction −θretro.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' It is important to notice that for other design methods, retro-reflection occurs at the angle +70◦, when three propagating channels are open.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' When the structure is illuminated from the normal direction, most of the energy couples to mode n = +1, where the reflection angle is θr = +70◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Therefore, channel n = −1 becomes decoupled from the other two, and when the structure is illuminated from the angle θi = −70◦, all the energy is reflected back to the source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Finally, a sweep of the incident angle reveals that the design method based on grid impedance is the solution that has the least sensitive response (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 7(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' It means that when the incident angle changes between −70◦ and +70◦, the power couples primarily to the same modes, unlike for other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Figure 8 illustrates and reports the results of the study for the best performance frequency for each method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The result is the same for the non-local optimization approach since in this case, the best performance frequency matches the design frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' FAR-FIELD SCATTERING FROM FINITE-SIZE STRUCTURES In the previous analysis, we considered infinite periodical structures excited by plane waves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Here, we study far-field scattering properties of finite-size structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' It is possible to study metasurfaces on the grid or sheet impedance levels using the mode-matching method for calculation of induced currents [7], [35] and the far-field approximation for the calculation of scattered fields [33], [36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' To do that, the following conditions have to be met: |r| ≫ λ, (7a) |r| ≫ L, (7b) L2/|r| ≪ λ, (7c) where |r| is the distance from the observation point to the center of the structure, and L is max(2a, 2b), in which a and b denote distances between the center of the metasur- face and the edges of the structure along the x and y- axes, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Considering TE polarized incident waves (Ei = E0e−jk(sin θix+cos θiz)ˆy) and selecting the observation point in the plane of incidence (xy-plane), the normalized scattering pattern in spherical coordinates can be determined by the following expressions: Fr(θ) = 1 2 cos θi � n rn(θi)(cos(θrn) + cos(θ))sinc(kaefn), (8) Fsh(θ) = 1 2 cos θi (cos(θ) − cos(θi))sinc(kaef), (9) where rn(θi) are the amplitudes of excited harmonics, de- termined by the mode matching technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Angle θrn shows the reflected angle for each harmonic, and sinc(x) is a sinc function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' In addition, aefn and aef can be represented by aefn = (sin θ − sin θrn)a and aefn = (sin θ − sin θi)a, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Finally, the total scattering pattern can be found as the sum Fsc = Fr + Fsh.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Worth to mention that normalization is performed with respect to the maximum of the reflected field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Alternatively to the analytical approach, one can study finite-size structures numerically using full-wave simulators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The result shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 9(a) corresponds to the analytical solution, and in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 9(b) to the full-wave simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' In both cases, the radiation pattern is calculated at 8 GHz for structures with the size 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='7λ × 7λ in the xy-plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The discrepancy between the two radiation patterns, which becomes more significant for side lobes (SL), is caused by the neglected current distortions near the edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Nevertheless, the general behavior is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Parameters of the patterns related to each method are shown in Table V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The beam- TABLE V: The normalized main lobe amplitude, the first side lobe amplitude in linear scale, and their directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Main lobe Amp/angle corresponds to n=+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' SL Amp/angle corresponds to n=0 SL Amp/angle corresponds to n=-1 Phase gradient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='94/68◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='52/-1◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='49/-67◦ Input impedance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='95/68◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='45/0◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='35/-67◦ Grid impedance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='99/69◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='13/-1◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='33/-67◦ Non-local 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='00/69◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='24/-2◦ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='39/-68◦ widths for all cases are similar and close to 9◦ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Moreover, as it is shown in the inset, the maximum of the scattered field is higher for the design methods based on grid impedance and non-local solution because the power efficiency is higher in these methods compared to the phase gradient design and optimization based on the input impedance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The most important difference corresponds to the side-lobe level (SLL).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' As it is expected, the highest side lobes occur along the θ = −70 and θ = 0, because there are two propagating Floquet harmonics along these directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Figure 10 shows the scattering patterns of all the methods at different frequencies, where all the patterns are normalized —n=0n=-1n=+1n=-2n=+2 50 0 50 50 0 50 Angle of incidence, :7 (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 7: Power distribution among different propagating modes depending on the incident angle at frequency 8 GHz for (a) phase gradient, (b) input impedance optimization, (c) grid impedance optimization, and (d) non-local optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 8: Power distribution among different propagating modes depending on the incident angle for the best-performance frequencies which are reported in Table IV, for (a) phase gradient, (b) input impedance optimization, (c) grid impedance optimization, and (d) non-local optimization method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' n=0 n=-1 n=+1n=-2 2n=+2 Eficiency, Mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 50 0 50 Angle of incidence, :n=0 n=-1 n=+1n=-2n=+2 Efficiency, Mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 50 0 50 Angle of incidence, :n=0 n=-1 n=+1-n=-2 一n=+2 Efficiency, Mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 50 0 50 Angle of incidence, :n=0 n=-1 n=+1n=-2 n=+2 Eficiency, Mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 50 0 50 Angle of incidence, :n=0 n=-1 n=+1 n=-2 n=+2 Efficiency, Mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 50 0 50 Angle of incidence, :n=0n=-1n=+1n=-2n=+2 Eficiency, Mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0 50 0 50 Angle of incidence, O:n=0 n=-1 n=+1n=-2 n=+2 Eficiency, Nn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 50 0 50 Angle of incidence, :n=0 n=-1 n=+1- n=-2 一n=+2 Efficiency, Mn 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0 50 0 50 Angle of incidence, :8 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 9: Normalized radiation pattern in linear scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' All pat- terns are normalized with respect to the main lobe amplitude of the non-local method which has the highest gain compared to the other approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' (a) analytical pattern based on the Huygens principle, (b) full-wave simulation in CST STUDIO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' to the main lobe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' It is important to notice that due to the fixed period of the structure (D = λ/(sin θr − sin θi)), by sweeping the frequency, the angle of reflection changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' This can be observed in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' By changing the frequency, the scattered n = +1 Floquet harmonic scans the space from the desired reflection angle (+70◦) at 8 GHz to smaller angles at higher frequencies and larger angles at lower frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Below 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 GHz there are no diffraction modes, therefore we plot the scattering patterns starting from 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='75 GHz, where most of the energy is reflected into the normal direction (see the blue line in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The red line in the figure corresponds to the scattering pattern at the design frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Eventually, the radiation patterns for 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='25 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 GHz are shown by yellow and purple lines, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' CONCLUSION We have presented a comprehensive analysis of four main design methods for anomalous reflectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' In order to provide a meaningful comparison we chose design methods that can be realized within the same topology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' At first, we performed an analysis of periodical infinite structures on the level of input and grid impedances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Then we proceeded to design actual implementations as supercells formed by six metal patches placed on top of a grounded dielectric substrate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Further, we analyzed the angular response of the designed metasurfaces (a) (b) (c) (d) Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 10: Frequency bandwidth patterns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' At 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 GHz, which is not plotted here, there is no diffracted mode, and all the energy goes back to the specular (normal) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' The patterns are plotted between 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='75 GHz to 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 GHz with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='25 GHz step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Designed based on (a) phase gradient, (b) input impedance optimization, (c) grid impedance optimization, (d) non-local optimization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' All patterns are normalized to the main lobe amplitude for each case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' and finally presented far-field radiation patterns of finite-size structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' In this work, we provide a comparative summary of the main features of previously introduced design methods as well as present an original study of a property that is frequently left unstudied: the angular response.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' This study can be considered referential for engineers working on reconfigurable intelligent surfaces, where similar design methods are utilized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' ACKNOWLEDGMENT This work was supported by the European Union’s Hori- zon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 956256 (project METAWIRELESS), and the Academy of Finland (grant 345178).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' REFERENCES [1] 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Tretyakov, “Independent control of multiple channels in metasurface devices,” Physical Review Applied, vol.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 14, no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 2, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 024089, 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='75GH 2 8GHz 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='25GH2 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5GHz0 30 30 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6 60 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 90 900 30 30 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6 60 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='2 90 900 30 30 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6 60 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='2 90 900 30 30 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='6 60 60 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='2 90 90-Phase grad Input imp Grid impNon-local Linear radiation pattern 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='8 68 70 72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 50 50Linear radiation pattern 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='95 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='9 66 68 70 72 50 0 509 [8] D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content='-H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE1T4oBgHgl3EQfBgIF/content/2301.02851v1.pdf'} +page_content=' Kwon, “Planar metasurface design for wide-angle refraction using interface field optimization,” IEEE Antennas and Wireless Propagation Letters, vol.' metadata={'source': 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Fink1, § +1Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria +2Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1040 Vienna, Austria +(Dated: January 10, 2023) +Entanglement is a genuine quantum mechanical property and the key resource in currently developed quantum technologies. +Sharing this fragile property between superconducting microwave circuits and optical or atomic systems would enable new +functionalities but has been hindered by the tremendous energy mismatch of ∼ 105 and the resulting mutually imposed loss and +noise. In this work we create and verify entanglement between microwave and optical fields in a millikelvin environment. Using +an optically pulsed superconducting electro-optical device, we deterministically prepare an itinerant microwave-optical state that +is squeezed by 0.72+0.31 +−0.25 dB and violates the Duan-Simon separability criterion by > 5 standard deviations. This establishes +the long-sought non-classical correlations between superconducting circuits and telecom wavelength light with wide-ranging +implications for hybrid quantum networks in the context of modularization, scaling, sensing and cross-platform verification. +Over the past decades we have witnessed spectacular +progress in our capabilities to manipulate and measure +genuine quantum mechanical properties, such as quan- +tum superpositions and entanglement, in a variety of +physical systems. These techniques serve now as the ba- +sis for the development of quantum technologies, where +the demonstration of quantum supremacy with tens of +superconducting qubits [1], an ultra-coherent quantum +memory with nuclear spins [2], and distributed quan- +tum entanglement over tens of kilometers using optical +photons [3] represent just a few of the highlights that +have already been achieved. +Going forward, combin- +ing these techniques [4–6] will enable the realization of +general-purpose quantum networks, where remote quan- +tum nodes, capable of storing and processing quantum +information, seamlessly communicate with each other by +distributing entanglement over optical channels [7]. As- +pects of this approach have already been adopted to con- +nect and entangle various quantum platforms remotely, +involving single atoms, ions, atomic ensembles, quantum +dots, rare-earth ions and nitrogen-vacancy centers [8]. +However, such long-distance quantum connectivity is +considerably more difficult to achieve with other promis- +ing platforms, such as semiconductor spin qubits or local +cryogenic networks of superconducting circuits [9, 10], +where no natural interface to room temperature noise- +resilient optical photons is available. +To overcome this limitation, a lot of effort is cur- +rently focused on the development of coherent quantum +transducers between microwave and optical photons [11– +16]. Direct noiseless conversion of a quantum state typ- +ically relies on a beam splitter process, where a strong +driving field mediates the conversion between weak mi- +crowave and optical signals - a deterministic approach +with exceptionally stringent requirements on conversion +† rsahu@ist.ac.at +‡ liu.qiu@ist.ac.at +§ jfink@ist.ac.at +⋆ These authors contributed equally to this work. +efficiency and added classical noise that are still out of +reach. Alternatively, the direct generation of quantum- +correlated microwave-optical photon pairs can also be +used as a resource for quantum teleportation and en- +tanglement distribution in the continuous and discrete +variable domain [17–19]. +In this paper we use an ultra-low noise cavity electro- +optical device to generate such non-classical correlations +in a deterministic protocol [20]. It consists of a 5 mil- +limeter diameter, 150 µm thick lithium niobate optical +resonator placed inside a superconducting aluminum mi- +crowave cavity at a temperature of 7 mK, described in +detail in Ref. [21]. As depicted in Fig. 1A, the microwave +mode ˆae is co-localized with and electro-optically coupled +to the optical whispering gallery modes at ωo/(2π) ≈ +193.46 THz via the Pockels effect. We match the tunable +microwave resonance frequency ωe/(2π) to the free spec- +tral range (FSR) of 8.799 GHz to realize a triply-resonant +system [22, 23] with the interaction Hamiltonian, +ˆHint = ℏg0ˆapˆa† +eˆa† +o + h.c., +(1) +with g0 the vacuum electro-optical coupling rate, and +ˆap (ˆao) the annihilation operator of the optical pump +(Stokes) mode [24]. Here we have ignored the interac- +tion with the suppressed optical anti-Stokes mode ˆat, as +shown in Fig. 1b (see supplementary information). +In this sideband suppressed situation, efficient two- +mode squeezing is achieved with a strong resonant opti- +cal pump tone, yielding the simple effective Hamiltonian, +ˆHeff = ℏg0√¯np(ˆa† +eˆa† +o + ˆaeˆao), where ¯np = ⟨ˆa† +pˆap⟩ is the +mean intra-cavity photon number of the optical pump +mode. Deterministic continuous-variable (CV) entangle- +ment between the out-propagating microwave and opti- +cal field can be generated below the parametric instability +threshold (C < 1) in the quantum back-action dominated +regime, where the quantum noise exceeds the microwave +thermal noise. Here C = 4¯npg2 +0/(κeκo) is the coopera- +tivity with the vacuum coupling rate g0/2π ≈ 37 Hz, and +the total loss rates of the microwave and optical Stokes +modes κe/2π ≈ 11 MHz and κo/2π ≈ 28 MHz. The re- +quired ultra-low noise operation is achieved despite the +arXiv:2301.03315v1 [quant-ph] 9 Jan 2023 + +2 +!p +^ap,in +^ao,out +^ae,out +!e +ae^ +ao^ +ao +ap +!p +!p +!e +- FSR +!p+ FSR += FSR += 1) +^at +^ +^ +ae^ +(me +- 1) +(mp ++ 1) +(mp +mp +b +a +FIG. 1. Physical and conceptual mode configuration. +a, Simulated microwave (left) and optical (right) mode dis- +tribution with azimuthal number me = 1 and mo = 17 (for +illustration, experimentally mo ≈ 20 000). Phase matching is +fulfilled due to the condition mo = mp − me and entangle- +ment is generated and verified between the out-propagating +microwave field ˆae,out and the optical Stokes field ˆao,out. b, +Sketch of the density of states of the relevant modes. +Un- +der the condition ωp − ωo = ωe the strong pump tone in ˆap +efficiently produces entangled pairs of microwave and optical +photons in ˆae and ˆao via spontaneous parametric downconver- +sion. Frequency up-conversion is suppressed via hybridization +of the anti-Stokes mode ˆat with an auxiliary mode. +required high power optical pump due to slow heating of +this millimeter sized device [21]. +In the following we characterize the microwave and op- +tical output fields via the dimensionless quadrature pairs +Xj and Pj (j = e, o for microwave and optics), which +satisfy the canonical commutation relations [Xj, Pj] = i. +A pair of Einstein-Podolsky-Rosen (EPR)-type opera- +tors X+ = +1 +√ +2 (Xe + Xo) and P− = +1 +√ +2 (Pe − Po) are +then constructed, and the microwave and optical output +fields are entangled, if the variance of the joint opera- +tors is reduced below the vacuum level, i.e. +∆− +EPR = +� +X2 ++ +� ++ +� +P 2 +− +� +< 1. +This is commonly referred to as +the Duan-Simon criterion [25, 26], which we apply to +each near-resonant frequency component of the two-mode +squeezed output mode (see SI). +For efficient entanglement generation, we use a 250 ns +long optical pump pulse (≈ 244 mW, C ≈ 0.18, ¯np ≈ +1.6 × 1010) at a 2 Hz repetition rate, cf. +pulse 1 in +Fig. 2a. The entangled output optical signal is filtered +via a Fabry-Perot cavity to reject the strong pump. The +entangled microwave output is amplified with a high- +electron-mobility transistor amplifier. Both outputs are +down-converted to an intermediate frequency of 40 MHz +with two local oscillators (LO) and the four quadra- +tures are extracted from heterodyne detection. +Long- +term phase stability between the two LOs is achieved via +extracting the relative phase drift by means of a second +phase alignment pump pulse that is applied 1 µs after +each entanglement pulse, together with a coherent reso- +nant microwave pulse, shown in Fig. 2(a). This generates +a high signal-to-noise coherent optical signal via stimu- +lated parametric down-conversion and allows for aligning +the phase of each individual measurement. +Figure 2b(c) shows the time-domain average power +over one million averages for the on-resonant microwave +(optics) signal with a spectrally under-sampled 40 MHz +bandwidth (hence not revealing the full pulse ampli- +tude). The two insets show the microwave (optical) signal +from spontaneous parametric down-conversion (SPDC) +due to pulse 1, cf. Fig. 2a, with an emission bandwidth +of ≈ 10 MHz. The larger signals during the second half +of the experiment are the reflected microwave pulse and +the generated optical tone (due to pulse 2) that is used +for LO phase alignment. The raw power measurements +are divided by the measurement bandwidth and rescaled +such that the off-resonant response matches the noise +photon number Nj,add + 0.5 of the measurement setup. +Ne,add = 13.1 ± 0.4 (2σ errors throughout the paper) +due to loss and amplifier noise and No,add = 5.5 ± 0.2 +due to optical losses are carefully determined using noise +thermometry of a temperature controlled 50 Ω load and +4-port calibration, respectively (see SI). Using this pro- +cedure ensures that the reported photon number units +correspond to the signals at the device outputs. +We continue the analysis in the frequency domain by +calculating the Fourier transform of each measurement +for three separate time intervals - before (2 µs), dur- +ing (200 ns) and right after the entangling pump pulse +(500 ns). +Figure 2d shows the resulting average mi- +crowave noise spectra for all three time intervals with +corresponding fit curves (dashed lines) and theory (solid +line). Before and after the pump pulse, the on-resonant +microwave output field takes on values above the vac- +uum level, with fitted intrinsic microwave bath occu- +pancies ¯ne,int = 0.03 ± 0.01 and 0.09 ± 0.03, respec- +tively. By fitting additional power dependent measure- +ments, we independently verify that the observed noise +floor corresponds to a waveguide bath occupancy of only +¯ne,wg = 0.001±0.002 at the very low average pump power +of ≈ 0.12 µW used in this experiment (see SI). The mea- +sured noise floor therefore corresponds to the shot noise +equivalent level Ne,add + 0.5 (gray dashed lines). Sim- +ilarly, Fig. 2e shows the obtained average optical noise +spectra during and after the pump, referenced to the +measured shot noise level before the pulse. As expected, +there is no visible increase of the optical noise level after +the pulse. +During the pump pulse, an approximately Lorentzian +shaped microwave and optical power spectrum are gen- +erated via the SPDC process. We perform a joint fit of +the microwave and optical power spectral density during +the pulse using a 5-mode theoretical model that includes + +3 +600 +400 +200 +0 +800 +1000 +13.6 +13.7 +1.5 +2.0 +2.5 +3.0 +0 +50 +100 +150 +1.5 +2.0 +2.5 +3.0 +6.05 +6.10 +!e +!o +!p +!e +!o +Input power +t +Output power +Before +pulse +Pulse +1 +Entangle +Pulse +2 +After +pulse +Phase +align +MW +signal +MW +reflection +b +a +c +0 +1 +2 +3 +t (μs) +4 +5 +0 +1 +2 +3 +4 +5 +13.75 +14.00 +6.00 +6.25 +-20 +0 +20 +-20 +0 +20 +Before-pulse +Theory +In-pulse +After-pulse +In-pulse +After-pulse +Theory +Δ!e=2π (MHz) +Δ!o=2π (MHz) +e +d +Ne,det (photons s-1 Hz-1) +No,det (photons s-1 Hz-1) +Ne,det (photons s-1 Hz-1) +No,det (photons s-1 Hz-1) +(μs) +t +Ne,add + 0.5 +No,add + 0.5 +FIG. 2. Measurement sequence and noise powers. a, Schematic pulse sequence of a single measurement. The optical +pulse 1 is applied at ωp and amplifies the vacuum (and any thermal noise) in the two modes ˆae and ˆao, thus generating the +SPDC signals. 1 µs later, a second optical pump with about 10 times lower power is applied together with a coherent microwave +pulse at ωe. The microwave photons stimulate the optical pump to down-convert, which generates a coherent pulse in the ˆao +mode that is used to extract slow LO phase drifts. b and c, Measured output power in the ˆae and ˆao mode in units of photons +per second in a 1 Hz bandwidth and averaged over a million experiments. The SPDC signals are shown in the insets with +the dashed gray lines indicating the calibrated detection noise floor Nj,add + 0.5. d, Corresponding microwave output power +spectral density vs. ∆ωe = ω − ωe centered on resonance right before the entanglement pulse, during the pulse and right +after the pulse, as indicated in panel a. Yellow and green dashed lines are fits to a Lorentzian function, which yields the +microwave bath occupancies before and after the entangling pulse. Error bars represent the 2σ statistical standard error and +the shaded regions represent the 95% confidence interval of the fit. (e), Corresponding optical output power spectral density +vs. ∆ωo = ωo − ω during and after the entanglement pulse, both normalized to the measured noise floor before the pulse. The +in-pulse noise spectra in panels d and e are fit jointly with theory, which yields C = 0.18 ± 0.01 and ¯ne,int = 0.07 ± 0.03. +the effects of measurement bandwidth. In this model, the +in-pulse microwave bath occupancy ¯ne,int = 0.07 ± 0.03 +and the cooperativity C = 0.18 ± 0.01 are the only free +fit parameters. Here the narrowed microwave linewidth +κe,eff/2π = 9.8 ± 1.8 MHz (taken from a Lorentzian +fit) agrees with coherent electro-optical dynamical back- +action [27], where κe,eff = (1 − C)κe. We conclude that +this cavity electro-optical device is deep in the quantum +back-action dominated regime, a prerequisite for efficient +microwave-optics entanglement generation. +For each frequency component the bipartite Gaussian +state of the propagating output fields can be fully char- +acterized by the 4 × 4 covariance matrix (CM) Vij = +⟨δuiδuj + δujδui⟩ /2, where δui = ui − ⟨ui⟩ and u ∈ +{Xe, Pe, Xo, Po} (see SI). The diagonal elements in V cor- +respond to the individual output field quadrature vari- +ances in dimensionless units. These are obtained from +the measured variances after subtracting the measured +detection noise offsets shown in Fig. 2, i.e. Vii(∆ω) = +Vii,meas(∆ωi) − Ni,add. The obtained CM from the data +in Fig. 2 at ∆ω = 0 is shown in Fig. 3a in its standard +form. It corresponds to the quantum state of the propa- +gating modes in the coaxial line and the coupling prism +attached to the device output, i.e. before setup losses or +amplification incur. The non-zero off-diagonal elements +indicate strong correlations between microwave and op- +tical quadratures. +The two-mode squeezed quadratures are more intu- +itively visualized in terms of the quasi-probability Wigner +function, +W(u) = exp[− 1 +2uV −1uT ] +π2� +det(V) +, +(2) + +4 +where u = (Xe, Pe, Xo, Po). Different marginals of this +Wigner function are shown in Fig. 3b, where the (Xe,Xo) +and (Pe,Po) marginals show two-mode squeezing in the +diagonal and off-diagonal directions. +The two cross- +quadrature marginals show a slightly different amount +of squeezing, which is due to the statistical uncertainty +in the measured CM. +Figure 3c shows the amount of two-mode squeezing +between microwave and optical photon pairs. Correla- +tions are observed at ∆ωj = ±(ω − ωj) around the reso- +nances due to energy conservation in the SPDC process +(see SI). The averaged microwave quadrature variance +(purple dots) ¯V11 = (V11+V22)/2 and the averaged optics +quadrature variance (green dots) ¯V33 = (V33 +V44)/2 are +shown in the top panel along with the prediction from +our five-mode theory (solid line) and a simple fit to a +Lorentzian function (dashed line), showing perfect agree- +ment. Measured microwave-optical correlations (yellow +dots) ¯V13 = (V13 − V24)/2 and the Lorentzian fit (dashed +line) lie slightly below the theoretical prediction (solid +line), which we assign to remaining imperfections in the +phase stability (see SI). +The bottom two panels of Fig. 3c show the squeezed +and anti-squeezed joint quadrature variances ∆∓ +EPR = +¯V11 + ¯V33 ∓ 2 ¯V13 (red and blue color respectively). We +observe two-mode squeezing below the vacuum level, i.e. +∆− +EPR < 1, with a bandwidth close to the effective mi- +crowave linewidth. The maximal on-resonant two-mode +squeezing is ∆− +EPR = 0.85+0.05 +−0.06 (2σ, 95% confidence) for +∼1 million pulses with ¯V11 = 0.93, ¯V33 = 0.84 and ¯V13 = +0.46. Hence, this deterministically generated microwave- +optical state violates the Duan-Simon separability crite- +rion by > 5σ. Note that this error also takes into account +systematics in the added noise calibration used for scal- +ing the raw data (see SI). These values correspond to a +state purity of ρ = 1/(4 +� +det[V ]) = 0.44 and demon- +strate microwave-optical entanglement between output +photons with a logarithmic negativity of EN = 0.17. The +supplementary material contains substantial additional +data for longer pulses and varying optical pump power, +which corroborates the presented results and findings, al- +beit with lower statistical significance for each individual +pump configuration (see SI). +Conclusions and outlook +In conclusion, we have demonstrated deterministic +quantum entanglement between propagating microwave +and optical photons,thus establishing a non-classical +communication channel between circuit quantum electro- +dynamics and quantum photonics. Our device can read- +ily be used for probabilistic heralding assisted protocols +[7, 28, 29] to mitigate optical setup losses and extend the +entanglement to room temperature fiber optics. We ex- +pect that the pulse repetition rate can be increased by or- +ders of magnitude with improved thermalization, higher +microwave and optical quality factors, and electro-optic +coupling enhancements that reduce the required pump +power and the associated thermal load. +Coupling effi- +ciency improvements will allow for higher levels of two- +mode squeezing and facilitate also deterministic entangle- +ment distribution schemes [30], teleportation-based state +transfer +[20, 31] and quantum-enhanced remote detec- +tion [32]. Being fully compatible with superconducting +qubits in a millikelvin environment such a device will fa- +cilitate the integration of remote superconducting quan- +tum processors into a single coherent optical quantum +network. +This is not only relevant for modularization +and scaling [33, 34], but also for efficient cross-platform +verification of classically intractable quantum processor +results [35]. +ACKNOWLEDGMENTS +L.Q. acknowledges fruitful discussions with Jie Li and +David Vitali. This work was supported by the European +Research Council under grant agreement no. +758053 +(ERC StG QUNNECT) and the European Union’s +Horizon 2020 research and innovation program under +grant agreement no. 899354 (FETopen SuperQuLAN). +L.Q. acknowledges generous support from the ISTFEL- +LOW programme. W.H. is the recipient of an ISTplus +postdoctoral fellowship with funding from the European +Union’s Horizon 2020 research and innovation program +under the Marie Sk�lodowska-Curie grant agreement no. +754411. G.A. is the recipient of a DOC fellowship of the +Austrian Academy of Sciences at IST Austria. J.M.F. +acknowledges support from the Austrian Science Fund +(FWF) through BeyondC (F7105) and the European +Union’s Horizon 2020 research and innovation programs +under grant agreement No 862644 (FETopen QUAR- +TET). +AUTHOR CONTRIBUTIONS +RS, WH, LQ, and GA worked on the setup. RS and +LQ performed measurements. LQ and RS did the data +analysis. +LQ developed the theory with contributions +from RS, YM and PR. RS and LQ wrote the manuscript +with contributions from all authors. JMF supervised the +project. +DATA AVAILABILITY STATEMENT +All data and code used to produce the figures in this +manuscript will be made available on Zenodo. + +5 +1.0 +2.0 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0 +2 +2 +-2 +-2 +0 +0 +-10 +-20 +10 +20 +c +b +(photons s-1 Hz-1) +(MHz) +0 +2 +-2 +2 +-2 +0 +2 +-2 +0 +2 +-2 +0 +2 +-2 +0 +2 +-2 +0 +a +Vij +Pe +Pe +Pe +Pe +Pe +Po +Po +Po +Po +Po +Xo +Xo +Xo +Xo +Xo +Xe +Xe +Xe +Xe +Xe +-0.5 +0.0 +0.5 +1.0 +(V11 + V22)/2 +(V33 + V44)/2 +(V13 − V24)/2 +V +ΔEPR +- +ΔEPR ++ +Δω/2π +FIG. 3. Characterization of the two-mode squeezed state. a, Measured covariance matrix Vij in its standard form plotted +for ∆ωj = 0 based on 925000 measurements. b, Corresponding Wigner function marginals of different output quadrature pairs +in comparison to vacuum. The contours in blue (grey) represent the 1/e fall-off from the maximum for the measured state +(vacuum). Middle two plots show two-mode squeezing below the vacuum level in the diagonal and off-diagonal directions. +c, Top panel, the measured average microwave output noise ¯V11 = (V11 + V22)/2 (purple), the average optical output noise +¯V33 = (V33 + V44)/2 (green) and the average correlations ¯V13 = (V11 − V24)/2 (yellow) as a function of the measurement +detunings. The solid lines represent the joint theory fit and the dashed lines are individual Lorentzian fits to serve as a guide +to eye. The middle (bottom) panel shows two-mode squeezing in red (anti-squeezing in blue) calculated from the top panels +as ∆± +EPR = ¯V11 + ¯V33 ± 2 ¯V13. The darker color error bars represent the 2σ statistical error and the outer (faint) 2σ error bars +also include the systematic error in calibrating the added noise of the measurement setup. +[1] F. Arute, K. Arya, R. Babbush, D. Bacon, J. C. +Bardin, R. Barends, R. Biswas, S. Boixo, F. G. S. L. +Brandao, D. A. Buell, B. Burkett, Y. Chen, Z. Chen, +B. 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Fink1 +1Institute of Science and Technology Austria, am Campus 1, 3400 Klosterneuburg, Austria +2Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1040 Vienna, Austria +(Dated: January 10, 2023) +CONTENTS +Page +I. Theory +3 +A +Covariance Matrix from Input-Output Theory +. . . . . . . . . . . . . . . . . . . . . . . . . +3 +1 +Quantum Langevin Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +3 +2 +Input-Output-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +4 +3 +Covariance Matrix of Filtered Output Fields . . . . . . . . . . . . . . . . . . . . . . . . +6 +B +Heterodyne Detection, Added Noise and Filtering . . . . . . . . . . . . . . . . . . . . . . . . +7 +1 +Heterodyne Measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +7 +2 +Realistic Measurements: Added Noise and Gain +. . . . . . . . . . . . . . . . . . . . . . +8 +3 +Covariance Matrix from Realistic Heterodyne Measurements +. . . . . . . . . . . . . . . . +9 +C +Entanglement Detection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +1 +Duan Criterion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +10 +2 +Logarithmic Negativity and Purity +. . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +II. Experimental Setup +11 +III. Setup Characterization and calibration +11 +A +Microwave added noise calibration . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +B +Optical added noise . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +IV. Data treatment +15 +A +Time-domain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +B +Pulse post-selection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +C +Frequency domain analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +16 +D +Joint-quadrature correlations +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +V. Quadrature histogram raw data +19 +VI. Non-classical correlations with 600 ns long optical pump pulses +20 +VII. Error analysis +20 +A +Statistical error . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +B +Systematic error +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +22 +References +22 +∗ These two authors contributed equally +arXiv:2301.03315v1 [quant-ph] 9 Jan 2023 + +2 +Introduced in Main Text +ˆae +microwave mode (annihilation operator) +me +microwave mode azimuthal number, me = 1 +ωe +microwave cavity frequency +κe, κe,eff, κe,in, κe,0 microwave total loss, effective total loss, waveguide coupling and intrinsic loss rates +¯ne,int, ¯ne,wg +microwave intrinsic and waveguide bath occupancy +Ne,add +added noise in the microwave detection +ˆao, ˆap, ˆat, ˆatm +optical Stokes, pump, anti-Stokes, and transverse-magnetic mode (annihilation operator) +ˆae/o,out +microwave and optical output field from the device +mp +optical pump mode azimuthal number, mp ∼ 20000 +κo +optical total loss rate +No,add +added noise in the optical detection +¯np +mean photon number of the optical pump mode +g0 +electro-optical vacuum coupling rate +g +photon enhanced electro-optical coupling rate (g = √¯npg0}) +C +cooperativity ( C = 4g2/κeκo ) +Xe,Pe +quadratures of the microwave output field +Xo,Po +quadratures of the optical Stokes output field +V +covariance matrix of the bipartite Gaussian state, Vij = ⟨∆ui∆uj + ∆uj∆ui⟩ /2, where +∆ui = ui − ⟨ui⟩ and u ∈ {Xe, Pe, Xo, Po}. +Nii,add +added noise in the quadrature variances measurements, N11,add = N22,add = Ne,add, +N33,add = N44,add = No,add +Vii,meas +diagonal covariance matrix elements from the calibrated measurement record, Vii += +Vii,meas − Nii,add +V11,V22, ¯V11 +quadrature variances of the microwave output field, ¯V11 = V11+V22 +2 +V33,V44, ¯V33 +quadrature variances of the optical Stokes output field, ¯V33 = V33+V44 +2 +V13,V24, ¯V13 +cross-correlation between microwave and optical quadratures, ¯V13 = V13−V24 +2 +∆∓ +EPR +squeezed and anti-squeezed joint quadrature variance between microwave and optical output +field, ∆∓ +EPR = ¯V11 + ¯V33 ∓ ¯V13 +Introduced in Supplementary Information +J +coupling rate between the optical anti-Stokes mode and TM mode +ˆae/o,in +input field (noise) operator for the microwave and optical mode +ˆae/o,0 +noise operator for the microwave and optical intrinsic loss +ηj +external cavity coupling efficiency of individual mode, j ∈ (e, o, p, t) +G(ω) +spectral filter of the output field +ˆA(Ω) +Fourier transform of operator ˆA(t), ˆA(Ω) = +� +dt eiΩt ˆA(t), ˆA†(Ω) = +� +dt ˆA†(t)eiΩt = [ ˆA(−Ω)]† +ˆX(ωn) +X quadrature of the output spectral mode, ˆX(ωn) = +1 +√ +2 +� ∞ +−∞ dω G(ωn − ω)ˆaout(ω) + h.c. +ˆP(ωn) +P quadrature of the output spectral mode, ˆP(ωn) = +1 +√ +2i +� ∞ +−∞ dω G(ωn − ω)ˆaout(ω) + h.c. +S ˆ +A ˆ +B(Ω) +Two-time correlation of two operators, S ˆ +A ˆ +B(Ω) = +1 +√ +2π +� ∞ +−∞ +� +ˆA(t) ˆB(t′) +� +eiΩtdt +∆LO +local oscillator and signal frequency difference in heterodyne measurement, ∆LO = ωLO−ωsig +Iout(t), Iout(ω) +unitless output field in the equivelant heterodyne detection +SII(ω) +double-sided noise spectrum of the output field in the equivelant heterodyne detection +Gdet(ω) +frequency dependent detection gain in the heterodyne detection +ˆIX/P,det(ωn) +detected output photocurrent quadratures in heterodyne detection, including detection gain +ˆIX/P,out(ωn) +unitless output field quadratures from Iout including added noise +D(ω) +covariance matrix of the detected quadratures from the heterodyne measurement record +Vmeas(ω) +covariance matrix of the total measured output field quadratures including added noise +ˆX+(ω) +joint quadrature of ˆXe(ω) and ˆXo(−ω), ˆX+(ω) = ( ˆXe(ω) + ˆXo(−ω))/ +√ +2 +ˆP−(ω) +joint quadrature of ˆPe(ω) and ˆPo(−ω), ˆP−(ω) = ( ˆPe(ω) − ˆPo(−ω))/ +√ +2 +EN +logarithm negativity +ρ +state purity + +3 +I. +THEORY +A. +Covariance Matrix from Input-Output Theory +1. +Quantum Langevin Equations +Our cavity electro-optical (CEO) device consists of a millimeter-sized lithium niobate optical resonator in a 3-D +superconducting microwave cavity at mK temperature [1]. The Pockels effect in lithium niobate allows for direct +coupling between the microwave and optical whispering gallery modes with maximal field overlap. The optical free +spectral range (FSR) matches the microwave cavity frequency, with microwave azimuthal mode number me = 1. As +shown in Fig. 1 in the main text, resonant three-wave mixing between the microwave mode (ˆae) and three adjacent +transverse-electric (TE) optical modes, i.e. Stokes (ˆao), pump (ˆap), and anti-Stokes (ˆat) mode, arises via the cavity +enhanced electro-optical interaction [2, 3]. In addition, the anti-Stokes mode is coupled to a transverse-magnetic (TM) +optical mode (ˆatm) of orthogonal polarization and similar frequency at rate of J [4]. This results in a total interaction +Hamiltonian, +ˆHI/ℏ = g0(ˆa† +pˆaeˆao + ˆa† +pˆa† +eˆat) + Jˆatˆa† +tm + h.c., +(1) +with g0 the vacuum electro-optical coupling rate. +For efficient entanglement generation, we drive the pump mode strongly with a short coherent input pulse ¯ap,in(t) +at frequency ωp [1], which results in a time-dependent mean intra-cavity field of the pump mode ¯ap(t), +˙¯ap = +� +i∆p − κp +2 +� +¯ap + √ηpκp¯ap,in, +(2) +where the pump tone is detuned from the pump mode by ∆p = ωp − ωo,p, with κp and ηp as the pump mode loss rate +and external coupling efficiency. In our experiments, we actively lock the laser frequency to the pump mode resonance, +with ∆p = 0. +The presence of the strong pump field results in an effective interaction Hamiltonian, +ˆHI,eff/ℏ = g(ˆaeˆao + ˆaeˆa† +t) + Jˆatˆa† +tm + h.c., +(3) +with multiphoton coupling rate g = ¯apg0. This includes the two-mode-squeezing (TMS) interaction between the +Stokes and microwave mode, and beam-splitter (BS) interaction between the anti-Stokes mode and microwave mode, +resulting in scattered Stokes and anti-Stokes sidebands that are located on the lower and upper side of the pump +tone by Ωe away. Microwave-optics entanglement between the microwave and optical Stokes output field can be +achieved via spontaneous parametric down-conversion (SPDC) process due to TMS interaction [5], which is further +facilitated by the suppressed anti-Stokes scattering due to the strong coupling between anti-Stokes and TM modes. +We can obtain the full dynamics of the intracavity fluctuation field in the rotating frame of the scattered sidebands +and microwave resonance, which can be described by the quantum Langevin equations (QLE), +˙ˆae = −κe +2 ˆae − igˆa† +o − ig∗ˆat + √ηeκeδˆae,in + +� +(1 − ηe) κeδˆae,0, +(4) +˙ˆao = +� +iδo − κo +2 +� +ˆao − igˆa† +e + √ηoκoδˆao,in + +� +(1 − ηo) κoδˆao,0, +(5) +˙ˆat = +� +iδt − κt +2 +� +ˆat − ig∗ˆae − iJˆatm + √κtδˆat,vac, +(6) +˙ˆatm = +� +iδtm − κtm +2 +� +ˆatm − iJˆat + √κtmδˆatm,vac, +(7) +with κj the total loss rate of the individual mode where j ∈ (e, o, t, tm), and ηk the external coupling efficiency +of the input field where k ∈ (e, o). We note that, the optical light is only coupled to the TE modes via efficient +prism coupling, with effective mode overlap Λ factor included in ηo for simplicity [1]. δj corresponds to the frequency +difference between mode j and scattered sidebands, with δo = ωo,p − ωe − ωo and δt/tm = ωo,p + ωe − ωt/tm, which +are mostly given by FSR and ωe mismatch, with additional contributions from optical mode dispersion and residual +optical mode coupling. We note that, for resonant pumping, we have δo = −δt in the case of absent optical mode +dispersion and residual mode coupling. In our experiments, we tune the microwave frequency to match the optical +FSR, i.e. ωe = ωo,p − ωo. + +4 +The equation of motion of all relevant modes may be represented more economically in the form +˙v(t) = M(t)v(t) + Kfin(t), +(8) +where we define the vectors of mode and noise operators +v = (ˆae, ˆa† +e, ˆao, ˆa† +o, ˆat, ˆa† +t, ˆatm, ˆa† +tm)⊤, +fin = (δˆae,0, δˆa† +e,0, δˆae,in, δˆa† +e,in, δˆao,0, δˆa† +o,0, δˆao,in, δˆa† +o,in, δˆat,vac, δˆa† +t,vac, δˆatm,vac, δˆa† +tm,vac)⊤, +(9) +as well as the matrices that encode the deterministic part of the QLE, +M(t) = +� +� +� +� +� +� +� +� +� +� +� +− κe +2 +0 +0 +−ig(t) +−ig∗(t) +0 +0 +0 +0 +− κe +2 ++ig∗(t) +0 +0 +ig(t) +0 +0 +0 +−ig(t) iδo − κo +2 +0 +0 +0 +0 +0 +ig∗(t) +0 +0 +−iδo − κo +2 +0 +0 +0 +0 +−ig(t) +0 +0 +0 +iδt − κt +2 +0 +−iJ +0 +0 +ig∗(t) +0 +0 +0 +−iδt − κt +2 +0 +iJ +0 +0 +0 +0 +−iJ +0 +iδtm − κtm +2 +0 +0 +0 +0 +0 +0 +iJ +0 +−iδtm − κtm +2 +� +� +� +� +� +� +� +� +� +� +� +, +(10) +and +K = +� +� +� +� +� +(1 − ηe)κe √ηeκe +0 +0 +0 +0 +0 +0 +� +(1 − ηo)κo √ηoκo +0 +0 +0 +0 +0 +0 +√κt +0 +0 +0 +0 +0 +0 +√κtm +� +� +� +� ⊗ 12, +(11) +which keeps track on which modes the noise acts. +2. +Input-Output-Theory +In the experiment the pump field is turned on at t = 0 and kept on until τpulse. For the optical pump pulse with +length τpulse =250 ns (600 ns, see main text), we reject a certain τdelay = 50 ns (100 ns) from the beginning of pulse +data. Since κpτdelay ≳ 1 we may assume that after τdelay the system has approached its steady state and especially +that the pump mode is in its steady state. Consequently we may assume that g(t > τdelay) ≃ g is constant over time. +One important figure of merit is the multiphoton cooperativity C = 4g2/κoκe, a measure for coherent coupling versus +the microwave and optical dissipation. Efficient entanglement generation can be achieved with complete anti-Stokes +scattering suppression, while below the parametric instability threshold, i.e. C < 1. +The output fields of the CEO device are +fout(t) = (ˆae,out(t), ˆa† +e,out(t), ˆao,out(t), ˆa† +o,out(t))⊤, +(12) +which consist of a contribution which was entangled via the coherent interactions v and a contribution which has +not interacted with the device fin. The output field fout will then propagate to the measurement device and is most +economically represented within the framework of input-output theory [6], +fout(t) = Lfin(t) − Nv(t), +(13) +where we define the matrices +N = (NJ, 04), +with +NJ = Diag(√ηeκe, √ηeκe, √ηoκo, √ηoκo), +(14) +and +L = +� +0 1 0 0 0 0 +0 0 0 1 0 0 +� +⊗ 12. +(15) +As all modes have reached steady state, the correlations in the output field may be obtained by going to Fourier + +5 +domain. Here we commit to following convention of the Fourier transformation +ˆA(ω) = +1 +√ +2π +� ∞ +−∞ +dω eiωt ˆA(t), +(16) +with the hermitian conjugate +( ˆA(ω))† = A†(−ω). +(17) +Note that in this convention e.g. [ae(ω), a† +e(ω′)] = δ(ω + ω′) are canonical pairs. +In our experiments, we focus on the correlations between the output propagating spectral modes of frequencies +ωe + ∆ωe and ωo − ∆ωo respectively for microwave and optical fields [7, 8]. We note that, due to energy conservation +in the SPDC process, we only focus on microwave and optical photon pairs around resonances with anti-correlated +frequencies, i.e. ∆ωe = ∆ωo = ∆ω. For this reason, we focus on the following vector of output fields in the rotating +frame, +fout(ω) = (ˆae,out(ω), ˆa† +e,out(−ω), ˆao,out(−ω), ˆa† +o,out(ω))⊤, +(18) +in the Fourier domain. From Eq. (8) we obtain +v(ω) = [iωO − M]−1 · K +� +�� +� +=S(ω) +·fin(ω), +(19) +with +O = Diag(1, −1, 1, 1) ⊗ σz. +(20) +Here we defined the vector of modes +v(ω) = (ˆae(ω), ˆa† +e(−ω), ˆao(−ω), ˆa† +o(ω), ˆat(ω), ˆa† +t(−ω), ˆatm(ω), ˆa† +tm(−ω))⊤, +(21) +as well as the vector of input fields +fin(ω) = (δˆae,0(ω), δˆa† +e,0(−ω), δˆae,in(ω), δˆa† +e,in(−ω), δˆao,0(−ω), δˆa† +o,0(ω), δˆao,in(−ω), δˆa† +o,in(ω), +δˆat,vac(ω), δˆa† +t,vac(−ω), δˆatm,vac(ω), δˆa† +tm,vac(−ω))⊤ +(22) +in the Fourier domain. +The output fields (see Eq. (13)) of the CEO device are straight forwardly obtained since in the Fourier domain +Eq. (13) is algebraic, +fout(ω) = Lfin(ω) + Nv(ω) = (L + N · [iωO − M]−1 · K)fin(ω). +(23) +The input noise operator correlations are given by, +⟨fin(ω)f † +in(ω′)⟩ = Dδ(ω + ω′), +(24) +with +D = Diag(¯ne,int + 1, ¯ne,int +� +�� +� +bath:e +, ¯ne,wg + 1, ¯ne,wg +� +�� +� +waveguide:e +, 1, 0 +���� +bath:o +, +1, 0 +���� +detector:o +, 1, 0 +���� +bath:t +, +1, 0 +���� +bath:tm +). +(25) +We note that, in our experiments, the microwave waveguide remains in the ground state, with ¯ne,wg = 0. The spectral +correlations of different output field can be simply obtained analytically from +⟨fout(ω)f † +out(ω′)⟩ = S(ω)DS†(−ω) +� +�� +� +˜ +Cff†(ω) +δ(ω + ω′). +(26) + +6 +Here we implicitly define the 4 × 4 matrix of output mode correlations with a single entry reading +⟨ˆaout(ω)ˆbout(ω′)⟩ = ˜Cab(ω)δ(ω + ω′), +(27) +where the operators ˆaout(ω),ˆbout(ω) were chosen from components of fout(ω) in Eq. (16). +3. +Covariance Matrix of Filtered Output Fields +We will now consider a situation where we define output field modes from a windowed Fourier transformation. Below +we will then show that these are indeed the experimentally observed signals. We start by defining the (dimensionless) +hermitian output field quadrature pair [8], +ˆXα(ωn) = +1 +√ +2T +� T/2 +−T/2 +dτ eiωnτˆaα,out(τ) + h.c., +(28) +ˆPα(ωn) = +1 +√ +2Ti +� T/2 +−T/2 +dτ eiωnτˆaα,out(τ) + h.c., +(29) +which meets the canonical commutation relation [ ˆXα(ωn), ˆPβ(ωm)] = iδnmδαβ where α = e, o. Due to the finite +window of the Fourier transformation, the frequencies ωn = 2π +T n becomes discrete. The quadrature modes at discrete +frequencies ωn can now be rewritten in terms of the (dimensionful) output fields fout(ω) from Eq. (23), which are +defined in the continuous Fourier domain. Therefore the quadrature operators may be obtained by convolution with +the a filter function G(ω) +ˆXα(ωn) = +1 +√ +2 +� ∞ +−∞ +dω G(ωn − ω)ˆaα,out(ω) + h.c. +(30) +ˆPα(ωn) = +1 +√ +2i +� ∞ +−∞ +dω G(ωn − ω)ˆaα,out(ω) + h.c. +(31) +Here the filter is +G(ω) = +1 +√ +2π +� ∞ +−∞ +dτ eiωτ 1[0,T ](τ) +√ +T += +� +2 +πT +sin(ωT/2) +ω +, +(32) +which is obtained from a Fourier transformation of the unit function 1[−T/2,T/2](t) = 1(0) for |t| ≤ T/2 (|t| > T/2). +A bipartite Gaussian state is characterized by the 4 × 4 covariance matrix (CM), +VAB(ωn) = 1 +2⟨{δ ˆA(ωn), δ ˆB(ωn)}⟩. +(33) +Here we defined δ ˆA = ˆA − ⟨ ˆA⟩ an operator with zero mean ⟨δ ˆA⟩ = 0 and the quadratures from +ˆA(ωn), ˆB(ωn) ∈ { ˆXe(ωn), ˆPe(ωn), ˆXo(−ωn), ˆPo(−ωn)} +(34) +and we also introduced the anti-commutator { ˆA, ˆB} = ˆA ˆB + ˆB ˆA. Note that the two-mode squeezing interaction +results in correlation between frequency reversed pairs on the microwave ωn and the optical side −ωn. Since in our +setting all first moments ⟨ ˆA⟩ = 0 the evaluation of the covariance matrix in Eq. (33) boils down to computing spectral +correlations which are rewritten as +⟨ ˆA(ωn) ˆB(ωn)⟩ = +� ∞ +−∞ +dω +� ∞ +−∞ +dω′ G(ωn − ω)G(ωn − ω′)⟨ ˆA(ω) ˆB(ω′)⟩ += +� ∞ +−∞ +dω +� ∞ +−∞ +dω′ G(ωn − ω)G(−ωn − ω′)CAB(ω)δ(ω + ω′) += +� ∞ +−∞ +dω F(ωn − ω)CAB(ω), +(35) + +7 +where we used the property G(−ω) = G(ω) and defined the effective filter F(ω) = G(ω)2. Similar to Eq. (26), we +defined the quadrature correlations +CAB(ω) = (C(ω))AB = 1 +2 +� +U ˜Cff †(ω)U † + (U ˜Cff †(ω)U †)⊤� +AB . +(36) +Here the unitary matrix U = u ⊕ u, with +u = +1 +√ +2 +� +1 +1 +−i i +� +, +(37) +corresponds to a rotation of the mode operators into quadrature operators ( ˆXα, ˆPα)⊤ = u · (ˆaα,out, ˆa† +α,out)⊤. The +covariance matrix of the quadrature modes at the discrete frequencies ωn is then obtained exactly by +VAB(ωn) = +� ∞ +−∞ +dω F(ωn − ω)CAB(ω), +(38) +where the quadrature correlations are convolved with an appropriate filter. +B. +Heterodyne Detection, Added Noise and Filtering +1. +Heterodyne Measurement +Here we discuss the quadrature extractions from the equivalent linear measurement, e.g. +balanced heterodyne +detection, with excess added noise [9]. In the heterodyne detection, the output field ˆaoute−iωjt (j ∈ e, o) is mixed with +a strong coherent local oscillator field ˆaLO(t) = αLOe−iωLOt at a 50:50 beam-splitter, where the output field from the +two ports are sent to a balanced photo-detector, which results in a photon current that is proportional to +ˆIout(t) = e−i∆LOtˆaout + ˆa† +outei∆LOt, +(39) +in the limit of strong LO (αLO ≫ 1) with ∆LO = ωLO − ωj. We consider finite measurement interval of time T, on +which we compute the windowed Fourier transformation of ˆIout(t), +ˆIout(ωn) = +1 +√ +T +� T +0 +dτ eiωnτ ˆIout(τ) = +1 +√ +T +� T +0 +dτ eiωnτ(e−i∆LOτˆaout(τ) + ei∆LOτˆa† +out(τ)) += aout(ωn − ∆LO) + a† +out(ωn + ∆LO), +(40) +where in a slight abuse of notation we define the dimensionless output fields aout(ωn). The reason why we are explicitly +working the windowed Fourier transformation is, that despite being in a steady state during the measurement (see +Sec. I B), the Fourier transformed data has a rather broad bandwidth (δωn = 2π +T ∼ 5MHz for a 200 ns time window) +due to the relatively short time of data collection T = τpulse−τdelay = 200 ns (especially for 250 ns optical pump pulse). +In the limit of long measurement times T → ∞, the bandwidth will tend to zero and the following discussion as well +as the results in Eq. (38) will coincide with standard Input-Output treatment in the continuous Fourier domain. In +our experiments, we extract the quadratures of microwave and optical output field, by decomposing the heterodyne +current spectra, in their real and imaginary parts which yields +ˆIout(ωn) = +1 +√ +2( ˆX(ωn − ∆LO) + ˆX(−ωn − ∆LO) +� +�� +� +ˆIX,out(ωn) ++i [ ˆP(ωn − ∆LO) − ˆP(−ωn − ∆LO)] +� +�� +� +ˆIP,out(ωn) +), +(41) +where we define the quadrature output fields ˆaout(ωn) = ( ˆX(ωn) + i ˆP(ωn))/ +√ +2, in the same way as in Eq. (28-29). +So far we have treated the photon current which result from a heterodyne measurement in terms of a time dependent +hermitian operator ˆIout(t). +In an actual experiment the heterodyne current is a real scalar I(t) quantity which +fluctuates in time and between different experimental runs. Taking taking the (fast) Fourier transform of this current +and decomposing it in its real and imaginary parts then yields I(ωn) = IX(ωn) + iIP (ωn). The theory of continuous +measurements and quantum trajectories [10, 11] tells us how to connect the measured scalar currents with the current + +8 +operators from input-output theory [6] +IA(ωn)IB(ωm) = 1 +2⟨{ˆIA,out(ωn), ˆIB,out(ωm)}⟩, +(42) +where we define the statistical average · · · over many experimental runs. +2. +Realistic Measurements: Added Noise and Gain +For the vacuum, the noise spectral density for both quadratures, are obtained by +SAA(ωn) = ⟨ ˆA(ωn) ˆA(ωn)⟩vac = 1 +2, +(43) +for the hermitian operator ˆA = ˆX, ˆP. Note that due to the discreteness of the Fourier domain we do not have a Dirac +delta as opposed to Eq. (27). The noise spectrum of the heterodyne current is defined by SII(ω) ≡ I(ωn)I(ωn) = +⟨ˆIout(ωn)ˆIout(ωn)⟩, where +SII(ωn) = 1 +2 (SXX (ωn − ∆LO) + SP P (ωn − ∆LO) + SXX (ωn + ∆LO) + SP P (ωn + ∆LO)) . +(44) +Focusing on the part of the spectrum located around ∆LO, +SII(ωn + ∆LO) = 1 +2 (SXX (ωn) + SP P (ωn) + 1) , +(45) +assuming ∆LO ≫ κe, κo. This indicates the simultaneous quadratures measurements and added shot noise in the +heterodyne measurements, even without experimental imperfections. +So far we have focused on the ideal theory +of the measurement and disregarded additional unknown sources of noise as well as the connection to the actually +measured quantities. In practice, the decomposed measured quadratures contain additional uncorrelated excess noise, +e.g. due to the added noise in the amplification or due to propagation losses [12]. We model this by phenomenologically +adding another uncorrelated noise process from an independent thermal reservoir and then multiplying by a gain factor +which converts the number of measured photons to the actually monitored voltage. To illustrate this we focus again +on a single output port, and with the added noise current ˆIX/P,add(ωn) and the frequency dependent calibration gain +Gdet(ωn), where +ˆIX,det(ωn) = +� +Gdet(ωn)(ˆIX,add(ωn) + ˆIX,out(ωn)), +(46) +ˆIP,det(ωn) = +� +Gdet(ωn)(ˆIP,add(ωn) + ˆIP,out(ωn)). +(47) +We thus obtain the detected heterodyne noise spectral density, +SII,det(ωn + ∆LO) =Gdet(ωn + ∆LO)[SXX (ωn) + SP P (ωn) ++ 1 + SIXIX,add(ωn + ∆LO) + SIP IP ,add(ωn + ∆LO) +� +�� +� +=2Nadd +], +(48) +where we define the spectra of the added noise SIOIO,add(ωn) = ⟨ˆIO,add(ωn)ˆIO,add(ωn)⟩. +The added noise Nadd +includes the excess vacuum noise from heterodyne measurement and the additional uncorrelated noise. Note that here +the factor 1 +2 was absorbed in the detections gains. The gain Gdet(ωn) can be simply obtained on both microwave and +optical side, from the cold measurements (optical pump off) with a known background. We note that, Eq. (48) lays +the foundation of microwave and optical calibrations in our CEO device. +In our experiments, we place the LO on opposite sites around the mode resonances, i.e., +∆LO,e = −ΩIF, +∆LO,o = ΩIF, +(49) +where ΩIF > 0 is the intermediate frequency for down-mixing. The heterodyne output field can be obtained similar + +9 +to Eq. (40), +ˆIout,e(ωn + ΩIF) = +1 +√ +2[( ˆXe(−ωn) + ˆXe(ωn + 2ΩIF) + i(− ˆPe(−ωn) + ˆPe(ωn + 2ΩIF))], +ˆIout,o(ωn + ΩIF) = +1 +√ +2[ ˆXo(−ωn − 2ΩIF) + ˆXo(ωn) + i(− ˆPo(−ωn − 2ΩIF) + ˆPo(ωn))], +(50) +with noise spectrum given by, +SII,e(ωn + ΩIF) = 1 +2(SXeXe (−ωn) + SPePe (−ωn)) + Ne,add, +SII,o(ωn + ΩIF) = 1 +2(SXoXo (ωn) + SPoPo (ωn)) + No,add. +(51) +We note that, Eq. (50) is adopted for field quadrature extraction (including the added noise) from the heterodyne +measurement, which reveals correlations in the quadrature histogram [cf. Fig.4 in the main text]. Despite of the +reversed sign in the expected field quaduratures, microwave and optical output photons appear at the same frequency +in the noise spectrum, i.e. ωn + ΩIF [cf. Fig.2 d,e in the main text]. +3. +Covariance Matrix from Realistic Heterodyne Measurements +Here we briefly explain the procedure of the covariance matrix reconstruction from the heterodyne measurements. +The cross correlations of the detected heterodyne current spectra can be obtained via, +DAB(ωn) = δIA,det(ωn + ΩIF)δIB,det(ωn + ΩIF), +(52) +where we define the centered current δIO,det = IO,det − IO,det, with +IO,det(ωn) ∈ {IXe,det(ωn), IPe,det(ωn), IXo,det(ωn), IPo,det(ωn)}. +(53) +Similar to Eq. (42)), we can obtain +DAB(ωn) = 1 +2⟨{δ ˆIA,det(ωn + ΩIF), δ ˆIB,det(ωn + ΩIF)}⟩ += +� +GA,det(ωn + ΩIF))GB,det(ωn + ΩIF)) +�1 +2⟨{δ ˆA(ωn), δ ˆB(ωn)}⟩ +� +�� +� +=VAB(ωn) ++NAB,add +� +, +(54) +where we define the diagonal added noise matrix NAB,add = (Nadd)AB = NA,addδAB with the calibrated added noise +Nadd and detection gain GA,det. +This equation establishes how the covariance matrix of the qudrature operators [cf. +Eq. (38)] is reconstruced from heterodyne measurements, and how they can be compared with the results from idealized +standard input-output theory Eq. (33). For simplicity, in the main text we define the total measured covariance matrix +including the added noise as, +VAB,meas(ωn) = DAB(ωn)/ +� +GA,det(ωn + ΩIF))GB,det(ωn + ΩIF)), +(55) +with VAB,meas(ωn) = VAB(ωn) + NAB,add. +We note that, in principle the location of both LOs can be arbitrary. As evident in Eq. 54, our choice of the LO +configuration, i.e. ∆LO,e = −∆LO,o = −ΩIF, offers a simple solution to the quantification of the broadband quantum +correlations, considering the limited detection bandwidth, frequency dependent gain, or microwave cavity frequency +shift, which may result in the loss of quantum correlations during quadrature extractions in heterodyne measurements +due to imperfect frequency matching. + +10 +C. +Entanglement Detection +1. +Duan Criterion +We will now discuss how show that the photons outgoing microwave and optical photons are indeed inseparable +or entangled. Our starting point is the covariance matrix which we defined in Eq. (33) and measured as outline in +Eq. (54). The experimentally measured covariance matrix is of the form +V = +� +Ve +Veo +Veo +Vo +� += +� +� +� +� +V11 +0 +˜V13 +˜V14 +0 +V11 +˜V14 − ˜V13 +˜V13 +˜V14 +V33 +0 +˜V14 − ˜V13 +0 +V33 +� +� +� +� . +(56) +Since there is no single mode squeezing we have V22 = V11 and V44 = V33. +For simplicity we have omitted the +frequency argument ωn of component. What we describe in the following will have to be repeated for every frequency +component. The off-diagonal part in the covariance matrix which encodes the two-mode squeezing can be written as +Veo ≃ V13(sin(θ)σx + cos(θ)σz), +(57) +where we define V13 = ( ˜V 2 +14 + ˜V 2 +13)1/2 and the mixing angle tan(θ) = ˜V14/ ˜V13. In our experimental setting ˜V14 maybe +non zero e.g. due to small finite detunings δo. For the detection of inseparability, we employ the criterion introduced +by Duan, Gidke, Cirac and Zoller [13]. This criterion states that if one can find local operations Uloc = Ue ⊗ Uo such +that the joint amplitude variance of ˆX+ = ( ˆXe + ˆXo)/ +√ +2 break the inequality, +∆X2 ++ = ⟨U † +loc ˆX2 ++Uloc⟩ < 1/2, +(58) +then the state is inseparable and, thus it must be concluded that it is entangled. +In this setting, it is enough to choose the local operations Uloc = UeUo to be a passive phase rotation on the optical +mode only, with Ue = 1 and Uo = e−iϕˆa† +oˆao, and phase rotation angle ϕ. In the space of covariance matrices, this +corresponds to the (symplectic) transformation Sϕ = 12 ⊕ Rϕ, where we define the rotation matrix, +Rϕ = +� +cos (ϕ) +sin (ϕ) +− sin (ϕ) cos (ϕ) +� +. +(59) +The local rotation of the phase V (ϕ) = SϕV S⊤ +ϕ will act on the off diagonal part of the covariance matrix as, +Vea(ϕ) = V13(cos(θ − ϕ)σz + sin(θ − ϕ)σx). +(60) +With these local rotations the joint amplitude variance becomes +∆X2 ++(ϕ) = ⟨( ˆXe + ˆXo cos(ϕ) + ˆPo sin(ϕ))2⟩/2 = V11 + V33 + 2V13 cos(θ − ϕ). +(61) +We can similarly define the joint quadrature ˆP− = ( ˆPe − ˆPo)/ +√ +2, where ∆P 2 +−(ϕ) = ∆X2 ++(ϕ). The variance of the +joint quadratures ∆X2 ++(ϕ) and ∆P 2 +−(ϕ) is minimized at the angle ϕ− = θ − π +∆− +EPR = ∆X2 ++(ϕ−) + ∆P 2 +−(ϕ−) = 2(V11 + V33 − 2V13), +(62) +which corresponds to the two-mode squeezing of microwave and optical output field, and the microwave-optics entan- +glement. In addition, the joint quadrature variance is maximized at the angle ϕ+ = θ and we obtain +∆+ +EPR = ∆X2 ++(ϕ+) + ∆P 2 +−(ϕ+) = 2(V11 + V33 + 2V13), +(63) +which corresponds to the anti-squeezing. + +11 +2. +Logarithmic Negativity and Purity +A mixed entangled state can be quantified by the logarithmic negativity, +EN = max [0, − log (2ζ−)] , +(64) +where ζ− is the smaller symplectic eigenvalue of the partially time reverse covariance matrix and can be obtained +analytically +ζ2 +− = S − +� +S2 − 4det(V ) +2 +(65) +where we defined the Seralian invariant S = det(Ve) + detVo + 2det(Veo). Furthermore the purity of a bipartite +Gaussian state is given by +ρ = +1 +4 +� +det(V ) +, +(66) +with ρ = 1 for a pure state i. e. the vacuum state. +II. +EXPERIMENTAL SETUP +The experimental setup is shown and described in SI Fig. 1. The laser is split into three parts, including an optical +pulsed pump at frequency ωp, a continuous signal at ωp −FSR for the 4-port calibration (cf. SI III), and a continuous +local oscillator (LO) at ωp − FSR + ΩIF for the optical heterodyne detection. The optical signal and pump pulse are +sent to the optical resonator of the electro-optical device (DUT) and the reflected light (with pump pulse rejected +by a filter cavity) is combined on a 50:50 beam splitter with the optical local oscillator with subsequent balanced +photodetection. +Microwave input signals are attenuated at different temperature stages of the dilution refrigerator (4 K: 20 dB, +800 mK: 10 dB, 10 mK: 20 dB), and sent to the coupling port of the microwave cavity of the DUT. The reflected +microwave signal is amplified and can then either be mixed with a microwave local oscillator of frequency FSR − ΩIF +and subsequently digitized, or directly measured by a vector network analyzer or a spectrum analyzer. +We note that, the optical LO is on the right side of optical mode, while the microwave LO is on the left side of the +microwave mode, with ΩIF/2π = 40MHz. More details are in the caption of Supplementary Fig. 1. +III. +SETUP CHARACTERIZATION AND CALIBRATION +In the main manuscript, we show results from two different sets of optical modes shown in Fig. 2. +The main +difference between these mode sets is the amount of suppression of the anti-Stokes scattering rate compared to Stokes +scattering rate given by scattering ratio S, which depends on the mode hybridisation of the anti-Stokes mode [5, 14]. +The first set of optical mode (Fig. 2a) from which we show most of our main results (main text Fig. 2, 3 and 5a) has +S =−10.3 dB on-resonance with an effective FSR = 8.799 GHz. The last power sweep shown in main text Fig. 5b +is measured with a second set of optical modes with a lower S =−3.1 dB and a different effective FSR = 8.791 GHz +(Fig. 2b). Despite it being the same optical resonator, the FSR for the second set of optical modes is slightly different, +because of partial hybridisation of the optical pump mode which alters the working FSR between the optical pump +and signal mode, see Fig. 2. +In the following, we carefully calibrate the added noise due to the microwave detection chain at both these working +FSRs (since microwave mode is parked at the working FSR). The added noise can be slightly different depending +on frequency of measurement due to impedance mismatch and reflections between components in the microwave +detection. +A. +Microwave added noise calibration +In the following, we carefully calibrate the slightly different added noise in the microwave detection chain at both +frequencies.The impedance mismatch and reflections between components in the microwave detection chain can vary + +12 +Dilution refrigerator +Microwave preparation +Optics preparation +Optics and microwave detection +MC1 +MS2 +MS3 +C1 +C2 +C3 +C5 +C4 +10mK +800mK +4K +300K +DC Microwave signal +(FSR) +DC Optical signal +(193.5 THz - FSR) +Optical pump +(193.5 THz) +Coherent optical signal +(193.5 THz - FSR) +Optical LO +(193.5 THz -FSR - ωlo) +Microwave LO +(FSR + ωlo) +IF +(ωlo) +RF lines +DUT +VNA +SA +DIGITIZER +LNA +RTA2 +RTA1 +Q +I +1 +1 +2 +90 +10 +1 +1 +2 +3 +3 +4 +PC1 +S3 +S4 +MS1 +DDG +25 +75 +BPD +Laser +Lock control +1550 nm +SSB +VOA1 +99 +1 +EDFA1 +EDFA2 +AOM1 +AOM2 +PC2 +PD2 +PD3 +PD4 +F3 +F1 +F2 +PD1 +S1 +S2 +ΩLO +OSA +PM +HEMT +Supplementary Fig. 1. Experimental setup for two-mode squeezing measurements. A tunable laser at frequency ωp +is initially divided equally in two parts, i.e. the optical pump and the optical signal together with the optical local oscillator +(LO). Light from the optical pump path is pulsed via an acousto-optic modulator (AOM1) which produces ns-pulses and shapes +them for amplification via an Erbium-doped fiber amplifier (EDFA). The output from the EDFA is first filtered in time via +AOM2 to remove the amplified spontaneous emission (ASE) noise and later in frequency via filter F1 (∼50 MHz linewidth with +15 GHz FSR) to remove any noise at the optical signal frequency (the reflected power is rejected by circulator C3). The filter +F1 is locked to the transmitted power by taking 1% of the filter transmission measured via photodiode PD3. The polarization +of the final output is controlled via polarization controller PC1 before being mixed with the optical signal via a 90-10 beam +splitter and sent to the dilution refrigerator (DR). The 10% output from the beam splitter is monitored on a fast detector PD2 +to measure the optical pump pulse power. The other half of the laser is again divided into two parts - 25% for the optical signal +and 75% for the optical LO. The signal part is sent first to a variable optical attenuator VOA1 to control the power and then +to a single sideband modulator SSB which produces the optical signal frequency at ωp − FSR and suppresses the tones at ωp +and ωp + FSR. 1% of the optical signal is used to monitor the SSB suppression ratio via an optical spectrum analyzer OSA +and 99% is sent to the DR after being polarization controlled via PC2. The optical LO is produced via a phase modulator PM +and detuned by ωIF/2π = 40 MHz. As the PM produces many sidebands, the undesired sidebands are suppressed via filter F3 +(∼50 MHz linewidth with 15 GHz FSR), reflection is rejected by circulator C5. F3 is temperature-stabilized and locked to the +transmitted power similar to F1. The optical LO is also amplified via EDFA2 before the optical balanced heterodyne. In the +DR, the light is focused via a gradient-index (GRIN) lens on the surface of the prism and coupled to the optical whispering +gallery mode resonator (WGMR) via evanescent coupling. Polarization controllers PC1 and PC2 are adjusted to efficiently +couple to the TE modes of the optical WGMR. The output light is sent in a similar fashion to the collection grin lens. Outside +the DR, the optical pump is filtered via filter F2 (similar to F3). The reflected light from F2 is redirected via C4 to be measured +with PD1 which produces the lock signal for the laser to be locked to optical WGMR. The filtered signal is finally mixed with +the optical LO and measured with a high speed balanced photo-diode BPD (400 MHz). The electrical signal from the BPD is +amplified via RTA1 before getting digitized. On the microwave side, the signal is sent from the microwave source S3 which is +connected to the DDG for accurately timed pulse generation (or from the VNA for microwave mode spectroscopy) to the fridge +input line via the microwave combiner (MC1). The input line is attenuated with attenuators distributed between 4 K and 10 mK +accumulating to 50 dB in order to suppress room temperature microwave noise. Circulator C1 and C2 shield the reflected tone +from the input signal and lead it to the amplified output line. The output line is amplified at 4 K by a HEMT-amplifier and +then at room temperature again with a low noise amplifier (LNA). The output line is connected to switch MS1 and MS2, to +select between an ESA, a VNA or a digitizer measurement via manual downconversion using MW LO S4 (40 MHz detuned). +Lastly, microwave switch MS3 allows to swap the device under test (DUT) for a temperature T50 Ω controllable load, which +serves as a broad band noise source in order to calibrate the microwave output line’s total gain and added noise. +the added noise slightly as a function of frequency. This added noise and corresponding gain due to a series of amplifiers + +13 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +a +b +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +|Soo|2 +|Soo|2 +ωp ++FSR +ωp +-FSR +ωp +ωp ++FSR +ωp +-FSR +ωp +Supplementary Fig. 2. Optical mode spectra in reflection. Normalized reflection intensity |Soo|2 spectra of optical modes +ˆao, ˆap and ˆat in red, green and blue respectively. a (b) shows the optical mode spectra of the first (second) set of modes with +the anti-Stokes and Stokes scattering ratio S = −10.3 dB (−3.1 dB). The dashed line marks the effective FSR between the +pump mode ˆap and the optical mode ˆao. The participation anti-Stokes optical mode ˆat is suppressed for this effective FSR as +marked by the dashed line over the blue mode. +and cable losses in the microwave detection chain is calibrated using a combination of a 50 Ω load, a thermometer +and a resistive heater that are thermally connected. The microwave detection chain is identical for the signals from +the 50 Ω load and the microwave cavity reflection, except for a small difference in cable length which we adjust for. +To calibrate the detection chain, we heat the 50 Ω load with the resistive heater and record the amplified noise +spectrum P50Ω(ω) as a function of temperature of 50 Ω load T50Ω. The output noise detected over a bandwidth B, +P50Ω, as a function of T50Ω is given as, +P50Ω = ℏωeGB +�1 +2 coth +� +ℏωe +2kBT50Ω +� ++ Ne,add +� +, +(67) +with ωe the center microwave frequency, Ne,add (G) the added noise (gain) of the microwave detection chain, and kB +the Boltzmann constant. +A bandwidth of 11 MHz is selected around the region of interest to calculate Ne,add and G. For ωe = 8.799 GHz, we +show the detected noise Ne,det = P50Ω/(ℏωeGB) as a function of T50Ω in SI Fig. 3 along with a fit using Eq. 67, with +two fitting parameters G and Ne,add. We note that, at T50Ω = 0 K, Ne,det = Ne,add + 0.5. Table 1 (third row) shows +the obtained added noise and gain for two frequencies of interest, i.e. ωe/2π = 8.799 GHz and ωe/2π = 8.791 GHz. +Next, we consider the difference in cable losses between the 50 Ω load and the microwave cavity, which are in- +dependently determined by measuring the microwave reflection from the microwave cavity and from the microwave +switch directly before it. Including the cable losses, the effective added noise increases while the gain decreases for +the reflected microwave detection, shown in Table 1 (fourth row). +Finally, we consider an additional error due to the temperature sensor inaccuracy of 2.5%. Although this does not +change the final Ne,add and G, it increases the uncertainty as shown in Table 1 (fifth row). The error calculated in +this section contributes to the systematic error reported in the main text. + +14 +Supplementary Tab. 1. The added noise and gain in microwave detection chain (1σ errors shown) +8.799 GHz +8.791 GHz +Detection Chain +Ne,add +G (dB) +Ne,add +G (dB) +50Ω load +(with fitting error) +11.74 ± 0.08 66.67 ± 0.02 11.76 ± 0.09 66.72 ± 0.03 +MW cavity +(including cable loss) +13.09 ± 0.09 66.20 ± 0.02 13.16 ± 0.10 66.23 ± 0.03 +MW cavity +(including temperature sensor uncertainty) 13.09 ± 0.33 66.20 ± 0.12 13.16 ± 0.34 66.23 ± 0.12 +0 +0.5 +1.0 +1.5 +13 +12 +14 +15 +16 +Experiment +Fit +(K) +Tf +Ne,det (photons s-1 Hz-1) +Supplementary Fig. 3. Characterization of the added noise in the microwave detection chain. Measured output +noise from a 50 Ω calibration load as a function of its temperature Tf. The measured noise is plotted in units of photons as +N 50Ω +det = P50Ω/(ℏωeGB). The dashed line at the bottom represents the fitted vacuum noise level in addition to the added noise. +The red line and shaded region represents the fit and the 95% confidence interval around it. +B. +Optical added noise +Optical added noise is calculated via 4-port calibration of our device [14]. In this calibration, we measure the +coherent response of our device through its 4 ports - optical input/output and microwave input/output. Sending +an optical (or microwave) signal to the DUT in combination with a strong pump leads to stimulated parametric +down-conversion (StPDC) process, which generates an amplified microwave (optical) coherent signal. We measure the +4 S-parameters of our device - microwave reflection (S11), optics reflection (S22), microwave to optics transmission +(S21) and optics to microwave transmission (S12). The mean transduction efficiency between microwave and optics +of the DUT is then calculated as, +η = +� +S12S21 +S11S22 +. +(68) +We use the transduction efficiency and Ne,add in the microwave detection chain from Sec. III A to calculate the +optical added noise. Ne,add is firstly used to calculate the effective microwave detection gain (different from the one +in Sec. III A, because the microwave detection line used for the 4-port calibration uses analog downconversion and +digitization, while the thermal calibration uses SA, see Fig. 1). The microwave gain, along with the (off-resonant) +microwave reflection measurement, is used to calculate the microwave input loss. +We can obtain the microwave +signal power at the DUT, which allows us to calculate the output optical power of the DUT using the transduction +efficiency. In conjunction with the measured output power at the end of the detection chain, the losses in the optical +detection path and hence, the effective added noise with respect to the optical port of the DUT can be calculated. +The calculated optical added noise is No,add = 5.54 ± 0.21(7.42 ± 0.22) for ωe = 8.799 GHz (8.791 GHz). + +15 +IV. +DATA TREATMENT +In this section, we describe all the steps for the data treatment in detail, which includes the time domain analysis +(Sec. IV A), the pulse post-selection due to setup drift (Sec. IV B), the frequency domain analysis (Sec. IV C), and +the quadrature correlations (Sec. IV D). +A. +Time-domain analysis +Both microwave and optical signals are detected via heterodyne detection by mixing with a strong local oscillator +that is ∼40 MHz detuned from respective mode resonance. The output heterodyne signals are digitized using a digitizer +at 1 GigaSamples/second. First, we digitally downconvert the digitized data at ωIF = 40 MHz. This yields the two +quadratures IXe/o,det(t) and IPe/o,det(t) of the microwave or optical output signal record with 40 MHz resolution +bandwidth (using 25 ns time resolution). +Supplementary Fig. +4 shows the calibrated output power (I2 +Xe/o,out + +I2 +Pe/o,out) [cf. +Eq. 50] and the phase (arctan(IXe/o,out/IPe/o,out)) from a single pulse sequence. +This includes the +stochastic SPDC signals from a strong pump pulse, and the coherent StPDC signal from a weaker pump pulse +together with a coherent microwave signal for calibration purposes. The SPDC signal produced by the first strong +pulse is labeled by the shaded region for one single pulse, and the averaged output power over many pulses is shown +in main text Fig. 2. The coherent microwave reflection and stimulated parametric downconverted optical signal are +adopted to obtain the phases during the pulse. We record this measured phase in both signal outputs during the +second optical pump pulse for phase-drift correction in later post processing. +0 +1 +2 +3 +4 +5 +0 +1 +2 +3 +4 +5 +1.0 +0 +200 +400 +0 +50 +100 +150 +200 +600 +800 +1000 +1200 +0.5 +0.0 +0.5 +-1.0 +1.0 +0.5 +0.0 +0.5 +-1.0 +a +b +(µs) +Phase (rad/π) +Phase (rad/π) + (photons/s/Hz) +Ne, + (photons/s/Hz) +No, +t +(µs) +t +Supplementary Fig. 4. Downconverted output signal for a single measured pulse sequence. a (b) show the measured +microwave (optical) output signal downconverted at 40 MHz. The shaded part in each case shows the region of the SPDC +signal (the first optical pump pulse). For a single pulse, the SNR of a SPDC signal is too small to be seen. However, during +the second optical pump pulse, a coherent response is seen in both signal outputs where the phase can be measured with high +SNR for each single shot. +In order to determine the accuracy of a phase correction for the first pump pulse based on the phase measurement +during the second pump pulse, we send a continuous microwave signal during both pump pulses and recorded the +phase of the converted optical pulse during the first and the second optical pump pulse. Supplementary Fig. 5 shows +the phase difference between the first and second optical pump pulse for 2500 trials along with a normal distribution +fit. +The fit variance for the distribution is 0.17 rad. +On a similar set of model data, applying a random phase +variation of 0.17 rad results in about 1.5-2.0% loss of correlations [cf. Sec. IV D], whereas, we observe about 6-8% loss +of correlations in the experiments. The imperfection in phase correction does not completely explain the decreased +quantum correlations, which might be due to other experimental instabilities, especially the optical pump laser lock. + +16 +0.0 +0.2 +0.4 +0.6 +0.8 +0.0 +0.2 +-0.2 +0.4 +-0.4 +0.6 +-0.6 +1.0 +Probability density +Phase difference (radians) +Supplementary Fig. 5. Accuracy of phase correction scheme. The histogram shows the difference in the measured phase +between the first and second optical pump pulse. Since we correct the phase in the first pump pulse based on the measured +optical phase of the second optical pump pulse, the difference shows the limitations of this method. The grey dashed line is a +normal distribution fit with variance 0.17 rad. +B. +Pulse post-selection +In our experiments, we use three temperature-stabilized optical filters, which may drift slowly in time. Two of +them are used in the optical heterodyne detection. In the signal path, one filter (F2 in Fig.1) is used to filter the +optical signal while reject the strong optical pump. In the LO path, one filter (F3 in Fig.1) is used to obtain a clean +optical LO tone, which is genearted by an electro-optic phase modulator (which produces multiple sidebands) and +then amplified using an EDFA (which produces excess amplified spontaneous emissions). +The slow filter drifts can be identified from the amplitude of the coherent optical signal produced via stimulated +parametric downconversion during the second optical pump pulse, which drops due to either the decreased transmission +after F2 or the reduced LO power after the F3. This is evident in the histogram of the converted optical power during +the second optical pump pulse as shown in Fig. 6a. The histogram is not symmetric and has a tail at the lower end. +To filter out the instances of drifted heterodyne detection, we select a threshold (in this case marked by a dashed +line in Fig 6) and remove all pulses below the selected threshold along with 20 neighboring pulses (10 s in total time) +before and after such instance. These numbers are chosen according to the filter drift and the filter temperature lock +time-scales. After such filtering, usually about 10% of the data is removed and the histogram of the converted optical +power during the second optical pump pulse becomes symmetric as shown in Fig. 6b. +C. +Frequency domain analysis +As already mentioned in the main text, with the help of time-domain analysis, we select three different time-snippets +to analyze the data in the frequency domain - before-pulse, on-pulse and post-pulse defined with respect to the first +optical pump pulse. The main challenge in processing the data in frequency domain is the proper normalization of +the measured output spectrum [cf. Eq. 48]. The microwave reflection baseline is not flat because of slight impedance +mismatches between different components in the microwave detection chain, with similar optical heterodyne shot noise +floor due to the frequency dependent balanced detector gain. In addition, we observe slight shift of a few millivolts each +time in the digitizer measurements when a new measurement is launched and the digitizer is reinitialized. Combined +with the fact that the amplifier gain in the microwave detection chain as well as the optical heterodyne gain (due to +optical LO power drift) may drift over a long time, an in-situ calibration of vacuum noise level is needed. +In case of microwave, we need to first correct for the microwave reflection baseline distortion from impedance +mismatch and then correct for the signal level shift caused by the digitizer. For the distorted baseline, we separately +measure the microwave output spectrum when the microwave cavity is in its ground state (thermalized to 7 mK at +mixing chamber). This measurement is shown in Supplementary Fig 4a (gray) along with the measured before-pulse +(cyan), on-pulse (purple) and after-pulse microwave noise spectrum (orange). Dividing the measured spectra with +the cold cavity spectrum reveals a flat baseline Lorentzian noise spectra, however with an offset due to the digitizer +drift. To correct for this offset, we perform an in-situ vacuum noise calibration using the off-resonance (waveguide) +noise in the before-pulse microwave noise spectrum. An independent measurement of the microwave waveguide noise +as a function of the average optical pump power (averaged over the full duty cycle) is shown in Supplementary Fig + +17 +50 +0 +100 +150 +200 +0.00 +0.01 +0.02 +50 +0 +100 +150 +200 +0.00 +0.01 +0.02 +a +b +Optics quanta (photons s-1 Hz-1) +Optics quanta (photons s-1 Hz-1) +Count (Normalized) +Count (Normalized) +Supplementary Fig. 6. Histogram of the converted optical power. The measured coherent optical power during the +second optical pump pulse depends on the optical heterodyne gain and the received optical signal power. Both of these values +can drift depending on the experimental setup’s stability. a shows the normalized histogram of this measured optical power +over all the collected pulses. The histogram has a tail on the lower end owing to the times when the heterodyne setup drifted. +Filtering the points which do not meet a selected threshold (shown by the grey dashed line in a), we remove the instances where +the setup had drifted and the optical heterodyne detection efficiency was compromised. The same histogram after removing +such points is shown in b. +8. The error bars (2σ deviation) result from the microwave detection chain gain and the measurement instrument +drift. The power law fit reveals that the microwave waveguide noise grows almost linearly with average optical pump +power, and only deviates significantly from 0 for optical pump power >3 µW. As we work with average optical pump +powers of ≪1 µW, we can safely assume the microwave waveguide noise to be zero. Therefore, we use the off-resonant +waveguide noise for before-pulse microwave noise spectrum as an in-situ vacuum noise calibration. +In case of optics, the optical detection is shot-noise limited, and the excess LO noise at the optical signal frequency +is suppressed by more than 40 dB using filter F1 in Fig.1. We use the before-pulse optical noise spectrum as the +vacuum noise level and normalize the optical on-pulse spectrum directly with the before-pulse in-situ calibration. +Supplementary Fig 4b shows noise spectrum (without normalization) of the optical off-pulse (cyan), on-pulse (green), +and the after-pulse (yellow). The signal during the optical pump pulse is clearly visible, and the noise level is identical +before and after the optical pulse. +The normalized noise spectra for both microwave and optics are shown in main text Fig. 2d and 2e., where we can +obtain the normalization gain [cf. Eq. 48]. +D. +Joint-quadrature correlations +The detected output quadratures including excess added noise, i.e. +ˆIXe,out(∆ω), +ˆIPe,out(∆ω), +ˆIXo,out(∆ω), +ˆIPo,out(∆ω), can be obtained from the real and imagrinary parts in the discret Fourier transform of the photocurrent +by normalizing to the detection gain [cf. Eq. 50]. +Similar to Sec. I C 1, we can define the joint detected quadratures, by applying phase rotation on the optical ones, +ˆIX,+(∆ω, φ) = +ˆIXe,out(∆ω) + +� +ˆIXo,out(∆ω) cos φ − ˆIPo,out(∆ω) sin φ +� +√ +2 +, +ˆIP,−(∆ω, φ) = +ˆIPe,out(∆ω) − +� +ˆIXo,out(∆ω) sin φ + ˆIPo,out(∆ω) cos φ +� +√ +2 +. +(69) +To verify the non-classical correlation between the unitless quadrature variables for output microwave and optics +field, i.e. +ˆXe(∆ω) & ˆXo(−∆ω) and ˆPe(∆ω) & ˆPo(−∆ω), we can calculate the phase dependent joint quadrature + +18 +40 +20 +60 +59 +60 +61 +0.30 +0.32 +0.16 +0.18 +0.20 +0.22 +0.24 +0.26 +0.28 +62 +63 +64 +65 +a +b + (MHz) +40 +20 +60 +MW output (nW/2MHz) +Opt output (µW/2MHz) +Off-pulse +In-pulse +Off-pulse +In-pulse +After-pulse +After-pulse +Cold cavity +ω/2π + (MHz) +ω/2π +Supplementary Fig. 7. +Spectra of output signals. +The microwave reflection baseline is not flat due to an impedance +mismatch between different components in the microwave detection chain. +As a result, the output power measured from +amplified vacuum noise (from the cold microwave cavity) is not flat (shown in gray in a). Additionally, the digitizer in our +setup has a different noise level each time it is started. As a result, the cold cavity baseline has an extra offset with respect to +all other measurements. a also shows the measured output spectra for time region before (during, after) the first optical pulse +shown in cyan (purple, orange). Similarly, b shows the output spectra for the optical output before (during, after) the first +optical pulse in cyan (green, orange). +100 +101 +0.0 +0.2 +0.4 +Experiment +0.01Pavg +0.97 +Pavg (µW) +Wavegide noise (photons/s/Hz) +Supplementary Fig. 8. Microwave waveguide noise as a function of the average optical pump power. The error bars +represent 2σ error. The solid line is a power law fit. We find the power law is actually quite close to a linear function. +variance [cf. Eq. 61], +� +ˆX2 ++(∆ω, φ) +� += +� +ˆI2 +X,+(∆ω, φ) +� +− Ne,add + No,add +2 +, +� +ˆP 2 +−(∆ω, φ) +� += +� +ˆI2 +P,−(∆ω, φ) +� +− Ne,add + No,add +2 +. +(70) +For ∆ω = 0, we plot the joint quadrature variance as a function of local oscillator phase in Fig. 9 (a). The shaded +region represent the 2σ statistical error in the calculated joint quadrature variances. We note that, the statistical 1σ +error of the variance for a Gaussian distributed data is given by +√ +2σ2/ +√ +N − 1, where N is the length of the dataset. +The obtained resonant ∆EPR(0, φ) is shown in Fig. 9(b). The minimum and maximum of ∆EPR(φ) over the local +oscillator phase are defined as min[∆EPR] = ∆− +EPR and max[∆EPR] = ∆+ +EPR. ∆− +EPR < 1 indicates non-classical joint +correlations and squeezing below vacuum levels. + +19 +-1.0 +-0.5 +0.0 +0.5 +1.0 +0.5 +1.0 +1.0 +2.0 +1.5 +2.5 +a +b +Quanta +Phase (rad/π) +-1.0 +-0.5 +0.0 +0.5 +1.0 +Phase (rad/π) +ΔEPR +X+ +P+ +Supplementary Fig. 9. Joint quadrature correlations and ∆EPR. +a. Joint quadratures at resonance X+(∆ω = 0) and +P+(∆ω = 0) are plotted as a function of the local oscillator phase φ. b. ∆EPR as a function of φ. The shaded region in both +plots represents the 2σ statistical error. +The broadband phase that minimizes ∆EPR(∆ω, φ), i.e. φmin(∆ω), reveals the difference in arrival times (group +delay) between the microwave and optical signal output (Supplementary Fig. 10a). After fixing the inferred time delay +between the in-pulse arrival time of the microwave and optical signal, φmin becomes independent of frequency detuning +from the mode resonances. Thus, we adjust for the differences in arrival times by ensuring that the slope of φmin with +respect to detuning ∆ω is minimized for all datasets we analyze, utilizing the broadband quantum correlations. +-1.0 +-0.5 +0.0 +0.5 +1.0 +a +b +0 +-20 +-30 +-10 +20 +10 +30 +(MHz) +-1.0 +-0.5 +0.0 +0.5 +1.0 +0 +-20 +-30 +-10 +20 +10 +30 + (rad/π) +φ + (rad/π) +φ +∆ω/2π +(MHz) +∆ω/2π +Supplementary Fig. 10. Effect of time delay between the microwave and optics signals. The plots show the local +oscillator phase φmin which minimizes ∆EPR(∆ω, φ) as a function of detuning frequency ∆ω. a (b) shows the case when the +time difference of arrival between the microwave and optics signals was 25 ns (≈ 0 ns). The solid lines are the linear fit to the +experimental data. +V. +QUADRATURE HISTOGRAM RAW DATA +Fig. 11 shows the normalized difference of the two-variable quadrature histograms obtained during and before the +optical pump pulse based on the data shown in Figs. 2 and 3 of main text. These unprocessed histograms directly +show the phase insensitive amplification in each channel as well as the correlations in (Xe,Xo) and (Pe,Po). Note +however that - in contrast to the analysis in the main text - taking this difference does not lead to a valid phase space +representation since also the vacuum noise of 0.5 together with the output noise of 0.026 ± 0.011 photons (due to the +residual microwave bath occupancy right before the pulse) are subtracted, hence the negative values. + +20 +20 +10 +0 +-10 +-20 +20 +10 +0 +-10 +-20 +20 +10 +0 +-10 +-20 +1.0 +0.5 +-1.0 +-0.5 +0.0 +10 +0 +-10 +10 +0 +-10 +10 +0 +-10 +-20 +-10 +0 +10 +-10 +0 +10 +-10 +0 +10 +-10 +0 +10 +20 +-20 +-10 +0 +10 +20 +-20 +-10 +0 +10 +20 +Pe +Pe +Pe +Po +Po +Po +Xo +Xo +Xo +Xe +Xe +Xe +Supplementary Fig. 11. Quadrature histogram raw data. Normalized difference of the two-variable quadrature histograms +obtained during and before the optical pump pulse based on the data shown in Figs. 2 and 3 of the main text. +VI. +NON-CLASSICAL CORRELATIONS WITH 600 ns LONG OPTICAL PUMP PULSES +Before experimenting with 250 ns long optical pump pulses, we used 600 ns long optical pump pulses. A sample +measurement with a 600 ns is shown in Fig. 12a similar to Fig. 3c of main text. Compared to 250 ns long pulses, the +main difference lies in the fact that ∆− +EPR in the middle panel exhibits a double-dip shape because the correlations +¯V13 have a wider bandwidth than the emitted noise spectra ( ¯V11 and ¯V33), which are narrowed due to dynamical back- +action [5]. Since in the measurement the correlations don’t clearly overwhelm the emitted noise, interference between +two Lorentzian functions of different widths (dashed line) leads to the specific shape of ∆− +EPR. Theory confirms this +even though the shown theory curve (solid red line) does not exhibit the specific line-shape due to higher expected +correlations compared to the experimentally observed values. These results indicate that ¯ne,int due to a 600 ns optical +pump pulse is large enough to prevent a clear observation of squeezing over the full bandwidth below the vacuum +level (∆− +EPR < 1). As a result, we switched to 250 ns optical pump pulses with higher statistics as shown in the main +text. +We also repeated the measurement with 600 ns long pulses with different optical pump powers. Fig. 12b shows the +measured pump power dependence with each data point based on 170000-412500 individual measurements each with +a 2 Hz repetition rate. The microwave mode thermal bath occupancy ¯ne,int changes little as a function of the peak +optical pump power at the device and is approximated with a constant function (solid maroon line in the top panel). +The on-resonance mean CM elements scale with cooperativity and are in excellent agreement with theory (solid lines) +based on the ¯ne,int. The on-resonance squeezing ∆− +EPR does not change significantly with cooperativity since both +excess noise and correlations scale together with cooperativity. The anti-squeezing ∆+ +EPR scales up with cooperativity +as expected. All but one measured mean values are below the vacuum level and three power settings show a > 2σ +significance for entanglement. Note that this power sweep was conducted on a different set of optical modes with a +different amount of anti-Stokes sideband suppression (see section III). +VII. +ERROR ANALYSIS +The covariance matrix of the output field quadratures V (ω) can be directly calculated from the extracted microwave +and optical quadratures from frequency domain analysis [cf. Eq. 54] We simply rotate the optical quadratures with the +phase that minimized the joint quadrature variance, and obtain the covariance matrix in the normal form. We note +that, the error in calculating the covariance matrix comes from two sources - the statistical error due to finite number +of pulses, and the systematic error in the vacuum noise level calibration. The detailed error analysis is described in +the following subsections. We note that, the uncertainty in all the reported numbers in the main text corresponds to +2 standard deviation. + +21 +0 +-10 +-20 +1.0 +2.0 +3.0 +4.0 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.2 +1.0 +1.4 +10 +20 +a +b +Interpolation +In-pulse +After-pulse +Before-pulse +0.1 +0.0 +1.0 +0.5 +0.0 +0.8 +1.0 +1.0 +2.0 +3.0 +0.2 +0.3 +140 +160 +180 +200 +220 +240 +Optical pump power (mW) +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +0.22 +Cooperativity +(V11 + V22)/2 +(V33 + V44)/2 +(V13 − V24)/2 +(V11 + V22)/2 +(V33 + V44)/2 +(V13 − V24)/2 +V +(photons s-1 Hz-1) +(photons s-1 Hz-1) +(MHz) +V +ΔEPR +- +ΔEPR ++ +ΔEPR +- +¯ne,int +ΔEPR ++ +Δω/2π +Supplementary Fig. 12. Non-classical correlations vs. optical pump power for 600 ns long pump pulses. a, Top +panel, the average microwave output noise ¯V11 (purple), the optical output noise ¯V33 (green) and correlations ¯V13 (yellow) as +a function detuning based on 412500 measurements with a 2 Hz repetition rate. The solid lines represent the joint theory +with fit parameters C = 0.22 and in-pulse microwave thermal bath occupancy ¯ne,int = 0.19 ± 0.03. The dashed lines are +individual Lorentzian fits to serve as a guide to the eye. ∆− +EPR (∆+ +EPR) in the middle (bottom) panel shown in red (blue) color +are calculated from the top panel data and fits as described in the main text. The darker color error bars represent the 2σ +statistical error and the outer (faint) error bars also include systematic errors. b, Power dependence of CM elements. The +top panel shows the microwave mode thermal bath occupancy ¯ne,int for before-pulse, after-pulse and in-pulse regimes (marked +in Fig. 2A) as a function of the peak optical pump power at the device and the corresponding cooperativity. The in-pulse +¯ne,int is obtained by the joint theory fit and approximated with a constant function (solid line). The middle panel shows +the on-resonance mean CM elements based on the ¯ne,int from the top panel. The bottom two panels show the on-resonance +squeezing ∆− +EPR and anti-squeezing ∆+ +EPR calculated from the middle panel along with theory (solid lines). The darker color +error bars represent the 2σ statistical error and the outer (faint) error bars also include systematic errors. All measured mean +values are below the vacuum level and three power settings show a > 2σ significance for entanglement. +A. +Statistical error +The error in the calculation of bivariate variances comes from the statistical uncertainties, arising from finite number +of observations of a random sample. This error is the major component of our total error in diagonal covariance matrix +elements. The error in calculating the variance of a sample distribution sampled from a Gaussian variable follows the +Chi-squared distribution and is given as, +Var(σ2) = +2σ2 +N − 1, +(71) +where, σ2 is the variance of sample distribution and N is its size. +In addition, the error in the covariance from a bivariate variable is given by the Wishart distribution [15]. For a +general bivariate covariance matrix Σ given as, +Σ = +� +σ2 +11 +ρσ11σ22 +ρσ11σ22 +σ2 +22 +� +, +(72) + +22 +the variance of the covariance matrix is given by, +Var(Σ) = +1 +N − 1 +� +2σ4 +11 +(1 + ρ2)σ2 +11σ2 +22 +(1 + ρ2)σ2 +11σ2 +22 +2σ4 +22 +� +. +(73) +B. +Systematic error +Although, the systematic error in our measurements are not as significant, they still are a noticeable source of +error. Here the error in calculating the covariance matrix results form the error in the estimation of the vacuum +noise levels. More specifically, the error in determining the added noise due to the microwave and optical detection +chain, as discussed in Sec. III. Propagating these systematic errors through the covariance matrix analysis is non- +trivial, since calculating the error in variance of erroneous quantities is challenging. Therefore, we use a worst-case +scenario approach to calculate the total error including the statistical error and the systematic error. We repeat +the full analysis, including the statistical errors, for the lower and upper bound of the uncertainty range from the +systematic errors for the microwave and optical added noise levels. Repeating the analysis expands the error bars in +the calculated quantities. We take the extremum of all the error bars from all the repetitions of analysis to get the +total error bar. We show both statistical error and the total error in the main text. +REFERENCES +[1] W. Hease, A. Rueda, R. Sahu, M. Wulf, G. Arnold, H. G. Schwefel, and J. M. Fink, PRX Quantum 1, 020315 (2020). +[2] M. Tsang, Phys. Rev. A 81, 063837 (2010). +[3] M. Tsang, Phys. Rev. A 84, 043845 (2011). +[4] A. Rueda, F. Sedlmeir, M. C. Collodo, U. Vogl, B. Stiller, G. Schunk, D. V. Strekalov, C. Marquardt, J. M. Fink, O. Painter, +G. Leuchs, and H. G. L. Schwefel, Optica 3, 597 (2016). +[5] L. Qiu, R. Sahu, W. Hease, G. Arnold, and J. M. Fink, arXiv:2210.12443 (2022). +[6] C. W. Gardiner and M. J. Collett, Physical Review A 31, 3761 (1985). +[7] S. L. Braunstein and P. van Loock, Rev. Mod. Phys. 77, 513 (2005). +[8] S. Zippilli, G. D. Giuseppe, and D. Vitali, New Journal of Physics 17, 043025 (2015). +[9] M. P. da Silva, D. Bozyigit, A. Wallraff, and A. Blais, Physical Review A 82, 043804 (2010). +[10] D. Walls and G. Milburn, Quantum optics (Springer Verlag, Berlin, 1994). +[11] H. M. Wiseman and G. J. Milburn, Quantum Measurement and Control (Cambridge University Press, 2009). +[12] C. M. Caves, Physical Review D 26, 1817 (1982). +[13] L.-M. Duan, G. Giedke, J. I. Cirac, and P. Zoller, Physical Review Letters 84, 2722 (2000). +[14] R. Sahu, W. Hease, A. Rueda, G. Arnold, L. Qiu, and J. M. Fink, Nature Communications 13, 1276 (2022). +[15] J. Wishart, Biometrika 20A, 32 (1928). + diff --git a/9tE1T4oBgHgl3EQfoARp/content/tmp_files/load_file.txt b/9tE1T4oBgHgl3EQfoARp/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..c78327c9d30d2270580637dbb634aceb447fea92 --- /dev/null +++ b/9tE1T4oBgHgl3EQfoARp/content/tmp_files/load_file.txt @@ -0,0 +1,2275 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf,len=2274 +page_content='Entangling microwaves with optical light Rishabh Sahu⋆,1, † Liu Qiu⋆,1, ‡ William Hease,1 Georg Arnold,1 Yuri Minoguchi,2 Peter Rabl,2 and Johannes M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Fink1, § 1Institute of Science and Technology Austria, Am Campus 1, 3400 Klosterneuburg, Austria 2Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1040 Vienna, Austria (Dated: January 10, 2023) Entanglement is a genuine quantum mechanical property and the key resource in currently developed quantum technologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Sharing this fragile property between superconducting microwave circuits and optical or atomic systems would enable new functionalities but has been hindered by the tremendous energy mismatch of ∼ 105 and the resulting mutually imposed loss and noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In this work we create and verify entanglement between microwave and optical fields in a millikelvin environment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Using an optically pulsed superconducting electro-optical device, we deterministically prepare an itinerant microwave-optical state that is squeezed by 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='72+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='31 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='25 dB and violates the Duan-Simon separability criterion by > 5 standard deviations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This establishes the long-sought non-classical correlations between superconducting circuits and telecom wavelength light with wide-ranging implications for hybrid quantum networks in the context of modularization, scaling, sensing and cross-platform verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Over the past decades we have witnessed spectacular progress in our capabilities to manipulate and measure genuine quantum mechanical properties, such as quan- tum superpositions and entanglement, in a variety of physical systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' These techniques serve now as the ba- sis for the development of quantum technologies, where the demonstration of quantum supremacy with tens of superconducting qubits [1], an ultra-coherent quantum memory with nuclear spins [2], and distributed quan- tum entanglement over tens of kilometers using optical photons [3] represent just a few of the highlights that have already been achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Going forward, combin- ing these techniques [4–6] will enable the realization of general-purpose quantum networks, where remote quan- tum nodes, capable of storing and processing quantum information, seamlessly communicate with each other by distributing entanglement over optical channels [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' As- pects of this approach have already been adopted to con- nect and entangle various quantum platforms remotely, involving single atoms, ions, atomic ensembles, quantum dots, rare-earth ions and nitrogen-vacancy centers [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' However, such long-distance quantum connectivity is considerably more difficult to achieve with other promis- ing platforms, such as semiconductor spin qubits or local cryogenic networks of superconducting circuits [9, 10], where no natural interface to room temperature noise- resilient optical photons is available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' To overcome this limitation, a lot of effort is cur- rently focused on the development of coherent quantum transducers between microwave and optical photons [11– 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Direct noiseless conversion of a quantum state typ- ically relies on a beam splitter process, where a strong driving field mediates the conversion between weak mi- crowave and optical signals - a deterministic approach with exceptionally stringent requirements on conversion † rsahu@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='at ‡ liu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='qiu@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='at § jfink@ist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='at ⋆ These authors contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' efficiency and added classical noise that are still out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Alternatively, the direct generation of quantum- correlated microwave-optical photon pairs can also be used as a resource for quantum teleportation and en- tanglement distribution in the continuous and discrete variable domain [17–19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In this paper we use an ultra-low noise cavity electro- optical device to generate such non-classical correlations in a deterministic protocol [20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' It consists of a 5 mil- limeter diameter, 150 µm thick lithium niobate optical resonator placed inside a superconducting aluminum mi- crowave cavity at a temperature of 7 mK, described in detail in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' As depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 1A, the microwave mode ˆae is co-localized with and electro-optically coupled to the optical whispering gallery modes at ωo/(2π) ≈ 193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='46 THz via the Pockels effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We match the tunable microwave resonance frequency ωe/(2π) to the free spec- tral range (FSR) of 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='799 GHz to realize a triply-resonant system [22, 23] with the interaction Hamiltonian, ˆHint = ℏg0ˆapˆa† eˆa† o + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=', (1) with g0 the vacuum electro-optical coupling rate, and ˆap (ˆao) the annihilation operator of the optical pump (Stokes) mode [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Here we have ignored the interac- tion with the suppressed optical anti-Stokes mode ˆat, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 1b (see supplementary information).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In this sideband suppressed situation, efficient two- mode squeezing is achieved with a strong resonant opti- cal pump tone, yielding the simple effective Hamiltonian, ˆHeff = ℏg0√¯np(ˆa† eˆa† o + ˆaeˆao), where ¯np = ⟨ˆa† pˆap⟩ is the mean intra-cavity photon number of the optical pump mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Deterministic continuous-variable (CV) entangle- ment between the out-propagating microwave and opti- cal field can be generated below the parametric instability threshold (C < 1) in the quantum back-action dominated regime, where the quantum noise exceeds the microwave thermal noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Here C = 4¯npg2 0/(κeκo) is the coopera- tivity with the vacuum coupling rate g0/2π ≈ 37 Hz, and the total loss rates of the microwave and optical Stokes modes κe/2π ≈ 11 MHz and κo/2π ≈ 28 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The re- quired ultra-low noise operation is achieved despite the arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='03315v1 [quant-ph] 9 Jan 2023 2 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='p ^ap,in ^ao,out ^ae,out !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e ae^ ao^ ao ap !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='p !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='p !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e FSR !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='p+ FSR = FSR = 1) ^at ^ ^ ae^ (me 1) (mp + 1) (mp mp b a FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Physical and conceptual mode configuration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' a, Simulated microwave (left) and optical (right) mode dis- tribution with azimuthal number me = 1 and mo = 17 (for illustration, experimentally mo ≈ 20 000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Phase matching is fulfilled due to the condition mo = mp − me and entangle- ment is generated and verified between the out-propagating microwave field ˆae,out and the optical Stokes field ˆao,out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' b, Sketch of the density of states of the relevant modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Un- der the condition ωp − ωo = ωe the strong pump tone in ˆap efficiently produces entangled pairs of microwave and optical photons in ˆae and ˆao via spontaneous parametric downconver- sion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Frequency up-conversion is suppressed via hybridization of the anti-Stokes mode ˆat with an auxiliary mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' required high power optical pump due to slow heating of this millimeter sized device [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In the following we characterize the microwave and op- tical output fields via the dimensionless quadrature pairs Xj and Pj (j = e, o for microwave and optics), which satisfy the canonical commutation relations [Xj, Pj] = i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' A pair of Einstein-Podolsky-Rosen (EPR)-type opera- tors X+ = 1 √ 2 (Xe + Xo) and P− = 1 √ 2 (Pe − Po) are then constructed, and the microwave and optical output fields are entangled, if the variance of the joint opera- tors is reduced below the vacuum level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ∆− EPR = � X2 + � + � P 2 − � < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This is commonly referred to as the Duan-Simon criterion [25, 26], which we apply to each near-resonant frequency component of the two-mode squeezed output mode (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' For efficient entanglement generation, we use a 250 ns long optical pump pulse (≈ 244 mW, C ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='18, ¯np ≈ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='6 × 1010) at a 2 Hz repetition rate, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' pulse 1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The entangled output optical signal is filtered via a Fabry-Perot cavity to reject the strong pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The entangled microwave output is amplified with a high- electron-mobility transistor amplifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Both outputs are down-converted to an intermediate frequency of 40 MHz with two local oscillators (LO) and the four quadra- tures are extracted from heterodyne detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Long- term phase stability between the two LOs is achieved via extracting the relative phase drift by means of a second phase alignment pump pulse that is applied 1 µs after each entanglement pulse, together with a coherent reso- nant microwave pulse, shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This generates a high signal-to-noise coherent optical signal via stimu- lated parametric down-conversion and allows for aligning the phase of each individual measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Figure 2b(c) shows the time-domain average power over one million averages for the on-resonant microwave (optics) signal with a spectrally under-sampled 40 MHz bandwidth (hence not revealing the full pulse ampli- tude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The two insets show the microwave (optical) signal from spontaneous parametric down-conversion (SPDC) due to pulse 1, cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2a, with an emission bandwidth of ≈ 10 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The larger signals during the second half of the experiment are the reflected microwave pulse and the generated optical tone (due to pulse 2) that is used for LO phase alignment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The raw power measurements are divided by the measurement bandwidth and rescaled such that the off-resonant response matches the noise photon number Nj,add + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 of the measurement setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Ne,add = 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='4 (2σ errors throughout the paper) due to loss and amplifier noise and No,add = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='2 due to optical losses are carefully determined using noise thermometry of a temperature controlled 50 Ω load and 4-port calibration, respectively (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Using this pro- cedure ensures that the reported photon number units correspond to the signals at the device outputs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We continue the analysis in the frequency domain by calculating the Fourier transform of each measurement for three separate time intervals - before (2 µs), dur- ing (200 ns) and right after the entangling pump pulse (500 ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Figure 2d shows the resulting average mi- crowave noise spectra for all three time intervals with corresponding fit curves (dashed lines) and theory (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Before and after the pump pulse, the on-resonant microwave output field takes on values above the vac- uum level, with fitted intrinsic microwave bath occu- pancies ¯ne,int = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='01 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='03, respec- tively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' By fitting additional power dependent measure- ments, we independently verify that the observed noise floor corresponds to a waveguide bath occupancy of only ¯ne,wg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='001±0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='002 at the very low average pump power of ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='12 µW used in this experiment (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The mea- sured noise floor therefore corresponds to the shot noise equivalent level Ne,add + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 (gray dashed lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Sim- ilarly, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2e shows the obtained average optical noise spectra during and after the pump, referenced to the measured shot noise level before the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' As expected, there is no visible increase of the optical noise level after the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' During the pump pulse, an approximately Lorentzian shaped microwave and optical power spectrum are gen- erated via the SPDC process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We perform a joint fit of the microwave and optical power spectral density during the pulse using a 5-mode theoretical model that includes 3 600 400 200 0 800 1000 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='6 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0 50 100 150 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='05 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='10 !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='o !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='p !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='o Input power t Output power Before pulse Pulse 1 Entangle Pulse 2 After pulse Phase align MW signal MW reflection b a c 0 1 2 3 t (μs) 4 5 0 1 2 3 4 5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='75 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='00 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='25 20 0 20 20 0 20 Before-pulse Theory In-pulse After-pulse In-pulse After-pulse Theory Δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e=2π (MHz) Δ!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='o=2π (MHz) e d Ne,det (photons s-1 Hz-1) No,det (photons s-1 Hz-1) Ne,det (photons s-1 Hz-1) No,det (photons s-1 Hz-1) (μs) t Ne,add + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 No,add + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Measurement sequence and noise powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' a, Schematic pulse sequence of a single measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The optical pulse 1 is applied at ωp and amplifies the vacuum (and any thermal noise) in the two modes ˆae and ˆao, thus generating the SPDC signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 1 µs later, a second optical pump with about 10 times lower power is applied together with a coherent microwave pulse at ωe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The microwave photons stimulate the optical pump to down-convert, which generates a coherent pulse in the ˆao mode that is used to extract slow LO phase drifts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' b and c, Measured output power in the ˆae and ˆao mode in units of photons per second in a 1 Hz bandwidth and averaged over a million experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The SPDC signals are shown in the insets with the dashed gray lines indicating the calibrated detection noise floor Nj,add + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' d, Corresponding microwave output power spectral density vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ∆ωe = ω − ωe centered on resonance right before the entanglement pulse, during the pulse and right after the pulse, as indicated in panel a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Yellow and green dashed lines are fits to a Lorentzian function, which yields the microwave bath occupancies before and after the entangling pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Error bars represent the 2σ statistical standard error and the shaded regions represent the 95% confidence interval of the fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (e), Corresponding optical output power spectral density vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ∆ωo = ωo − ω during and after the entanglement pulse, both normalized to the measured noise floor before the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The in-pulse noise spectra in panels d and e are fit jointly with theory, which yields C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='01 and ¯ne,int = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' the effects of measurement bandwidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In this model, the in-pulse microwave bath occupancy ¯ne,int = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='03 and the cooperativity C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='18 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='01 are the only free fit parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Here the narrowed microwave linewidth κe,eff/2π = 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='8 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='8 MHz (taken from a Lorentzian fit) agrees with coherent electro-optical dynamical back- action [27], where κe,eff = (1 − C)κe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We conclude that this cavity electro-optical device is deep in the quantum back-action dominated regime, a prerequisite for efficient microwave-optics entanglement generation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' For each frequency component the bipartite Gaussian state of the propagating output fields can be fully char- acterized by the 4 × 4 covariance matrix (CM) Vij = ⟨δuiδuj + δujδui⟩ /2, where δui = ui − ⟨ui⟩ and u ∈ {Xe, Pe, Xo, Po} (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The diagonal elements in V cor- respond to the individual output field quadrature vari- ances in dimensionless units.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' These are obtained from the measured variances after subtracting the measured detection noise offsets shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Vii(∆ω) = Vii,meas(∆ωi) − Ni,add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The obtained CM from the data in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2 at ∆ω = 0 is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 3a in its standard form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' It corresponds to the quantum state of the propa- gating modes in the coaxial line and the coupling prism attached to the device output, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' before setup losses or amplification incur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The non-zero off-diagonal elements indicate strong correlations between microwave and op- tical quadratures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The two-mode squeezed quadratures are more intu- itively visualized in terms of the quasi-probability Wigner function, W(u) = exp[− 1 2uV −1uT ] π2� det(V) , (2) 4 where u = (Xe, Pe, Xo, Po).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Different marginals of this Wigner function are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 3b, where the (Xe,Xo) and (Pe,Po) marginals show two-mode squeezing in the diagonal and off-diagonal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The two cross- quadrature marginals show a slightly different amount of squeezing, which is due to the statistical uncertainty in the measured CM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Figure 3c shows the amount of two-mode squeezing between microwave and optical photon pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Correla- tions are observed at ∆ωj = ±(ω − ωj) around the reso- nances due to energy conservation in the SPDC process (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The averaged microwave quadrature variance (purple dots) ¯V11 = (V11+V22)/2 and the averaged optics quadrature variance (green dots) ¯V33 = (V33 +V44)/2 are shown in the top panel along with the prediction from our five-mode theory (solid line) and a simple fit to a Lorentzian function (dashed line), showing perfect agree- ment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Measured microwave-optical correlations (yellow dots) ¯V13 = (V13 − V24)/2 and the Lorentzian fit (dashed line) lie slightly below the theoretical prediction (solid line), which we assign to remaining imperfections in the phase stability (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The bottom two panels of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 3c show the squeezed and anti-squeezed joint quadrature variances ∆∓ EPR = ¯V11 + ¯V33 ∓ 2 ¯V13 (red and blue color respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We observe two-mode squeezing below the vacuum level, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ∆− EPR < 1, with a bandwidth close to the effective mi- crowave linewidth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The maximal on-resonant two-mode squeezing is ∆− EPR = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='85+0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='05 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='06 (2σ, 95% confidence) for ∼1 million pulses with ¯V11 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='93, ¯V33 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='84 and ¯V13 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Hence, this deterministically generated microwave- optical state violates the Duan-Simon separability crite- rion by > 5σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Note that this error also takes into account systematics in the added noise calibration used for scal- ing the raw data (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' These values correspond to a state purity of ρ = 1/(4 � det[V ]) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='44 and demon- strate microwave-optical entanglement between output photons with a logarithmic negativity of EN = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The supplementary material contains substantial additional data for longer pulses and varying optical pump power, which corroborates the presented results and findings, al- beit with lower statistical significance for each individual pump configuration (see SI).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Conclusions and outlook In conclusion, we have demonstrated deterministic quantum entanglement between propagating microwave and optical photons,thus establishing a non-classical communication channel between circuit quantum electro- dynamics and quantum photonics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Our device can read- ily be used for probabilistic heralding assisted protocols [7, 28, 29] to mitigate optical setup losses and extend the entanglement to room temperature fiber optics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We ex- pect that the pulse repetition rate can be increased by or- ders of magnitude with improved thermalization, higher microwave and optical quality factors, and electro-optic coupling enhancements that reduce the required pump power and the associated thermal load.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Coupling effi- ciency improvements will allow for higher levels of two- mode squeezing and facilitate also deterministic entangle- ment distribution schemes [30], teleportation-based state transfer [20, 31] and quantum-enhanced remote detec- tion [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Being fully compatible with superconducting qubits in a millikelvin environment such a device will fa- cilitate the integration of remote superconducting quan- tum processors into a single coherent optical quantum network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This is not only relevant for modularization and scaling [33, 34], but also for efficient cross-platform verification of classically intractable quantum processor results [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ACKNOWLEDGMENTS L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' acknowledges fruitful discussions with Jie Li and David Vitali.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This work was supported by the European Research Council under grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 758053 (ERC StG QUNNECT) and the European Union’s Horizon 2020 research and innovation program under grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 899354 (FETopen SuperQuLAN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' acknowledges generous support from the ISTFEL- LOW programme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' is the recipient of an ISTplus postdoctoral fellowship with funding from the European Union’s Horizon 2020 research and innovation program under the Marie Sk�lodowska-Curie grant agreement no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 754411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' is the recipient of a DOC fellowship of the Austrian Academy of Sciences at IST Austria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' acknowledges support from the Austrian Science Fund (FWF) through BeyondC (F7105) and the European Union’s Horizon 2020 research and innovation programs under grant agreement No 862644 (FETopen QUAR- TET).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' AUTHOR CONTRIBUTIONS RS, WH, LQ, and GA worked on the setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' RS and LQ performed measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' LQ and RS did the data analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' LQ developed the theory with contributions from RS, YM and PR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' RS and LQ wrote the manuscript with contributions from all authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' JMF supervised the project.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' DATA AVAILABILITY STATEMENT All data and code used to produce the figures in this manuscript will be made available on Zenodo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0 2 2 2 2 0 0 10 20 10 20 c b (photons s-1 Hz-1) (MHz) 0 2 2 2 2 0 2 2 0 2 2 0 2 2 0 2 2 0 a Vij Pe Pe Pe Pe Pe Po Po Po Po Po Xo Xo Xo Xo Xo Xe Xe Xe Xe Xe 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 (V11 + V22)/2 (V33 + V44)/2 (V13 − V24)/2 V ΔEPR ΔEPR + Δω/2π FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Characterization of the two-mode squeezed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' a, Measured covariance matrix Vij in its standard form plotted for ∆ωj = 0 based on 925000 measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' b, Corresponding Wigner function marginals of different output quadrature pairs in comparison to vacuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The contours in blue (grey) represent the 1/e fall-off from the maximum for the measured state (vacuum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Middle two plots show two-mode squeezing below the vacuum level in the diagonal and off-diagonal directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' c, Top panel, the measured average microwave output noise ¯V11 = (V11 + V22)/2 (purple), the average optical output noise ¯V33 = (V33 + V44)/2 (green) and the average correlations ¯V13 = (V11 − V24)/2 (yellow) as a function of the measurement detunings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The solid lines represent the joint theory fit and the dashed lines are individual Lorentzian fits to serve as a guide to eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The middle (bottom) panel shows two-mode squeezing in red (anti-squeezing in blue) calculated from the top panels as ∆± EPR = ¯V11 + ¯V33 ± 2 ¯V13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The darker color error bars represent the 2σ statistical error and the outer (faint) 2σ error bars also include the systematic error in calibrating the added noise of the measurement setup.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' [1] F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Arute, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Arya, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Babbush, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Bacon, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Bardin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Barends, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Biswas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Boixo, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Brandao, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Buell, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Burkett, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Chen, Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Chen, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Chiaro, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Collins, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Courtney, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Dunsworth, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Farhi, B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Foxen, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Fowler, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Gidney, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Giustina, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Graff, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Guerin, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Habegger, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Harrigan, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Hartmann, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Ho, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Hoffmann, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Huang, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Humble, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Isakov, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Jeffrey, Z.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Korotkov, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Kostritsa, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Landhuis, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Lind- mark, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Lucero, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Lyakh, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Zhou, Architectures for Multinode Superconduct- ing Quantum Computers (2022), arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='06167.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' [35] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Kn¨orzer, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Malz, and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Cirac, Cross-Platform Ver- ification in Quantum Networks (2022), arXiv:2212.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='07789.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Supplementary Information for: ”Entangling microwaves with optical light” Rishabh Sahu,1, ∗ Liu Qiu,1, ∗ William Hease,1 Georg Arnold,1 Yuri Minoguchi,2 Peter Rabl,2 and Johannes M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Fink1 1Institute of Science and Technology Austria, am Campus 1, 3400 Klosterneuburg, Austria 2Vienna Center for Quantum Science and Technology, Atominstitut, TU Wien, 1040 Vienna, Austria (Dated: January 10, 2023) CONTENTS Page I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Theory 3 A Covariance Matrix from Input-Output Theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 3 1 Quantum Langevin Equations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 3 2 Input-Output-Theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 4 3 Covariance Matrix of Filtered Output Fields .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 6 B Heterodyne Detection, Added Noise and Filtering .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 7 1 Heterodyne Measurement .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 7 2 Realistic Measurements: Added Noise and Gain .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 8 3 Covariance Matrix from Realistic Heterodyne Measurements .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 9 C Entanglement Detection .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 10 1 Duan Criterion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 10 2 Logarithmic Negativity and Purity .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 11 II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Experimental Setup 11 III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Setup Characterization and calibration 11 A Microwave added noise calibration .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 11 B Optical added noise .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 14 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Data treatment 15 A Time-domain analysis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 15 B Pulse post-selection .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 16 C Frequency domain analysis .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 16 D Joint-quadrature correlations .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 17 V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Quadrature histogram raw data 19 VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Non-classical correlations with 600 ns long optical pump pulses 20 VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Error analysis 20 A Statistical error .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 21 B Systematic error .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 22 References 22 ∗ These two authors contributed equally arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='03315v1 [quant-ph] 9 Jan 2023 2 Introduced in Main Text ˆae microwave mode (annihilation operator) me microwave mode azimuthal number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' me = 1 ωe microwave cavity frequency κe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' κe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='eff,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' κe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='in,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' κe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 microwave total loss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' effective total loss,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' waveguide coupling and intrinsic loss rates ¯ne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='int,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ¯ne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='wg microwave intrinsic and waveguide bath occupancy Ne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='add added noise in the microwave detection ˆao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆap,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆatm optical Stokes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' pump,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' anti-Stokes,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' and transverse-magnetic mode (annihilation operator) ˆae/o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='out microwave and optical output field from the device mp optical pump mode azimuthal number,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' mp ∼ 20000 κo optical total loss rate No,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='add added noise in the optical detection ¯np mean photon number of the optical pump mode g0 electro-optical vacuum coupling rate g photon enhanced electro-optical coupling rate (g = √¯npg0}) C cooperativity ( C = 4g2/κeκo ) Xe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='Pe quadratures of the microwave output field Xo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='Po quadratures of the optical Stokes output field V covariance matrix of the bipartite Gaussian state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Vij = ⟨∆ui∆uj + ∆uj∆ui⟩ /2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' where ∆ui = ui − ⟨ui⟩ and u ∈ {Xe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Pe,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Xo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Po}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Nii,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='add added noise in the quadrature variances measurements,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' N11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='add = N22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='add = Ne,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='add,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' N33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='add = N44,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='add = No,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='add Vii,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='meas diagonal covariance matrix elements from the calibrated measurement record,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Vii = Vii,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='meas − Nii,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='add V11,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='V22,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ¯V11 quadrature variances of the microwave output field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ¯V11 = V11+V22 2 V33,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='V44,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ¯V33 quadrature variances of the optical Stokes output field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ¯V33 = V33+V44 2 V13,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='V24,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ¯V13 cross-correlation between microwave and optical quadratures,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ¯V13 = V13−V24 2 ∆∓ EPR squeezed and anti-squeezed joint quadrature variance between microwave and optical output field,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ∆∓ EPR = ¯V11 + ¯V33 ∓ ¯V13 Introduced in Supplementary Information J coupling rate between the optical anti-Stokes mode and TM mode ˆae/o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='in input field (noise) operator for the microwave and optical mode ˆae/o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 noise operator for the microwave and optical intrinsic loss ηj external cavity coupling efficiency of individual mode,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' j ∈ (e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' t) G(ω) spectral filter of the output field ˆA(Ω) Fourier transform of operator ˆA(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆA(Ω) = � dt eiΩt ˆA(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆA†(Ω) = � dt ˆA†(t)eiΩt = [ ˆA(−Ω)]† ˆX(ωn) X quadrature of the output spectral mode,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆX(ωn) = 1 √ 2 � ∞ −∞ dω G(ωn − ω)ˆaout(ω) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆP(ωn) P quadrature of the output spectral mode, ˆP(ωn) = 1 √ 2i � ∞ −∞ dω G(ωn − ω)ˆaout(ω) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' S ˆ A ˆ B(Ω) Two-time correlation of two operators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' S ˆ A ˆ B(Ω) = 1 √ 2π � ∞ −∞ � ˆA(t) ˆB(t′) � eiΩtdt ∆LO local oscillator and signal frequency difference in heterodyne measurement,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ∆LO = ωLO−ωsig Iout(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Iout(ω) unitless output field in the equivelant heterodyne detection SII(ω) double-sided noise spectrum of the output field in the equivelant heterodyne detection Gdet(ω) frequency dependent detection gain in the heterodyne detection ˆIX/P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='det(ωn) detected output photocurrent quadratures in heterodyne detection,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' including detection gain ˆIX/P,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='out(ωn) unitless output field quadratures from Iout including added noise D(ω) covariance matrix of the detected quadratures from the heterodyne measurement record Vmeas(ω) covariance matrix of the total measured output field quadratures including added noise ˆX+(ω) joint quadrature of ˆXe(ω) and ˆXo(−ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆX+(ω) = ( ˆXe(ω) + ˆXo(−ω))/ √ 2 ˆP−(ω) joint quadrature of ˆPe(ω) and ˆPo(−ω),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆP−(ω) = ( ˆPe(ω) − ˆPo(−ω))/ √ 2 EN logarithm negativity ρ state purity 3 I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' THEORY A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Covariance Matrix from Input-Output Theory 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Quantum Langevin Equations Our cavity electro-optical (CEO) device consists of a millimeter-sized lithium niobate optical resonator in a 3-D superconducting microwave cavity at mK temperature [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The Pockels effect in lithium niobate allows for direct coupling between the microwave and optical whispering gallery modes with maximal field overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The optical free spectral range (FSR) matches the microwave cavity frequency, with microwave azimuthal mode number me = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 1 in the main text, resonant three-wave mixing between the microwave mode (ˆae) and three adjacent transverse-electric (TE) optical modes, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Stokes (ˆao), pump (ˆap), and anti-Stokes (ˆat) mode, arises via the cavity enhanced electro-optical interaction [2, 3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In addition, the anti-Stokes mode is coupled to a transverse-magnetic (TM) optical mode (ˆatm) of orthogonal polarization and similar frequency at rate of J [4].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This results in a total interaction Hamiltonian, ˆHI/ℏ = g0(ˆa† pˆaeˆao + ˆa† pˆa† eˆat) + Jˆatˆa† tm + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=', (1) with g0 the vacuum electro-optical coupling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' For efficient entanglement generation, we drive the pump mode strongly with a short coherent input pulse ¯ap,in(t) at frequency ωp [1], which results in a time-dependent mean intra-cavity field of the pump mode ¯ap(t), ˙¯ap = � i∆p − κp 2 � ¯ap + √ηpκp¯ap,in, (2) where the pump tone is detuned from the pump mode by ∆p = ωp − ωo,p, with κp and ηp as the pump mode loss rate and external coupling efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In our experiments, we actively lock the laser frequency to the pump mode resonance, with ∆p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The presence of the strong pump field results in an effective interaction Hamiltonian, ˆHI,eff/ℏ = g(ˆaeˆao + ˆaeˆa† t) + Jˆatˆa† tm + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=', (3) with multiphoton coupling rate g = ¯apg0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This includes the two-mode-squeezing (TMS) interaction between the Stokes and microwave mode, and beam-splitter (BS) interaction between the anti-Stokes mode and microwave mode, resulting in scattered Stokes and anti-Stokes sidebands that are located on the lower and upper side of the pump tone by Ωe away.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Microwave-optics entanglement between the microwave and optical Stokes output field can be achieved via spontaneous parametric down-conversion (SPDC) process due to TMS interaction [5], which is further facilitated by the suppressed anti-Stokes scattering due to the strong coupling between anti-Stokes and TM modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We can obtain the full dynamics of the intracavity fluctuation field in the rotating frame of the scattered sidebands and microwave resonance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' which can be described by the quantum Langevin equations (QLE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˙ˆae = −κe 2 ˆae − igˆa† o − ig∗ˆat + √ηeκeδˆae,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='in + � (1 − ηe) κeδˆae,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (4) ˙ˆao = � iδo − κo 2 � ˆao − igˆa† e + √ηoκoδˆao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='in + � (1 − ηo) κoδˆao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (5) ˙ˆat = � iδt − κt 2 � ˆat − ig∗ˆae − iJˆatm + √κtδˆat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='vac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (6) ˙ˆatm = � iδtm − κtm 2 � ˆatm − iJˆat + √κtmδˆatm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='vac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (7) with κj the total loss rate of the individual mode where j ∈ (e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' tm),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' and ηk the external coupling efficiency of the input field where k ∈ (e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' o).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We note that, the optical light is only coupled to the TE modes via efficient prism coupling, with effective mode overlap Λ factor included in ηo for simplicity [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' δj corresponds to the frequency difference between mode j and scattered sidebands, with δo = ωo,p − ωe − ωo and δt/tm = ωo,p + ωe − ωt/tm, which are mostly given by FSR and ωe mismatch, with additional contributions from optical mode dispersion and residual optical mode coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We note that, for resonant pumping, we have δo = −δt in the case of absent optical mode dispersion and residual mode coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In our experiments, we tune the microwave frequency to match the optical FSR, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ωe = ωo,p − ωo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 4 The equation of motion of all relevant modes may be represented more economically in the form ˙v(t) = M(t)v(t) + Kfin(t),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (8) where we define the vectors of mode and noise operators v = (ˆae,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆa† e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆa† o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆa† t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆatm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆa† tm)⊤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' fin = (δˆae,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' δˆa† e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' δˆae,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='in,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' δˆa† e,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='in,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' δˆao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' δˆa† o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' δˆao,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='in,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' δˆa† o,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='in,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' δˆat,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='vac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' δˆa† t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='vac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' δˆatm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='vac,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' δˆa† tm,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='vac)⊤,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (9) as well as the matrices that encode the deterministic part of the QLE,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' M(t) = � � � � � � � � � � � − κe 2 0 0 −ig(t) −ig∗(t) 0 0 0 0 − κe 2 +ig∗(t) 0 0 ig(t) 0 0 0 −ig(t) iδo − κo 2 0 0 0 0 0 ig∗(t) 0 0 −iδo − κo 2 0 0 0 0 −ig(t) 0 0 0 iδt − κt 2 0 −iJ 0 0 ig∗(t) 0 0 0 −iδt − κt 2 0 iJ 0 0 0 0 −iJ 0 iδtm − κtm 2 0 0 0 0 0 0 iJ 0 −iδtm − κtm 2 � � � � � � � � � � � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (10) and K = � � � � � (1 − ηe)κe √ηeκe 0 0 0 0 0 0 � (1 − ηo)κo √ηoκo 0 0 0 0 0 0 √κt 0 0 0 0 0 0 √κtm � � � � ⊗ 12,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (11) which keeps track on which modes the noise acts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Input-Output-Theory In the experiment the pump field is turned on at t = 0 and kept on until τpulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' For the optical pump pulse with length τpulse =250 ns (600 ns, see main text), we reject a certain τdelay = 50 ns (100 ns) from the beginning of pulse data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Since κpτdelay ≳ 1 we may assume that after τdelay the system has approached its steady state and especially that the pump mode is in its steady state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Consequently we may assume that g(t > τdelay) ≃ g is constant over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' One important figure of merit is the multiphoton cooperativity C = 4g2/κoκe, a measure for coherent coupling versus the microwave and optical dissipation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Efficient entanglement generation can be achieved with complete anti-Stokes scattering suppression, while below the parametric instability threshold, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' C < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The output fields of the CEO device are fout(t) = (ˆae,out(t), ˆa† e,out(t), ˆao,out(t), ˆa† o,out(t))⊤, (12) which consist of a contribution which was entangled via the coherent interactions v and a contribution which has not interacted with the device fin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The output field fout will then propagate to the measurement device and is most economically represented within the framework of input-output theory [6], fout(t) = Lfin(t) − Nv(t), (13) where we define the matrices N = (NJ, 04), with NJ = Diag(√ηeκe, √ηeκe, √ηoκo, √ηoκo), (14) and L = � 0 1 0 0 0 0 0 0 0 1 0 0 � ⊗ 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (15) As all modes have reached steady state, the correlations in the output field may be obtained by going to Fourier 5 domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Here we commit to following convention of the Fourier transformation ˆA(ω) = 1 √ 2π � ∞ −∞ dω eiωt ˆA(t), (16) with the hermitian conjugate ( ˆA(ω))† = A†(−ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (17) Note that in this convention e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' [ae(ω), a† e(ω′)] = δ(ω + ω′) are canonical pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In our experiments, we focus on the correlations between the output propagating spectral modes of frequencies ωe + ∆ωe and ωo − ∆ωo respectively for microwave and optical fields [7, 8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We note that, due to energy conservation in the SPDC process, we only focus on microwave and optical photon pairs around resonances with anti-correlated frequencies, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ∆ωe = ∆ωo = ∆ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' For this reason, we focus on the following vector of output fields in the rotating frame, fout(ω) = (ˆae,out(ω), ˆa† e,out(−ω), ˆao,out(−ω), ˆa† o,out(ω))⊤, (18) in the Fourier domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (8) we obtain v(ω) = [iωO − M]−1 · K � �� � =S(ω) fin(ω), (19) with O = Diag(1, −1, 1, 1) ⊗ σz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (20) Here we defined the vector of modes v(ω) = (ˆae(ω), ˆa† e(−ω), ˆao(−ω), ˆa† o(ω), ˆat(ω), ˆa† t(−ω), ˆatm(ω), ˆa† tm(−ω))⊤, (21) as well as the vector of input fields fin(ω) = (δˆae,0(ω), δˆa† e,0(−ω), δˆae,in(ω), δˆa† e,in(−ω), δˆao,0(−ω), δˆa† o,0(ω), δˆao,in(−ω), δˆa† o,in(ω), δˆat,vac(ω), δˆa† t,vac(−ω), δˆatm,vac(ω), δˆa† tm,vac(−ω))⊤ (22) in the Fourier domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The output fields (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (13)) of the CEO device are straight forwardly obtained since in the Fourier domain Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (13) is algebraic, fout(ω) = Lfin(ω) + Nv(ω) = (L + N · [iωO − M]−1 · K)fin(ω).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (23) The input noise operator correlations are given by, ⟨fin(ω)f † in(ω′)⟩ = Dδ(ω + ω′), (24) with D = Diag(¯ne,int + 1, ¯ne,int � �� � bath:e , ¯ne,wg + 1, ¯ne,wg � �� � waveguide:e , 1, 0 ���� bath:o , 1, 0 ���� detector:o , 1, 0 ���� bath:t , 1, 0 ���� bath:tm ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (25) We note that, in our experiments, the microwave waveguide remains in the ground state, with ¯ne,wg = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The spectral correlations of different output field can be simply obtained analytically from ⟨fout(ω)f † out(ω′)⟩ = S(ω)DS†(−ω) � �� � ˜ Cff†(ω) δ(ω + ω′).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (26) 6 Here we implicitly define the 4 × 4 matrix of output mode correlations with a single entry reading ⟨ˆaout(ω)ˆbout(ω′)⟩ = ˜Cab(ω)δ(ω + ω′), (27) where the operators ˆaout(ω),ˆbout(ω) were chosen from components of fout(ω) in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Covariance Matrix of Filtered Output Fields We will now consider a situation where we define output field modes from a windowed Fourier transformation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Below we will then show that these are indeed the experimentally observed signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We start by defining the (dimensionless) hermitian output field quadrature pair [8], ˆXα(ωn) = 1 √ 2T � T/2 −T/2 dτ eiωnτˆaα,out(τ) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=', (28) ˆPα(ωn) = 1 √ 2Ti � T/2 −T/2 dτ eiωnτˆaα,out(τ) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=', (29) which meets the canonical commutation relation [ ˆXα(ωn), ˆPβ(ωm)] = iδnmδαβ where α = e, o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Due to the finite window of the Fourier transformation, the frequencies ωn = 2π T n becomes discrete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The quadrature modes at discrete frequencies ωn can now be rewritten in terms of the (dimensionful) output fields fout(ω) from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (23), which are defined in the continuous Fourier domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Therefore the quadrature operators may be obtained by convolution with the a filter function G(ω) ˆXα(ωn) = 1 √ 2 � ∞ −∞ dω G(ωn − ω)ˆaα,out(ω) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (30) ˆPα(ωn) = 1 √ 2i � ∞ −∞ dω G(ωn − ω)ˆaα,out(ω) + h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (31) Here the filter is G(ω) = 1 √ 2π � ∞ −∞ dτ eiωτ 1[0,T ](τ) √ T = � 2 πT sin(ωT/2) ω , (32) which is obtained from a Fourier transformation of the unit function 1[−T/2,T/2](t) = 1(0) for |t| ≤ T/2 (|t| > T/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' A bipartite Gaussian state is characterized by the 4 × 4 covariance matrix (CM), VAB(ωn) = 1 2⟨{δ ˆA(ωn), δ ˆB(ωn)}⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (33) Here we defined δ ˆA = ˆA − ⟨ ˆA⟩ an operator with zero mean ⟨δ ˆA⟩ = 0 and the quadratures from ˆA(ωn), ˆB(ωn) ∈ { ˆXe(ωn), ˆPe(ωn), ˆXo(−ωn), ˆPo(−ωn)} (34) and we also introduced the anti-commutator { ˆA, ˆB} = ˆA ˆB + ˆB ˆA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Note that the two-mode squeezing interaction results in correlation between frequency reversed pairs on the microwave ωn and the optical side −ωn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Since in our setting all first moments ⟨ ˆA⟩ = 0 the evaluation of the covariance matrix in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (33) boils down to computing spectral correlations which are rewritten as ⟨ ˆA(ωn) ˆB(ωn)⟩ = � ∞ −∞ dω � ∞ −∞ dω′ G(ωn − ω)G(ωn − ω′)⟨ ˆA(ω) ˆB(ω′)⟩ = � ∞ −∞ dω � ∞ −∞ dω′ G(ωn − ω)G(−ωn − ω′)CAB(ω)δ(ω + ω′) = � ∞ −∞ dω F(ωn − ω)CAB(ω), (35) 7 where we used the property G(−ω) = G(ω) and defined the effective filter F(ω) = G(ω)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (26), we defined the quadrature correlations CAB(ω) = (C(ω))AB = 1 2 � U ˜Cff †(ω)U † + (U ˜Cff †(ω)U †)⊤� AB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (36) Here the unitary matrix U = u ⊕ u, with u = 1 √ 2 � 1 1 −i i � , (37) corresponds to a rotation of the mode operators into quadrature operators ( ˆXα, ˆPα)⊤ = u · (ˆaα,out, ˆa† α,out)⊤.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The covariance matrix of the quadrature modes at the discrete frequencies ωn is then obtained exactly by VAB(ωn) = � ∞ −∞ dω F(ωn − ω)CAB(ω), (38) where the quadrature correlations are convolved with an appropriate filter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Heterodyne Detection, Added Noise and Filtering 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Heterodyne Measurement Here we discuss the quadrature extractions from the equivalent linear measurement, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' balanced heterodyne detection, with excess added noise [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In the heterodyne detection, the output field ˆaoute−iωjt (j ∈ e, o) is mixed with a strong coherent local oscillator field ˆaLO(t) = αLOe−iωLOt at a 50:50 beam-splitter, where the output field from the two ports are sent to a balanced photo-detector, which results in a photon current that is proportional to ˆIout(t) = e−i∆LOtˆaout + ˆa† outei∆LOt, (39) in the limit of strong LO (αLO ≫ 1) with ∆LO = ωLO − ωj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We consider finite measurement interval of time T, on which we compute the windowed Fourier transformation of ˆIout(t), ˆIout(ωn) = 1 √ T � T 0 dτ eiωnτ ˆIout(τ) = 1 √ T � T 0 dτ eiωnτ(e−i∆LOτˆaout(τ) + ei∆LOτˆa† out(τ)) = aout(ωn − ∆LO) + a† out(ωn + ∆LO), (40) where in a slight abuse of notation we define the dimensionless output fields aout(ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The reason why we are explicitly working the windowed Fourier transformation is, that despite being in a steady state during the measurement (see Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' I B), the Fourier transformed data has a rather broad bandwidth (δωn = 2π T ∼ 5MHz for a 200 ns time window) due to the relatively short time of data collection T = τpulse−τdelay = 200 ns (especially for 250 ns optical pump pulse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In the limit of long measurement times T → ∞, the bandwidth will tend to zero and the following discussion as well as the results in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (38) will coincide with standard Input-Output treatment in the continuous Fourier domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In our experiments, we extract the quadratures of microwave and optical output field, by decomposing the heterodyne current spectra, in their real and imaginary parts which yields ˆIout(ωn) = 1 √ 2( ˆX(ωn − ∆LO) + ˆX(−ωn − ∆LO) � �� � ˆIX,out(ωn) +i [ ˆP(ωn − ∆LO) − ˆP(−ωn − ∆LO)] � �� � ˆIP,out(ωn) ), (41) where we define the quadrature output fields ˆaout(ωn) = ( ˆX(ωn) + i ˆP(ωn))/ √ 2, in the same way as in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (28-29).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' So far we have treated the photon current which result from a heterodyne measurement in terms of a time dependent hermitian operator ˆIout(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In an actual experiment the heterodyne current is a real scalar I(t) quantity which fluctuates in time and between different experimental runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Taking taking the (fast) Fourier transform of this current and decomposing it in its real and imaginary parts then yields I(ωn) = IX(ωn) + iIP (ωn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The theory of continuous measurements and quantum trajectories [10, 11] tells us how to connect the measured scalar currents with the current 8 operators from input-output theory [6] IA(ωn)IB(ωm) = 1 2⟨{ˆIA,out(ωn), ˆIB,out(ωm)}⟩, (42) where we define the statistical average · · · over many experimental runs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Realistic Measurements: Added Noise and Gain For the vacuum, the noise spectral density for both quadratures, are obtained by SAA(ωn) = ⟨ ˆA(ωn) ˆA(ωn)⟩vac = 1 2, (43) for the hermitian operator ˆA = ˆX, ˆP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Note that due to the discreteness of the Fourier domain we do not have a Dirac delta as opposed to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (27).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The noise spectrum of the heterodyne current is defined by SII(ω) ≡ I(ωn)I(ωn) = ⟨ˆIout(ωn)ˆIout(ωn)⟩, where SII(ωn) = 1 2 (SXX (ωn − ∆LO) + SP P (ωn − ∆LO) + SXX (ωn + ∆LO) + SP P (ωn + ∆LO)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (44) Focusing on the part of the spectrum located around ∆LO, SII(ωn + ∆LO) = 1 2 (SXX (ωn) + SP P (ωn) + 1) , (45) assuming ∆LO ≫ κe, κo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This indicates the simultaneous quadratures measurements and added shot noise in the heterodyne measurements, even without experimental imperfections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' So far we have focused on the ideal theory of the measurement and disregarded additional unknown sources of noise as well as the connection to the actually measured quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In practice, the decomposed measured quadratures contain additional uncorrelated excess noise, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' due to the added noise in the amplification or due to propagation losses [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We model this by phenomenologically adding another uncorrelated noise process from an independent thermal reservoir and then multiplying by a gain factor which converts the number of measured photons to the actually monitored voltage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' To illustrate this we focus again on a single output port, and with the added noise current ˆIX/P,add(ωn) and the frequency dependent calibration gain Gdet(ωn), where ˆIX,det(ωn) = � Gdet(ωn)(ˆIX,add(ωn) + ˆIX,out(ωn)), (46) ˆIP,det(ωn) = � Gdet(ωn)(ˆIP,add(ωn) + ˆIP,out(ωn)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (47) We thus obtain the detected heterodyne noise spectral density, SII,det(ωn + ∆LO) =Gdet(ωn + ∆LO)[SXX (ωn) + SP P (ωn) + 1 + SIXIX,add(ωn + ∆LO) + SIP IP ,add(ωn + ∆LO) � �� � =2Nadd ], (48) where we define the spectra of the added noise SIOIO,add(ωn) = ⟨ˆIO,add(ωn)ˆIO,add(ωn)⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The added noise Nadd includes the excess vacuum noise from heterodyne measurement and the additional uncorrelated noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Note that here the factor 1 2 was absorbed in the detections gains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The gain Gdet(ωn) can be simply obtained on both microwave and optical side, from the cold measurements (optical pump off) with a known background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We note that, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (48) lays the foundation of microwave and optical calibrations in our CEO device.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In our experiments, we place the LO on opposite sites around the mode resonances, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=', ∆LO,e = −ΩIF, ∆LO,o = ΩIF, (49) where ΩIF > 0 is the intermediate frequency for down-mixing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The heterodyne output field can be obtained similar 9 to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (40), ˆIout,e(ωn + ΩIF) = 1 √ 2[( ˆXe(−ωn) + ˆXe(ωn + 2ΩIF) + i(− ˆPe(−ωn) + ˆPe(ωn + 2ΩIF))], ˆIout,o(ωn + ΩIF) = 1 √ 2[ ˆXo(−ωn − 2ΩIF) + ˆXo(ωn) + i(− ˆPo(−ωn − 2ΩIF) + ˆPo(ωn))], (50) with noise spectrum given by, SII,e(ωn + ΩIF) = 1 2(SXeXe (−ωn) + SPePe (−ωn)) + Ne,add, SII,o(ωn + ΩIF) = 1 2(SXoXo (ωn) + SPoPo (ωn)) + No,add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (51) We note that, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (50) is adopted for field quadrature extraction (including the added noise) from the heterodyne measurement, which reveals correlations in the quadrature histogram [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='4 in the main text].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Despite of the reversed sign in the expected field quaduratures, microwave and optical output photons appear at the same frequency in the noise spectrum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ωn + ΩIF [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='2 d,e in the main text].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Covariance Matrix from Realistic Heterodyne Measurements Here we briefly explain the procedure of the covariance matrix reconstruction from the heterodyne measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The cross correlations of the detected heterodyne current spectra can be obtained via, DAB(ωn) = δIA,det(ωn + ΩIF)δIB,det(ωn + ΩIF), (52) where we define the centered current δIO,det = IO,det − IO,det, with IO,det(ωn) ∈ {IXe,det(ωn), IPe,det(ωn), IXo,det(ωn), IPo,det(ωn)}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (53) Similar to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (42)), we can obtain DAB(ωn) = 1 2⟨{δ ˆIA,det(ωn + ΩIF), δ ˆIB,det(ωn + ΩIF)}⟩ = � GA,det(ωn + ΩIF))GB,det(ωn + ΩIF)) �1 2⟨{δ ˆA(ωn), δ ˆB(ωn)}⟩ � �� � =VAB(ωn) +NAB,add � , (54) where we define the diagonal added noise matrix NAB,add = (Nadd)AB = NA,addδAB with the calibrated added noise Nadd and detection gain GA,det.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This equation establishes how the covariance matrix of the qudrature operators [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (38)] is reconstruced from heterodyne measurements, and how they can be compared with the results from idealized standard input-output theory Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' For simplicity, in the main text we define the total measured covariance matrix including the added noise as, VAB,meas(ωn) = DAB(ωn)/ � GA,det(ωn + ΩIF))GB,det(ωn + ΩIF)), (55) with VAB,meas(ωn) = VAB(ωn) + NAB,add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We note that, in principle the location of both LOs can be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' As evident in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 54, our choice of the LO configuration, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ∆LO,e = −∆LO,o = −ΩIF, offers a simple solution to the quantification of the broadband quantum correlations, considering the limited detection bandwidth, frequency dependent gain, or microwave cavity frequency shift, which may result in the loss of quantum correlations during quadrature extractions in heterodyne measurements due to imperfect frequency matching.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 10 C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Entanglement Detection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Duan Criterion We will now discuss how show that the photons outgoing microwave and optical photons are indeed inseparable or entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Our starting point is the covariance matrix which we defined in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (33) and measured as outline in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (54).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The experimentally measured covariance matrix is of the form V = � Ve Veo Veo Vo � = � � � � V11 0 ˜V13 ˜V14 0 V11 ˜V14 − ˜V13 ˜V13 ˜V14 V33 0 ˜V14 − ˜V13 0 V33 � � � � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (56) Since there is no single mode squeezing we have V22 = V11 and V44 = V33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' For simplicity we have omitted the frequency argument ωn of component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' What we describe in the following will have to be repeated for every frequency component.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The off-diagonal part in the covariance matrix which encodes the two-mode squeezing can be written as Veo ≃ V13(sin(θ)σx + cos(θ)σz), (57) where we define V13 = ( ˜V 2 14 + ˜V 2 13)1/2 and the mixing angle tan(θ) = ˜V14/ ˜V13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In our experimental setting ˜V14 maybe non zero e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' due to small finite detunings δo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' For the detection of inseparability, we employ the criterion introduced by Duan, Gidke, Cirac and Zoller [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This criterion states that if one can find local operations Uloc = Ue ⊗ Uo such that the joint amplitude variance of ˆX+ = ( ˆXe + ˆXo)/ √ 2 break the inequality, ∆X2 + = ⟨U † loc ˆX2 +Uloc⟩ < 1/2, (58) then the state is inseparable and, thus it must be concluded that it is entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In this setting, it is enough to choose the local operations Uloc = UeUo to be a passive phase rotation on the optical mode only, with Ue = 1 and Uo = e−iϕˆa† oˆao, and phase rotation angle ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In the space of covariance matrices, this corresponds to the (symplectic) transformation Sϕ = 12 ⊕ Rϕ, where we define the rotation matrix, Rϕ = � cos (ϕ) sin (ϕ) − sin (ϕ) cos (ϕ) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (59) The local rotation of the phase V (ϕ) = SϕV S⊤ ϕ will act on the off diagonal part of the covariance matrix as, Vea(ϕ) = V13(cos(θ − ϕ)σz + sin(θ − ϕ)σx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (60) With these local rotations the joint amplitude variance becomes ∆X2 +(ϕ) = ⟨( ˆXe + ˆXo cos(ϕ) + ˆPo sin(ϕ))2⟩/2 = V11 + V33 + 2V13 cos(θ − ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (61) We can similarly define the joint quadrature ˆP− = ( ˆPe − ˆPo)/ √ 2, where ∆P 2 −(ϕ) = ∆X2 +(ϕ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The variance of the joint quadratures ∆X2 +(ϕ) and ∆P 2 −(ϕ) is minimized at the angle ϕ− = θ − π ∆− EPR = ∆X2 +(ϕ−) + ∆P 2 −(ϕ−) = 2(V11 + V33 − 2V13), (62) which corresponds to the two-mode squeezing of microwave and optical output field, and the microwave-optics entan- glement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In addition, the joint quadrature variance is maximized at the angle ϕ+ = θ and we obtain ∆+ EPR = ∆X2 +(ϕ+) + ∆P 2 −(ϕ+) = 2(V11 + V33 + 2V13), (63) which corresponds to the anti-squeezing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 11 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Logarithmic Negativity and Purity A mixed entangled state can be quantified by the logarithmic negativity, EN = max [0, − log (2ζ−)] , (64) where ζ− is the smaller symplectic eigenvalue of the partially time reverse covariance matrix and can be obtained analytically ζ2 − = S − � S2 − 4det(V ) 2 (65) where we defined the Seralian invariant S = det(Ve) + detVo + 2det(Veo).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Furthermore the purity of a bipartite Gaussian state is given by ρ = 1 4 � det(V ) , (66) with ρ = 1 for a pure state i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' the vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' EXPERIMENTAL SETUP The experimental setup is shown and described in SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The laser is split into three parts, including an optical pulsed pump at frequency ωp, a continuous signal at ωp −FSR for the 4-port calibration (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' SI III), and a continuous local oscillator (LO) at ωp − FSR + ΩIF for the optical heterodyne detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The optical signal and pump pulse are sent to the optical resonator of the electro-optical device (DUT) and the reflected light (with pump pulse rejected by a filter cavity) is combined on a 50:50 beam splitter with the optical local oscillator with subsequent balanced photodetection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Microwave input signals are attenuated at different temperature stages of the dilution refrigerator (4 K: 20 dB, 800 mK: 10 dB, 10 mK: 20 dB), and sent to the coupling port of the microwave cavity of the DUT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The reflected microwave signal is amplified and can then either be mixed with a microwave local oscillator of frequency FSR − ΩIF and subsequently digitized, or directly measured by a vector network analyzer or a spectrum analyzer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We note that, the optical LO is on the right side of optical mode, while the microwave LO is on the left side of the microwave mode, with ΩIF/2π = 40MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' More details are in the caption of Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' SETUP CHARACTERIZATION AND CALIBRATION In the main manuscript, we show results from two different sets of optical modes shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The main difference between these mode sets is the amount of suppression of the anti-Stokes scattering rate compared to Stokes scattering rate given by scattering ratio S, which depends on the mode hybridisation of the anti-Stokes mode [5, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The first set of optical mode (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2a) from which we show most of our main results (main text Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2, 3 and 5a) has S =−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='3 dB on-resonance with an effective FSR = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='799 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The last power sweep shown in main text Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 5b is measured with a second set of optical modes with a lower S =−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='1 dB and a different effective FSR = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='791 GHz (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Despite it being the same optical resonator, the FSR for the second set of optical modes is slightly different, because of partial hybridisation of the optical pump mode which alters the working FSR between the optical pump and signal mode, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In the following, we carefully calibrate the added noise due to the microwave detection chain at both these working FSRs (since microwave mode is parked at the working FSR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The added noise can be slightly different depending on frequency of measurement due to impedance mismatch and reflections between components in the microwave detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Microwave added noise calibration In the following, we carefully calibrate the slightly different added noise in the microwave detection chain at both frequencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='The impedance mismatch and reflections between components in the microwave detection chain can vary 12 Dilution refrigerator Microwave preparation Optics preparation Optics and microwave detection MC1 MS2 MS3 C1 C2 C3 C5 C4 10mK 800mK 4K 300K DC Microwave signal (FSR) DC Optical signal (193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 THz - FSR) Optical pump (193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 THz) Coherent optical signal (193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 THz - FSR) Optical LO (193.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 THz -FSR - ωlo) Microwave LO (FSR + ωlo) IF (ωlo) RF lines DUT VNA SA DIGITIZER LNA RTA2 RTA1 Q I 1 1 2 90 10 1 1 2 3 3 4 PC1 S3 S4 MS1 DDG 25 75 BPD Laser Lock control 1550 nm SSB VOA1 99 1 EDFA1 EDFA2 AOM1 AOM2 PC2 PD2 PD3 PD4 F3 F1 F2 PD1 S1 S2 ΩLO OSA PM HEMT Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Experimental setup for two-mode squeezing measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' A tunable laser at frequency ωp is initially divided equally in two parts, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' the optical pump and the optical signal together with the optical local oscillator (LO).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Light from the optical pump path is pulsed via an acousto-optic modulator (AOM1) which produces ns-pulses and shapes them for amplification via an Erbium-doped fiber amplifier (EDFA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The output from the EDFA is first filtered in time via AOM2 to remove the amplified spontaneous emission (ASE) noise and later in frequency via filter F1 (∼50 MHz linewidth with 15 GHz FSR) to remove any noise at the optical signal frequency (the reflected power is rejected by circulator C3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The filter F1 is locked to the transmitted power by taking 1% of the filter transmission measured via photodiode PD3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The polarization of the final output is controlled via polarization controller PC1 before being mixed with the optical signal via a 90-10 beam splitter and sent to the dilution refrigerator (DR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The 10% output from the beam splitter is monitored on a fast detector PD2 to measure the optical pump pulse power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The other half of the laser is again divided into two parts - 25% for the optical signal and 75% for the optical LO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The signal part is sent first to a variable optical attenuator VOA1 to control the power and then to a single sideband modulator SSB which produces the optical signal frequency at ωp − FSR and suppresses the tones at ωp and ωp + FSR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 1% of the optical signal is used to monitor the SSB suppression ratio via an optical spectrum analyzer OSA and 99% is sent to the DR after being polarization controlled via PC2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The optical LO is produced via a phase modulator PM and detuned by ωIF/2π = 40 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' As the PM produces many sidebands, the undesired sidebands are suppressed via filter F3 (∼50 MHz linewidth with 15 GHz FSR), reflection is rejected by circulator C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' F3 is temperature-stabilized and locked to the transmitted power similar to F1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The optical LO is also amplified via EDFA2 before the optical balanced heterodyne.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In the DR, the light is focused via a gradient-index (GRIN) lens on the surface of the prism and coupled to the optical whispering gallery mode resonator (WGMR) via evanescent coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Polarization controllers PC1 and PC2 are adjusted to efficiently couple to the TE modes of the optical WGMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The output light is sent in a similar fashion to the collection grin lens.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Outside the DR, the optical pump is filtered via filter F2 (similar to F3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The reflected light from F2 is redirected via C4 to be measured with PD1 which produces the lock signal for the laser to be locked to optical WGMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The filtered signal is finally mixed with the optical LO and measured with a high speed balanced photo-diode BPD (400 MHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The electrical signal from the BPD is amplified via RTA1 before getting digitized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' On the microwave side, the signal is sent from the microwave source S3 which is connected to the DDG for accurately timed pulse generation (or from the VNA for microwave mode spectroscopy) to the fridge input line via the microwave combiner (MC1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The input line is attenuated with attenuators distributed between 4 K and 10 mK accumulating to 50 dB in order to suppress room temperature microwave noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Circulator C1 and C2 shield the reflected tone from the input signal and lead it to the amplified output line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The output line is amplified at 4 K by a HEMT-amplifier and then at room temperature again with a low noise amplifier (LNA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The output line is connected to switch MS1 and MS2, to select between an ESA, a VNA or a digitizer measurement via manual downconversion using MW LO S4 (40 MHz detuned).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Lastly, microwave switch MS3 allows to swap the device under test (DUT) for a temperature T50 Ω controllable load, which serves as a broad band noise source in order to calibrate the microwave output line’s total gain and added noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' the added noise slightly as a function of frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This added noise and corresponding gain due to a series of amplifiers 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 a b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 |Soo|2 |Soo|2 ωp +FSR ωp FSR ωp ωp +FSR ωp FSR ωp Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Optical mode spectra in reflection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Normalized reflection intensity |Soo|2 spectra of optical modes ˆao, ˆap and ˆat in red, green and blue respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' a (b) shows the optical mode spectra of the first (second) set of modes with the anti-Stokes and Stokes scattering ratio S = −10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='3 dB (−3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='1 dB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The dashed line marks the effective FSR between the pump mode ˆap and the optical mode ˆao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The participation anti-Stokes optical mode ˆat is suppressed for this effective FSR as marked by the dashed line over the blue mode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' and cable losses in the microwave detection chain is calibrated using a combination of a 50 Ω load, a thermometer and a resistive heater that are thermally connected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The microwave detection chain is identical for the signals from the 50 Ω load and the microwave cavity reflection, except for a small difference in cable length which we adjust for.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' To calibrate the detection chain, we heat the 50 Ω load with the resistive heater and record the amplified noise spectrum P50Ω(ω) as a function of temperature of 50 Ω load T50Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The output noise detected over a bandwidth B, P50Ω, as a function of T50Ω is given as, P50Ω = ℏωeGB �1 2 coth � ℏωe 2kBT50Ω � + Ne,add � , (67) with ωe the center microwave frequency, Ne,add (G) the added noise (gain) of the microwave detection chain, and kB the Boltzmann constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' A bandwidth of 11 MHz is selected around the region of interest to calculate Ne,add and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' For ωe = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='799 GHz, we show the detected noise Ne,det = P50Ω/(ℏωeGB) as a function of T50Ω in SI Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 3 along with a fit using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 67, with two fitting parameters G and Ne,add.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We note that, at T50Ω = 0 K, Ne,det = Ne,add + 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Table 1 (third row) shows the obtained added noise and gain for two frequencies of interest, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ωe/2π = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='799 GHz and ωe/2π = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='791 GHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Next, we consider the difference in cable losses between the 50 Ω load and the microwave cavity, which are in- dependently determined by measuring the microwave reflection from the microwave cavity and from the microwave switch directly before it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Including the cable losses, the effective added noise increases while the gain decreases for the reflected microwave detection, shown in Table 1 (fourth row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Finally, we consider an additional error due to the temperature sensor inaccuracy of 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Although this does not change the final Ne,add and G, it increases the uncertainty as shown in Table 1 (fifth row).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The error calculated in this section contributes to the systematic error reported in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 14 Supplementary Tab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The added noise and gain in microwave detection chain (1σ errors shown) 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='799 GHz 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='791 GHz Detection Chain Ne,add G (dB) Ne,add G (dB) 50Ω load (with fitting error) 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='74 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='08 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='67 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='02 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='76 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='09 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='03 MW cavity (including cable loss) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='09 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='02 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='10 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='03 MW cavity (including temperature sensor uncertainty) 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='33 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='20 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='12 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='34 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='23 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='12 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 13 12 14 15 16 Experiment Fit (K) Tf Ne,det (photons s-1 Hz-1) Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Characterization of the added noise in the microwave detection chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Measured output noise from a 50 Ω calibration load as a function of its temperature Tf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The measured noise is plotted in units of photons as N 50Ω det = P50Ω/(ℏωeGB).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The dashed line at the bottom represents the fitted vacuum noise level in addition to the added noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The red line and shaded region represents the fit and the 95% confidence interval around it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Optical added noise Optical added noise is calculated via 4-port calibration of our device [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In this calibration, we measure the coherent response of our device through its 4 ports - optical input/output and microwave input/output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Sending an optical (or microwave) signal to the DUT in combination with a strong pump leads to stimulated parametric down-conversion (StPDC) process, which generates an amplified microwave (optical) coherent signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We measure the 4 S-parameters of our device - microwave reflection (S11), optics reflection (S22), microwave to optics transmission (S21) and optics to microwave transmission (S12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The mean transduction efficiency between microwave and optics of the DUT is then calculated as, η = � S12S21 S11S22 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (68) We use the transduction efficiency and Ne,add in the microwave detection chain from Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' III A to calculate the optical added noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Ne,add is firstly used to calculate the effective microwave detection gain (different from the one in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' III A, because the microwave detection line used for the 4-port calibration uses analog downconversion and digitization, while the thermal calibration uses SA, see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The microwave gain, along with the (off-resonant) microwave reflection measurement, is used to calculate the microwave input loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We can obtain the microwave signal power at the DUT, which allows us to calculate the output optical power of the DUT using the transduction efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In conjunction with the measured output power at the end of the detection chain, the losses in the optical detection path and hence, the effective added noise with respect to the optical port of the DUT can be calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The calculated optical added noise is No,add = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='54 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='21(7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='42 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='22) for ωe = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='799 GHz (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='791 GHz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 15 IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' DATA TREATMENT In this section, we describe all the steps for the data treatment in detail, which includes the time domain analysis (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' IV A), the pulse post-selection due to setup drift (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' IV B), the frequency domain analysis (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' IV C), and the quadrature correlations (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' IV D).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Time-domain analysis Both microwave and optical signals are detected via heterodyne detection by mixing with a strong local oscillator that is ∼40 MHz detuned from respective mode resonance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The output heterodyne signals are digitized using a digitizer at 1 GigaSamples/second.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' First, we digitally downconvert the digitized data at ωIF = 40 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This yields the two quadratures IXe/o,det(t) and IPe/o,det(t) of the microwave or optical output signal record with 40 MHz resolution bandwidth (using 25 ns time resolution).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 4 shows the calibrated output power (I2 Xe/o,out + I2 Pe/o,out) [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 50] and the phase (arctan(IXe/o,out/IPe/o,out)) from a single pulse sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This includes the stochastic SPDC signals from a strong pump pulse, and the coherent StPDC signal from a weaker pump pulse together with a coherent microwave signal for calibration purposes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The SPDC signal produced by the first strong pulse is labeled by the shaded region for one single pulse, and the averaged output power over many pulses is shown in main text Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The coherent microwave reflection and stimulated parametric downconverted optical signal are adopted to obtain the phases during the pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We record this measured phase in both signal outputs during the second optical pump pulse for phase-drift correction in later post processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 0 1 2 3 4 5 0 1 2 3 4 5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0 200 400 0 50 100 150 200 600 800 1000 1200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 a b (µs) Phase (rad/π) Phase (rad/π) (photons/s/Hz) Ne, (photons/s/Hz) No, t (µs) t Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Downconverted output signal for a single measured pulse sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' a (b) show the measured microwave (optical) output signal downconverted at 40 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The shaded part in each case shows the region of the SPDC signal (the first optical pump pulse).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' For a single pulse, the SNR of a SPDC signal is too small to be seen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' However, during the second optical pump pulse, a coherent response is seen in both signal outputs where the phase can be measured with high SNR for each single shot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In order to determine the accuracy of a phase correction for the first pump pulse based on the phase measurement during the second pump pulse, we send a continuous microwave signal during both pump pulses and recorded the phase of the converted optical pulse during the first and the second optical pump pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 5 shows the phase difference between the first and second optical pump pulse for 2500 trials along with a normal distribution fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The fit variance for the distribution is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='17 rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' On a similar set of model data, applying a random phase variation of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='17 rad results in about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5-2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0% loss of correlations [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' IV D], whereas, we observe about 6-8% loss of correlations in the experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The imperfection in phase correction does not completely explain the decreased quantum correlations, which might be due to other experimental instabilities, especially the optical pump laser lock.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 Probability density Phase difference (radians) Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Accuracy of phase correction scheme.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The histogram shows the difference in the measured phase between the first and second optical pump pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Since we correct the phase in the first pump pulse based on the measured optical phase of the second optical pump pulse, the difference shows the limitations of this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The grey dashed line is a normal distribution fit with variance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='17 rad.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Pulse post-selection In our experiments, we use three temperature-stabilized optical filters, which may drift slowly in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Two of them are used in the optical heterodyne detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In the signal path, one filter (F2 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='1) is used to filter the optical signal while reject the strong optical pump.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In the LO path, one filter (F3 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='1) is used to obtain a clean optical LO tone, which is genearted by an electro-optic phase modulator (which produces multiple sidebands) and then amplified using an EDFA (which produces excess amplified spontaneous emissions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The slow filter drifts can be identified from the amplitude of the coherent optical signal produced via stimulated parametric downconversion during the second optical pump pulse, which drops due to either the decreased transmission after F2 or the reduced LO power after the F3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This is evident in the histogram of the converted optical power during the second optical pump pulse as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 6a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The histogram is not symmetric and has a tail at the lower end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' To filter out the instances of drifted heterodyne detection, we select a threshold (in this case marked by a dashed line in Fig 6) and remove all pulses below the selected threshold along with 20 neighboring pulses (10 s in total time) before and after such instance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' These numbers are chosen according to the filter drift and the filter temperature lock time-scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' After such filtering, usually about 10% of the data is removed and the histogram of the converted optical power during the second optical pump pulse becomes symmetric as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 6b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Frequency domain analysis As already mentioned in the main text, with the help of time-domain analysis, we select three different time-snippets to analyze the data in the frequency domain - before-pulse, on-pulse and post-pulse defined with respect to the first optical pump pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The main challenge in processing the data in frequency domain is the proper normalization of the measured output spectrum [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The microwave reflection baseline is not flat because of slight impedance mismatches between different components in the microwave detection chain, with similar optical heterodyne shot noise floor due to the frequency dependent balanced detector gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In addition, we observe slight shift of a few millivolts each time in the digitizer measurements when a new measurement is launched and the digitizer is reinitialized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Combined with the fact that the amplifier gain in the microwave detection chain as well as the optical heterodyne gain (due to optical LO power drift) may drift over a long time, an in-situ calibration of vacuum noise level is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In case of microwave, we need to first correct for the microwave reflection baseline distortion from impedance mismatch and then correct for the signal level shift caused by the digitizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' For the distorted baseline, we separately measure the microwave output spectrum when the microwave cavity is in its ground state (thermalized to 7 mK at mixing chamber).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This measurement is shown in Supplementary Fig 4a (gray) along with the measured before-pulse (cyan), on-pulse (purple) and after-pulse microwave noise spectrum (orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Dividing the measured spectra with the cold cavity spectrum reveals a flat baseline Lorentzian noise spectra, however with an offset due to the digitizer drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' To correct for this offset, we perform an in-situ vacuum noise calibration using the off-resonance (waveguide) noise in the before-pulse microwave noise spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' An independent measurement of the microwave waveguide noise as a function of the average optical pump power (averaged over the full duty cycle) is shown in Supplementary Fig 17 50 0 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='02 50 0 100 150 200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='02 a b Optics quanta (photons s-1 Hz-1) Optics quanta (photons s-1 Hz-1) Count (Normalized) Count (Normalized) Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Histogram of the converted optical power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The measured coherent optical power during the second optical pump pulse depends on the optical heterodyne gain and the received optical signal power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Both of these values can drift depending on the experimental setup’s stability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' a shows the normalized histogram of this measured optical power over all the collected pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The histogram has a tail on the lower end owing to the times when the heterodyne setup drifted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Filtering the points which do not meet a selected threshold (shown by the grey dashed line in a), we remove the instances where the setup had drifted and the optical heterodyne detection efficiency was compromised.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The same histogram after removing such points is shown in b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The error bars (2σ deviation) result from the microwave detection chain gain and the measurement instrument drift.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The power law fit reveals that the microwave waveguide noise grows almost linearly with average optical pump power, and only deviates significantly from 0 for optical pump power >3 µW.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' As we work with average optical pump powers of ≪1 µW, we can safely assume the microwave waveguide noise to be zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Therefore, we use the off-resonant waveguide noise for before-pulse microwave noise spectrum as an in-situ vacuum noise calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In case of optics, the optical detection is shot-noise limited, and the excess LO noise at the optical signal frequency is suppressed by more than 40 dB using filter F1 in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We use the before-pulse optical noise spectrum as the vacuum noise level and normalize the optical on-pulse spectrum directly with the before-pulse in-situ calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Supplementary Fig 4b shows noise spectrum (without normalization) of the optical off-pulse (cyan), on-pulse (green), and the after-pulse (yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The signal during the optical pump pulse is clearly visible, and the noise level is identical before and after the optical pulse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The normalized noise spectra for both microwave and optics are shown in main text Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2d and 2e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=', where we can obtain the normalization gain [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Joint-quadrature correlations The detected output quadratures including excess added noise, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆIXe,out(∆ω), ˆIPe,out(∆ω), ˆIXo,out(∆ω), ˆIPo,out(∆ω), can be obtained from the real and imagrinary parts in the discret Fourier transform of the photocurrent by normalizing to the detection gain [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 50].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Similar to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' I C 1, we can define the joint detected quadratures, by applying phase rotation on the optical ones, ˆIX,+(∆ω, φ) = ˆIXe,out(∆ω) + � ˆIXo,out(∆ω) cos φ − ˆIPo,out(∆ω) sin φ � √ 2 , ˆIP,−(∆ω, φ) = ˆIPe,out(∆ω) − � ˆIXo,out(∆ω) sin φ + ˆIPo,out(∆ω) cos φ � √ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (69) To verify the non-classical correlation between the unitless quadrature variables for output microwave and optics field, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ˆXe(∆ω) & ˆXo(−∆ω) and ˆPe(∆ω) & ˆPo(−∆ω), we can calculate the phase dependent joint quadrature 18 40 20 60 59 60 61 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='24 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='26 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='28 62 63 64 65 a b (MHz) 40 20 60 MW output (nW/2MHz) Opt output (µW/2MHz) Off-pulse In-pulse Off-pulse In-pulse After-pulse After-pulse Cold cavity ω/2π (MHz) ω/2π Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Spectra of output signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The microwave reflection baseline is not flat due to an impedance mismatch between different components in the microwave detection chain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' As a result, the output power measured from amplified vacuum noise (from the cold microwave cavity) is not flat (shown in gray in a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Additionally, the digitizer in our setup has a different noise level each time it is started.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' As a result, the cold cavity baseline has an extra offset with respect to all other measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' a also shows the measured output spectra for time region before (during, after) the first optical pulse shown in cyan (purple, orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Similarly, b shows the output spectra for the optical output before (during, after) the first optical pulse in cyan (green, orange).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 100 101 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='4 Experiment 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='01Pavg 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='97 Pavg (µW) Wavegide noise (photons/s/Hz) Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Microwave waveguide noise as a function of the average optical pump power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The error bars represent 2σ error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The solid line is a power law fit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We find the power law is actually quite close to a linear function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' variance [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 61], � ˆX2 +(∆ω, φ) � = � ˆI2 X,+(∆ω, φ) � − Ne,add + No,add 2 , � ˆP 2 −(∆ω, φ) � = � ˆI2 P,−(∆ω, φ) � − Ne,add + No,add 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (70) For ∆ω = 0, we plot the joint quadrature variance as a function of local oscillator phase in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 9 (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The shaded region represent the 2σ statistical error in the calculated joint quadrature variances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We note that, the statistical 1σ error of the variance for a Gaussian distributed data is given by √ 2σ2/ √ N − 1, where N is the length of the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The obtained resonant ∆EPR(0, φ) is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 9(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The minimum and maximum of ∆EPR(φ) over the local oscillator phase are defined as min[∆EPR] = ∆− EPR and max[∆EPR] = ∆+ EPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ∆− EPR < 1 indicates non-classical joint correlations and squeezing below vacuum levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 19 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 a b Quanta Phase (rad/π) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 Phase (rad/π) ΔEPR X+ P+ Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Joint quadrature correlations and ∆EPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Joint quadratures at resonance X+(∆ω = 0) and P+(∆ω = 0) are plotted as a function of the local oscillator phase φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ∆EPR as a function of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The shaded region in both plots represents the 2σ statistical error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The broadband phase that minimizes ∆EPR(∆ω, φ), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' φmin(∆ω), reveals the difference in arrival times (group delay) between the microwave and optical signal output (Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 10a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' After fixing the inferred time delay between the in-pulse arrival time of the microwave and optical signal, φmin becomes independent of frequency detuning from the mode resonances.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Thus, we adjust for the differences in arrival times by ensuring that the slope of φmin with respect to detuning ∆ω is minimized for all datasets we analyze, utilizing the broadband quantum correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 a b 0 20 30 10 20 10 30 (MHz) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0 20 30 10 20 10 30 (rad/π) φ (rad/π) φ ∆ω/2π (MHz) ∆ω/2π Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Effect of time delay between the microwave and optics signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The plots show the local oscillator phase φmin which minimizes ∆EPR(∆ω, φ) as a function of detuning frequency ∆ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' a (b) shows the case when the time difference of arrival between the microwave and optics signals was 25 ns (≈ 0 ns).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The solid lines are the linear fit to the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' QUADRATURE HISTOGRAM RAW DATA Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 11 shows the normalized difference of the two-variable quadrature histograms obtained during and before the optical pump pulse based on the data shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2 and 3 of main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' These unprocessed histograms directly show the phase insensitive amplification in each channel as well as the correlations in (Xe,Xo) and (Pe,Po).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Note however that - in contrast to the analysis in the main text - taking this difference does not lead to a valid phase space representation since also the vacuum noise of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 together with the output noise of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='026 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='011 photons (due to the residual microwave bath occupancy right before the pulse) are subtracted, hence the negative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 20 20 10 0 10 20 20 10 0 10 20 20 10 0 10 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 10 0 10 10 0 10 10 0 10 20 10 0 10 10 0 10 10 0 10 10 0 10 20 20 10 0 10 20 20 10 0 10 20 Pe Pe Pe Po Po Po Xo Xo Xo Xe Xe Xe Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Quadrature histogram raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Normalized difference of the two-variable quadrature histograms obtained during and before the optical pump pulse based on the data shown in Figs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2 and 3 of the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' VI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' NON-CLASSICAL CORRELATIONS WITH 600 ns LONG OPTICAL PUMP PULSES Before experimenting with 250 ns long optical pump pulses, we used 600 ns long optical pump pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' A sample measurement with a 600 ns is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 12a similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 3c of main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Compared to 250 ns long pulses, the main difference lies in the fact that ∆− EPR in the middle panel exhibits a double-dip shape because the correlations ¯V13 have a wider bandwidth than the emitted noise spectra ( ¯V11 and ¯V33), which are narrowed due to dynamical back- action [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Since in the measurement the correlations don’t clearly overwhelm the emitted noise, interference between two Lorentzian functions of different widths (dashed line) leads to the specific shape of ∆− EPR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Theory confirms this even though the shown theory curve (solid red line) does not exhibit the specific line-shape due to higher expected correlations compared to the experimentally observed values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' These results indicate that ¯ne,int due to a 600 ns optical pump pulse is large enough to prevent a clear observation of squeezing over the full bandwidth below the vacuum level (∆− EPR < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' As a result, we switched to 250 ns optical pump pulses with higher statistics as shown in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We also repeated the measurement with 600 ns long pulses with different optical pump powers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 12b shows the measured pump power dependence with each data point based on 170000-412500 individual measurements each with a 2 Hz repetition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The microwave mode thermal bath occupancy ¯ne,int changes little as a function of the peak optical pump power at the device and is approximated with a constant function (solid maroon line in the top panel).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The on-resonance mean CM elements scale with cooperativity and are in excellent agreement with theory (solid lines) based on the ¯ne,int.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The on-resonance squeezing ∆− EPR does not change significantly with cooperativity since both excess noise and correlations scale together with cooperativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The anti-squeezing ∆+ EPR scales up with cooperativity as expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' All but one measured mean values are below the vacuum level and three power settings show a > 2σ significance for entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Note that this power sweep was conducted on a different set of optical modes with a different amount of anti-Stokes sideband suppression (see section III).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' VII.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ERROR ANALYSIS The covariance matrix of the output field quadratures V (ω) can be directly calculated from the extracted microwave and optical quadratures from frequency domain analysis [cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 54] We simply rotate the optical quadratures with the phase that minimized the joint quadrature variance, and obtain the covariance matrix in the normal form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We note that, the error in calculating the covariance matrix comes from two sources - the statistical error due to finite number of pulses, and the systematic error in the vacuum noise level calibration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The detailed error analysis is described in the following subsections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We note that, the uncertainty in all the reported numbers in the main text corresponds to 2 standard deviation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 21 0 10 20 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='4 10 20 a b Interpolation In-pulse After-pulse Before-pulse 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='3 140 160 180 200 220 240 Optical pump power (mW) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='22 Cooperativity (V11 + V22)/2 (V33 + V44)/2 (V13 − V24)/2 (V11 + V22)/2 (V33 + V44)/2 (V13 − V24)/2 V (photons s-1 Hz-1) (photons s-1 Hz-1) (MHz) V ΔEPR ΔEPR + ΔEPR ¯ne,int ΔEPR + Δω/2π Supplementary Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Non-classical correlations vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' optical pump power for 600 ns long pump pulses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' a, Top panel, the average microwave output noise ¯V11 (purple), the optical output noise ¯V33 (green) and correlations ¯V13 (yellow) as a function detuning based on 412500 measurements with a 2 Hz repetition rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The solid lines represent the joint theory with fit parameters C = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='22 and in-pulse microwave thermal bath occupancy ¯ne,int = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='19 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content='03.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The dashed lines are individual Lorentzian fits to serve as a guide to the eye.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' ∆− EPR (∆+ EPR) in the middle (bottom) panel shown in red (blue) color are calculated from the top panel data and fits as described in the main text.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The darker color error bars represent the 2σ statistical error and the outer (faint) error bars also include systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' b, Power dependence of CM elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The top panel shows the microwave mode thermal bath occupancy ¯ne,int for before-pulse, after-pulse and in-pulse regimes (marked in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' 2A) as a function of the peak optical pump power at the device and the corresponding cooperativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The in-pulse ¯ne,int is obtained by the joint theory fit and approximated with a constant function (solid line).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The middle panel shows the on-resonance mean CM elements based on the ¯ne,int from the top panel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The bottom two panels show the on-resonance squeezing ∆− EPR and anti-squeezing ∆+ EPR calculated from the middle panel along with theory (solid lines).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The darker color error bars represent the 2σ statistical error and the outer (faint) error bars also include systematic errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' All measured mean values are below the vacuum level and three power settings show a > 2σ significance for entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Statistical error The error in the calculation of bivariate variances comes from the statistical uncertainties, arising from finite number of observations of a random sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' This error is the major component of our total error in diagonal covariance matrix elements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' The error in calculating the variance of a sample distribution sampled from a Gaussian variable follows the Chi-squared distribution and is given as, Var(σ2) = 2σ2 N − 1, (71) where, σ2 is the variance of sample distribution and N is its size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' In addition, the error in the covariance from a bivariate variable is given by the Wishart distribution [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' For a general bivariate covariance matrix Σ given as, Σ = � σ2 11 ρσ11σ22 ρσ11σ22 σ2 22 � , (72) 22 the variance of the covariance matrix is given by, Var(Σ) = 1 N − 1 � 2σ4 11 (1 + ρ2)σ2 11σ2 22 (1 + ρ2)σ2 11σ2 22 2σ4 22 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' (73) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Systematic error Although, the systematic error in our measurements are not as significant, they still are a noticeable source of error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Here the error in calculating the covariance matrix results form the error in the estimation of the vacuum noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' More specifically, the error in determining the added noise due to the microwave and optical detection chain, as discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Propagating these systematic errors through the covariance matrix analysis is non- trivial, since calculating the error in variance of erroneous quantities is challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Therefore, we use a worst-case scenario approach to calculate the total error including the statistical error and the systematic error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We repeat the full analysis, including the statistical errors, for the lower and upper bound of the uncertainty range from the systematic errors for the microwave and optical added noise levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' Repeating the analysis expands the error bars in the calculated quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We take the extremum of all the error bars from all the repetitions of analysis to get the total error bar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/9tE1T4oBgHgl3EQfoARp/content/2301.03315v1.pdf'} +page_content=' We show both statistical error and the total error in the main text.' metadata={'source': 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100644 index 0000000000000000000000000000000000000000..21d9def7e3124e1dd83df05cc2d595e28964c9fc --- /dev/null +++ b/H9FKT4oBgHgl3EQfdS4O/content/tmp_files/2301.11819v1.pdf.txt @@ -0,0 +1,3085 @@ +Boltzmann equation and its cosmological applications +Seishi Enomoto∗, Yu-Hang Su, Man-Zhu Zheng, and Hong-Hao Zhang† +School of Physics, Sun Yat-sen University, Guangzhou 510275, China +Abstract +We review the derivation of the Boltzmann equation and its cosmological applica- +tions. Our paper derives the Boltzmann equation by the language of quantum field +theory without any assumption of the finite temperature system. We also introduce +two examples of cosmological applications, dark matter abundance and baryogenesis, +with techniques in their calculations. +1 +Introduction +The early Universe, composed of hot plasma, evolves in time while maintaining thermal +equilibrium at most epochs, and thus the evolution is described simply by thermodynam- +ics. The cosmologically important events, therefore, are focused on the various turning +epochs in which some particle species depart from the thermal bath as seen in, e.g., Big +Bang nucleosynthesis (BBN) and recombination confirmed by both the theory and obser- +vations, and the dark matter (DM) relic abundance and the baryogenesis scenario derived +from some hypotheses. Qualitatively, the turning epochs can be estimated by comparing +the reaction rate and the spatial expansion rate of the Universe, but more quantitative +treatment to ensure accurate predictions is required by the present cosmological obser- +vations. The Boltzmann equation is a powerful tool for following the out-of-equilibrium +dynamics in detail, and can describe the evolution of the particle distribution due to the +spatial expansion and the momentum exchanges through interactions. Since the Boltz- +mann equation provides the abstract relation between the macroscopic and the microscopic +evolution of the particle distributions, it is applicable to various situations, and hence, the +solving equations differ for each system. +The most successful application of the Boltzmann equation is for the theory of BBN +[1, 2, 3, 4, 5, 6, 7] (see also, e.g., [8, 9, 10]), which describes how the initial protons and +neutrons form the light elements at the final. As the result, the abundances of the light +elements are predicted, and they are well consistent with the present observations. The +other well-motivated and formulated example is the Kompaneets equation [11, 12], which +is derived from the Boltzmann equation in the photon-electron system. The evolution +equation is applied to the various analysis in cosmological or astronomical situations, +e.g., the last scattering surface in the recombination epoch, the Sunyaev-Zel’dovich effect +[13, 14, 15, 16] that is a distortion of the cosmic microwave background radiation by hot +electrons in galaxies, etc. +On the other side, exploring Beyond the Standard Model (BSM), there are many +studies on the DM candidate realizing the present density parameter through the relic +abundance, which can be predicted using the Boltzmann equation. +The most famous +and simple scenario for DM candidates is described by weakly interacting massive par- +ticles (WIMPs) [17, 18], but the current observation requires a more extended scenario +or the other mechanism, e.g., feebly interacting massive particles (FIMPs) [19], strongly +interacting massive particles (SIMPs) [20], processing into forbidden channels [21, 22], co- +annihilations [21], co-scattering [23, 24], zombie [25, 26], and inverse decays [27, 28], etc. +∗seishi@mail.sysy.edu.cn +†zhh98@mail.sysu.edu.cn +1 +arXiv:2301.11819v1 [hep-ph] 27 Jan 2023 + +(See also [29] for their summarized analysis.) The more complicated models are consid- +ered, the more technical treatment in the Boltzmann equation is required. As the other +topic for exploring BSM with the Boltzmann equation, there are also many studies for +baryogenesis scenarios explaining the origin of (baryonic) matter-antimatter asymmetry +in our Universe, e.g., GUT baryogenesis [30, 31, 32], leptogenesis [33], electroweak baryo- +genesis [34, 35], and Affleck-Dine baryogenesis [36, 37], etc. The analysis tends to be more +complicated than the case of DM abundance and requires more technical treatment. +The studies using the Boltzmann equation have been more popular and important as +some approaches to confirm the current physics in detail and explore BSM in the early +Universe. We aim to provide a clearer understanding of the Boltzmann equation and its +techniques with some cosmological examples. This paper is organized as follows. First, +we derive the Boltzmann equation by the language of the quantum field theory in section +2. Next, we demonstrate application examples how the Boltzmann equation is applied for +the DM abundance in section 3 and the baryogenesis scenario in section 4 with simple toy +models. Finally, we summarize our discussion in section 5. +2 +Boltzmann equation +The Boltzmann equation is a quite powerful tool to describe the evolution of particles in +Cosmology. Although the equation is applicable to various situations, the used formulae +are also dependent on the circumstance. In this section, we derive the evolution formula +of the distribution function f = f(xµ, pµ) from the basic statement +L[f] = C[f] +(1) +where L and C are called the Liouville operator and the collision operator, respectively. +The Liouville operator describes the variation of the distribution of a particle along a +dynamical parameter, and the collision operator describes the source of the variation +through the microscopic processes. +2.1 +Liouville operator +We define the Liouville operator to describe the variation of the distribution along the +geodesic line parametrized by the affine parameter λ. Using the momentum relations +pµ = dxµ +dλ , +pµpµ = m2 +(2) +where pµ is a four-momentum and m is the mass, and the geodesic equation +0 = dpµ +dλ + Γµ +νρpνpρ +(3) +where Γµ +νρ is the affine connection, the Liouville operator can be written as +L[f] ≡ +df +dλ +���� +geodesic line += +dxµ +dλ ∂µf + dpµ +dλ +∂f +∂pµ +(4) += +pµ∂µf − Γµ +νρpνpρ ∂f +∂pµ . +(5) +In the FLRW space-time, gµν = diag(1, −a2, −a2, −a2) where a = a(t) is the scale +factor, the distribution function is spatially homogeneous and isotropic: f = f(t, E). +Then the concrete representation of the Liouville operator term can be written as +L[f] = E ˙f − H|⃗p|2 ∂f +∂E +(6) +2 + +where H = ˙a/a is the Hubble parameter and (⃗p)i ≡ api is the physical momentum. +2.2 +Collision operator +We define the collision operator C[f] as the variation rate by the microscopic processes: +C[f] ≡ +df +dλ +���� +microscopic process += +E df +dt +���� +microscopic process +. +(7) +In order to evaluate the variation of the distribution function, we assume the followings. +First, the process can be described by the quantum field theory. Second, the microscopic +process can be evaluated on the Minkowski space since the gravitational effect is already +evaluated in the Liouville operator part. +With the above assumption, the distribution function f can be regarded as the expec- +tation value of all possible occupation numbers with their probabilities, as we will see later. +Also the probabilities can be evaluated by the quantum field theory on the flat space. To +derive the concrete representation of (7), we need to construct the corresponding quantum +state and then evaluate the transition probability. +2.2.1 +Eigenstate for occupation number +At first, we construct a multi-particle state |{n}⟩ in order to include all information about +the particle occupations. We impose |{n}⟩ to be the eigenstate satisfying +ˆna(⃗k)|{n}⟩ = na(⃗k)|{n}⟩, +⟨{n}|{n}⟩ = 1, +(8) +where ˆna(⃗k) and na(⃗k) are the occupation operator and its corresponding occupation +number for species a ∈ {n} in the unit phase space, respectively. Here the occupation +operator is defined by +ˆna(⃗k) ≡ 1 +V a(a)† +⃗k +a(a) +⃗k , +(9) +where V ≡ +� +d3x = (2π)3δ3(⃗k = 0) is a volume of the system, and a(a) +⃗k +is an annihilation +operator for species a which satisfies +[a(a) +⃗k , a(b)† +⃗p +] = δab · (2π)3δ3(⃗k − ⃗p), +(others) = 0. +(10) +In the case of the fermionic species, the commutation relations are replaced with the anti- +commutation relations. Then one can obtain the representation of the eigenstate |{n}⟩ +by +|{n}⟩ ≡ +� +a∈{n} +� +�� +⃗p +1 +√na! · +√ +V na (a(a)† +⃗p +)na +� +� |0⟩. +(11) +Note that the occupation number na(⃗k) must be an integer. +Furthermore, it is convenient to define the increased/decreased state from |{n}⟩ for +3 + +later discussion. We define them by1 +|{n};⃗k(+1) +a +⟩ = +1 +√1 ± na +√ +V +a(a)† +⃗k +|{n}⟩, +( + : bosons, +− : fermions ) , +(14) +|{n};⃗k(−1) +a +⟩ = +1 +√na +√ +V +a(a) +⃗k |{n}⟩. +(15) +The coefficients are chosen to be unit vectors +⟨{n};⃗k(±1) +a +|{n};⃗k(±1) +a +⟩ = 1. +(16) +These increased/decreased states also become the eigenstate of the occupation operator: +ˆna(⃗k)|{n};⃗k(+1) +a +⟩ = +� +1 ± na(⃗k) +� +|{n};⃗k(+1) +a +⟩, +( + : bosons, +− : fermions ) , +(17) +ˆna(⃗k)|{n};⃗k(−1) +a +⟩ = ± +� +na(⃗k) − 1 +� +|{n};⃗k(−1) +a +⟩, +( + : bosons, +− : fermions ) . +(18) +2.2.2 +Transition probability +Using the eigenstates discussed in the previous section, let us consider the transition +probability of the process +A, B, · · · → X, Y, · · · +(19) +in the background in which other particles ({n}) exist. +For simplicity, we consider a +case that each species in the process are different. Taking the initial state as |{n}⟩ in +order to begin the given occupation numbers, the final state through the process (19) +can be represented as |{n};⃗k(−1) +A +,⃗k(−1) +B +, · · · ,⃗k(+1) +X +,⃗k(+1) +Y +, · · · ⟩. The probability from the +infinite past (in-state) to the infinite future (out-state) on the background particles can +be evaluated by2 +P(A, B, · · · → X, Y, · · · ){n} += +� +a=A,B,··· ,X,Y,··· +� +V +� +d3⃗ka +(2π)3 +� +ga +� +× +���⟨{n};⃗k(−1) +A +,⃗k(−1) +B +, · · · ,⃗k(+1) +X +,⃗k(+1) +Y +, · · · | ˆS|{n}⟩ +��� +2 +(20) +where ga denotes the internal degrees of freedom for species a, and ˆS is the S-matrix oper- +ator. The S-matrix element describing the process (19) without the background particles +can be represented by the invariant scattering amplitude as +inv⟨kX, kY , · · · | ˆS|kA, kB, · · · ⟩inv += +iM(kA, kB, · · · → kX, kY , · · · ) +×(2π)4δ4(kA + kB + · · · − kX − kY − · · · )(21) +1For the bosonic state, the N-increased/decreased state can be defined by +|{n};⃗k(+N) +a +⟩ += +� +na! +(na + N)! +1 +V N · (a(a)† +⃗k +)N|{n}⟩, +(12) +|{n};⃗k(−N) +a +⟩ += +� +(na − 1)! +(na + N − 1)! +1 +V N · (a(a) +⃗k )N|{n}⟩. +(13) +2If the initial or the final state includes N of the same species, the extra factor +1 +N! for each duplicated +species is needed. +4 + +where +|ka, kb, · · · ⟩inv ≡ +� +2Eka2Ekb · · · a† +⃗kaa† +⃗kb · · · |0⟩ +(22) +is a Lorentz invariant particle state. The representation (21) indicates that the S-matrix +operator includes +ˆS +⊃ +� +a=A,B,··· ,X,Y,··· +�� +d3⃗k′ +a +(2π)3 +1 +� +2E′a +� +ga +� +× a† +⃗k′ +Xa† +⃗k′ +Y · · · a⃗k′ +Aa⃗k′ +B · · · +×iM(k′ +A, k′ +B, · · · → k′ +X, k′ +Y , · · · ) · (2π)4δ4(k′ +A + k′ +B + · · · − k′ +X − k′ +Y − · · · ).(23) +Using the above expression, the S-matrix element on the particle background can be +written as +⟨{n};⃗k(−1) +A +,⃗k(−1) +B +, · · ·⃗k(+1) +X +,⃗k(+1) +Y +, · · · | ˆS|{n}⟩ +(24) += +� +a=A,B,··· ,X,Y,··· +�� +d3⃗k′ +a +(2π)3 +1 +� +2E′ +A +� +ga +� +×iM(k′ +A, k′ +B, · · · → k′ +X, k′ +Y , · · · ) · (2π)4δ4(k′ +A + k′ +B + · · · − k′ +X − k′ +Y − · · · ) +×⟨{n};⃗k(−1) +A +,⃗k(−1) +B +, · · ·⃗k(+1) +X +,⃗k(+1) +Y +, · · · |a† +⃗k′ +Xa† +⃗k′ +Y · · · a⃗k′ +Aa⃗k′ +B · · · |{n}⟩ +(25) += +1 +√2EAV · 2EBV · · · · 2EXV · 2EY V · · · · +×iM(kA, kB, · · · → kX, kY , · · · ) · (2π)4δ4(kA + kB + · · · − kX − kY − · · · ) +× +� +nAnB · · · (1 ± nX)(1 ± nY ) · · · +(26) +where +/− is for bosonic/fermionic particles of the produced species X, Y, · · · . As substi- +tuting the above form into (20), one can obtain +P(A, B, · · · → X, Y, · · · ){n} += +� +a=A,B··· ,X,Y,··· +�� +d3⃗ka +(2π)3 +1 +2Eka +� +ga +� +× |M(kA, kB, · · · → kX, kY , · · · )|2 +×(2π)4δ4(kA + kB + · · · − kX − kY − · · · ) · V T +×nAnB · · · (1 ± nX)(1 ± nY ) · · · +(27) +where T = +� +dt = 2πδ(E = 0) is the transition time scale. +Although the expression of the probability is derived, the result (27) is constructed by +the exact information of quanta represented by the microscopic occupation numbers per a +unit phase space na that is an integer. Since it is impossible to know the exact quantum +state, the statistical average should be considered. The probability to realize the state +|{n}⟩ can be represented by +P{n} ≡ +� +a∈{n} +� +⃗k +p(na(⃗k)), +∞ +� +na=0 +p(na(⃗k)) = 1 +(28) +where p(na(⃗k)) is a probability to be the occupation na on the momentum ⃗k. Multiplying +(28) into (27) and summing over by each occupation numbers3, we can obtained the +3This procedure is equivalent to +|{n}⟩⟨{n}| → +� +{n} +P{n}|{n}⟩⟨{n}| +(29) +in (20), that is, the initial state is considered by the density operator. +5 + +statistical probability as +⟨P(A, B, · · · → X, Y, · · · ){n}⟩ +≡ +� +{n} +P{n} · P(A, B, · · · → X, Y, · · · ){n} += +� +a=A,B,··· ,X,Y,··· +�� +d3⃗ka +(2π)3 +1 +2Ea +� +ga +� +× |M(kA, kB, · · · → kX, kY , · · · )|2 +×(2π)4δ4(kA + kB + · · · − kX − kY − · · · ) · V T +×fAfB · · · (1 ± fX)(1 ± fY ) · · · +(30) +where we denoted +fa ≡ +∞ +� +na=0 +p(na(⃗k)) na(⃗k). +(31) +The important thing is that the expectation value fa can be interpreted as the distribution +function even though the exact forms of both the probability p(na(⃗k)) and the relating +occupation na(⃗k) are unknown. +Using the result of the total probability (30), one can also define the partial probability, +as an example, for the species A of the momentum ⃗kA by +pA(A, B, · · · → X, Y, · · · ) +≡ +d⟨P(A, B, · · · → X, Y, · · · ){n}⟩ +V d3⃗kA +(2π)3 +� +gA +(32) += +T +2EA +� +a̸=A +�� +d3⃗ka +(2π)3 +1 +2Ea +� +ga +� +× |M(kA, kB, · · · → kX, kY , · · · )|2 +×(2π)4δ4(kA + kB + · · · − kX − kY − · · · ) +×fAfB · · · (1 ± fX)(1 ± fY ) · · · . +(33) +2.2.3 +Expression of collision term +The variation of the distribution ∆f through the microscopic process can be evaluated by +∆fφ ∼ +� +all processes +∆Nφ · [−pφ(φ, A, B, · · · → X, Y, · · · ) + pφ(X, Y, · · · → φ, A, B, · · · )] (34) +where φ is the focusing species, ∆Nφ is a changing number of the quantum φ in the process +φ, A, B, · · · ↔ X, Y, · · · (∆Nφ = 1 in this case), and pφ is the partial transition probability +6 + +for φ derived in (33). Finally, the collision term can be evaluated as +C[fφ] +∼ +Eφ +∆fφ +∆t +���� +microscopic process +(35) += +−1 +2 +� +all processes +� +a̸=φ +�� +d3⃗ka +(2π)3 +1 +2Ea +� +ga +� +×(2π)4δ4(kA + kB + · · · − kX − kY − · · · ) +×∆Nφ +� +|M(kφ, kA, kB, · · · → kX, kY , · · · )|2 +×fφfAfB · · · (1 ± fX)(1 ± fY ) · · · +− |M(kX, kY , · · · → kφ, kA, kB, · · · )|2 +×fXfY · · · (1 ± fφ)(1 ± fA)(1 ± fB) · · · ] . +(36) +In the above derivation, we set ∆t = T. We derived the above result with the single-particle +state for all species for simplicity. In the case of including the N-duplicated species in +φ, A, B, · · · or X, Y, · · · , one needs to multiply an extra factor 1/N! for the species. +Note that all the squared amplitudes in (36) must be regarded as the subtracted state +in which the contribution of on-shell particles in the intermediate processes is subtracted +in order to avoid the double-counting of the processes. Such a situation will be faced in +which the leading contributions of the amplitude consist of the loop diagrams or higher +order of couplings, e.g., the baryogenesis scenario as we discuss later. +2.3 +Full and integrated Boltzmann equation +Eqs. (6) and (36) lead the full Boltzmann equation for a species φ on the FLRW space-time +as +˙fφ − H |⃗kφ|2 +Eφ +∂fφ +∂Eφ += +− 1 +2Eφ +� +all processes +� +a̸=φ +�� +d3⃗ka +(2π)3 +1 +2Ea +� +ga +� +×(2π)4δ4(kA + kB + · · · − kX − kY − · · · ) +×∆Nφ +� +|M(kφ, kA, kB, · · · → kX, kY , · · · )|2 +×fφfAfB · · · (1 ± fX)(1 ± fY ) · · · +− |M(kX, kY , · · · → kφ, kA, kB, · · · )|2 +×fXfY · · · (1 ± fφ)(1 ± fA)(1 ± fB) · · · ] .(37) +Eqs. (37) for all species describe the detail evolution of the distribution functions, but +they are not useful to solve because of a lot of variables. To simplify the equations, the +integrated Boltzmann equation is useful and convenient. The momentum integral of the +left hand side of (37) leads +� +d3⃗kφ +(2π)3 +� +gφ +(LHS of (37)) += +˙nφ + 3Hnφ +(38) +where +nφ ≡ +� +d3⃗kφ +(2π)3 +� +gφ +fφ +(39) +7 + +is the number density of φ. Note in the integration (38) that the variables t and |⃗k| = +√ +E2 − m2 are independent and a property of the total derivative +� +d3⃗k +(2π)3 +|⃗k|2 +E +∂ +∂E (· · · ) = +� +d3⃗k +(2π)3 ⃗k · ∂ +∂⃗k +(· · · ) = −3 +� +d3⃗k +(2π)3 (· · · ) +(40) +is used. As the result, the number of the dynamical variables are reduced from (#species)×(#t)× +(#E) to (#species)×(#t), while the right hand side of the integrated (37) still includes +the E-dependent distribution functions. An useful approximation is to apply the Maxwell- +Boltzmann similarity distribution4 +fa, fMB +a +≪ 1 +and +fa(t, Ea) ∼ +na(t) +nMB +a +(t)fMB +a +(t, Ea), +(41) +where +fMB +a += exp +� +−Ea − µa +Ta +� +, +nMB +a += +� +d3⃗kφ +(2π)3 +� +gφ +fMB +a +(42) +are the Maxwell-Boltzmann distribution and its number density, respectively. Supposing +the identical temperature for all species Tφ = TA = · · · ≡ T, and substituting (41) into +the right hand side of (38) and assuming the chemical equilibrium +µφ + µA + µB + · · · = µX + µY + · · · +(43) +where µa is the chemical potential of species a, one can obtain the integrated Boltzmann +equation as +˙nφ + 3Hnφ += +� +d3⃗kφ +(2π)3 +� +gφ +(RHS of (37)) += +− +� +all processes +∆Nφ [nφnAnB · · · × ⟨R(φ, A, B, · · · → X, Y, · · · )⟩ +−nMB +φ +nMB +A nMB +B +· · · +nXnY · · · +nMB +X nMB +Y +· · · × ⟨R(¯φ, ¯A, ¯B, · · · → ¯X, ¯Y , · · · )⟩ +� +(44) +where the bar “ ¯ ” denotes its anti-particle state, +R(A, B, · · · → X, Y, · · · ) +≡ +1 +2EA2EB · · · +� d3kX +(2π)3 +d3kY +(2π)3 · · · +1 +2EX2EY · · · +× +� +gA,gB,··· ,gX,gY ,··· |M(kA, kB · · · → kX, kY , · · · )|2 +� +gA,gB,··· +×(2π)4δ4(kA + kB + · · · − kX − kY − · · · ) +(45) +is a reaction rate integrated over the final state, and +⟨R(A, B, · · · → X, Y, · · · )⟩ +≡ +1 +nMB +A nMB +B +· · · +� +d3kA +(2π)3 +d3kB +(2π)3 · · · fMB +A +fMB +B +· · · +× +� +gA,gB,··· +R(A, B, · · · → X, Y, · · · ) (46) +4The well-used approximation fa ∼ +na +nMB +a +f MB +a +can be justified in the case that species a is in the kinetic +equilibrium through interacting with the thermal bath. See appendix A for detail. The case in deviating +from the kinetic equilibrium is discussed in section 2.4. +8 + +is the thermally averaged reaction rate by the Maxwell-Boltzmann distributions of the +species appearing in the initial state. We used the property of the CPT-invariance of the +amplitude +M(X, Y, · · · → A, B, · · · ) = M( ¯A, ¯B, · · · → ¯X, ¯Y , · · · ) +(47) +to derive the last term in (44). +Although eq. (44) describing the evolution of number densities is obtained by integra- +tion of (37) directly, in general, the other evolution equations of the statistical quantities +Qa(t) ≡ +� +d3⃗ka +(2π)3 +� +ga +fa(⃗ka)qa(t, Ea) +(48) +can also be derived through the same procedure with the corresponding coefficient qa(t, Ea), +e.g., the energy density Q = ρ for q = Ea, and the pressure Q = P for q = |⃗k|2/3Ea. Such +equations help to extract more detailed thermodynamic variables, e.g., to determine the +independent temperatures for each species, as we will see in the next subsection. +2.4 +Temperature parameter +The integrated Boltzmann equation (44) is quite useful and can be applied to many +situations. However, it might not be suitable for some situations in which the kinetic +equilibrium is highly violated because the formula is based on the approximation by the +Maxwell-Boltzmann similarity distribution, which is justified by the kinetic equilibrium of +the target particles with the thermal bath as discussed in appendix A. Although to obtain +the most appropriate solution is to solve the full Boltzmann equation, it takes a lot of +costs to the calculation. In this section, we introduce an alternative method based on the +integrated Boltzmann equation. +Instead of using the similarity distribution (41), we introduce more generalized simi- +9 + +larity distribution by5 6 +fa(t, Ea) ∼ na(t) +nneq +a +(t)fneq +a +(t, Ea), +fneq +a +(t, Ea) ≡ exp +� +−Ea − µa +Ta(t) +� +. +(54) +Here nneq +a +is the number density evaluated by the “non-equilibrium” Maxwell-Boltzmann +distribution fneq +a +that is parametrized by the temperature parameter Ta(t). In general, the +temperature parameter is independent of the thermal bath temperature T(t). +Especially as the property of the Maxwell-Boltzmann distribution form, the tempera- +ture parameter can be expressed by the ratio of the pressure and the number density +Ta = +P neq +a +nneq +a += +Pa +na +(55) +because of +P neq +a += +� +d3⃗k +(2π)3 +� +ga +fneq +a +|⃗ka|2 +3Ea += +� +d3⃗k +(2π)3 +� +ga +Ta +3 +� +−⃗pa · +∂ +∂⃗pa +� +fneq +a += +Ta +� +d3⃗p +(2π)3 +� +ga +fneq +a += nneq +a +Ta +(56) +and +Pa = +� +d3⃗k +(2π)3 +� +ga +fa +|⃗ka|2 +3Ea += na +nneq +a +� +d3⃗p +(2π)3 +� +ga +fneq +a +|⃗ka|2 +3Ea += na +nneq +a +P neq +a +. +(57) +Therefore, the evolution equation for the temperature parameter can be derived from the +pressure’s one which can be constructed from the original full Boltzmann equation (37) +multiplied by |⃗ka|2/3Ea. +After the integration by the momentum, one can obtain the +5The normalization factor na(t)/nneq +a +(t) can be regarded as a corresponding quantity to the chemical +potential parameter ˜µa(t): +na(t) +nneq +a +(t) += +exp +� ˜µa(t) − µa +Ta(t) +� +. +(49) +6In the case of the Bose-Einstein/Fermi-Dirac type distribution +fa(t, Ea) = +� +e(Ea−˜µa(t))/Ta(t) ∓ 1 +�−1 +( − : boson, ++ : fermion) +(50) +where Ta(t) and ˜µa(t) are the temperature and the chemical potential parameters respectively, the tem- +perature parameter can be represented as +˜Ta(t) = +1 +ρa(t) + Pa(t) +� +d3⃗ka +(2π)3 +⃗k2 +a +3 fa(t, Ea) (1 ± fa(t, Ea)) +( + : boson, +− : fermion) +(51) +with energy density ρa and pressure Pa evaluated by (50). The above representation is consistent with (55) +in the nonrelativistic limit: fa ≪ 1, Ea ∼ ma, and ρa ∼ mana ≫ Pa. The chemical potential parameter +can be obtained by +˜µa(t) = ρa(t) + pa(t) − Ta(t)sa(t) +na(t) +(52) +where sa(t) is the entropy density defined by +sa(t) = +� +d3⃗ka +(2π)3 [±(1 ± fa) ln(1 ± fa) − fa ln fa] . +( + : boson, +− : fermion) +(53) +10 + +coupled equations for species φ as +˙nφ + 3Hnφ += − +� +all processes +∆Nφ [nφnAnB · · · × ⟨R(φ, A, B, · · · → X, Y, · · · )⟩neq +−nXnY · · · × ⟨R(X, Y, · · · → φ, A, B, · · · )⟩neq] , +(58) +nφ ˙Tφ + Hnφ +� +2Tφ − +� +|⃗kφ|4 +3E3 +φ +�neq� += − +� +all processes +∆Nφ +� +nφnAnB · · · × +�� +|⃗kφ|2 +3Eφ +− Tφ +� +R(φ, A, B, · · · → X, Y, · · · ) +�neq +−nXnY · · · × +� +⟨RTφ(X, Y, · · · → φ, A, B, · · · )⟩neq +−Tφ⟨R(X, Y, · · · → φ, A, B, · · · )⟩neq)] , +(59) +where R is a rate defined in (45) and +RTφ(X, Y, · · · → φ, A, B, · · · ) += +1 +2EX2EY · · · +� +d3⃗kφ +(2π)3 +d3⃗kA +(2π)3 +d3⃗kB +(2π)3 · · · +× +� +gφ,gA,gB,··· ,gX,gY ,··· |M(kX, kY , · · · → kφ, kA, kB, · · · )|2 +� +gX,gY ,··· +· |⃗kφ|2 +3Eφ +×(2π)4δ4(kφ + kA + kB + · · · − kX − kY − · · · ) +(60) +is a “temperature weighted” rate, and +� +|⃗kφ|4 +3E3 +φ +�neq += +1 +nneq +φ +� +d3⃗kφ +(2π)3 +� +gφ +fneq +φ +· |⃗kφ|4 +3E3 +φ +, +(61) +⟨R(a, b, · · · → i, j, · · · )⟩neq += +1 +nneq +a +nneq +b +· · · +� +d3⃗ka +(2π)3 +d3⃗kb +(2π)3 · · · +� +ga,gb,··· +×fneq +a +fneq +b +· · · R(a, b, · · · → i, j, · · · ), +(62) +are the thermally averaged quantities by the non-equilibrium distribution fneq including +only the initial species a, b, · · · , not the final species i, j, · · · . Solving the coupled equations +(58) and (59) for all the species can be expected to obtain more accurate results than +the former integrated Boltzmann equation (44). Following the evolution in practice, the +combined quantity +y = +mφTφ +s2/3 +∝ +Tφ +T 2 +(63) +instead of the solo Tφ, where s is the entropy density, is convenient for the non-relativistic +φ because of the asymptotic behavior Tφ(t) ∝ a(t)−2 ∝ T(t)2 after freezing out. +3 +Application to DM abundance +One of the cosmological application of the Boltzmann equation is for the estimation of the +DM abundance. Because DM is stable, the main process changing the particle number is +not decay/inverse-decay but the 2-2 annihilation/creation scatterings +χ, ¯χ ↔ ψ, ¯ψ +(64) +11 + +where χ is a DM and ψ are a standard model particle. Since the rate in the 2-2 scattering +can be represented by the annihilation cross section as +R(χ, ¯χ → ψ, ¯ψ) = σv +(65) +where v is the Møller velocity7 for the pair of the DM particles, the dynamics can be solve +as the annihilation cross section is given. Assuming the symmetric DM nχ = n¯χ and the +thermal distribution for the standard model particles nψ = n ¯ψ = nMB +ψ , the Boltzmann +equation (44) for the DM leads a simple form +˙nχ + 3Hnχ = − +� +n2 +χ − (nMB +χ +)2� +⟨σv⟩. +(67) +Instead of the particle number to follow its evolution by time, it is convenient to use the +yield Yχ ≡ nχ/s with a dynamical variable x ≡ mχ/T, where s = 2π2 +45 heff(T)T 3 is the +entropy density and heff(T) ∼ 100 for T ≳ 100 GeV is the effective degrees of freedom +defined by the entropy density. In the case of no creation/annihilation process, the yield +Yχ becomes a constant since the number and the entropy in the comoving volume is +conserved. With these variables, the Boltzmann equation (67) can be represented as +Y ′ +χ = −(1 + δh)s⟨σv⟩ +xH +� +Y 2 +χ − (Y MB +χ +)2� +(68) +where we denote ′ ≡ d/dx, and +δh ≡ +T +3heff +dheff +dT . +(69) +Since the adiabatic parameter δh tends to be negligible in the almost era of the thermal +history8, we set δh = 0 in the later discussion for simplicity. Moreover, we denote +Y MB +χ +≡ +nMB +χ +s +∼ +gχ +heff +45 +25/2π7/2 x3/2e−x +(x ≫ 1), +(70) +where gχ is the degrees of freedom for the DM particle. +3.1 +Relic abundance in freeze-out +As a simple and reasonable setup, we assume that the DM particles χ are in thermal +equilibrium initially. Then, the dynamics described by (68) can be explained as follow. At +first, the system is in the thermal equilibrium due to the stronger scattering effect than +the spatial expansion9, but the yield has a small deviation from the thermal value due to +7The definition with the 4-momenta is given by +v12 = +� +(k1 · k2)2 − m2 +1m2 +2 +k0 +1k0 +2 +, +(66) +which can be identical to the relative velocity only in case of the parallel 3-momenta; ⃗k1 · ⃗k2 = ±|⃗k1||⃗k2|. +8If the DM mass scale is around O(10) GeV, the freeze-out occurs around the QCD transition scale +T ∼ O(100) MeV, in which |δh| ∼ O(1). Thus, there is a few percent level contribution from the adiabatic +parameter δh even in the WIMP model. See Refs. [38, 39, 40, 41, 42, 43, 44, 45] for the determination of +that parameter in detail. +9If the interaction rate becomes lower than the Hubble rate at the relativistic regime x ≲ 1, the +abundance freezes out with the massless abundance (hot relic): +Y∞ ∼ Yhot = 45ζ(3) +2π4 +gχ +heff(Tf ) +× +� +1 +(boson) +3/4 +(fermion) +(71) +where ζ(3) = 1.202 · · · . +12 + +the expansion effect as +Yχ ∼ Y MB +χ ++ ∆(x), +∆(x) = +xH +s⟨σv⟩ +−Y ′ +χ +Y MB +χ ++ Y ∼ +xH +2s⟨σv⟩ ≪ Y MB +χ +(72) +as long as nMB⟨σv⟩ ≫ xH. The deviation ∆ continues growing in later time, and finally +the evolution of the yield freezes out because the expansion rate exceeds the scattering +rate. The freeze-out occurs when ∆(xf) = cY MB(xf), c ∼ O(1). The freeze-out time +x = xf and the final abundance Y∞ = Y (x = ∞) can be estimated by [8, 46] +xf += +ln +� +c(c + 2) +√ +90 +(2π)3 +gχ +� +geff(Tf)mχMplσn +� +− +� +n + 1 +2 +� +ln xf, +(73) += +ln +� +c(c + 2) +√ +90 +(2π)3 +gχ +� +geff(Tf)mχMplσn +� +− +� +n + 1 +2 +� +ln +� +ln +� +c(c + 2) +√ +90 +(2π)3 +gχ +� +geff(Tf)mχMplσn +�� ++ · · · , +(74) +Y∞ += +(n + 1) +� +45 +π +gχ +� +geff(Tf) +xn+1 +f +Mplmχσn +(75) +where Mpl = 1.22 × 1019 GeV is the Planck mass, geff(T) is the effective degrees of +freedom defined by the energy density ρ = π2 +30geff(T)T 4, and Tf = mχ/xf is the freeze- +out temperature. In the derivation of the analytic results (73) and (75), the temperature +dependence of the cross section is approximated by the most dominant part as +⟨σv⟩ = σnx−n, +(76) +where σn is a constant10. +Especially, n = 0 and 1 correspond to s-wave and p-wave +scattering, respectively. +Although a numerical factor c still has uncertainty, choosing +c(c + 2) = n + 1 leads to better analysis for the final abundance Y∞ within 5% accuracy +for xf ≳ 3 [8]. +As an example, let us consider a WIMP model. Choosing the parameters as mχ = 120 +GeV, n = 1, σn = α2 +W +m2χ , αW = +1 +30, gχ = 2, heff = geff = 90, one can obtain the analytic +results +xf = 23.2, +Y∞ = 3.81 × 10−12. +(77) +The actual evolution is depicted in Figure 1. +Finally, we need to mention the validity of the approximated results (73) and (75). +Their behaviors can deviate easily if the master equation (67) includes the significant ex- +tra processes by other species or the singular behavior of the cross sections. Especially it +is known some exceptional cases; (i) mutual annihilations of multiple species (coannihi- +lations), (ii) annihilations into heaver states (forbidden channels), (iii) annihilations near +a pole in the cross section [21], and (iv) simultaneous chemical and kinetic decoupling +(coscattering) [23]. In these cases, the analysis should be performed more carefully. See +[21, 22, 23, 24, 47, 48, 49] as their example cases, and also [50, 51, 52] as examples of the +evaluation with the temperature parameter. +10See also Appendix B for the actual analysis of the thermally averaged cross section. +13 + + 1x10-13 + 1x10-12 + 1x10-11 + 1x10-10 + 1x10-9 + 1x10-8 + 15 + 20 + 25 + 30 + 35 + 40 + 45 +Yχ +YχMB +Ylow +Yhigh +Y +x +Evolution of yields +Figure 1: The numerical plots of the evolution for each yield with parameters mχ = 120 +GeV, n = 1, σn = α2 +W +m2χ , αW = +1 +30, gχ = 2, heff = geff = 90. The red and the blue lines +show the actual evolution of Yχ and the thermal yield Y MB +χ +, respectively. The dashed +lines of green and purple show the approximated solutions Ylow ≡ Y MB +χ ++ ∆ and Yhigh ≡ +Y∞ +� +1 − Y∞ +n+1 +s⟨σv⟩ +H +�−1 +, respectively. +3.2 +Constraint on relic abundance +The relic abundance for the stable particles through the freeze-out of their annihilation +processes, as similar to χ particles discussed in the above, are restricted by the cosmological +observation results. An useful parameter relating to the relic abundance is the density +parameter defined by +Ωχ ≡ +ρχ +3M2 +pl +8π H2 +∼ 16π3 +135 +heff(T)T 3 +M2 +plH2 +· mχYχ. +(78) +Since the yield maintains the constant after the freeze-out unless the additional entropy +production occurs in the later era, one can estimate the present density parameter of +χ with the present values. The current observation through the the Cosmic Microwave +Background [53] provides T0 = 2.726 K, heff(T0) = 3.91, H0 = 100h2 km/s/Mpc, h = +0.677, therefore one can estimate to +Ωχ,now ∼ +mχYχ +3.64h2 × 10−9 GeV. +(79) +Because the present density parameter for the cold matter component is observed as +Ωch2 = 0.119 and it must be larger than the χ’s component, one can obtain a bound as +mχYχ < 4.36 × 10−10 GeV. +(80) +The set of parameters shown in (77) is seemingly suitable for the above constraint with +a bit of the modification. +However, that would fail by taking into account the direct +detections of the DM that focuses on the process of χ, ψ ↔ χ, ψ, where ψ is a standard +model particle. If the annihilation process occurs through a similar interaction to the +14 + +electroweak gauge interaction, the cross section for χ-ψ elastic scattering also relates to +the same gauge interaction. One can estimate σχψ→χψ ∼ G2 +F m2 +χ ∼ 10−36 cm2, but it is +already excluded by the direct detection [54]. +3.3 +Relic abundance in freeze-in +The discussion and the result in the previous subsections are based on the freeze-out +scenario in which the DM particles are in thermal equilibrium initially. However, it is not +satisfied if the interaction between the DM particles and the thermal bath is too small, +so-called FIMP (feebly interacting massive particle) scenario [19, 55]. In this situation, +the yield of DM evolves from zero through the thermal production from the thermal bath. +Although the DM never reaches the thermal equilibrium, the yield freezes in with a non- +thermal yield at last. +We discuss here the relic abundance by the freeze-in scenario in two cases of the pair- +creation of DM by (1) scattering from thermal scattering and (2) decay from a heavier +particle. +3.3.1 +Pair-creation by scattering +In the case that DM-pair (χ¯χ) is produced by thermal pair particles (ψ ¯ψ), the Boltzmann +equation is given by (68) as +Y ′ +χ +∼ +s⟨σv⟩ +xH (Y MB +χ +)2, +(81) +where we approximated Yχ ≪ Y MB +χ +and the adiabatic degrees δh ∼ 0 until the freezing-in. +For simplicity, we consider the simple interaction described by +Lint = λ(χ†χ)(ψ†ψ). +(82) +where χ is the bosonic DM, ψi labeled i are the massless bosons in the thermal bath, and +y is a coupling constant. The thermally averaged cross section is given by +⟨σv⟩ += +g2 +χg2 +ψ +(nMB +χ +)2 +λ2 +(2π)5 T +� ∞ +4m2χ +ds +� +s − 4m2χ K1(√s/T). +(83) +where gχ and gψ are the degrees of freedom for each species. Therefore, one can estimate +the final yield at x = mχ/T = ∞ as +Yχ(∞) +∼ +� ∞ +0 +dx s⟨σv⟩ +xH (Y MB +χ +)2 +(84) += +3π2 +128 · g2 +χg2 +ψ +λ2 +(2π)5 · +m4 +χ +H(T = mχ) s(T = mχ). +(85) +This result implies that the yield freezing occurs around the earlier stage x ∼ O(1) because +(85) can be regarded as Yχ(∞) ∼ +nMB +χ +⟨σv⟩ +H +Y MB +χ +��� +x∼1. +Applying the obtained relic abundance (85) to the relation of the present density +parameter (79), one can obtain the required strength of the coupling as +λ = 1.0 × 10−12 · +1 +gχgψ +· +�geff(T = mχ) +100 +�1/4 �heff(T = mχ) +100 +�1/2 +· +�Ωχ,nowh2 +0.119 +�1/2 +. (86) +15 + +3.3.2 +Pair-creation by decay +The other possible freeze-in scenario is due to the pair production from a heavier particle: +σ → χ¯χ [19, 56]. The original Boltzmann equation for DM is given by +dYχ +dxσ += +(1 + δh)Γσ→χ¯χ +xσH +K1(xσ) +K2(xσ) +� +Yσ − +� Yχ +Y MB +χ +�2 +Y MB +σ +� +(87) +∼ +Γσ→χ¯χ +xσH +K1(xσ) +K2(xσ)Y MB +σ +(88) +where Γσ→χ¯χ is a decay constant and xσ ≡ mσ/T. We also approximated Yσ ∼ Y MB +σ +, +Yχ ≪ Y MB +χ +, and δh ∼ 0 in the second line. Therefore, the final yield can be estimated as +Yχ(∞) +∼ +� ∞ +0 +dxσ +Γσ→χ¯χ +xσH +K1(xσ) +K2(xσ)Y MB +σ +(89) += +3gσ +4π · +Γσ→χ¯χ +H(T = mσ) +m3 +σ +s(T = mσ). +(90) +where gσ is the degrees of freedom for σ. If the decay constant can be represented by the +coupling constant y as +Γσ→χ¯χ = gχ · y2 +8πmσ, +(91) +the required magnitude of the coupling with the relation formula to the density parameter +(79) can be estimated as +y2 ∼ 2.7 × 10−24 · mσ +mχ +· +1 +gσgχ +· +�geff(T = mσ) +100 +�1/2 heff(T = mσ) +100 +· Ωχ,nowh2 +0.119 +. +(92) +4 +Application to baryogenesis +The other popular application of the Boltzmann equation in cosmology is the baryogenesis +scenario that describes the dynamical evolution of the baryon number in the Universe from +zero at the beginning to the non-zero at present. The present abundance of the baryons +can be estimated from (79). Replacing χ’s mass mχ into the nucleon mass mN = 939 +MeV and using the present density parameter for the baryon Ωbh2 = 0.0224 [53], one can +obtain +YB,now = 8.69 × 10−11. +(93) +There are three conditions suggested by A. D. Sakharov [57] in order to develop the +baryon abundance from YB = 0 to non-zero: (1) baryon number (B) violation, (2) C +and CP violation, (3) non-equilibrium condition. Their brief reasons are as follows. The +B violation is trivial by definition. If the baryon number violating processes conserve +C or CP, their anti-particle processes happen with the same rate. +As the result, the +net baryon number is always zero. +Even if the processes violate the baryon number, +C and CP, the thermal equilibrium reduces the baryon asymmetry due to their inverse +processes. Especially, the Boltzmann equation provides a powerful tool to quantify the +third condition. +To see how to construct the Boltzmann equations for the baryogenesis, let us consider +with a toy model11 as shown in Table 1. The model includes Majorana-type of chiral +11Replacing X, ψ, φ into the right-handed neutrino, left-handed neutrino, Higgs doublet in the standard +model, respectively, one can obtain the type-I seesaw model that can realize the well-known leptogenesis +scenario [33]. However, the correspondence is not complete: the type-I seesaw model includes the gauge +interactions that induces ∆L = 1 scattering process. +16 + +Species +Particle statistic +#B +Xa +Chiral fermion (Majorana) +− +ψi +Chiral fermion +b +φ +Complex scalar +0 +Process +∆B +Xa → ψi, φ +b +Xa → ¯ψi, ¯φ +−b +ψi, ψj → ¯φ, ¯φ +−2b +ψi, φ → ¯ψj, ¯φ +−2b +Table 1: Left: the matter contents and their baryon number. All the anti-particles have +the opposite sign of the baryon number. Right: Possible processes up to the 4-body B- +violating interactions and their variation of the baryon number. The bar (¯) on each species +denotes the anti-particle. In addition to the shown processes here, their inverse processes +are also possible. Although there are elastic scatterings Xa, ψi, → Xb, ψj, Xa, φ → Xb, φ, +and ψi, φ → ψj, φ, we omited them because they do not change the baryon number. +fermions Xa for a = 1, · · · , NX, baryonic chiral fermions ψi for i = 1, · · · , Nψ with the +common baryon number b, and a non-baryonic complex scalar φ. +For simplicity, the +baryonic fermions ψi and the scalar φ are massless and they are always in the thermal +equilibrium. Because Xa are the Majorana fermion, Xa and ¯Xa can be identified. Thus, +once we set the fundamental interaction to provide a decay/inverse-decay processes Xa ↔ +ψi, φ, their anti-particle processes Xa ↔ ¯ψi, ¯φ also exist. These 3-body interactions also +induce the B-violating 2-2 scatterings exchanging Xa fermions. +4.1 +Mean net baryon number +At first, we consider only the decay processes for simplicity. This situation is realized +when Xa particles start to decay after the scattering processes freeze out. Here we define +the mean net baryon number by +ϵa += +� +f +∆Bf +� +rXa→f − r ¯ +Xa→ ¯f +� +(94) +where the summation runs for all decay processes, ∆Bf and rXa→f are the generated +baryon number through the process of Xa → f and its branching ratio, respectively. The +physical meaning of the mean net baryon number ϵa is an average of the produced baryon +number by a single quantum of Xa. In the case of our toy model, this quantity can be +represented as +ϵa += +b +� +i +� +rXa→ψi,φ − rXa→ ¯ψi,¯φ +� +(95) +This result reflect the requirements of B-violation and C, CP violation. +If the decay +processes are B-conserving b = 0 or C, CP conserving processes rXa→ψi,φ = rXa→ ¯ψi,¯φ, the +mean net baryon number is vanished. +Supposing that only a single flavour X1 survives and all of X1 particles decay into +the baryonic fermions ψi, the generated baryon number can be estimated by YB ∼ ϵ1YX1. +Especially, the baryon abundance can be maximized if X1 particles are the hot relic YX1 ∼ +Yhot ∼ +45 +2π4 +gX1 +heff(Tf): +YB ∼ 45 +2π4 · +ϵ1gX1 +heff(Tf), +(96) +where gX1 = 2 is the degrees of freedom of the Majorana-type fermion X1. +17 + +4.2 +Boltzmann equations in baryogenesis scenario +Although we considered quite simplified situation in the previous subsection, in reality, +the situation is more complicated since the system includes the dynamical decay/inverse +decay and scattering processes. In order to quantify the actual evolution of the baryon +abundance including the scattering effects, we need to construct the Boltzmann equations +in this system and solve them. +For simplicity, we suppose again that only a single flavour X1 affects to the evolu- +tion of the net baryon number. +Using the definition of the net baryon density nB = +b � +i +� +nψi − n ¯ψi +� +, the evolution of the system including the processes in Table 1 are de- +scribed by12 +˙nX1 + 3HnX1 += +− +�MX1 +EX1 +� +ΓX1(nX1 − nMB +X1 ) + · · · , +(97) +˙nB + 3HnB += +ϵ1 +�MX1 +EX1 +� +ΓX1 +� +nX1 − nMB +X1 +� +− 2ΓS nB + · · · , +(98) +where MX1 (EX1) is the mass (energy) of X1, ΓX1 ∼ � +i +� +ΓX1→ψiφ + ΓX1→ ¯ψi,¯φ +� +is the +total width of X1 and +ϵ1 ≡ b · +� +i +� +ΓX1→ψiφ − ΓX1→ ¯ψi,¯φ +� +ΓX1 +(99) +is the mean net baryon number corresponding to (95), and +ΓS += +nMB +φ +⟨σψφ→ ¯ψ ¯φv⟩ + nMB +ψ ⟨σψψ→¯φ¯φv⟩ +(100) +is the reaction rates through the B-violating scatterings up to the tree level. The omitted +parts “· · · ” denote the sub-leading processes in terms of the order of couplings. Eq. (97) +describes the dissipation of Xa, and it converts to the baryon with the rate ϵa and flows +into the baryon sector. However, the produced baryons also wash themselves out through +the B-violating scattering processes due to the last term in (98). Therefore, the smaller +B-violating scattering effect is favored for remaining the more net baryons as long as Xa +can be thermalized enough at the initial. +To solve the equations of motion (97) and (98), it is convenient to use the yields +YX1 = nX1/s, YB = nB/s and the variable x = MX1/T as +Y ′ +X1 += +−γD(YX1 − Y MB +X1 ), +(101) +Y ′ +B += +ϵ1γD(YX1 − Y MB +X1 ) − 2γSYB, +(102) +where +Y MB +X1 += +nMB +X1 +s += +45 +4π4 +gX1 +heff +x2K2(x), +(103) +γD += +1 +x · ΓX1 +H(T) +�mX1 +EX1 +� += +� +45 +4π3geff +Mpl +MX1 +· K1(x) +K2(x) +ΓX1 +T , +(104) +γS += +1 +x · +ΓS +H(T) += +� +45 +4π3geff +Mpl +MX1 +· ΓS(T) +T +, +(105) +with the n-th order of the modified Bessel function Kn(x). We assumed the adiabatic evo- +lution of the relativistic degrees h′ +eff/heff ∼ 0 to obtain (101) and (102). The dimensionless +12See appendix B for the treatment of the thermally averaged quantities. And also see Appendix C for +the detail of the derivation of the equations, especially, the treatment of the real intermediate state (RIS) +to avoid the double-counting. +18 + +parameters γD,S are the reaction rates normalized by the Hubble parameter. In general, +γD is proportional to x2 (x1) at the limit of x ≪ 1 (x ≫ 1), whereas the behavior of γS +depends on the detail of the interaction as we will see its concrete form with an example +model later. +Eqs.(101) and (102) can provide the analytic form of YB as +YB(∞) += +−ϵ1 +� ∞ +0 +dx Y ′ +X1(x) exp +� +−2 +� ∞ +x +dx′ γS(x′) +� +(106) +Especially in the weakly scattering case, +� ∞ +0 dx γS ≲ 1, one can approximate the above +result as +YB(∞) +∼ +−ϵ1 +� ∞ +0 +dx Y ′ +X1(x) = ϵ1Yhot. +(107) +The physical interpretation is that the whole X1 particles existing from the beginning +can convert to the net baryons without any wash-out process in this case. Hence the +approximated result does not depend on the detail of the decay process γD. The result +(107) is consistent with the former estimation in (96). On the other hand, the strongly +scattering case causes the wash-out process significantly, and thus the final net baryon +abundance is strongly suppressed from the result of (107). +To see the concrete evolution dynamics, we consider the following interaction +Lint += +− +� +a,i +yaiφXaψi + (h.c.) +(108) +with the Yukawa coupling yai and the two-component spinors Xa and ψi. This interaction +leads the concrete representation of the decay width and the scattering rate as +ΓX1 += +˜αMX1, +(109) +ΓS += +T · 8˜α2 +πgψ +˜γS(x) +(110) +where we denoted +˜α += +� +i +gψigφ +|y1i|2 +32π , +(111) +˜γS += +1 +8 +� ∞ +0 +dz K1(z) +� +2 +� +z4 +z2 + x2 + +x2z2 +z2 + 2x2 ln +� +1 + z2 +x2 +�� ++ +x2z4 +(z2 − x2)2 + ˜α2x4 + 2 +� +z2 − x2 ln +� +1 + z2 +x2 +�� ++ +4x2(z2 − x2) +(z2 − x2)2 + ˜α2x4 +� +z2 − +� +z2 + x2� +ln +� +1 + z2 +x2 +��� +(112) +∼ +� +1 +(x ≪ 1) +8/x2 +(x ≫ 1) , +(113) +and gψ ≡ � +i gψi = Nψgψi. Here gψi and gφ are the degrees of freedom of the chiral fermion +ψi and the scalar φ, not including their anti-particle state. The asymptotic behaviors for +each reaction rate are governed by +γD(x) ∝ +� x2 +(x ≪ 1) +x1 +(x ≫ 1) , +γS(x) ∝ +� (constant) +(x ≪ 1) +x−2 +(x ≫ 1) . +(114) +19 + +1e-06 +1e-04 +1e-02 +1e+00 +1e+02 +1e+04 +1e+06 + 0.01 + 0.1 + 1 + 10 + 100 +For decays (γD) +For scatterings (γS) +MX1 = 1016 GeV +MX1 = 1015 GeV +MX1 = 1014 GeV +MX1 = 1013 GeV +MX1 = 1013 GeV +MX1 = 1014 GeV +MX1 = 1015 GeV +MX1 = 1016 GeV +γ = Γ/xH +x = MX1/T +Evolution of rate of reactions +1e-12 +1e-10 +1e-08 +1e-06 +1e-04 +1e-02 +1e+00 + 0.01 + 0.1 + 1 + 10 + 100 +MX1 = 1016 GeV +MX1 = 1015 GeV +MX1 = 1014 GeV +MX1 = 1013 GeV +Analytic (hot relic decay) +YB/ε1 +x = MX1/T +Evolution of YB/ε1 +Figure 2: The numerical plots of the evolution of the interaction rates (upper) and YB/ϵ1 +(lower) for each mass of X. The numerical parameters are chosen as gX1 = 2, gψi = gφ = 1, +Nψ = 3, ˜α = 0.01, heff = geff = 100, and assumed the thermal distribution for X1 and +YB = 0 at the initial. +The solid lines in red, yellow, green, and blue correspond to +MX1 = 1016, 1015, 1014, and 1013 GeV, respectively. In the upper figure, “For decays” and +“For scatterings” depict γD(x) and γS(x), respectively. In the lower figure, the dashed +line in purple shows the approximated solution (96) due to the decay of the hot relic, +YB/ϵ1 ∼ Yhot = 45 +2π · +gX1 +heff . +20 + +The actual behavior of γD and γS with concrete parameters are shown in the upper side +of Figure 2. The asymptotic behaviors at x ≪ 1 and x ≫ 1 are consistent with (114). The +enhancement structures for each γS seen around x ∼ O(1) are induced by the resonant +process through the on-shell s-channel shown in (112). +The lower side in Figure 2 shows the evolution of YB/ϵ1, which is the numerical result +from the coupled equations (101) and (102). The result shows that the heavier mass of X +can generate more the net baryon number because the reaction rates are reduced for the +heavier case, and hence the generated baryons can avoid the wash-out process. Especially, +the plot for MX1 = 1016 GeV leads the close result to the hot relic approximation (107), +whereas the plot for MX1 = 1013 GeV shows the dumping by the wash-out effect at the +late stage. The milder decrease at the middle stage is caused by the decay of X particles +that supplies the net baryons to compensate for the wash-out effect. +Finally, the obtained yield of the net baryon number YB should be compared with +the current bound (93), YB,now ∼ 10−10. Since the mean net baryon number can roughly +be estimated by ϵ1 ∼ ˜α2 sin2 θCP where θCP is a CP phase in the considered model, one +can obtain the constraint from the current observation as YB,now/ϵ1 ≳ 10−10/˜α2 ∼ 10−6, +where we used ˜α = 0.01. Therefore, one can find that MX1 ≳ 1014 GeV is allowed by +compared with the lower plot in Figure 2. +5 +Summary +In this paper we have demonstrated the derivation of the Boltzmann equation from the +microscopic point of view with the quantum field theory, in which the transition probabil- +ity has been constructed with the statistically averaged quantum states. Although both +results of the full and the integrated Boltzmann equation (37) and (44) are consistent with +the well-known results, our derivation ensures that especially the full Boltzmann equation +is widely applicable even in the non-equilibrium state since the derivation does not as- +sume any distribution type nor the temperature of the system. Especially the integrated +Boltzmann equation (44) is quite convenient and applicable for wide situations. In the +particular case that the kinetic equilibrium cannot be ensured, the coupled equations with +the temperature parameter (58) and (59) are better for following the dynamics. +As the application examples of the (integrated) Boltzmann equation in cosmology, we +have reviewed two cases, the relic abundance of the DM and the baryogenesis scenario. +For the former case, we have shown the Boltzmann equation and its analysis. The analytic +results (73) and (75) are quite helpful for estimating the final relic abundance of the DM +and its freeze-out epoch. For the latter case, we have derived the Boltzmann equation +with a specific model and show the numerical analysis. The final net baryon number can +be estimated by the analytic result (107) in the case of the weakly interacting system, +whereas that is strongly suppressed by the wash-out effect in the case of the strongly +interacting system. +The Boltzmann equation is a powerful tool for following the evolution of the particle +number or other thermal quantities, and thus it will be applied for many more situations +in future and will open a new frontier of the current physics. We hope this paper helps +you to use the Boltzmann equation and its techniques thoughtfully. +Acknowledgment +We thank Chengfeng Cai, Yi-Lei Tang, and Masato Yamanaka for useful discussions and +comments. This work is supported in part by the National Natural Science Foundation of +21 + +China under Grant No. 12275367, and the Sun Yat-Sen University Science Foundation. +A +Validity of the Maxwell-Boltzmann similarity approxi- +mation +Although the approximation of the distribution function by the Maxwell Boltzmann simi- +larity distribution is used well in many situations, such approximation is not always valid. +In this appendix, we show that the approximation is valid if the focusing species is in the +kinetic equilibrium through interacting with the thermal bath. +Let us consider the situation of the particle number conserving process a(k1)+b(k2) ↔ +a(k3)+b(k4), where a and b denote the particle species. If this process happens fast enough +and the species b maintains the thermal distribution, the condition of the detailed balance +leads +0 += +fa(t, E1)fMB +b +(t, E2) − fa(t, E3)fMB +b +(t, E4) +(115) += +� fa(t, E1) +fMB +a +(t, E1) − +fa(t, E3) +fMB +a +(t, E3) +� +fMB +a +(t, E1)fMB +b +(t, E2), +(116) +where we assumed the common temperature to the thermal bath and the energy conserva- +tion law: fMB +a +(t, E1)fMB +b +(t, E2) = fMB +a +(t, E3)fMB +b +(t, E4). Because the above relation must +be satisfied by arbitrary energy, one can obtain +fa(t, E) +fMB +a +(t, E) += C(t) +(117) +where C(t) is a function which is dependent on time but independent of the energy. +The function C(t) can be determined by integrating over the momentum of fa(t, E) = +C(t)fMB +a +(t, E1), i.e., na(t) = C(t)nMB +a +(t). Finally, one can obtain the desired form of the +distribution: +fa(t, E) = C(t)fMB +a +(t, E) = +na(t) +nMB +a +(t)fMB +a +(t, E). +(118) +B +Formulae for thermal average by Boltzmann-Maxwell dis- +tribution +In this section, we summarize the convenient formulae used in the various thermally av- +eraged quantities by the Maxwell-Boltzmann distribution, especially for number density, +decay rate, and cross section. +B.1 +Number density and modified Bessel function +The number density with the Maxwell-Boltzmann distribution is given by +nMB += +g +� +d3k +(2π)3 fMB +(119) += +g +2π2 m2TK2(m/T)eµ/T +(120) += +g × +� +� +� +� +� +� +� +1 +π2 T 3eµ/T + · · · +(T ≫ m) +�mT +2π +�3/2 +e−(m−µ)/T +� +1 + 15T +8m + · · · +� +(T ≪ m) +, +(121) +22 + +where Kn is the n-th order of the modified Bessel function given by +Kn(x) += +� ∞ +0 +dθ e−x cosh θ cosh nθ +(122) += +� +� +� +� +� +� +� +Γ(n) +2 +�2 +x +�n ++ · · · +(0 < x ≪ √1 + n) +� π +2x e−x +� +1 + 4n2 − 1 +8x ++ · · · +� +(x ≫ 1) +(123) +Especially, the following relations are helpful in analysis: +Kn(x) += +x +2n (Kn+1(x) − Kn−1(x)) , +(124) +d +dx (xnKn(x)) += +−xnKn−1(x). +(125) +B.2 +Thermally averaged decay rate +The rate defined in (45) for the single initial species relates to the decay rate, R(A → +X, Y, · · · ) = mA +2EA ΓA→X,Y,···, where +ΓA→X,Y,··· += +1 +2mA +� d3kX +(2π)3 +d3kY +(2π)3 · · · +1 +2EX2EY +· · · +×(2π)4δ4(kA − kX − kY − · · · ) +× 1 +gA +� +gA,gX,gY ,··· +|M(A → X, Y, · · · )|2 +(126) +is the partial width for the process A → X, Y, · · · . The factor mA/EA in R corresponds to +the inverse Lorentz gamma factor describing the life-time dilation. The thermal average +of the rate is given by +⟨R(A → X, Y, · · · )⟩ += +1 +2ΓA→X,Y,··· +�mA +EA +� +, +(127) +�mA +EA +� += +gA +nMB +A +� +d3kA +(2π)3 +mA +EA +fMB +A +(128) += +K1(mA/T) +K2(mA/T) +(129) += +� +� +� +mA +2T + · · · +(T ≫ mA) +1 − 3T +2mA ++ · · · +(T ≪ mA) +. +(130) +B.3 +Thermal averaged cross section +The rate averaged by the initial 2-species relates to the scattering rate, +R(A, B → X, Y, · · · ) += +σv +(131) += +1 +2EA2EB +� d3kX +(2π)3 +d3kY +(2π)3 · · · +1 +2EX2EY +· · · +×(2π)4δ4(kA + kB − kX − kY − · · · ) +× +1 +gAgB +� +gA,gB,gX,gY ,··· +|M(A → X, Y, · · · )|2, +(132) +23 + +where σ = σ(s) is the cross section for the process A, B → X, Y, · · · dependent on the +Mandelstam variable s = (kA + kB)2, and v is the Møller velocity +v = +� +(kA · kB)2 − m2 +Am2 +B +EAEB += +� +(s − (mA + mB)2)(s − (mA − mB)2) +2EAEB +. +(133) +The thermal average of the rate can be obtained by +⟨R(A, B → X, Y, · · · )⟩ += +⟨σv⟩ +(134) += +gAgB +nMB +A nMB +B +� +d3kA +(2π)3 +d3kB +(2π)3 σv · fMB +A +fMB +B +(135) += +gAgB +nMB +A nMB +B +� ∞ +0 +d|⃗kA| d|⃗kB| +� π +0 +dθ · +1 +4π2 +|⃗kA|2|⃗kB|2 sin θ +EAEB +×σ(s) · +� +(s − (mA + mB)2)(s − (mA − mB)2) +× exp +� +−EA + EB +T ++ µA + µB +T +� +, +(136) +where the integral variable θ denotes the angle between ⃗kA and ⃗kB, i.e., ⃗kA · ⃗kB = +|⃗kA||⃗kB| cos θ. +In order to perform the integral in (136), it is convenient to change the integral variables +(|⃗kA|, |⃗kB|, θ) to (E+, E−, s), where E± ≡ EA ± EB [39]. The Jacobian is given by +����� +∂(|⃗kA|, |⃗kB|, θ) +∂(E+, E−, s) +����� += +EAEB +4|⃗kA|2|⃗kB|2 sin θ +. +(137) +The integral region can be obtained from the expression of the Mandelstam variable, +s = m2 +A + m2 +B + 2 +� +EAEB + |⃗kA||⃗kB| cos θ +� +, +(138) +which leads +(s − m2 +A − m2 +B − 2EAEB)2 ≤ 4|⃗kA|2|⃗kB|2 = 4(E2 +A − m2 +A)(E2 +B − m2 +B). +(139) +The above inequality is equivalent to +� +E− − m2 +A − m2 +B +s +E+ +�2 +≤ (E2 ++ − s) +� +1 − (mA + mB)2 +s +� � +1 − (mA − mB)2 +s +� +(140) +Therefore, the integral region can be obtained as +e− ≤ E− ≤ e+, +(141) +E+ ≥ √s, +(142) +s ≥ (mA + mB)2, +(143) +where +e± ≡ m2 +A − m2 +B +s +E+ ± +� +(E2 ++ − s) +� +1 − (mA + mB)2 +s +� � +1 − (mA − mB)2 +s +� +. +(144) +24 + +Using the above results, the integral (136) can be performed as +⟨σv⟩ += +gAgB +nMB +A nMB +B +1 +2(2π)4 e(µA+µB)/T +× +� ∞ +(mA+mB)2 ds · σ(s) · (s − (mA + mB)2)(s − (mA − mB)2) +× T +√sK1(√s/T) +(145) += +1 +4m2 +Am2 +BT +� ∞ +mA+mB +d√s · σ(s) · (s − (mA + mB)2) +×(s − (mA − mB)2) · +K1(√s/T) +K2(mA/T)K2(mB/T), +(146) +where we used the representation of the number density (120). +Especially in the case of the non-relativistic limit, mA, mB ≫ T, it is convenient to +use the representation13 +� +σv(s) +≡ +σ(s) · vNR(s), +vNR(s) ≡ +� +s − (mA + mB)2 +mAmB +(147) +and the replacement of the integral variable s to y defined by +√s += +mA + mB + Ty. +(148) +Since the integral parameter y corresponds to ⃗k2/mT naively, we can expect that the +significant integral interval is on y ≲ O(1). Then (146) can be approximated as +⟨σv⟩ +∼ +2 +√π +� +1 − 15T +8mA +− 15T +8mB ++ +3T +8(mA + mB) + · · · +� +× +� ∞ +0 +dy · +� +(� +σv)0 + Ty · (� +σv)′ +0 + · · · +� +×e−y · √y +� +1 + Ty +2mA ++ Ty +2mB +− +Ty +4(mA + mB) + · · · +� +(149) += +(� +σv)0 + 3 +2T +� +−3 +4 +� 1 +mA ++ +1 +mB +� +(� +σv)0 + (� +σv)′ +0 +� ++ O(T 2), +(150) +where we used the asymptotic expansion (123) and the Taylor series around √s = mA + +mB, +� +σv = (� +σv)0 + (√s − mA − mB) · (� +σv)′ +0 + · · · +(151) +(� +σv)0 ≡ � +σv(√s = mA + mB), +(� +σv)′ +0 ≡ +d � +σv +d√s +����√s=mA+mB +. +(152) +C +Derivation of the Boltzmann equations in baryogenesis +scenario +In this section, we demonstrate the derivation of the Boltzmann equation in the baryoge- +nesis scenario with the processes listed in Table 1. Indeed, the straightforward derivation +13The Lorentz-invariant “velocity” vNR behaves as vNR ∼ +��� +⃗kA +mA − +⃗kB +mB +��� at the non-relativistic limit. Note +that +lim +vNR→0 � +σv remains non-zero (s-wave contribution) in general. +25 + +of the Boltzmann equations leads to the over-counting problem in the amplitudes. For +example, once a contribution of the decay/inverse-decay process X ↔ ψ, φ is included +in the Boltzmann equation, the straightforward contribution from the scattering process +¯ψ, ¯φ ↔ ψ, φ is over-counted because such process can be divided into ¯ψ, ¯φ ↔ X and +X ↔ ψ, φ if the intermediate state X is on-shell. Therefore, in general, one must regard +the straightforward contribution in the scattering processes as the subtracted state of the +real intermediated state (RIS) from the full contribution [8, 58]: +|M|2 +Boltzmann eq. += +|M|2 +subtracted ≡ |M|2 +full − |M|2 +RIS. +In a case of the scattering process ¯ψ, ¯φ → ψ, φ, the full amplitude part can be represented +as +iM( ¯ψ, ¯φ → ψ, φ)full +∼ +iM(X → ψ, φ) · +i +s − M2 +X + iMXΓX +· iM( ¯ψ, ¯φ → X) (153) +where s is the Mandelstam variable, MX and ΓX are X’s mass and total decay width, +respectively. On the other hand, the RIS part can be evaluated as the limit of the narrow +width by +��M( ¯ψ, ¯φ → ψ, φ) +��2 +RIS += +lim +ΓX→0 +��M( ¯ψ, ¯φ → ψ, φ) +��2 +full +(154) += +lim +ΓX→0 |M(X → ψ, φ)|2 +1 +(s − M2 +X)2 + (MXΓX)2 |M( ¯ψ, ¯φ → X)|2 +∼ +|M(X → ψ, φ)|2 · +π +MXΓX +δ(s − M2 +X) · |M( ¯ψ, ¯φ → X)|2. +(155) +In the last line, the narrow width approximation is applied. Since the contribution of both +amplitudes in (155) is the order of ΓX, the RIS contribution is also the order of ΓX in +total. Therefore, RIS part in the scattering process contributes to the decay/inverse-decay +process. +Taking into account the above notice, we derive the Boltzmann equation. For simplic- +ity, we suppose that only a single flavour X1 affects to the evolution of the net baryon +number. Because of no scattering processes associated with Xa, the equation governing +nXa is simply written as +˙nX1 + 3HnX1 += +� d3kX1 +(2π)3 +d3kψi +(2π)3 +d3kφ +(2π)3 +1 +2EX12Eψi2Eφ +(2π)4δ4(kX1 − kψi − kφ) +× 1 +gX1 +� +gX1,gψi,gφ +� +−fX1|M(X1 → ψiφ)|2 + fψifφ|M(ψiφ → X1)|2 +−fX1|M(X1 → ¯ψi ¯φ)|2 + f ¯ψif¯φ|M( ¯ψi ¯φ → X1)|2� ++ · · · +(156) += +− +� +i +⟨ΓX1⟩(nX1 − nMB +X1 ) + · · · +(157) +where +ΓX1 += +ΓX1→ψiφ + ΓX1→ ¯ψi ¯φ + · · · +(158) +is the total decay width of X1 and +⟨ΓX1⟩ +≡ +1 +nMB +X1 +� +gX1 +� d3kX1 +(2π)3 +MX1 +EX1 +ΓX1fMB +X1 +(159) += +ΓX1 · K1(MX1/T) +K2(MX1/T) +∼ ΓX1 × +� MX1/2T +(MX1 ≪ T) +1 +(MX1 ≫ T) +(160) +26 + +is the thermally averaged width, Kn(x) is the modified Bessel function. To derive (157), +we assumed the universal distributions for ψi (fψi ∼ +1 +Nψ fψ, fψ ≡ � +i fψi) and ignored the +chemical potentials in the thermal distributions (fMB +ψ += fMB +¯ψ +, fMB +φ += fMB +¯φ +). Besides, +we assumed φ is always in the thermal equilibrium (fφ = fMB +φ +). +On the other hand, +ψi’s equation should be derived with the consideration of the subtracted state in some +scattering processes to avoid the over-counting of the decay/inverse-decay processes: +˙nψ + 3Hnψ += +� +i +� d3kX1 +(2π)3 +d3kψi +(2π)3 +d3kφ +(2π)3 +1 +2EX12Eψi2Eφ +(2π)4δ4(kX1 − kψi − kφ) +× +� +gX1,gψi,gφ +� +fX1|M(X1 → ψiφ)|2 − fψifφ|M(ψiφ → X1)|2� ++ +� +i,j +� d3kψi +(2π)3 +d3kψj +(2π)3 +d3kφ1 +(2π)3 +d3kφ2 +(2π)3 +1 +2Eψi2Eψj2Eφ12Eφ2 +� +gψi,gψj ,gφ1,gφ2 +× +� +(2π)4δ4(kψi + kψj − kφ1 − kφ2) +× +� +−fψifψj|M(ψiψj → ¯φ1 ¯φ2)|2 + f¯φ1f¯φ2|M(¯φ1 ¯φ2 → ψiψj)|2� ++(2π)4δ4(kψi + kφ1 − kψj − kφ2) +× +� +−fψifφ1|M(ψiφ1 → ¯ψj ¯φ2)|2 +sub + f ¯ψjf¯φ2|M( ¯ψj ¯φ2 → ψiφ1)|2 +sub +�� ++ · · · +(161) += +� +nX1⟨ΓX1→ψφ⟩ − nMB +X1 +nψ +nMB +ψ +⟨ΓX1→ ¯ψ ¯φ⟩ +� +−(nψ)2⟨σψψ→¯φ¯φv⟩ + (nMB +ψ )2⟨σ ¯ψ ¯ψ→φφv⟩ +−nMB +φ +� +nψ⟨σψφ→ ¯ψ ¯φv⟩ − n ¯ψ⟨σ ¯ψ ¯φ→ψφv⟩ +� ++ +� +nψ +nMB +ψ +⟨(ΓX1→ ¯ψ ¯φ)2⟩ +ΓX1 +− n ¯ψ +nMB +ψ +⟨(ΓX1→ψφ)2⟩ +ΓX1 +� +nMB +X1 ++ · · · +(162) +where we used the notations nψ ≡ � +i nψi, ΓXa→ψφ ≡ � +i ΓXa→ψiφ, and +⟨σψφ→ ¯ψ ¯φv⟩ +≡ +1 +nMB +ψ nMB +φ +· gψgφ +� d3kψi +(2π)3 +d3kφ +(2π)3 σψφ→ ¯ψ ¯φv · fMB +ψi fMB +φ +(163) +⟨σψψ→¯φ¯φv⟩ +≡ +1 +(nMB +ψ )2 · g2 +ψ +� d3kψi +(2π)3 +d3kψj +(2π)3 σψψ→¯φ¯φv · fMB +ψi fMB +ψj +(164) +with gψ ≡ � +i gψi are the thermally averaged cross sections. +The fourth line in (162) +corresponds to the RIS contribution that makes the thermal balance to the first line, +while the processes in the third line includes the resonant structure as seen in (153). With +the expression of the net baryon number density nB = b(nψ − n ¯ψ), one can finally obtain +the equation for the net baryons using (162) as +˙nB + 3HnB += +ϵ1⟨ΓX1⟩ +� +nX1 − nMB +X1 +� +−nMB +φ +� +2bnMB +ψ +� +⟨σψφ→ ¯ψ ¯φv⟩ − ⟨σ ¯ψ ¯φ→ψφv⟩ +� ++nB +� +⟨σψiφ→ ¯ψ ¯φv⟩ + ⟨σ ¯ψi ¯φ→ψφv⟩ +�� +−nMB +ψ +� +2bnMB +ψ +� +⟨σψψ→¯φ¯φv⟩ − ⟨σ ¯ψ ¯ψ→φφv⟩ +� ++nB +� +⟨σψψ→¯φ¯φv⟩ + ⟨σ ¯ψ ¯ψ→φφv⟩ +�� ++ · · · . +(165) +27 + +where ϵ1 is the mean net number defined in (99), and we used the approximation +nψ + n ¯ψ ∼ 2nMB +ψ +≫ |nB| = b|nψ − n ¯ψ|. +(166) +Note that the tree level contribution of cross sections and their anti-state are same in +general. Therefore, the terms in second and the fourth lines in (165) are cancelled in the +leading order, respectively. +References +[1] R. A. Alpher, H. Bethe and G. Gamow, “The origin of chemical elements,” Phys. Rev. +73, 803-804 (1948) doi:10.1103/PhysRev.73.803 +[2] C. 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Plumacher, Annals Phys. 315, 305-351 (2005) +doi:10.1016/j.aop.2004.02.003 [arXiv:hep-ph/0401240 [hep-ph]]. +31 + diff --git a/H9FKT4oBgHgl3EQfdS4O/content/tmp_files/load_file.txt b/H9FKT4oBgHgl3EQfdS4O/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..820f3c3199bdebf620bc4863fada62e425334fa7 --- /dev/null +++ b/H9FKT4oBgHgl3EQfdS4O/content/tmp_files/load_file.txt @@ -0,0 +1,1162 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf,len=1161 +page_content='Boltzmann equation and its cosmological applications Seishi Enomoto∗, Yu-Hang Su, Man-Zhu Zheng, and Hong-Hao Zhang† School of Physics, Sun Yat-sen University, Guangzhou 510275, China Abstract We review the derivation of the Boltzmann equation and its cosmological applica- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Our paper derives the Boltzmann equation by the language of quantum field theory without any assumption of the finite temperature system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' We also introduce two examples of cosmological applications, dark matter abundance and baryogenesis, with techniques in their calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 1 Introduction The early Universe, composed of hot plasma, evolves in time while maintaining thermal equilibrium at most epochs, and thus the evolution is described simply by thermodynam- ics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The cosmologically important events, therefore, are focused on the various turning epochs in which some particle species depart from the thermal bath as seen in, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=', Big Bang nucleosynthesis (BBN) and recombination confirmed by both the theory and obser- vations, and the dark matter (DM) relic abundance and the baryogenesis scenario derived from some hypotheses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Qualitatively, the turning epochs can be estimated by comparing the reaction rate and the spatial expansion rate of the Universe, but more quantitative treatment to ensure accurate predictions is required by the present cosmological obser- vations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The Boltzmann equation is a powerful tool for following the out-of-equilibrium dynamics in detail, and can describe the evolution of the particle distribution due to the spatial expansion and the momentum exchanges through interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Since the Boltz- mann equation provides the abstract relation between the macroscopic and the microscopic evolution of the particle distributions, it is applicable to various situations, and hence, the solving equations differ for each system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The most successful application of the Boltzmann equation is for the theory of BBN [1, 2, 3, 4, 5, 6, 7] (see also, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=', [8, 9, 10]), which describes how the initial protons and neutrons form the light elements at the final.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' As the result, the abundances of the light elements are predicted, and they are well consistent with the present observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The other well-motivated and formulated example is the Kompaneets equation [11, 12], which is derived from the Boltzmann equation in the photon-electron system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The evolution equation is applied to the various analysis in cosmological or astronomical situations, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=', the last scattering surface in the recombination epoch, the Sunyaev-Zel’dovich effect [13, 14, 15, 16] that is a distortion of the cosmic microwave background radiation by hot electrons in galaxies, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' On the other side, exploring Beyond the Standard Model (BSM), there are many studies on the DM candidate realizing the present density parameter through the relic abundance, which can be predicted using the Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The most famous and simple scenario for DM candidates is described by weakly interacting massive par- ticles (WIMPs) [17, 18], but the current observation requires a more extended scenario or the other mechanism, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=', feebly interacting massive particles (FIMPs) [19], strongly interacting massive particles (SIMPs) [20], processing into forbidden channels [21, 22], co- annihilations [21], co-scattering [23, 24], zombie [25, 26], and inverse decays [27, 28], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' ∗seishi@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='sysy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='cn †zhh98@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='sysu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='cn 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='11819v1 [hep-ph] 27 Jan 2023 (See also [29] for their summarized analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=') The more complicated models are consid- ered, the more technical treatment in the Boltzmann equation is required.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' As the other topic for exploring BSM with the Boltzmann equation, there are also many studies for baryogenesis scenarios explaining the origin of (baryonic) matter-antimatter asymmetry in our Universe, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=', GUT baryogenesis [30, 31, 32], leptogenesis [33], electroweak baryo- genesis [34, 35], and Affleck-Dine baryogenesis [36, 37], etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The analysis tends to be more complicated than the case of DM abundance and requires more technical treatment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The studies using the Boltzmann equation have been more popular and important as some approaches to confirm the current physics in detail and explore BSM in the early Universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' We aim to provide a clearer understanding of the Boltzmann equation and its techniques with some cosmological examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' This paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' First, we derive the Boltzmann equation by the language of the quantum field theory in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Next, we demonstrate application examples how the Boltzmann equation is applied for the DM abundance in section 3 and the baryogenesis scenario in section 4 with simple toy models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Finally, we summarize our discussion in section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 2 Boltzmann equation The Boltzmann equation is a quite powerful tool to describe the evolution of particles in Cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Although the equation is applicable to various situations, the used formulae are also dependent on the circumstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In this section, we derive the evolution formula of the distribution function f = f(xµ, pµ) from the basic statement L[f] = C[f] (1) where L and C are called the Liouville operator and the collision operator, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The Liouville operator describes the variation of the distribution of a particle along a dynamical parameter, and the collision operator describes the source of the variation through the microscopic processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='1 Liouville operator We define the Liouville operator to describe the variation of the distribution along the geodesic line parametrized by the affine parameter λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Using the momentum relations pµ = dxµ dλ , pµpµ = m2 (2) where pµ is a four-momentum and m is the mass, and the geodesic equation 0 = dpµ dλ + Γµ νρpνpρ (3) where Γµ νρ is the affine connection, the Liouville operator can be written as L[f] ≡ df dλ ���� geodesic line = dxµ dλ ∂µf + dpµ dλ ∂f ∂pµ (4) = pµ∂µf − Γµ νρpνpρ ∂f ∂pµ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (5) In the FLRW space-time, gµν = diag(1, −a2, −a2, −a2) where a = a(t) is the scale factor, the distribution function is spatially homogeneous and isotropic: f = f(t, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Then the concrete representation of the Liouville operator term can be written as L[f] = E ˙f − H|⃗p|2 ∂f ∂E (6) 2 where H = ˙a/a is the Hubble parameter and (⃗p)i ≡ api is the physical momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='2 Collision operator We define the collision operator C[f] as the variation rate by the microscopic processes: C[f] ≡ df dλ ���� microscopic process = E df dt ���� microscopic process .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (7) In order to evaluate the variation of the distribution function, we assume the followings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' First, the process can be described by the quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Second, the microscopic process can be evaluated on the Minkowski space since the gravitational effect is already evaluated in the Liouville operator part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' With the above assumption, the distribution function f can be regarded as the expec- tation value of all possible occupation numbers with their probabilities, as we will see later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Also the probabilities can be evaluated by the quantum field theory on the flat space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' To derive the concrete representation of (7), we need to construct the corresponding quantum state and then evaluate the transition probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='1 Eigenstate for occupation number At first, we construct a multi-particle state |{n}⟩ in order to include all information about the particle occupations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' We impose |{n}⟩ to be the eigenstate satisfying ˆna(⃗k)|{n}⟩ = na(⃗k)|{n}⟩, ⟨{n}|{n}⟩ = 1, (8) where ˆna(⃗k) and na(⃗k) are the occupation operator and its corresponding occupation number for species a ∈ {n} in the unit phase space, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Here the occupation operator is defined by ˆna(⃗k) ≡ 1 V a(a)† ⃗k a(a) ⃗k , (9) where V ≡ � d3x = (2π)3δ3(⃗k = 0) is a volume of the system, and a(a) ⃗k is an annihilation operator for species a which satisfies [a(a) ⃗k , a(b)† ⃗p ] = δab · (2π)3δ3(⃗k − ⃗p), (others) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (10) In the case of the fermionic species, the commutation relations are replaced with the anti- commutation relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Then one can obtain the representation of the eigenstate |{n}⟩ by |{n}⟩ ≡ � a∈{n} � �� ⃗p 1 √na!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · √ V na (a(a)† ⃗p )na � � |0⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (11) Note that the occupation number na(⃗k) must be an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Furthermore, it is convenient to define the increased/decreased state from |{n}⟩ for 3 later discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' We define them by1 |{n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⃗k(+1) a ⟩ = 1 √1 ± na √ V a(a)† ⃗k |{n}⟩, ( + : bosons, − : fermions ) , (14) |{n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⃗k(−1) a ⟩ = 1 √na √ V a(a) ⃗k |{n}⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (15) The coefficients are chosen to be unit vectors ⟨{n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⃗k(±1) a |{n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⃗k(±1) a ⟩ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (16) These increased/decreased states also become the eigenstate of the occupation operator: ˆna(⃗k)|{n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⃗k(+1) a ⟩ = � 1 ± na(⃗k) � |{n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⃗k(+1) a ⟩, ( + : bosons, − : fermions ) , (17) ˆna(⃗k)|{n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⃗k(−1) a ⟩ = ± � na(⃗k) − 1 � |{n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⃗k(−1) a ⟩, ( + : bosons, − : fermions ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (18) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='2 Transition probability Using the eigenstates discussed in the previous section, let us consider the transition probability of the process A, B, · · · → X, Y, · · · (19) in the background in which other particles ({n}) exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' For simplicity, we consider a case that each species in the process are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Taking the initial state as |{n}⟩ in order to begin the given occupation numbers, the final state through the process (19) can be represented as |{n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⃗k(−1) A ,⃗k(−1) B , · · · ,⃗k(+1) X ,⃗k(+1) Y , · · · ⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The probability from the infinite past (in-state) to the infinite future (out-state) on the background particles can be evaluated by2 P(A, B, · · · → X, Y, · · · ){n} = � a=A,B,··· ,X,Y,··· � V � d3⃗ka (2π)3 � ga � × ���⟨{n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⃗k(−1) A ,⃗k(−1) B , · · · ,⃗k(+1) X ,⃗k(+1) Y , · · · | ˆS|{n}⟩ ��� 2 (20) where ga denotes the internal degrees of freedom for species a, and ˆS is the S-matrix oper- ator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The S-matrix element describing the process (19) without the background particles can be represented by the invariant scattering amplitude as inv⟨kX, kY , · · · | ˆS|kA, kB, · · · ⟩inv = iM(kA, kB, · · · → kX, kY , · · · ) ×(2π)4δ4(kA + kB + · · · − kX − kY − · · · )(21) 1For the bosonic state, the N-increased/decreased state can be defined by |{n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⃗k(+N) a ⟩ = � na!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (na + N)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 1 V N · (a(a)† ⃗k )N|{n}⟩, (12) |{n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⃗k(−N) a ⟩ = � (na − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (na + N − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 1 V N · (a(a) ⃗k )N|{n}⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (13) 2If the initial or the final state includes N of the same species, the extra factor 1 N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' for each duplicated species is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 4 where |ka, kb, · · · ⟩inv ≡ � 2Eka2Ekb · · · a† ⃗kaa† ⃗kb · · · |0⟩ (22) is a Lorentz invariant particle state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The representation (21) indicates that the S-matrix operator includes ˆS ⊃ � a=A,B,··· ,X,Y,··· �� d3⃗k′ a (2π)3 1 � 2E′a � ga � × a† ⃗k′ Xa† ⃗k′ Y · · · a⃗k′ Aa⃗k′ B · · · ×iM(k′ A, k′ B, · · · → k′ X, k′ Y , · · · ) · (2π)4δ4(k′ A + k′ B + · · · − k′ X − k′ Y − · · · ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (23) Using the above expression, the S-matrix element on the particle background can be written as ⟨{n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⃗k(−1) A ,⃗k(−1) B , · · ·⃗k(+1) X ,⃗k(+1) Y , · · · | ˆS|{n}⟩ (24) = � a=A,B,··· ,X,Y,··· �� d3⃗k′ a (2π)3 1 � 2E′ A � ga � ×iM(k′ A, k′ B, · · · → k′ X, k′ Y , · · · ) · (2π)4δ4(k′ A + k′ B + · · · − k′ X − k′ Y − · · · ) ×⟨{n};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⃗k(−1) A ,⃗k(−1) B , · · ·⃗k(+1) X ,⃗k(+1) Y , · · · |a† ⃗k′ Xa† ⃗k′ Y · · · a⃗k′ Aa⃗k′ B · · · |{n}⟩ (25) = 1 √2EAV · 2EBV · · · · 2EXV · 2EY V · · · · ×iM(kA, kB, · · · → kX, kY , · · · ) · (2π)4δ4(kA + kB + · · · − kX − kY − · · · ) × � nAnB · · · (1 ± nX)(1 ± nY ) · · · (26) where +/− is for bosonic/fermionic particles of the produced species X, Y, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' As substi- tuting the above form into (20), one can obtain P(A, B, · · · → X, Y, · · · ){n} = � a=A,B··· ,X,Y,··· �� d3⃗ka (2π)3 1 2Eka � ga � × |M(kA, kB, · · · → kX, kY , · · · )|2 ×(2π)4δ4(kA + kB + · · · − kX − kY − · · · ) · V T ×nAnB · · · (1 ± nX)(1 ± nY ) · · · (27) where T = � dt = 2πδ(E = 0) is the transition time scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Although the expression of the probability is derived, the result (27) is constructed by the exact information of quanta represented by the microscopic occupation numbers per a unit phase space na that is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Since it is impossible to know the exact quantum state, the statistical average should be considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The probability to realize the state |{n}⟩ can be represented by P{n} ≡ � a∈{n} � ⃗k p(na(⃗k)), ∞ � na=0 p(na(⃗k)) = 1 (28) where p(na(⃗k)) is a probability to be the occupation na on the momentum ⃗k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Multiplying (28) into (27) and summing over by each occupation numbers3, we can obtained the 3This procedure is equivalent to |{n}⟩⟨{n}| → � {n} P{n}|{n}⟩⟨{n}| (29) in (20), that is, the initial state is considered by the density operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 5 statistical probability as ⟨P(A, B, · · · → X, Y, · · · ){n}⟩ ≡ � {n} P{n} · P(A, B, · · · → X, Y, · · · ){n} = � a=A,B,··· ,X,Y,··· �� d3⃗ka (2π)3 1 2Ea � ga � × |M(kA, kB, · · · → kX, kY , · · · )|2 ×(2π)4δ4(kA + kB + · · · − kX − kY − · · · ) · V T ×fAfB · · · (1 ± fX)(1 ± fY ) · · · (30) where we denoted fa ≡ ∞ � na=0 p(na(⃗k)) na(⃗k).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (31) The important thing is that the expectation value fa can be interpreted as the distribution function even though the exact forms of both the probability p(na(⃗k)) and the relating occupation na(⃗k) are unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Using the result of the total probability (30), one can also define the partial probability, as an example, for the species A of the momentum ⃗kA by pA(A, B, · · · → X, Y, · · · ) ≡ d⟨P(A, B, · · · → X, Y, · · · ){n}⟩ V d3⃗kA (2π)3 � gA (32) = T 2EA � a̸=A �� d3⃗ka (2π)3 1 2Ea � ga � × |M(kA, kB, · · · → kX, kY , · · · )|2 ×(2π)4δ4(kA + kB + · · · − kX − kY − · · · ) ×fAfB · · · (1 ± fX)(1 ± fY ) · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (33) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='3 Expression of collision term The variation of the distribution ∆f through the microscopic process can be evaluated by ∆fφ ∼ � all processes ∆Nφ · [−pφ(φ, A, B, · · · → X, Y, · · · ) + pφ(X, Y, · · · → φ, A, B, · · · )] (34) where φ is the focusing species, ∆Nφ is a changing number of the quantum φ in the process φ, A, B, · · · ↔ X, Y, · · · (∆Nφ = 1 in this case), and pφ is the partial transition probability 6 for φ derived in (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Finally, the collision term can be evaluated as C[fφ] ∼ Eφ ∆fφ ∆t ���� microscopic process (35) = −1 2 � all processes � a̸=φ �� d3⃗ka (2π)3 1 2Ea � ga � ×(2π)4δ4(kA + kB + · · · − kX − kY − · · · ) ×∆Nφ � |M(kφ, kA, kB, · · · → kX, kY , · · · )|2 ×fφfAfB · · · (1 ± fX)(1 ± fY ) · · · − |M(kX, kY , · · · → kφ, kA, kB, · · · )|2 ×fXfY · · · (1 ± fφ)(1 ± fA)(1 ± fB) · · · ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (36) In the above derivation, we set ∆t = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' We derived the above result with the single-particle state for all species for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In the case of including the N-duplicated species in φ, A, B, · · · or X, Y, · · · , one needs to multiply an extra factor 1/N!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' for the species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Note that all the squared amplitudes in (36) must be regarded as the subtracted state in which the contribution of on-shell particles in the intermediate processes is subtracted in order to avoid the double-counting of the processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Such a situation will be faced in which the leading contributions of the amplitude consist of the loop diagrams or higher order of couplings, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=', the baryogenesis scenario as we discuss later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='3 Full and integrated Boltzmann equation Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (6) and (36) lead the full Boltzmann equation for a species φ on the FLRW space-time as ˙fφ − H |⃗kφ|2 Eφ ∂fφ ∂Eφ = − 1 2Eφ � all processes � a̸=φ �� d3⃗ka (2π)3 1 2Ea � ga � ×(2π)4δ4(kA + kB + · · · − kX − kY − · · · ) ×∆Nφ � |M(kφ, kA, kB, · · · → kX, kY , · · · )|2 ×fφfAfB · · · (1 ± fX)(1 ± fY ) · · · − |M(kX, kY , · · · → kφ, kA, kB, · · · )|2 ×fXfY · · · (1 ± fφ)(1 ± fA)(1 ± fB) · · · ] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (37) Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (37) for all species describe the detail evolution of the distribution functions, but they are not useful to solve because of a lot of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' To simplify the equations, the integrated Boltzmann equation is useful and convenient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The momentum integral of the left hand side of (37) leads � d3⃗kφ (2π)3 � gφ (LHS of (37)) = ˙nφ + 3Hnφ (38) where nφ ≡ � d3⃗kφ (2π)3 � gφ fφ (39) 7 is the number density of φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Note in the integration (38) that the variables t and |⃗k| = √ E2 − m2 are independent and a property of the total derivative � d3⃗k (2π)3 |⃗k|2 E ∂ ∂E (· · · ) = � d3⃗k (2π)3 ⃗k · ∂ ∂⃗k (· · · ) = −3 � d3⃗k (2π)3 (· · · ) (40) is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' As the result, the number of the dynamical variables are reduced from (#species)×(#t)× (#E) to (#species)×(#t), while the right hand side of the integrated (37) still includes the E-dependent distribution functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' An useful approximation is to apply the Maxwell- Boltzmann similarity distribution4 fa, fMB a ≪ 1 and fa(t, Ea) ∼ na(t) nMB a (t)fMB a (t, Ea), (41) where fMB a = exp � −Ea − µa Ta � , nMB a = � d3⃗kφ (2π)3 � gφ fMB a (42) are the Maxwell-Boltzmann distribution and its number density, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Supposing the identical temperature for all species Tφ = TA = · · · ≡ T,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' and substituting (41) into the right hand side of (38) and assuming the chemical equilibrium µφ + µA + µB + · · · = µX + µY + · · · (43) where µa is the chemical potential of species a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' one can obtain the integrated Boltzmann equation as ˙nφ + 3Hnφ = � d3⃗kφ (2π)3 � gφ (RHS of (37)) = − � all processes ∆Nφ [nφnAnB · · · × ⟨R(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · → X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · )⟩ −nMB φ nMB A nMB B · · nXnY · · · nMB X nMB Y · · × ⟨R(¯φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' ¯A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' ¯B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · → ¯X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' ¯Y ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · )⟩ � (44) where the bar “ ¯ ” denotes its anti-particle state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' R(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · → X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · ) ≡ 1 2EA2EB · · · � d3kX (2π)3 d3kY (2π)3 · · · 1 2EX2EY · · · × � gA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='··· |M(kA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' kB · · · → kX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' kY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · )|2 � gA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='··· ×(2π)4δ4(kA + kB + · · · − kX − kY − · · · ) (45) is a reaction rate integrated over the final state,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' and ⟨R(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · → X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · )⟩ ≡ 1 nMB A nMB B · · � d3kA (2π)3 d3kB (2π)3 · · · fMB A fMB B · · × � gA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='··· R(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · → X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · ) (46) 4The well-used approximation fa ∼ na nMB a f MB a can be justified in the case that species a is in the kinetic equilibrium through interacting with the thermal bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' See appendix A for detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The case in deviating from the kinetic equilibrium is discussed in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 8 is the thermally averaged reaction rate by the Maxwell-Boltzmann distributions of the species appearing in the initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' We used the property of the CPT-invariance of the amplitude M(X, Y, · · · → A, B, · · · ) = M( ¯A, ¯B, · · · → ¯X, ¯Y , · · · ) (47) to derive the last term in (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Although eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (44) describing the evolution of number densities is obtained by integra- tion of (37) directly, in general, the other evolution equations of the statistical quantities Qa(t) ≡ � d3⃗ka (2π)3 � ga fa(⃗ka)qa(t, Ea) (48) can also be derived through the same procedure with the corresponding coefficient qa(t, Ea), e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=', the energy density Q = ρ for q = Ea, and the pressure Q = P for q = |⃗k|2/3Ea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Such equations help to extract more detailed thermodynamic variables, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=', to determine the independent temperatures for each species, as we will see in the next subsection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='4 Temperature parameter The integrated Boltzmann equation (44) is quite useful and can be applied to many situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' However, it might not be suitable for some situations in which the kinetic equilibrium is highly violated because the formula is based on the approximation by the Maxwell-Boltzmann similarity distribution, which is justified by the kinetic equilibrium of the target particles with the thermal bath as discussed in appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Although to obtain the most appropriate solution is to solve the full Boltzmann equation, it takes a lot of costs to the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In this section, we introduce an alternative method based on the integrated Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Instead of using the similarity distribution (41), we introduce more generalized simi- 9 larity distribution by5 6 fa(t, Ea) ∼ na(t) nneq a (t)fneq a (t, Ea), fneq a (t, Ea) ≡ exp � −Ea − µa Ta(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (54) Here nneq a is the number density evaluated by the “non-equilibrium” Maxwell-Boltzmann distribution fneq a that is parametrized by the temperature parameter Ta(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In general, the temperature parameter is independent of the thermal bath temperature T(t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Especially as the property of the Maxwell-Boltzmann distribution form,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' the tempera- ture parameter can be expressed by the ratio of the pressure and the number density Ta = P neq a nneq a = Pa na (55) because of P neq a = � d3⃗k (2π)3 � ga fneq a |⃗ka|2 3Ea = � d3⃗k (2π)3 � ga Ta 3 � −⃗pa · ∂ ∂⃗pa � fneq a = Ta � d3⃗p (2π)3 � ga fneq a = nneq a Ta (56) and Pa = � d3⃗k (2π)3 � ga fa |⃗ka|2 3Ea = na nneq a � d3⃗p (2π)3 � ga fneq a |⃗ka|2 3Ea = na nneq a P neq a .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (57) Therefore, the evolution equation for the temperature parameter can be derived from the pressure’s one which can be constructed from the original full Boltzmann equation (37) multiplied by |⃗ka|2/3Ea.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' After the integration by the momentum, one can obtain the 5The normalization factor na(t)/nneq a (t) can be regarded as a corresponding quantity to the chemical potential parameter ˜µa(t): na(t) nneq a (t) = exp � ˜µa(t) − µa Ta(t) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (49) 6In the case of the Bose-Einstein/Fermi-Dirac type distribution fa(t, Ea) = � e(Ea−˜µa(t))/Ta(t) ∓ 1 �−1 ( − : boson, + : fermion) (50) where Ta(t) and ˜µa(t) are the temperature and the chemical potential parameters respectively, the tem- perature parameter can be represented as ˜Ta(t) = 1 ρa(t) + Pa(t) � d3⃗ka (2π)3 ⃗k2 a 3 fa(t, Ea) (1 ± fa(t, Ea)) ( + : boson, − : fermion) (51) with energy density ρa and pressure Pa evaluated by (50).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The above representation is consistent with (55) in the nonrelativistic limit: fa ≪ 1, Ea ∼ ma, and ρa ∼ mana ≫ Pa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The chemical potential parameter can be obtained by ˜µa(t) = ρa(t) + pa(t) − Ta(t)sa(t) na(t) (52) where sa(t) is the entropy density defined by sa(t) = � d3⃗ka (2π)3 [±(1 ± fa) ln(1 ± fa) − fa ln fa] .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' ( + : boson,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' − : fermion) (53) 10 coupled equations for species φ as ˙nφ + 3Hnφ = − � all processes ∆Nφ [nφnAnB · · · × ⟨R(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · → X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · )⟩neq −nXnY · · · × ⟨R(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · → φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · )⟩neq] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (58) nφ ˙Tφ + Hnφ � 2Tφ − � |⃗kφ|4 3E3 φ �neq� = − � all processes ∆Nφ � nφnAnB · · · × �� |⃗kφ|2 3Eφ − Tφ � R(φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · → X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · ) �neq −nXnY · · · × � ⟨RTφ(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · → φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · )⟩neq −Tφ⟨R(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · → φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · )⟩neq)] ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (59) where R is a rate defined in (45) and RTφ(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · → φ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' B,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · ) = 1 2EX2EY · · · � d3⃗kφ (2π)3 d3⃗kA (2π)3 d3⃗kB (2π)3 · · · × � gφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='··· ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='··· |M(kX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' kY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · → kφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' kA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' kB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · )|2 � gX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='··· |⃗kφ|2 3Eφ ×(2π)4δ4(kφ + kA + kB + · · · − kX − kY − · · · ) (60) is a “temperature weighted” rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' and � |⃗kφ|4 3E3 φ �neq = 1 nneq φ � d3⃗kφ (2π)3 � gφ fneq φ |⃗kφ|4 3E3 φ ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (61) ⟨R(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · → i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · )⟩neq = 1 nneq a nneq b · · � d3⃗ka (2π)3 d3⃗kb (2π)3 · · · � ga,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gb,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='··· ×fneq a fneq b · · R(a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · → i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (62) are the thermally averaged quantities by the non-equilibrium distribution fneq including only the initial species a,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' not the final species i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' j,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Solving the coupled equations (58) and (59) for all the species can be expected to obtain more accurate results than the former integrated Boltzmann equation (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Following the evolution in practice, the combined quantity y = mφTφ s2/3 ∝ Tφ T 2 (63) instead of the solo Tφ, where s is the entropy density, is convenient for the non-relativistic φ because of the asymptotic behavior Tφ(t) ∝ a(t)−2 ∝ T(t)2 after freezing out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 3 Application to DM abundance One of the cosmological application of the Boltzmann equation is for the estimation of the DM abundance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Because DM is stable, the main process changing the particle number is not decay/inverse-decay but the 2-2 annihilation/creation scatterings χ, ¯χ ↔ ψ, ¯ψ (64) 11 where χ is a DM and ψ are a standard model particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Since the rate in the 2-2 scattering can be represented by the annihilation cross section as R(χ, ¯χ → ψ, ¯ψ) = σv (65) where v is the Møller velocity7 for the pair of the DM particles, the dynamics can be solve as the annihilation cross section is given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Assuming the symmetric DM nχ = n¯χ and the thermal distribution for the standard model particles nψ = n ¯ψ = nMB ψ , the Boltzmann equation (44) for the DM leads a simple form ˙nχ + 3Hnχ = − � n2 χ − (nMB χ )2� ⟨σv⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (67) Instead of the particle number to follow its evolution by time, it is convenient to use the yield Yχ ≡ nχ/s with a dynamical variable x ≡ mχ/T, where s = 2π2 45 heff(T)T 3 is the entropy density and heff(T) ∼ 100 for T ≳ 100 GeV is the effective degrees of freedom defined by the entropy density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In the case of no creation/annihilation process, the yield Yχ becomes a constant since the number and the entropy in the comoving volume is conserved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' With these variables, the Boltzmann equation (67) can be represented as Y ′ χ = −(1 + δh)s⟨σv⟩ xH � Y 2 χ − (Y MB χ )2� (68) where we denote ′ ≡ d/dx, and δh ≡ T 3heff dheff dT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (69) Since the adiabatic parameter δh tends to be negligible in the almost era of the thermal history8, we set δh = 0 in the later discussion for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Moreover, we denote Y MB χ ≡ nMB χ s ∼ gχ heff 45 25/2π7/2 x3/2e−x (x ≫ 1), (70) where gχ is the degrees of freedom for the DM particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='1 Relic abundance in freeze-out As a simple and reasonable setup, we assume that the DM particles χ are in thermal equilibrium initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Then, the dynamics described by (68) can be explained as follow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' At first, the system is in the thermal equilibrium due to the stronger scattering effect than the spatial expansion9, but the yield has a small deviation from the thermal value due to 7The definition with the 4-momenta is given by v12 = � (k1 · k2)2 − m2 1m2 2 k0 1k0 2 , (66) which can be identical to the relative velocity only in case of the parallel 3-momenta;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' ⃗k1 · ⃗k2 = ±|⃗k1||⃗k2|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 8If the DM mass scale is around O(10) GeV, the freeze-out occurs around the QCD transition scale T ∼ O(100) MeV, in which |δh| ∼ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Thus, there is a few percent level contribution from the adiabatic parameter δh even in the WIMP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' See Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' [38, 39, 40, 41, 42, 43, 44, 45] for the determination of that parameter in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 9If the interaction rate becomes lower than the Hubble rate at the relativistic regime x ≲ 1, the abundance freezes out with the massless abundance (hot relic): Y∞ ∼ Yhot = 45ζ(3) 2π4 gχ heff(Tf ) × � 1 (boson) 3/4 (fermion) (71) where ζ(3) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='202 · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 12 the expansion effect as Yχ ∼ Y MB χ + ∆(x), ∆(x) = xH s⟨σv⟩ −Y ′ χ Y MB χ + Y ∼ xH 2s⟨σv⟩ ≪ Y MB χ (72) as long as nMB⟨σv⟩ ≫ xH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The deviation ∆ continues growing in later time, and finally the evolution of the yield freezes out because the expansion rate exceeds the scattering rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The freeze-out occurs when ∆(xf) = cY MB(xf), c ∼ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The freeze-out time x = xf and the final abundance Y∞ = Y (x = ∞) can be estimated by [8, 46] xf = ln � c(c + 2) √ 90 (2π)3 gχ � geff(Tf)mχMplσn � − � n + 1 2 � ln xf, (73) = ln � c(c + 2) √ 90 (2π)3 gχ � geff(Tf)mχMplσn � − � n + 1 2 � ln � ln � c(c + 2) √ 90 (2π)3 gχ � geff(Tf)mχMplσn �� + · · · , (74) Y∞ = (n + 1) � 45 π gχ � geff(Tf) xn+1 f Mplmχσn (75) where Mpl = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='22 × 1019 GeV is the Planck mass, geff(T) is the effective degrees of freedom defined by the energy density ρ = π2 30geff(T)T 4, and Tf = mχ/xf is the freeze- out temperature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In the derivation of the analytic results (73) and (75), the temperature dependence of the cross section is approximated by the most dominant part as ⟨σv⟩ = σnx−n, (76) where σn is a constant10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Especially, n = 0 and 1 correspond to s-wave and p-wave scattering, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Although a numerical factor c still has uncertainty, choosing c(c + 2) = n + 1 leads to better analysis for the final abundance Y∞ within 5% accuracy for xf ≳ 3 [8].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' As an example, let us consider a WIMP model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Choosing the parameters as mχ = 120 GeV, n = 1, σn = α2 W m2χ , αW = 1 30, gχ = 2, heff = geff = 90, one can obtain the analytic results xf = 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='2, Y∞ = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='81 × 10−12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (77) The actual evolution is depicted in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Finally, we need to mention the validity of the approximated results (73) and (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Their behaviors can deviate easily if the master equation (67) includes the significant ex- tra processes by other species or the singular behavior of the cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Especially it is known some exceptional cases;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (i) mutual annihilations of multiple species (coannihi- lations), (ii) annihilations into heaver states (forbidden channels), (iii) annihilations near a pole in the cross section [21], and (iv) simultaneous chemical and kinetic decoupling (coscattering) [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In these cases, the analysis should be performed more carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' See [21, 22, 23, 24, 47, 48, 49] as their example cases, and also [50, 51, 52] as examples of the evaluation with the temperature parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 10See also Appendix B for the actual analysis of the thermally averaged cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 13 1x10-13 1x10-12 1x10-11 1x10-10 1x10-9 1x10-8 15 20 25 30 35 40 45 Yχ YχMB Ylow Yhigh Y x Evolution of yields Figure 1: The numerical plots of the evolution for each yield with parameters mχ = 120 GeV, n = 1, σn = α2 W m2χ , αW = 1 30, gχ = 2, heff = geff = 90.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The red and the blue lines show the actual evolution of Yχ and the thermal yield Y MB χ , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The dashed lines of green and purple show the approximated solutions Ylow ≡ Y MB χ + ∆ and Yhigh ≡ Y∞ � 1 − Y∞ n+1 s⟨σv⟩ H �−1 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='2 Constraint on relic abundance The relic abundance for the stable particles through the freeze-out of their annihilation processes, as similar to χ particles discussed in the above, are restricted by the cosmological observation results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' An useful parameter relating to the relic abundance is the density parameter defined by Ωχ ≡ ρχ 3M2 pl 8π H2 ∼ 16π3 135 heff(T)T 3 M2 plH2 mχYχ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (78) Since the yield maintains the constant after the freeze-out unless the additional entropy production occurs in the later era, one can estimate the present density parameter of χ with the present values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The current observation through the the Cosmic Microwave Background [53] provides T0 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='726 K, heff(T0) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='91, H0 = 100h2 km/s/Mpc, h = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='677, therefore one can estimate to Ωχ,now ∼ mχYχ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='64h2 × 10−9 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (79) Because the present density parameter for the cold matter component is observed as Ωch2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='119 and it must be larger than the χ’s component, one can obtain a bound as mχYχ < 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='36 × 10−10 GeV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (80) The set of parameters shown in (77) is seemingly suitable for the above constraint with a bit of the modification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' However, that would fail by taking into account the direct detections of the DM that focuses on the process of χ, ψ ↔ χ, ψ, where ψ is a standard model particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' If the annihilation process occurs through a similar interaction to the 14 electroweak gauge interaction, the cross section for χ-ψ elastic scattering also relates to the same gauge interaction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' One can estimate σχψ→χψ ∼ G2 F m2 χ ∼ 10−36 cm2, but it is already excluded by the direct detection [54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='3 Relic abundance in freeze-in The discussion and the result in the previous subsections are based on the freeze-out scenario in which the DM particles are in thermal equilibrium initially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' However, it is not satisfied if the interaction between the DM particles and the thermal bath is too small, so-called FIMP (feebly interacting massive particle) scenario [19, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In this situation, the yield of DM evolves from zero through the thermal production from the thermal bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Although the DM never reaches the thermal equilibrium, the yield freezes in with a non- thermal yield at last.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' We discuss here the relic abundance by the freeze-in scenario in two cases of the pair- creation of DM by (1) scattering from thermal scattering and (2) decay from a heavier particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='1 Pair-creation by scattering In the case that DM-pair (χ¯χ) is produced by thermal pair particles (ψ ¯ψ), the Boltzmann equation is given by (68) as Y ′ χ ∼ s⟨σv⟩ xH (Y MB χ )2, (81) where we approximated Yχ ≪ Y MB χ and the adiabatic degrees δh ∼ 0 until the freezing-in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' For simplicity, we consider the simple interaction described by Lint = λ(χ†χ)(ψ†ψ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (82) where χ is the bosonic DM, ψi labeled i are the massless bosons in the thermal bath, and y is a coupling constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The thermally averaged cross section is given by ⟨σv⟩ = g2 χg2 ψ (nMB χ )2 λ2 (2π)5 T � ∞ 4m2χ ds � s − 4m2χ K1(√s/T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (83) where gχ and gψ are the degrees of freedom for each species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Therefore, one can estimate the final yield at x = mχ/T = ∞ as Yχ(∞) ∼ � ∞ 0 dx s⟨σv⟩ xH (Y MB χ )2 (84) = 3π2 128 · g2 χg2 ψ λ2 (2π)5 · m4 χ H(T = mχ) s(T = mχ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (85) This result implies that the yield freezing occurs around the earlier stage x ∼ O(1) because (85) can be regarded as Yχ(∞) ∼ nMB χ ⟨σv⟩ H Y MB χ ��� x∼1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Applying the obtained relic abundance (85) to the relation of the present density parameter (79), one can obtain the required strength of the coupling as λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='0 × 10−12 · 1 gχgψ �geff(T = mχ) 100 �1/4 �heff(T = mχ) 100 �1/2 �Ωχ,nowh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='119 �1/2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (86) 15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='2 Pair-creation by decay The other possible freeze-in scenario is due to the pair production from a heavier particle: σ → χ¯χ [19, 56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The original Boltzmann equation for DM is given by dYχ dxσ = (1 + δh)Γσ→χ¯χ xσH K1(xσ) K2(xσ) � Yσ − � Yχ Y MB χ �2 Y MB σ � (87) ∼ Γσ→χ¯χ xσH K1(xσ) K2(xσ)Y MB σ (88) where Γσ→χ¯χ is a decay constant and xσ ≡ mσ/T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' We also approximated Yσ ∼ Y MB σ , Yχ ≪ Y MB χ , and δh ∼ 0 in the second line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Therefore, the final yield can be estimated as Yχ(∞) ∼ � ∞ 0 dxσ Γσ→χ¯χ xσH K1(xσ) K2(xσ)Y MB σ (89) = 3gσ 4π · Γσ→χ¯χ H(T = mσ) m3 σ s(T = mσ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (90) where gσ is the degrees of freedom for σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' If the decay constant can be represented by the coupling constant y as Γσ→χ¯χ = gχ · y2 8πmσ, (91) the required magnitude of the coupling with the relation formula to the density parameter (79) can be estimated as y2 ∼ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='7 × 10−24 · mσ mχ 1 gσgχ �geff(T = mσ) 100 �1/2 heff(T = mσ) 100 Ωχ,nowh2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='119 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (92) 4 Application to baryogenesis The other popular application of the Boltzmann equation in cosmology is the baryogenesis scenario that describes the dynamical evolution of the baryon number in the Universe from zero at the beginning to the non-zero at present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The present abundance of the baryons can be estimated from (79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Replacing χ’s mass mχ into the nucleon mass mN = 939 MeV and using the present density parameter for the baryon Ωbh2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='0224 [53], one can obtain YB,now = 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='69 × 10−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (93) There are three conditions suggested by A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Sakharov [57] in order to develop the baryon abundance from YB = 0 to non-zero: (1) baryon number (B) violation, (2) C and CP violation, (3) non-equilibrium condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Their brief reasons are as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The B violation is trivial by definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' If the baryon number violating processes conserve C or CP, their anti-particle processes happen with the same rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' As the result, the net baryon number is always zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Even if the processes violate the baryon number, C and CP, the thermal equilibrium reduces the baryon asymmetry due to their inverse processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Especially, the Boltzmann equation provides a powerful tool to quantify the third condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' To see how to construct the Boltzmann equations for the baryogenesis, let us consider with a toy model11 as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The model includes Majorana-type of chiral 11Replacing X, ψ, φ into the right-handed neutrino, left-handed neutrino, Higgs doublet in the standard model, respectively, one can obtain the type-I seesaw model that can realize the well-known leptogenesis scenario [33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' However, the correspondence is not complete: the type-I seesaw model includes the gauge interactions that induces ∆L = 1 scattering process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 16 Species Particle statistic #B Xa Chiral fermion (Majorana) − ψi Chiral fermion b φ Complex scalar 0 Process ∆B Xa → ψi, φ b Xa → ¯ψi, ¯φ −b ψi, ψj → ¯φ, ¯φ −2b ψi, φ → ¯ψj, ¯φ −2b Table 1: Left: the matter contents and their baryon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' All the anti-particles have the opposite sign of the baryon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Right: Possible processes up to the 4-body B- violating interactions and their variation of the baryon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The bar (¯) on each species denotes the anti-particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In addition to the shown processes here, their inverse processes are also possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Although there are elastic scatterings Xa, ψi, → Xb, ψj, Xa, φ → Xb, φ, and ψi, φ → ψj, φ, we omited them because they do not change the baryon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' fermions Xa for a = 1, · · · , NX, baryonic chiral fermions ψi for i = 1, · · · , Nψ with the common baryon number b, and a non-baryonic complex scalar φ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' For simplicity, the baryonic fermions ψi and the scalar φ are massless and they are always in the thermal equilibrium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Because Xa are the Majorana fermion, Xa and ¯Xa can be identified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Thus, once we set the fundamental interaction to provide a decay/inverse-decay processes Xa ↔ ψi, φ, their anti-particle processes Xa ↔ ¯ψi, ¯φ also exist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' These 3-body interactions also induce the B-violating 2-2 scatterings exchanging Xa fermions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='1 Mean net baryon number At first, we consider only the decay processes for simplicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' This situation is realized when Xa particles start to decay after the scattering processes freeze out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Here we define the mean net baryon number by ϵa = � f ∆Bf � rXa→f − r ¯ Xa→ ¯f � (94) where the summation runs for all decay processes, ∆Bf and rXa→f are the generated baryon number through the process of Xa → f and its branching ratio, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The physical meaning of the mean net baryon number ϵa is an average of the produced baryon number by a single quantum of Xa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In the case of our toy model, this quantity can be represented as ϵa = b � i � rXa→ψi,φ − rXa→ ¯ψi,¯φ � (95) This result reflect the requirements of B-violation and C, CP violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' If the decay processes are B-conserving b = 0 or C, CP conserving processes rXa→ψi,φ = rXa→ ¯ψi,¯φ, the mean net baryon number is vanished.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Supposing that only a single flavour X1 survives and all of X1 particles decay into the baryonic fermions ψi, the generated baryon number can be estimated by YB ∼ ϵ1YX1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Especially, the baryon abundance can be maximized if X1 particles are the hot relic YX1 ∼ Yhot ∼ 45 2π4 gX1 heff(Tf): YB ∼ 45 2π4 · ϵ1gX1 heff(Tf), (96) where gX1 = 2 is the degrees of freedom of the Majorana-type fermion X1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='2 Boltzmann equations in baryogenesis scenario Although we considered quite simplified situation in the previous subsection, in reality, the situation is more complicated since the system includes the dynamical decay/inverse decay and scattering processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In order to quantify the actual evolution of the baryon abundance including the scattering effects, we need to construct the Boltzmann equations in this system and solve them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' For simplicity, we suppose again that only a single flavour X1 affects to the evolu- tion of the net baryon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Using the definition of the net baryon density nB = b � i � nψi − n ¯ψi � ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' the evolution of the system including the processes in Table 1 are de- scribed by12 ˙nX1 + 3HnX1 = − �MX1 EX1 � ΓX1(nX1 − nMB X1 ) + · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (97) ˙nB + 3HnB = ϵ1 �MX1 EX1 � ΓX1 � nX1 − nMB X1 � − 2ΓS nB + · · · ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (98) where MX1 (EX1) is the mass (energy) of X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' ΓX1 ∼ � i � ΓX1→ψiφ + ΓX1→ ¯ψi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='¯φ � is the total width of X1 and ϵ1 ≡ b · � i � ΓX1→ψiφ − ΓX1→ ¯ψi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='¯φ � ΓX1 (99) is the mean net baryon number corresponding to (95),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' and ΓS = nMB φ ⟨σψφ→ ¯ψ ¯φv⟩ + nMB ψ ⟨σψψ→¯φ¯φv⟩ (100) is the reaction rates through the B-violating scatterings up to the tree level.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The omitted parts “· · · ” denote the sub-leading processes in terms of the order of couplings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (97) describes the dissipation of Xa, and it converts to the baryon with the rate ϵa and flows into the baryon sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' However, the produced baryons also wash themselves out through the B-violating scattering processes due to the last term in (98).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Therefore, the smaller B-violating scattering effect is favored for remaining the more net baryons as long as Xa can be thermalized enough at the initial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' To solve the equations of motion (97) and (98),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' it is convenient to use the yields YX1 = nX1/s,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' YB = nB/s and the variable x = MX1/T as Y ′ X1 = −γD(YX1 − Y MB X1 ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (101) Y ′ B = ϵ1γD(YX1 − Y MB X1 ) − 2γSYB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (102) where Y MB X1 = nMB X1 s = 45 4π4 gX1 heff x2K2(x),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (103) γD = 1 x · ΓX1 H(T) �mX1 EX1 � = � 45 4π3geff Mpl MX1 K1(x) K2(x) ΓX1 T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (104) γS = 1 x · ΓS H(T) = � 45 4π3geff Mpl MX1 ΓS(T) T ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (105) with the n-th order of the modified Bessel function Kn(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' We assumed the adiabatic evo- lution of the relativistic degrees h′ eff/heff ∼ 0 to obtain (101) and (102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The dimensionless 12See appendix B for the treatment of the thermally averaged quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' And also see Appendix C for the detail of the derivation of the equations, especially, the treatment of the real intermediate state (RIS) to avoid the double-counting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 18 parameters γD,S are the reaction rates normalized by the Hubble parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In general, γD is proportional to x2 (x1) at the limit of x ≪ 1 (x ≫ 1), whereas the behavior of γS depends on the detail of the interaction as we will see its concrete form with an example model later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (101) and (102) can provide the analytic form of YB as YB(∞) = −ϵ1 � ∞ 0 dx Y ′ X1(x) exp � −2 � ∞ x dx′ γS(x′) � (106) Especially in the weakly scattering case, � ∞ 0 dx γS ≲ 1, one can approximate the above result as YB(∞) ∼ −ϵ1 � ∞ 0 dx Y ′ X1(x) = ϵ1Yhot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (107) The physical interpretation is that the whole X1 particles existing from the beginning can convert to the net baryons without any wash-out process in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Hence the approximated result does not depend on the detail of the decay process γD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The result (107) is consistent with the former estimation in (96).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' On the other hand, the strongly scattering case causes the wash-out process significantly, and thus the final net baryon abundance is strongly suppressed from the result of (107).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' To see the concrete evolution dynamics, we consider the following interaction Lint = − � a,i yaiφXaψi + (h.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=') (108) with the Yukawa coupling yai and the two-component spinors Xa and ψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' This interaction leads the concrete representation of the decay width and the scattering rate as ΓX1 = ˜αMX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (109) ΓS = T · 8˜α2 πgψ ˜γS(x) (110) where we denoted ˜α = � i gψigφ |y1i|2 32π ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (111) ˜γS = 1 8 � ∞ 0 dz K1(z) � 2 � z4 z2 + x2 + x2z2 z2 + 2x2 ln � 1 + z2 x2 �� + x2z4 (z2 − x2)2 + ˜α2x4 + 2 � z2 − x2 ln � 1 + z2 x2 �� + 4x2(z2 − x2) (z2 − x2)2 + ˜α2x4 � z2 − � z2 + x2� ln � 1 + z2 x2 ��� (112) ∼ � 1 (x ≪ 1) 8/x2 (x ≫ 1) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (113) and gψ ≡ � i gψi = Nψgψi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Here gψi and gφ are the degrees of freedom of the chiral fermion ψi and the scalar φ, not including their anti-particle state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The asymptotic behaviors for each reaction rate are governed by γD(x) ∝ � x2 (x ≪ 1) x1 (x ≫ 1) , γS(x) ∝ � (constant) (x ≪ 1) x−2 (x ≫ 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (114) 19 1e-06 1e-04 1e-02 1e+00 1e+02 1e+04 1e+06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='1 1 10 100 For decays (γD) For scatterings (γS) MX1 = 1016 GeV MX1 = 1015 GeV MX1 = 1014 GeV MX1 = 1013 GeV MX1 = 1013 GeV MX1 = 1014 GeV MX1 = 1015 GeV MX1 = 1016 GeV γ = Γ/xH x = MX1/T Evolution of rate of reactions 1e-12 1e-10 1e-08 1e-06 1e-04 1e-02 1e+00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='1 1 10 100 MX1 = 1016 GeV MX1 = 1015 GeV MX1 = 1014 GeV MX1 = 1013 GeV Analytic (hot relic decay) YB/ε1 x = MX1/T Evolution of YB/ε1 Figure 2: The numerical plots of the evolution of the interaction rates (upper) and YB/ϵ1 (lower) for each mass of X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The numerical parameters are chosen as gX1 = 2, gψi = gφ = 1, Nψ = 3, ˜α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='01, heff = geff = 100, and assumed the thermal distribution for X1 and YB = 0 at the initial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The solid lines in red, yellow, green, and blue correspond to MX1 = 1016, 1015, 1014, and 1013 GeV, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In the upper figure, “For decays” and “For scatterings” depict γD(x) and γS(x), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In the lower figure, the dashed line in purple shows the approximated solution (96) due to the decay of the hot relic, YB/ϵ1 ∼ Yhot = 45 2π · gX1 heff .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 20 The actual behavior of γD and γS with concrete parameters are shown in the upper side of Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The asymptotic behaviors at x ≪ 1 and x ≫ 1 are consistent with (114).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The enhancement structures for each γS seen around x ∼ O(1) are induced by the resonant process through the on-shell s-channel shown in (112).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The lower side in Figure 2 shows the evolution of YB/ϵ1, which is the numerical result from the coupled equations (101) and (102).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The result shows that the heavier mass of X can generate more the net baryon number because the reaction rates are reduced for the heavier case, and hence the generated baryons can avoid the wash-out process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Especially, the plot for MX1 = 1016 GeV leads the close result to the hot relic approximation (107), whereas the plot for MX1 = 1013 GeV shows the dumping by the wash-out effect at the late stage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The milder decrease at the middle stage is caused by the decay of X particles that supplies the net baryons to compensate for the wash-out effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Finally, the obtained yield of the net baryon number YB should be compared with the current bound (93), YB,now ∼ 10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Since the mean net baryon number can roughly be estimated by ϵ1 ∼ ˜α2 sin2 θCP where θCP is a CP phase in the considered model, one can obtain the constraint from the current observation as YB,now/ϵ1 ≳ 10−10/˜α2 ∼ 10−6, where we used ˜α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='01.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Therefore, one can find that MX1 ≳ 1014 GeV is allowed by compared with the lower plot in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 5 Summary In this paper we have demonstrated the derivation of the Boltzmann equation from the microscopic point of view with the quantum field theory, in which the transition probabil- ity has been constructed with the statistically averaged quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Although both results of the full and the integrated Boltzmann equation (37) and (44) are consistent with the well-known results, our derivation ensures that especially the full Boltzmann equation is widely applicable even in the non-equilibrium state since the derivation does not as- sume any distribution type nor the temperature of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Especially the integrated Boltzmann equation (44) is quite convenient and applicable for wide situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In the particular case that the kinetic equilibrium cannot be ensured, the coupled equations with the temperature parameter (58) and (59) are better for following the dynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' As the application examples of the (integrated) Boltzmann equation in cosmology, we have reviewed two cases, the relic abundance of the DM and the baryogenesis scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' For the former case, we have shown the Boltzmann equation and its analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The analytic results (73) and (75) are quite helpful for estimating the final relic abundance of the DM and its freeze-out epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' For the latter case, we have derived the Boltzmann equation with a specific model and show the numerical analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The final net baryon number can be estimated by the analytic result (107) in the case of the weakly interacting system, whereas that is strongly suppressed by the wash-out effect in the case of the strongly interacting system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The Boltzmann equation is a powerful tool for following the evolution of the particle number or other thermal quantities, and thus it will be applied for many more situations in future and will open a new frontier of the current physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' We hope this paper helps you to use the Boltzmann equation and its techniques thoughtfully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Acknowledgment We thank Chengfeng Cai, Yi-Lei Tang, and Masato Yamanaka for useful discussions and comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' This work is supported in part by the National Natural Science Foundation of 21 China under Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 12275367, and the Sun Yat-Sen University Science Foundation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' A Validity of the Maxwell-Boltzmann similarity approxi- mation Although the approximation of the distribution function by the Maxwell Boltzmann simi- larity distribution is used well in many situations, such approximation is not always valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In this appendix, we show that the approximation is valid if the focusing species is in the kinetic equilibrium through interacting with the thermal bath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Let us consider the situation of the particle number conserving process a(k1)+b(k2) ↔ a(k3)+b(k4), where a and b denote the particle species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' If this process happens fast enough and the species b maintains the thermal distribution, the condition of the detailed balance leads 0 = fa(t, E1)fMB b (t, E2) − fa(t, E3)fMB b (t, E4) (115) = � fa(t, E1) fMB a (t, E1) − fa(t, E3) fMB a (t, E3) � fMB a (t, E1)fMB b (t, E2), (116) where we assumed the common temperature to the thermal bath and the energy conserva- tion law: fMB a (t, E1)fMB b (t, E2) = fMB a (t, E3)fMB b (t, E4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Because the above relation must be satisfied by arbitrary energy, one can obtain fa(t, E) fMB a (t, E) = C(t) (117) where C(t) is a function which is dependent on time but independent of the energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The function C(t) can be determined by integrating over the momentum of fa(t, E) = C(t)fMB a (t, E1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=', na(t) = C(t)nMB a (t).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Finally, one can obtain the desired form of the distribution: fa(t, E) = C(t)fMB a (t, E) = na(t) nMB a (t)fMB a (t, E).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (118) B Formulae for thermal average by Boltzmann-Maxwell dis- tribution In this section, we summarize the convenient formulae used in the various thermally av- eraged quantities by the Maxwell-Boltzmann distribution, especially for number density, decay rate, and cross section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='1 Number density and modified Bessel function The number density with the Maxwell-Boltzmann distribution is given by nMB = g � d3k (2π)3 fMB (119) = g 2π2 m2TK2(m/T)eµ/T (120) = g × � � � � � � � 1 π2 T 3eµ/T + · · · (T ≫ m) �mT 2π �3/2 e−(m−µ)/T � 1 + 15T 8m + · · · � (T ≪ m) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (121) 22 where Kn is the n-th order of the modified Bessel function given by Kn(x) = � ∞ 0 dθ e−x cosh θ cosh nθ (122) = � � � � � � � Γ(n) 2 �2 x �n + · · · (0 < x ≪ √1 + n) � π 2x e−x � 1 + 4n2 − 1 8x + · · · � (x ≫ 1) (123) Especially,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' the following relations are helpful in analysis: Kn(x) = x 2n (Kn+1(x) − Kn−1(x)) ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (124) d dx (xnKn(x)) = −xnKn−1(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (125) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='2 Thermally averaged decay rate The rate defined in (45) for the single initial species relates to the decay rate, R(A → X, Y, · · · ) = mA 2EA ΓA→X,Y,···, where ΓA→X,Y,··· = 1 2mA � d3kX (2π)3 d3kY (2π)3 · · · 1 2EX2EY · · ×(2π)4δ4(kA − kX − kY − · · · ) × 1 gA � gA,gX,gY ,··· |M(A → X, Y, · · · )|2 (126) is the partial width for the process A → X, Y, · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The factor mA/EA in R corresponds to the inverse Lorentz gamma factor describing the life-time dilation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The thermal average of the rate is given by ⟨R(A → X, Y, · · · )⟩ = 1 2ΓA→X,Y,··· �mA EA � , (127) �mA EA � = gA nMB A � d3kA (2π)3 mA EA fMB A (128) = K1(mA/T) K2(mA/T) (129) = � � � mA 2T + · · · (T ≫ mA) 1 − 3T 2mA + · · · (T ≪ mA) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (130) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='3 Thermal averaged cross section The rate averaged by the initial 2-species relates to the scattering rate,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' R(A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' B → X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · ) = σv (131) = 1 2EA2EB � d3kX (2π)3 d3kY (2π)3 · · · 1 2EX2EY · · ×(2π)4δ4(kA + kB − kX − kY − · · · ) × 1 gAgB � gA,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gX,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gY ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='··· |M(A → X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · )|2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (132) 23 where σ = σ(s) is the cross section for the process A,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' B → X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Y,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' · · · dependent on the Mandelstam variable s = (kA + kB)2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' and v is the Møller velocity v = � (kA · kB)2 − m2 Am2 B EAEB = � (s − (mA + mB)2)(s − (mA − mB)2) 2EAEB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (133) The thermal average of the rate can be obtained by ⟨R(A, B → X, Y, · · · )⟩ = ⟨σv⟩ (134) = gAgB nMB A nMB B � d3kA (2π)3 d3kB (2π)3 σv · fMB A fMB B (135) = gAgB nMB A nMB B � ∞ 0 d|⃗kA| d|⃗kB| � π 0 dθ · 1 4π2 |⃗kA|2|⃗kB|2 sin θ EAEB ×σ(s) · � (s − (mA + mB)2)(s − (mA − mB)2) × exp � −EA + EB T + µA + µB T � , (136) where the integral variable θ denotes the angle between ⃗kA and ⃗kB, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=', ⃗kA · ⃗kB = |⃗kA||⃗kB| cos θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In order to perform the integral in (136), it is convenient to change the integral variables (|⃗kA|, |⃗kB|, θ) to (E+, E−, s), where E± ≡ EA ± EB [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The Jacobian is given by ����� ∂(|⃗kA|, |⃗kB|, θ) ∂(E+, E−, s) ����� = EAEB 4|⃗kA|2|⃗kB|2 sin θ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (137) The integral region can be obtained from the expression of the Mandelstam variable, s = m2 A + m2 B + 2 � EAEB + |⃗kA||⃗kB| cos θ � , (138) which leads (s − m2 A − m2 B − 2EAEB)2 ≤ 4|⃗kA|2|⃗kB|2 = 4(E2 A − m2 A)(E2 B − m2 B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (139) The above inequality is equivalent to � E− − m2 A − m2 B s E+ �2 ≤ (E2 + − s) � 1 − (mA + mB)2 s � � 1 − (mA − mB)2 s � (140) Therefore, the integral region can be obtained as e− ≤ E− ≤ e+, (141) E+ ≥ √s, (142) s ≥ (mA + mB)2, (143) where e± ≡ m2 A − m2 B s E+ ± � (E2 + − s) � 1 − (mA + mB)2 s � � 1 − (mA − mB)2 s � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (144) 24 Using the above results, the integral (136) can be performed as ⟨σv⟩ = gAgB nMB A nMB B 1 2(2π)4 e(µA+µB)/T × � ∞ (mA+mB)2 ds · σ(s) · (s − (mA + mB)2)(s − (mA − mB)2) × T √sK1(√s/T) (145) = 1 4m2 Am2 BT � ∞ mA+mB d√s · σ(s) · (s − (mA + mB)2) ×(s − (mA − mB)2) · K1(√s/T) K2(mA/T)K2(mB/T), (146) where we used the representation of the number density (120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Especially in the case of the non-relativistic limit, mA, mB ≫ T, it is convenient to use the representation13 � σv(s) ≡ σ(s) · vNR(s), vNR(s) ≡ � s − (mA + mB)2 mAmB (147) and the replacement of the integral variable s to y defined by √s = mA + mB + Ty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (148) Since the integral parameter y corresponds to ⃗k2/mT naively, we can expect that the significant integral interval is on y ≲ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Then (146) can be approximated as ⟨σv⟩ ∼ 2 √π � 1 − 15T 8mA − 15T 8mB + 3T 8(mA + mB) + · · · � × � ∞ 0 dy · � (� σv)0 + Ty · (� σv)′ 0 + · · · � ×e−y · √y � 1 + Ty 2mA + Ty 2mB − Ty 4(mA + mB) + · · · � (149) = (� σv)0 + 3 2T � −3 4 � 1 mA + 1 mB � (� σv)0 + (� σv)′ 0 � + O(T 2),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (150) where we used the asymptotic expansion (123) and the Taylor series around √s = mA + mB,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' � σv = (� σv)0 + (√s − mA − mB) · (� σv)′ 0 + · · · (151) (� σv)0 ≡ � σv(√s = mA + mB),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (� σv)′ 0 ≡ d � σv d√s ����√s=mA+mB .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (152) C Derivation of the Boltzmann equations in baryogenesis scenario In this section, we demonstrate the derivation of the Boltzmann equation in the baryoge- nesis scenario with the processes listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Indeed, the straightforward derivation 13The Lorentz-invariant “velocity” vNR behaves as vNR ∼ ��� ⃗kA mA − ⃗kB mB ��� at the non-relativistic limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Note that lim vNR→0 � σv remains non-zero (s-wave contribution) in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' 25 of the Boltzmann equations leads to the over-counting problem in the amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' For example, once a contribution of the decay/inverse-decay process X ↔ ψ, φ is included in the Boltzmann equation, the straightforward contribution from the scattering process ¯ψ, ¯φ ↔ ψ, φ is over-counted because such process can be divided into ¯ψ, ¯φ ↔ X and X ↔ ψ, φ if the intermediate state X is on-shell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Therefore, in general, one must regard the straightforward contribution in the scattering processes as the subtracted state of the real intermediated state (RIS) from the full contribution [8, 58]: |M|2 Boltzmann eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' = |M|2 subtracted ≡ |M|2 full − |M|2 RIS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' In a case of the scattering process ¯ψ, ¯φ → ψ, φ, the full amplitude part can be represented as iM( ¯ψ, ¯φ → ψ, φ)full ∼ iM(X → ψ, φ) · i s − M2 X + iMXΓX iM( ¯ψ, ¯φ → X) (153) where s is the Mandelstam variable, MX and ΓX are X’s mass and total decay width, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' On the other hand, the RIS part can be evaluated as the limit of the narrow width by ��M( ¯ψ, ¯φ → ψ, φ) ��2 RIS = lim ΓX→0 ��M( ¯ψ, ¯φ → ψ, φ) ��2 full (154) = lim ΓX→0 |M(X → ψ, φ)|2 1 (s − M2 X)2 + (MXΓX)2 |M( ¯ψ, ¯φ → X)|2 ∼ |M(X → ψ, φ)|2 · π MXΓX δ(s − M2 X) · |M( ¯ψ, ¯φ → X)|2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (155) In the last line, the narrow width approximation is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Since the contribution of both amplitudes in (155) is the order of ΓX, the RIS contribution is also the order of ΓX in total.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Therefore, RIS part in the scattering process contributes to the decay/inverse-decay process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Taking into account the above notice, we derive the Boltzmann equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' For simplic- ity, we suppose that only a single flavour X1 affects to the evolution of the net baryon number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Because of no scattering processes associated with Xa,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' the equation governing nXa is simply written as ˙nX1 + 3HnX1 = � d3kX1 (2π)3 d3kψi (2π)3 d3kφ (2π)3 1 2EX12Eψi2Eφ (2π)4δ4(kX1 − kψi − kφ) × 1 gX1 � gX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gψi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gφ � −fX1|M(X1 → ψiφ)|2 + fψifφ|M(ψiφ → X1)|2 −fX1|M(X1 → ¯ψi ¯φ)|2 + f ¯ψif¯φ|M( ¯ψi ¯φ → X1)|2� + · · · (156) = − � i ⟨ΓX1⟩(nX1 − nMB X1 ) + · · · (157) where ΓX1 = ΓX1→ψiφ + ΓX1→ ¯ψi ¯φ + · · · (158) is the total decay width of X1 and ⟨ΓX1⟩ ≡ 1 nMB X1 � gX1 � d3kX1 (2π)3 MX1 EX1 ΓX1fMB X1 (159) = ΓX1 · K1(MX1/T) K2(MX1/T) ∼ ΓX1 × � MX1/2T (MX1 ≪ T) 1 (MX1 ≫ T) (160) 26 is the thermally averaged width,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Kn(x) is the modified Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' To derive (157), we assumed the universal distributions for ψi (fψi ∼ 1 Nψ fψ, fψ ≡ � i fψi) and ignored the chemical potentials in the thermal distributions (fMB ψ = fMB ¯ψ , fMB φ = fMB ¯φ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Besides, we assumed φ is always in the thermal equilibrium (fφ = fMB φ ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' On the other hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' ψi’s equation should be derived with the consideration of the subtracted state in some scattering processes to avoid the over-counting of the decay/inverse-decay processes: ˙nψ + 3Hnψ = � i � d3kX1 (2π)3 d3kψi (2π)3 d3kφ (2π)3 1 2EX12Eψi2Eφ (2π)4δ4(kX1 − kψi − kφ) × � gX1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gψi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gφ � fX1|M(X1 → ψiφ)|2 − fψifφ|M(ψiφ → X1)|2� + � i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='j � d3kψi (2π)3 d3kψj (2π)3 d3kφ1 (2π)3 d3kφ2 (2π)3 1 2Eψi2Eψj2Eφ12Eφ2 � gψi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gψj ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gφ1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='gφ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='(2π)4δ4(kψi + kψj − kφ1 − kφ2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='−fψifψj|M(ψiψj → ¯φ1 ¯φ2)|2 + f¯φ1f¯φ2|M(¯φ1 ¯φ2 → ψiψj)|2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='+(2π)4δ4(kψi + kφ1 − kψj − kφ2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='× ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='−fψifφ1|M(ψiφ1 → ¯ψj ¯φ2)|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='sub + f ¯ψjf¯φ2|M( ¯ψj ¯φ2 → ψiφ1)|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='sub ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='�� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='+ · · · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='(161) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='nX1⟨ΓX1→ψφ⟩ − nMB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='X1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='nψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='nMB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⟨ΓX1→ ¯ψ ¯φ⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='−(nψ)2⟨σψψ→¯φ¯φv⟩ + (nMB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='ψ )2⟨σ ¯ψ ¯ψ→φφv⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='−nMB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='φ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='nψ⟨σψφ→ ¯ψ ¯φv⟩ − n ¯ψ⟨σ ¯ψ ¯φ→ψφv⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='nψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='nMB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⟨(ΓX1→ ¯ψ ¯φ)2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='ΓX1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='− n ¯ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='nMB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='ψ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='⟨(ΓX1→ψφ)2⟩ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='ΓX1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='nMB ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='X1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='+ · · · ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='(162) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='where we used the notations nψ ≡ � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content='i nψi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' ΓXa→ψφ ≡ � i ΓXa→ψiφ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' and ⟨σψφ→ ¯ψ ¯φv⟩ ≡ 1 nMB ψ nMB φ gψgφ � d3kψi (2π)3 d3kφ (2π)3 σψφ→ ¯ψ ¯φv · fMB ψi fMB φ (163) ⟨σψψ→¯φ¯φv⟩ ≡ 1 (nMB ψ )2 · g2 ψ � d3kψi (2π)3 d3kψj (2π)3 σψψ→¯φ¯φv · fMB ψi fMB ψj (164) with gψ ≡ � i gψi are the thermally averaged cross sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' The fourth line in (162) corresponds to the RIS contribution that makes the thermal balance to the first line, while the processes in the third line includes the resonant structure as seen in (153).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' With the expression of the net baryon number density nB = b(nψ − n ¯ψ), one can finally obtain the equation for the net baryons using (162) as ˙nB + 3HnB = ϵ1⟨ΓX1⟩ � nX1 − nMB X1 � −nMB φ � 2bnMB ψ � ⟨σψφ→ ¯ψ ¯φv⟩ − ⟨σ ¯ψ ¯φ→ψφv⟩ � +nB � ⟨σψiφ→ ¯ψ ¯φv⟩ + ⟨σ ¯ψi ¯φ→ψφv⟩ �� −nMB ψ � 2bnMB ψ � ⟨σψψ→¯φ¯φv⟩ − ⟨σ ¯ψ ¯ψ→φφv⟩ � +nB � ⟨σψψ→¯φ¯φv⟩ + ⟨σ ¯ψ ¯ψ→φφv⟩ �� + · · · .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (165) 27 where ϵ1 is the mean net number defined in (99), and we used the approximation nψ + n ¯ψ ∼ 2nMB ψ ≫ |nB| = b|nψ − n ¯ψ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' (166) Note that the tree level contribution of cross sections and their anti-state are same in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' Therefore, the terms in second and the fourth lines in (165) are cancelled in the leading order, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/H9FKT4oBgHgl3EQfdS4O/content/2301.11819v1.pdf'} +page_content=' References [1] R.' metadata={'source': 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Degenc, Karen Veroyd +aOden Institute for Computational Engineering and Sciences, University of Texas at Austin, 201 E 24th St, +Austin, TX 78712, USA +bSchool of Computational Science and Engineering, Georgia Institute of Technology, 756 W Peachtree St +NW, Atlanta, GA 30308, USA +cComputational Geoscience, Geothermics, and Reservoir Geophysics, RWTH Aachen University, +Mathieustr. 30, 52074 Aachen, Germany +dCenter for Analysis, Scientific Computing and Applications, Department of Mathematics and Computer +Science, Eindhoven University of Technology, 5612 AZ Eindhoven, The Netherlands +Abstract +We consider optimal sensor placement for a family of linear Bayesian inverse prob- +lems characterized by a deterministic hyper-parameter. The hyper-parameter describes +distinct configurations in which measurements can be taken of the observed physical +system. To optimally reduce the uncertainty in the system’s model with a single set of +sensors, the initial sensor placement needs to account for the non-linear state changes +of all admissible configurations. We address this requirement through an observabil- +ity coefficient which links the posteriors’ uncertainties directly to the choice of sensors. +We propose a greedy sensor selection algorithm to iteratively improve the observability +coefficient for all configurations through orthogonal matching pursuit. The algorithm +allows explicitly correlated noise models even for large sets of candidate sensors, and +remains computationally efficient for high-dimensional forward models through model +order reduction. We demonstrate our approach on a large-scale geophysical model +of the Perth Basin, and provide numerical studies regarding optimality and scalability +with regard to classic optimal experimental design utility functions. +1. Introduction +In the Bayesian approach to inverse problems (c.f. [1]), the uncertainty in a param- +eter is described via a probability distribution. With Bayes’ Theorem, the prior belief in +a parameter is updated when new information is revealed such that the posterior distri- +bution describes the parameter with improved certainty. Bayes’ posterior is optimal in +the sense that it is the unique minimizer of the sum of the relative entropy between the +posterior and the prior, and the mean squared error between the model prediction and +the experimental data. The noise model drives, along with the measurements, how the +posterior’s uncertainty is reduced in comparison to the prior. A critical aspect – espe- +arXiv:2301.12019v1 [math.NA] 27 Jan 2023 + +cially for expensive experimental data1 – is how to select the measurements to improve +the posterior’s credibility best. The selection of adequate sensors meeting individual +applications’ needs is, therefore, a big goal of the optimal experimental design (OED) +research field and its surrounding community. We refer to the literature (e.g., [3, 4, 5]) +for introductions. +The analysis and algorithm presented in this work significantly extend our initial +ideas presented in [6] in which we seek to generalize the 3D-VAR stability results +from [7] to the probabilistic Bayesian setting. Our proposed algorithm is directly re- +lated to the orthogonal matching pursuit (OMP) algorithm [8, 9] for the parameterized- +background data-weak (PBDW) method and the empirical interpolation method (EIM) +([10, 11]). Closely related OED methods for linear Bayesian inverse problems over +partial differential equations (PDEs) include [12, 13, 14, 15, 16, 17], mostly for A- and +D-OED and uncorrelated noise. In recent years, these methods have also been extended +to non-linear Bayesian inverse problems, e.g., [18, 19, 20, 21, 22], while an advance to +correlated noise has been made in [23]. In particular, [21, 22] use similar algorithmic +approaches to this work by applying a greedy algorithm to maximize the expected in- +formation gain. Common strategies for dealing with the high dimensions imposed by +the PDE model use the framework in [24] for discretization, combined with parameter +reduction methods (e.g., [25, 26, 27, 28, 29, 30, 31]) and model order reduction (MOR) +methods for uncertainty quantification (UQ) problems (e.g., [32, 33, 34, 35, 36]). +In this paper, we consider inverse problem settings, in which a deterministic hyper- +parameter describes anticipated system configurations such as material properties or +loading conditions. Each configuration changes the model non-linearly, so we obtain +a family of possible posterior distributions for any measurement data. Supposing data +can only be obtained with a single set of sensors regardless of the system’s configu- +ration, the OED task becomes to reduce the uncertainty in each posterior uniformly +over all hyper-parameters. This task is challenging for high-dimensional models since +1) each configuration requires its own computationally expensive model solve, and 2) +for large sets of admissible measurements, the comparison between sensors requires +the inversion of the associated, possibly dense noise covariance matrix. By building +upon [6], this paper addresses both challenges and proposes in detail a sensor selection +algorithm that remains efficient even for correlated noise models. +The main contributions are as follows: First, we identify an observability coeffi- +cient as a link between the sensor choice and the maximum eigenvalue of each poste- +rior distribution. We also provide an analysis of its sensitivity to model approximations. +Second, we decompose the noise covariance matrix for any observation operator to al- +low fast computation of the observability gain under expansion with additional sensors. +Third, we propose a sensor selection algorithm that iteratively constructs an observa- +tion operator from a large set of sensors to increase the observability coefficient over +all hyper-parameters. The algorithm is applicable to correlated noise models, and re- +quires, through the efficient use of MOR techniques, only a single full-order model +1For instance, for projects harvesting geothermal energy, the development costs (e.g., drilling, stimula- +tion, and tests) take up 50 − 70% of the total budget ([2]). As each borehole can cost several million dollars, +it is essential to plan their location carefully. +2 + +evaluation per selected sensor. +While the main idea and derivation of the observability coefficient are similar to +[6], this work additionally features 1) an analysis of the observability coefficient re- +garding model approximations, 2) explicit computational details for treating correlated +noise models, and 3) a comprehensive discussion of the individual steps in the sen- +sor selection algorithm. Moreover, the proposed method is tested using a large-scale +geophysical model of the Perth Basin. +This paper is structured as follows: In Section 2 we introduce the hyper-parameterized +inverse problem setting, including all assumptions for the prior distribution, the noise +model, and the forward model. In Section 3, we then establish and analyze the con- +nection between the observability coefficient and the posterior uncertainty. +We fi- +nally propose our sensor selection algorithm in Section 4 which exploits the presented +analysis to choose sensors that improve the observability coefficient even in a hyper- +parameterized setting. We demonstrate the applicability and scalability of our approach +on a high-dimensional geophysical model in Section 5 before concluding in Section 6. +2. Problem setting +Let X be a Hilbert space with inner product ⟨·, ·⟩X and induced norm ∥x∥2 +X := +⟨x, x⟩X. We consider the problem of identifying unknown states xtrue(θ) ∈ X of a single +physical system under changeable configurations θ from noisy measurements +d(θ) ≈ [ℓ1(xtrue(θ)), . . . , ℓK(xtrue(θ))]T ∈ RK. +The measurements are obtained by a set of K unique sensors (or experiments) ℓ1, . . . , ℓK ∈ +X′. Our goal is to choose these sensors from a large sensor library L ⊂ X′ of options +in a way that optimizes how much information is gained from their measurements for +any configurations θ. +Hyper-parameterized forward model +We consider the unknown state xtrue to be uniquely characterized by two sources of +information: +• an unknown parameter utrue ∈ RM describing uncertainties in the governing +physical laws, and +• a hyper-parameter (or configuration2) θ ∈ P ⊂ Rp describing dependencies on +controllable configurations under which the system may be observed (such as +material properties or loading conditions) where P is a given compact set en- +closing all possible configurations. +For any given u ∈ RM and θ ∈ P, we let xθ(u) ∈ X be the solution of an abstract model +equation Mθ(xθ(u); u) = 0 and assume that the map u → xθ(u) is well-defined, linear, +2We call θ interchangeably hyper-parameter or configuration to either stress its role in the mathematical +model or physical interpretation. +3 + +and uniformly continuous in u, i.e. +∃ ¯η > 0 : +η(θ) := sup +u∈RM +∥xθ(u)∥X +∥u∥Σ−1 +pr +< ¯η +∀ θ ∈ P. +(1) +Remark 1. Although we assumed that utrue lies in the Euclidean space RM, any other +linear space can be considered via an affine transformation onto an appropriate basis +(see [12, 37]). For infinite-dimensional spaces, we first discretize with appropriate +treatment of the adjoint operator (c.f. [24]). +Remark 2. By keeping the model equation general, we stress the applicability of our +approach to a wide range of problems. For instance, time-dependent states can be +treated by choosing X as a Bochner space or its discretization (c.f. [38]). We also +do not formally restrict the dimension of X, though any implementation relies on the +ability to compute xθ(u) with sufficient accuracy. To this end, we note that the analysis +in Section 3.2 can be applied to determine how discretization errors affect the observ- +ability criterion in the sensor selection. +Following a probabilistic approach to inverse problems, we express the initial un- +certainty in utrue = utrue(θ) of any xtrue = xθ(utrue) in configuration θ through a random +variable u with Gaussian prior µpr = N +� +upr, Σpr +� +, where upr ∈ RM is the prior mean +and Σpr ∈ RM×M is a symmetric positive definite (s.p.d.) covariance matrix. The latter +defines the inner product ⟨·, ·⟩Σ−1 +pr and its induced norm ∥ · ∥Σ−1 +pr through +⟨u, v⟩Σ−1 +pr := uTΣ−1 +pr ˜u, +∥u∥2 +Σ−1 +pr := ⟨u, u⟩Σ−1 +pr , +∀ u, v ∈ RM. +(2) +With these definitions, the probability density function (pdf) for µpr is +πpr(u) = +1 +� +(2π)M det Σpr +exp +� +−1 +2∥u − upr∥2 +Σ−1 +pr +� +. +For simplicity, we assume {utrue(θ)}θ∈P to be independent realizations of u such that +we may consider the same prior for all θ without accounting for a possible history of +measurements at different configurations. +Sensor library and noise model +For taking measurements of the unknown states {xtrue(θ)}θ, we call any linear func- +tional ℓ ∈ X′ a sensor, and its application to a state x ∈ X its measurement ℓ(x) ∈ R. +We model experimental measurements dℓ ∈ R of the actual physical state xtrue as +dℓ = ℓ(xtrue) + εℓ where εℓ ∼ N(0, cov(εℓ, εℓ)) is a Gaussian random variable. We +permit noise in different sensor measurements to be correlated with a known covari- +ance function cov. In a slight overload of notation, we write cov : L × L → R, +cov(ℓi, ℓj) := cov(εℓi, εℓj) as a symmetric bilinear form over the sensor library. Any +ordered subset S = {ℓ1, . . . , ℓK} ⊂ L of sensors can then form a (linear and continuous) +observation operator through +L := [ℓ1, . . . , ℓK]T : X → RK, +Lx := [ℓ1(x), . . . , ℓK(x)]T . +4 + +The experimental measurements of L have the form +d = �ℓ1(xtrue) + εℓ1, . . . , ℓK(xtrue) + εℓK +�T = Lxtrue + ε +with +ε = �εℓ1, . . . , εℓK +�T ∼ N(0, σ2ΣL), +(3) +where σ2ΣL is the noise covariance matrix defined through +ΣL ∈ RK×K, +such that +� +σ2ΣL +� +i, j := cov(ℓj, ℓi) = cov(εℓj, εℓi) +(4) +with an auxiliary scaling parameter3 σ2 > 0. We assume that the library L and the noise +covariance function cov have been chosen such that ΣL is s.p.d. for any combination +of sensors in L. This assumption gives rise to the L-dependent inner product and its +induced norm +� +d, ˜d +� +Σ−1 +L := dTΣ−1 +L ˜d, +∥d∥2 +Σ−1 +L := ⟨d, d⟩Σ−1 +L , +∀ d, ˜d ∈ RK. +(5) +Measured with respect to this norm, the largest observation of any (normalized) state +is thus +γL := sup +∥x∥X=1 +∥Lx∥Σ−1 +L = sup +x∈X +∥Lx∥Σ−1 +L +∥x∥X +. +(6) +We show in Section 4.1 that γL increases under expansion of L with additional sensors +despite the change in norm, and is therefore bounded by γL ≤ γL. +We also define the parameter-to-observable map +GL,θ : RM → RK, +such that +GL,θ (u) := Lxθ(u). +(7) +With the assumptions above – in particular the linearity and uniform continuity (1) of +x in u – the map GL,θ is linear and uniformly bounded in u. We let GL,θ ∈ RK×M +denote its matrix representation with respect to the unit basis {em}M +m=1. The likelihood +of d ∈ RK obtained through the observation operator L for the parameter u ∈ RM and +the system configuration θ is then +ΦL +� +d +��� u, θ +� +:= +1 +� +2K det ΣL +exp +� +− 1 +2σ2 +���d − GL,θ (u) +���2 +Σ−1 +L +� +. +Note that GL,θ and GL,θ may depend non-linearly on θ. +Posterior distribution +Once noisy measurement data d ≈ Lxtrue(θ) is available, Bayes’ theorem yields the +posterior pdf as +πL,θ +post(u | d) = +1 +Z(θ) exp +� +− 1 +2σ2 +���GL,θ (u) − d +���2 +Σ−1 +L − 1 +2∥u − upr∥2 +Σ−1 +pr +� +∝ πpr(u)·ΦL +� +d +��� u, θ +� +, +(8) +3We introduce σ2 here as an additional variable to ease the discussion of scaling in Section 13. However, +we can set σ2 = 1 without loss of generality (w.l.o.g.). +5 + +with normalization constant +Z(θ) := +� +Rp exp +� +− 1 +2σ2 +���GL,θ (u) − d +���2 +Σ−1 +L +� +dµpr. +Due to the linearity of the parameter-to-observable map, the posterior measure µL,θ +post is +a Gaussian +µL,θ +post = N(uL,θ +post(d), ΣL,θ +post) +with known (c.f. [1]) mean and covariance matrix +uL,θ +post(d) = ΣL,θ +post +� 1 +σ2 GT +L,θΣ−1 +L d + Σ−1 +pr upr +� +∈ RM, +(9) +ΣL,θ +post = +� 1 +σ2 GT +L,θΣ−1 +L GL,θ + Σ−1 +pr +�−1 +∈ RM×M. +(10) +The posterior µL,θ +post thus depends not only on the choice of sensors, but also on the con- +figuration θ under which their measurements were obtained. Therefore, to decrease the +uncertainty in all possible posteriors with a single, θ-independent observation operator +L, the construction of L should account for all admissible configurations θ ∈ P under +which xtrue may be observed. +Remark 3. The linearity of xθ(u) in u is a strong assumption that dictates the Gaussian +posterior. However, in combination with the hyper-parameter θ, our setting here can +be re-interpreted as the Laplace-approximation for a non-linear state map θ �→ x(θ) +(c.f. [39, 21, 40]). The sensor selection presented here is then an intermediary step for +OED over non-linear forward models. +3. The Observability Coefficient +In this section, we characterize how the choice of sensors in the observation op- +erator L and its associated noise covariance matrix ΣL influence the uncertainty in the +posteriors µL,θ +post, θ ∈ P. We identify an observability coefficient that bounds the eigen- +values of the posterior covariance matrices ΣL,θ +post, θ ∈ P with respect to L, and facilitates +the sensor selection algorithm presented in Section 4. +3.1. Eigenvalues of the Posterior Covariance Matrix +The uncertainty in the posterior πL,θ +post for any configuration θ ∈ P is uniquely char- +acterized by the posterior covariance matrix ΣL,θ +post, which is in turn connected to the +observation operator L through the parameter-to-observable map GL,θ and the noise co- +variance matrix ΣL. To measure the uncertainty in ΣL,θ +post, the OED literature suggests a +variety of different utility functions to be minimized over L in order to optimize the sen- +sor choice. Many of these utility functions can be expressed in terms of the eigenvalues +6 + +λθ,1 +L ≥ · · · ≥ λθ,M +L +> 0 of ΣL,θ +post, e.g., +A-OED: +trace(ΣL,θ +post) = +M +� +m=1 +λθ,m +L +(mean variance) +D-OED: +det(ΣL,θ +post) = +M +� +m=1 +λθ,m +L +(volume) +E-OED: +λmax(ΣL,θ +post) = λθ,1 +L +(spectral radius). +In practice, the choice of the utility function is dictated by the application. In E-optimal +experimental design (E-OED), for instance, posteriors whose uncertainty ellipsoids +stretch out into any one direction are avoided, whereas D-OED minimizes the overall +volume of the uncertainty ellipsoid regardless of the uncertainty in any one parameter +direction. We refer to [3] for a detailed introduction and other OED criteria. +Considering the hyper-parameterized setting where each configuration θ influences +the posterior uncertainty, we seek to choose a single observation operator L such that +the selected utility function remains small for all configurations θ ∈ P, e.g., for E-OED, +minimizing +min +ℓ1,...,ℓK∈L max +θ∈P λmax(ΣL,θ +post) +such that +L = [ℓ1, . . . , ℓK]T +guarantees that the longest axis of each posterior covariance matrix ΣL,θ +post for any θ ∈ P +has the same guaranteed upper bound. The difficulty here is that the minimization over +P necessitates repeated, cost-intensive model evaluations to compute the utility func- +tion for many different configurations θ. In the following, we therefore introduce an +upper bound to the posterior eigenvalues that can be optimized through an observabil- +ity criterion with far fewer model solves. The bound’s optimization indirectly reduces +the different utility functions through the posterior eigenvalues. +Recalling that ΣL,θ +post is s.p.d., let {ψm}M +m=1 be an orthonormal eigenvector basis of +ΣL,θ +post, i.e. ψT +mψn = δm,n and +ΣL,θ +postψm = λθ,m +L ψm +m = 1, . . . , M. +(11) +Using the representation (10), any eigenvalue λθ,m +L +can be written in the form +1 +λθ,m +L += ψT +m +� +ΣL,θ +post +�−1 ψm = ψT +m +� 1 +σ2 GT +L,θΣ−1 +L GL,θ + Σ−1 +pr +� +ψm = 1 +σ2 +���GL,θ (ψm) +���2 +Σ−1 +L + ∥ψm∥2 +Σ−1 +pr . +(12) +Since ψm depends implicitly on L and θ through (11), we cannot use this representation +directly to optimize over L. To take out the dependency on ψm, we bound ∥ψm∥2 +Σ−1 +pr ≥ +1 +λmax +pr +in terms of the maximum eigenvalue of the prior covariance matrix Σpr. Likewise, we +define +βG(θ) := inf +u∈RM +���GL,θ (u) +���Σ−1 +L +∥u∥Σ−1 +pr += inf +u∈RM +∥Lxθ(u)∥Σ−1 +L +∥u∥Σ−1 +pr +, +(13) +7 + +as the minimum ratio between an observation for a parameter u relative to the prior’s +covariance norm. From (12) and (13) we obtain the upper bound +λθ,m +L += +����������� +1 +σ2 +���GL,θ (ψm) +���2 +Σ−1 +L +∥ψm∥2 +Σ−1 +pr ++ 1 +����������� +−1 +∥ψm∥−2 +Σ−1 +pr ≤ +� 1 +σ2 βG(θ)2 + 1 +�−1 +λmax +pr . +Geometrically, this bound means that the radius λθ,1 +L of the outer ball around the pos- +terior uncertainty ellipsoid is smaller than that of the prior uncertainty ellipsoid by at +least the factor +� 1 +σ2 βG(θ)2 + 1 +�−1. By choosing L to maximize minθ βG(θ), we therefore +minimize this outer ball containing all uncertainty ellipsoids (i.e., for any θ ∈ P). As +expected, the influence of L is strongest when the measurement noise is small such that +data can be trusted (σ2 ≪ 1), and diminishes with increasing noise levels (σ2 ≫ 1). +3.2. Parameter Restriction +An essential property of βG(θ) is that βG(θ) = 0 if K < M, i.e., the number of sen- +sors in L is smaller than the number of parameter dimensions. In this case, βG(θ) cannot +distinguish between sensors during the first M − 1 steps of an iterative algorithm, or in +general when less than a total of M sensors are supposed to be chosen. For medium- +dimensional parameter spaces (M ∈ O(10)), we mitigate this issue by restricting u to +the subspace span{ϕ1, . . . , ϕmin{K,M}} ⊂ RM spanned by the first min{K, M} eigenvec- +tors of Σpr corresponding to its largest eigenvalues, i.e., the subspace with the largest +prior uncertainty. For high-dimensional parameter spaces or when the model Mθ has a +non-trivial null-space, we bound βG(θ) further +βG(θ) = inf +u∈RM +∥Lxθ(u)∥Σ−1 +L +∥xθ(u)∥X +∥xθ(u)∥X +∥u∥Σ−1 +pr +≥ inf +x∈Wθ +∥Lx∥Σ−1 +L +∥x∥X +inf +u∈RM +∥xθ(u)∥X +∥u∥Σ−1 +pr += βL|W(θ) η(θ) +(14) +where we define the linear space Wθ of all achievable states +Wθ := {xθ(u) ∈ X : u ∈ RM} +and the coefficients +βL|W(θ) := inf +x∈Wθ +∥Lx∥Σ−1 +L +∥x∥X +, +η(θ) := inf +u∈RM +∥xθ(u)∥X +∥u∥Σ−1 +pr +. +(15) +The value of η(θ) describes the minimal state change that a parameter u can achieve +relative to its prior-induced norm ∥u∥Σ−1 +pr . It can filter out parameter directions that have +little influence on the states xθ(u). In contrast, the observability coefficient βL|W(θ) +depends on the prior only implicitly via Wθ; it quantifies the minimum amount of +information (measured with respect to the noise model) that can be obtained on any +state in Wθ relative to its norm. Future work will investigate how to optimally restrict +the parameter space based on η(θ) before choosing sensors that maximize βL|W(θ). +Existing parameter reduction approaches in a similar context include [28, 41, 42, 27]. +In this work, however, we solely focus on the maximization of βG(θ) and, by extension, +βL|W(θ) and henceforth assume that M is sufficiently small and η := infθ∈P η(θ) > 0 is +bounded away from zero. +8 + +3.3. Observability under model approximations +To optimize the observability coefficient βG(θ) or βL|W(θ), it must be computed for +many different configurations θ ∈ P. The accumulating computational cost motivates +the use of reduced-order surrogate models, which typically yield considerable com- +putational savings versus the original full-order model. However, this leads to errors +in the state approximation. In the following, we thus quantify the influence of state +approximation error on the observability coefficients βG(θ) and βL|W(θ). An analysis +of the change in posterior distributions when the entire model Mθ is substituted in the +inverse problem can be found in [1]. +Suppose a reduced-order surrogate model ˜ +Mθ(˜xθ(u); u) = 0 is available that yields +for any configuration θ ∈ P and parameter u ∈ RM a unique solution ˜xθ(u) ∈ X such +that +∥xθ(u) − ˜xθ(u)∥X ≤ εθ ∥xθ(u)∥X +with accuracy +0 ≤ εθ ≤ ε < 1. +(16) +Analogously to (13) and (15), we define the reduced-order observability coefficients +˜βG(θ) := inf +u∈RM +∥L˜xθ(u)∥Σ−1 +L +∥u∥Σ−1 +pr +, +˜βL|W(θ) := inf +u∈RM +∥L˜xθ(u)∥Σ−1 +L +∥˜xθ(u)∥X +(17) +to quantify the smallest observations of the surrogate states. For many applications, +it is possible to choose a reduced-order model whose solution can be computed at a +significantly reduced cost such that ˜βG(θ) and ˜βL|W(θ) are much cheaper to compute +than their full-order counterparts βG(θ) and βL|W(θ). Since the construction of such a +surrogate model depends strongly on the application itself, we refer to the literature +(e.g., [43, 44, 45, 46, 47]) for tangible approaches. +Recalling the definition of γL in (6), we start by bounding how closely the surrogate +observability coefficient ˜βL|W(θ) approximates the full-order βL|W(θ). +Proposition 1. Let η(θ) > 0 hold, and let ˜xθ(u) ∈ X be an approximation to xθ(u) such +that (16) holds for all θ ∈ P, u ∈ RM. Then +(1 − εθ) ˜βL|W(θ) − γLεθ ≤ βL|W(θ) ≤ (1 + εθ) ˜βL|W(θ) + γLεθ. +(18) +Proof. Let u ∈ RM \ {0} be arbitrary. Using (16) and the (reversed) triangle inequality, +we obtain the bound +∥˜xθ(u)∥X +∥xθ(u)∥X +≥ ∥xθ(u)∥X − ∥xθ(u) − ˜xθ(u)∥X +∥xθ(u)∥X +≥ 1 − εθ. +(19) +Note here that η(θ) > 0 implies ∥xθ(u)∥X > 0 so the quotient is indeed well defined. +The ratio of observation to state can now be bounded from below by +∥Lxθ(u)∥Σ−1 +L +∥xθ(u)∥X +≥ +∥L˜xθ(u)∥Σ−1 +L +∥xθ(u)∥X +− +∥L(xθ(u) − ˜xθ(u))∥Σ−1 +L +∥xθ(u)∥X +≥ ∥˜xθ(u)∥X +∥xθ(u)∥X +∥L˜xθ(u)∥Σ−1 +L +∥˜xθ(u)∥X +− γL +∥xθ(u) − ˜xθ(u)∥X +∥xθ(u)∥X +≥ (1 − εθ) +∥L˜xθ(u)∥Σ−1 +L +∥˜xθ(u)∥X +− γLεθ +≥ (1 − εθ)˜βL|W(θ) − γLεθ, +9 + +where we have applied the reverse triangle inequality, definition (6), the bounds (16), +(19), and definition (17) of ˜βL|W(θ). Since u is arbitrary, the lower bound in (18) follows +from definition (13) of βL|W(θ). The upper bound in (18) follows analogously. +For the observability of the parameter-to-observable map GL,θ and its approxima- +tion u �→ L˜xθ(u), we obtain a similar bound. It uses the norm η(θ) of xθ : u �→ xθ(u) as +a map from the parameter to the state space, see (1). +Proposition 2. Let ˜xθ(u) ∈ X be an approximation to xθ(u) such that (16) holds for all +θ ∈ P, u ∈ RM. Then +˜βG(θ) − γLη(θ)εθ ≤ βG(θ) ≤ ˜βG(θ) + γLη(θ)εθ. +(20) +Proof. Let u ∈ RM \ {0} be arbitrary. Then +∥Lxθ(u)∥Σ−1 +L ≥ ∥L˜xθ(u)∥Σ−1 +L − ∥L(xθ(u) − ˜xθ(u))∥Σ−1 +L +≥ ∥L˜xθ(u)∥Σ−1 +L − γL ∥xθ(u) − ˜xθ(u)∥X +≥ ∥L˜xθ(u)∥Σ−1 +L − γLεθ ∥xθ(u)∥X +≥ ∥L˜xθ(u)∥Σ−1 +L − γLεθη(θ)∥u∥Σ−1 +pr , +where we have used the reverse triangle inequality, followed by (6), (16), and (1). We +divide by ∥u∥Σ−1 +pr and take the infimum over u to obtain +βG(θ) = inf +u∈RM +∥Lxθ(u)∥Σ−1 +L +∥u∥Σ−1 +pr +≥ inf +u∈RM +∥L˜xθ(u)∥Σ−1 +L +∥u∥Σ−1 +pr +− γL η(θ) εθ = ˜βG(θ) − γL η(θ) εθ. +The upper bound in (20) follows analogously. +If εθ is sufficiently small, Propositions 1 and 2 justify employing the surrogates +˜βL|W(θ) and ˜βG(θ) instead of the original full-order observability coefficients βL|W(θ) +and βG(θ). This substitution becomes especially necessary when the computation of +xθ(u) is too expensive to evaluate βL|W(θ) or βG(θ) repeatedly for different configura- +tions θ. +Another approximation step in our sensor selection algorithm relies on the identifi- +cation of a parameter direction v ∈ RM with comparatively small observability, i.e. +∥Lxθ(v)∥Σ−1 +L +∥v∥Σ−1 +pr +≈ inf +u∈RM +∥Lxθ(u)∥Σ−1 +L +∥u∥Σ−1 +pr += βG(θ) +or +∥Lxθ(v)∥Σ−1 +L +∥xθ(v)∥X +≈ inf +x∈Wθ +∥Lx∥Σ−1 +L +∥x∥X += βL|W(θ). +The ideal choice would be the infimizer of respectively βG(θ) or βL|W(θ), but its compu- +tation involves M full-order model evaluations (c.f. Section 4.2). To avoid these costly +computations, we instead choose v as the infimizer of the respective reduced-order +observability coefficient. This choice is justified for small εθ < 1 by the following +proposition: +10 + +Proposition 3. Let η(θ) > 0 hold, and let ˜xθ(u) ∈ X be an approximation to xθ(u) such +that (16) holds for all θ ∈ P, u ∈ RM. Suppose v ∈ arg infu∈RM ∥u∥−1 +Σ−1 +pr ∥L˜xθ(u)∥Σ−1 +L , then +βG(θ) ≤ +∥Lxθ(v)∥Σ−1 +L +∥v∥Σ−1 +pr +≤ βG(θ) + 2γLη(θ)εθ. +(21) +Likewise, if v ∈ arg infu∈RM ∥˜xθ(u)∥−1 +X ∥L˜xθ(u)∥Σ−1 +L , then +βL|W(θ) ≤ +∥Lxθ(v)∥Σ−1 +L +∥xθ(v)∥X +≤ 1 + εθ +1 − εθ +�βL|W(θ) + γLεθ +� + γLεθ. +(22) +Proof. For both (21) and (22) the lower bound follows directly from definitions (13) +and (15). To prove the upper bound in (21), let v ∈ arg infu∈RM ∥u∥−1 +Σ−1 +pr ∥L˜xθ(u)∥Σ−1 +L . +Following the same steps as in the proof of Proposition 2, we can then bound +∥Lxθ(v)∥Σ−1 +L +∥v∥Σ−1 +pr +≤ +∥L˜xθ(v)∥Σ−1 +L +∥v∥Σ−1 +pr ++ +∥L(xθ(v) − ˜xθ(v))∥Σ−1 +L +∥v∥Σ−1 +pr +≤ ˜βG(θ) + γLη(θ)εθ. +The upper bound in (21) then follows with Proposition 2. +To prove the upper bound in (22), let v ∈ arg infu∈RM ∥˜xθ(u)∥−1 +X ∥L˜xθ(u)∥Σ−1 +L . Then +∥Lxθ(v)∥Σ−1 +L +∥xθ(v)∥X +≤ +∥L˜xθ(v)∥Σ−1 +L +∥˜xθ(v)∥X +∥˜xθ(v)∥X +∥xθ(v)∥X ++ +∥L(xθ(v) − ˜xθ(v))∥Σ−1 +L +∥xθ(v)∥X +≤ (1 + ε) ˜βL|W(θ) + γLεθ. +The result then follows with Proposition 1. +4. Sensor selection +In the following, we present a sensor selection algorithm that iteratively increases +the minimal observability coefficient minθ∈P βG(θ) and thereby decreases the upper +bound for the eigenvalues of the posterior covariance matrix for all admissible system +configurations θ ∈ P. The iterative approach is relatively easy to implement, allows a +simple way of dealing with combinatorial restrictions, and can deal with large4 sensor +libraries. +4.1. Cholesky decomposition +The covariance function cov connects an observation operator L to its observabil- +ity coefficients βG(θ) and βL|W(θ) through the noise covariance matrix ΣL. Its inverse +enters the norm ∥·∥Σ−1 +L and the posterior covariance matrix ΣL,θ +post. The inversion poses +a challenge when the noise is correlated, i.e., when ΣL is not diagonal, as even the +expansion of L with a single sensor ℓ ∈ L changes each entry of Σ−1 +L . In naive compu- +tations of the observability coefficients and the posterior covariance matrix, this leads +4For instance, in Section 5.3 we apply the presented algorithm to a library with KL = 11, 045 available +sensor positions. +11 + +Algorithm 1: CholeskyExpansion +Input: observation operator L = [ℓ1, . . . , ℓK]T, noise covariance matrix ΣL, +Cholesky matrix CL, new sensor ℓ ∈ X′ +L ← [ℓ1, . . . , ℓK, ℓ]T +// operator expansion +if K = 0 then +ΣL ← (cov(ℓ, ℓ)) , CL ← +� √cov(ℓ, ℓ) +� +∈ R1×1 +// first sensor +else +v ← [cov(ℓ1, ℓ), . . . , cov(ℓK, ℓ)]T ∈ RK +// matrix expansion +w ← C−1 +L v ∈ RK, s ← cov(ℓ, ℓ), c ← s − wTw ∈ R +ΣL ← +� ΣL +v +vT +s +� +, CL ← +� CL +0 +wT +c +� +∈ R(K+1)×(K+1) +return L, ΣL, CL +to M dense linear system solves of order O((K + 1)3) each time the observation oper- +ator is expanded. In the following, we therefore expound on how Σ−1 +L changes under +expansion of L to exploit its structure when comparing potential sensor choices. +Suppose L = [ℓ1, . . . , ℓK]T has already been chosen with sensors ℓk ∈ X′, but shall +be expanded by another sensor ℓ to +[L, ℓ] := [ℓ1, . . . , ℓK, ℓ]T : X → RK+1. +Following definition (4), the noise covariance matrix Σ[L,ℓ] of the expanded operator +[L, ℓ] has the form +Σ[L,ℓ] = +� ΣL +vL,ℓ +vT +L,ℓ +vℓ,ℓ +� += +� CL +0 +cT +L,ℓ +cℓ,ℓ +� � CT +L +cL,ℓ +0 +cℓ,ℓ +� +, +where CLCT +L = ΣL ∈ RK×K is the Cholesky decomposition of the s.p.d. noise covariance +matrix ΣL for the original observation operator L, and vL,ℓ, cL,ℓ ∈ RK, vℓ,ℓ, cℓ,ℓ ∈ R are +defined through +�vL,ℓ +� +i := cov(ℓi, ℓ), +cL,ℓ := C−1 +L vL,ℓ, +vℓ,ℓ := cov(ℓ, ℓ), +cℓ,ℓ := +� +vℓ,ℓ − cT +L,ℓcL,ℓ. +Note that Σ[L,ℓ] is s.p.d. by the assumptions posed on cov in Section 2; consequently, +cℓ,ℓ is well-defined and strictly positive. With this factorization, the expanded Cholesky +matrix C[L,ℓ] with C[L,ℓ]CT +[L,ℓ] = Σ[L,ℓ] can be computed in O(K2), dominated by the +linear system solve with the triangular CL for obtaining cL,ℓ. It is summarized in Algo- +rithm 1 for later use in the sensor selection algorithm. +Using the Cholesky decomposition, the inverse of Σ[L,ℓ] factorizes to +Σ−1 +[L,ℓ] = +� CT +L +cL,ℓ +0 +cℓ,ℓ +�−1 � CL +0 +cT +L,ℓ +cℓ,ℓ +�−1 += +� C−T +L +rL,ℓ +0 +1/cℓ,ℓ +� � C−1 +L +0 +rT +L,ℓ +1/cℓ,ℓ +� +, +12 + +Algorithm 2: ObservabilityGain +Input: observation operator L = [ℓ1, . . . , ℓK]T, Cholesky matrix CL, sensor +candidate ℓ ∈ X′, state x ∈ X +d ← Lx, z ← C−1 +L d +// preparation +if K = 0 then +return ℓ(xK)2/cov(ℓ, ℓ) +// one sensor only +else +v ← [cov(ℓ1, ℓ), . . . , cov(ℓK, ℓ)]T ∈ RK +// general case +w ← C−1 +L v ∈ RK +return (ℓ(xK)−wT z) +2 +cov(ℓ,ℓ)−wT w +where +rL,ℓ := − 1 +cℓ,ℓ +C−T +L cL,ℓ = − 1 +cℓ,ℓ +C−T +L C−1 +L vL,ℓ = − 1 +cℓ,ℓ +Σ−1 +L vL,ℓ. +For an arbitrary state x ∈ X, the norm of the extended observation [L, ℓ](x) = +� +LxT, ℓ(x) +�T ∈ +RK+1 in the corresponding norm ∥·∥Σ−1 +[L,ℓ] is hence connected to the original observation +Lx ∈ RK in the original norm ∥·∥Σ−1 +L via +∥ [L, ℓ](x) ∥2 +Σ−1 +[L,ℓ] = +� Lx +ℓ(x) +�T � ΣL +vL,ℓ +vT +L,ℓ +vℓ,ℓ +�−1 � Lx +ℓ(x) +� += +� Lx +ℓ(x) +�T � C−T +L +rL,ℓ +0 +1/cℓ,ℓ +� � C−1 +L +0 +rT +L,ℓ +1/cℓ,ℓ +� � Lx +ℓ(x) +� += +� +C−1 +L Lx +rT +L,ℓLx + ℓ(x)/cℓ,ℓ +�T � +C−1 +L Lx +rT +L,ℓLx + ℓ(x)/cℓ,ℓ +� += (Lx)TC−T +L C−1 +L Lx + (rT +L,ℓLx + ℓ(x)/cℓ,ℓ)2 += ∥Lx∥2 +Σ−1 +L + (rT +L,ℓLx + ℓK+1(x)/cℓ,ℓ)2 +≥ ∥Lx∥2 +Σ−1 +L . +(23) +We conclude from this result that the norm ∥Lx∥Σ−1 +L of any observation, and therefore +also the continuity coefficient γL defined in (6), is increasing under expansion of L +despite the change in norms. For any configuration θ, the observability coefficients +βG(θ) and βL|W(θ) are thus non-decreasing when sensors are selected iteratively. +Given a state x ∈ X and an observation operator L, we can determine the sen- +sor ℓK+1 ∈ L that increases the observation of x the most by comparing the increase +(rT +L,ℓLx + ℓ(x)/cℓ,ℓ)2 for all ℓ ∈ L. Algorithm 2 summarizes the computation of this +observability gain for use in the sensor selection algorithm (see Section 4.3). Its gen- +eral runtime is determined by K + 1 sensor evaluations and two linear solves with the +triangular Cholesky matrix CL in O(K2). When called with the same L and the same +13 + +state x for different candidate sensors ℓ, the preparation step must only be performed +once, which reduces the runtime to one sensor evaluation and one linear system solve +in all subsequent calls. Compared to computing ∥ [L, ℓ](x) ∥2 +Σ−1 +[L,ℓ] for all KL candidate +sensors in the library L, we save O(KLK2). +4.2. Computation of the observability coefficient +We next discuss the computation of the observability coefficient βG(θ) for a given +configuration θ and observation operator L. +Let Σpr = UTDprU be the eigenvalue decomposition of the s.p.d. prior covariance +matrix with U = �ϕ1, . . . , ϕM +� ∈ RM×M, ϕ j ∈ RM orthonormal in the Euclidean inner +product, and Dpr = diag(λ1 +pr, . . . , λM +pr) a diagonal matrix containing the eigenvalues +λ1 +pr ≥ · · · ≥ λM +pr > 0 in decreasing order. Using the eigenvector basis {ϕm}M +m=1, we define +the matrix +M(θ) := �Lxθ(ϕ1), . . . , Lxθ(ϕM)� ∈ RK×M +(24) +featuring all observations of the associated states xθ(ϕ j) for the configuration θ. The +observability coefficient βG(θ) can then be computed as the square root of the minimum +eigenvalue λmin of the generalized eigenvalue problem +M(θ)TC−T +L C−1 +L M(θ)umin = λminD−1 +pr umin. +(25) +Note that (25) has M real, non-negative eigenvalues because the matrix on the left is +symmetric positive semi-definite, and Dpr is s.p.d. (c.f. [48]). The eigenvector umin +contains the basis coefficients in the eigenvector basis {ϕm}M +m=1 of the “worst-case” pa- +rameter, i.e. the infimizer of βG(θ). +Remark 4. For computing βL|W(θ), we exchange the right-hand side matrix D−1 +pr in +(25) with the X-inner-product matrix for the states xθ(ϕ1), . . . , xθ(ϕM). +The solution of the eigenvalue problem can be computed in O(M3), with an addi- +tional O(MK2 + M2K) for the computation of the left-hand side matrix in (25). The +dominating cost is hidden in M(θ) since it requires KM sensor observations and K full- +order model solves. To reduce the computational cost, we therefore approximate βG(θ) +with ˜βG(θ) by exchanging the full-order states xθ(ϕ j) in (24) with their reduced-order +approximations ˜xθ(ϕ j). The procedure is summarized in Algorithm 3. +Remark 5. If K < M, Algorithm 3 restricts the parameter space, as discussed in Sec- +tion 3.2, to the span of the first K eigenvectors ϕ1, . . . , ϕK encoding the least certain +directions in the prior. A variation briefly discussed in [8] in the context of the PBDW +method to prioritize the least certain parameters even further is to only expand the +parameter space once the observability coefficient on the subspace surpasses a prede- +termined threshold. +4.3. Sensor selection +In our sensor selection algorithm, we iteratively expand the observation operator +L and thereby increase the observability coefficient L for all θ ∈ P. Although this +14 + +Algorithm 3: SurrogateObservability +Input: configuration θ ∈ P, observation operator L = [ℓ1, . . . , ℓK]T with +K > 0, Cholesky matrix CL +N ← min{M, K} +// parameter restriction +M ← �L˜xθ(ϕ1), . . . , L˜xθ(ϕN)�, S ← +�� +xθ(ϕi), xθ(ϕ j) +� +X +�N +i, j=1 // matrix setup +Find (λmin, umin) of +� +C−1 +L M +�T � +C−1 +L M +� +umin = λminSumin +// eigenvalue +problem +return +√ +λmin, umin +procedure cannot guarantee finding the maximum observability over all sensor com- +binations, the underlying greedy searches are well-established in practice, and can be +shown to perform with exponentially decreasing error rates in closely related settings, +see [49, 8, 50, 51, 52]. In each iteration, the algorithm performs two main steps: +• A greedy search over a training set Ξtrain ⊂ P to identify the configuration +θ ∈ Ξtrain for which the observability coefficient βG(θ) is minimal; +• A data-matching step to identify the sensor in the library that maximizes the +observation of the “worst-case” parameter at the selected configuration θ. +The procedure is summarized in Algorithm 4. It terminates when Kmax ≤ KL sensors +have been selected.5 In the following, we explain its computational details. +Preparations +In order to increase βG(θ) uniformly over the hyper-parameter domain P, we con- +sider a finite training set, Ξtrain ⊂ P, that is chosen to be fine enough to capture the θ- +dependent variations in xθ(u). We assume a reduced-order model is available such that +we can compute approximations ˜xθ(ϕm) ≈ xθ(ϕm) for each θ ∈ Ξtrain and 1 ≤ m ≤ M +within an acceptable computation time while guaranteeing the accuracy requirement +(16). If necessary, the two criteria can be balanced via adaptive training domains (e.g., +[53, 54]). +Remark 6. If storage allows (e.g., with projection-based surrogate models), we only +compute the surrogate states once and avoid unnecessary re-computations when up- +dating the surrogate observability coefficients ˜βG(θ) in each iteration. +As a first “worst-case” parameter direction, u0, we choose the vector ϕ1 with the +largest prior uncertainty. Likewise, we choose the “worst-case” configuration θK ∈ P +as the one for which the corresponding state ˜xθ(ϕ1) is the largest. +5This termination criterion can easily be adapted to prescribe a minimum value of the observability +coefficient. This value should be chosen with respect to the observability βG(L) achieved with the entire +sensor library. +15 + +Algorithm 4: SensorSelection +Input: sensor library L ⊂ X′, training set Ξtrain ⊂ P, maximum number of +sensors Kmax ≤ |KL|, surrogate model ˜ +Mθ, covariance function +cov : L × L → R +Compute Σpr = �ϕ1, . . . , ϕM +�T Dpr +�ϕ1, . . . , ϕM +� +// eigenvalue +decomposition +For all θ ∈ Ξtrain, 1 ≤ m ≤ M, compute ˜xθ(ϕm) +// preparation +K ← 0, θ0 ← arg maxθ∈Ξtrain ∥˜xθ(ϕ1)∥X, u0 ← ϕ1 +// initialization +while K < Kmax do +Solve full-order equation MθK(xK, uK) for xK +// "worst-case" state +ℓK+1 ← arg maxℓ∈L ObservabilityGain(L, CL, ℓ) +// sensor +selection +L, ΣL, CL ← CholeskyExpansion(L, ΣL, CL, ℓK+1) +// expansion +K ← K + 1 +for θ ∈ Ξtrain do +˜βL|W(θ), umin(θ) ← SurrogateObservability(θ, L, CL) // update +coefficients +θK ← arg minθ∈Ξtrain ˜βL|W(θ) +// greedy step +uK ← �min{M,K} +m=1 +[umin(θK)]m ϕm +return L, CL +Data-matching step +In each iteration, we first compute the full-order state xK = xθK(uK) at the “worst- +case” parameter uK and configuration θK. We then choose the sensor ℓK+1 which most +improves the observation of the “worst-case” state xK under the expanded observation +operator [LT, ℓK+1]T and its associated norm. We thereby iteratively approximate the +information that would be obtained by measuring with all sensors in the library L. For +fixed θK and in combination with selecting x to have the smallest observability in Wθ, +we arrive at an algorithm similar to worst-case orthogonal matching pursuit (c.f. [8, 9]) +but generalized to deal with the covariance function cov in the noise model (3). +Remark 7. We use the full-order state xθK(uK) rather than its reduced-order approxi- +mation in order to avoid training on local approximation inaccuracies in the reduced- +order model. Here, by using the “worst-case” parameter direction uK, we only require +a single full-order solve per iteration instead of the M required for setting up the entire +posterior covariance matrix ΣL,θ +post. +Greedy step +We train the observation operator L on all configurations θ ∈ Ξtrain by varying for +which θ the “worst-case” state is computed. Specifically, we follow a greedy approach +16 + +where, in iteration K, we choose the minimizer θK of βG(θ) over the training domain +Ξtrain, i.e., the configuration for which the current observation operator L is the least +advantageous. The corresponding “worst-case” parameter uK is the parameter direction +for which the least significant observation is achieved. By iteratively increasing the +observability at the “worst-case” parameters and hyper-parameters, we increase the +minimum of βG(θ) throughout the training domain. +Remark 8. Since the computation of ˜βG(θ) requires as many reduced-order model +solves as needed for the posterior covariance matrix over the surrogate model, it is +possible to directly target an (approximated) OED utility function in the greedy step +in place of ˜βL|W(θ) without major concessions in the computational efficiency. The +OMP step can then still be performed for the “worst-case” parameter with only one +full-order model solve, though its benefit for the utility function should be evaluated +carefully. +Runtime +Assuming the dominating computational restriction is the model evaluation to solve +for xθ(u) – as is usually the case for PDE models – then the runtime of each iteration in +Algorithm 4 is determined by one full-order model evaluation, and KL sensor measure- +ments of the full-order state. Compared to computed the posterior covariance matrix +for the chosen configuration, the OMP step saves N − 1 full-order model solves. +The other main factor in the runtime of Algorithm 4 is the |Ξtrain|M reduced-order +model evaluations with KL sensor evaluations each that need to be performed in each +iteration (unless they can be pre-computed). The parameter dimension M not only en- +ters as a scaling factor, but also affects the cost of the reduced-order model itself since +larger values of M generally require larger or more complicated reduced-order models +to achieve the desired accuracy (16). In turn, the computational cost of the reduced- +order model indicates how large Ξtrain may be chosen for a given computational budget. +While some cost can be saved through adaptive training sets and models, overall, this +connection to M stresses the need for an adequate initial parameter reduction as dis- +cussed in Section 3.2. +5. Numerical Results +We numerically confirm the validity of our sensor selection approach using a geo- +physical model of a section of the Perth Basin in Western Australia. The basin has +raised interest in the geophysics community due to its high potential for geothermal +energy, e.g., [55, 56, 57, 58, 59]. We focus on a subsection that spans an area of +63 km × 70 km and reaches 19 km below the surface. The model was introduced in +[60] and the presented section of the model was discussed extensively in the context +of MOR in [61, 62]. In particular, the subsurface temperature distribution is described +through a steady-state heat conduction problem with different subdomains for the geo- +logical layers, and local measurements may be obtained through boreholes. The bore- +hole locations need to be chosen carefully due to their high costs (typically several +million dollars, [63]), which in turn motivates our application of Algorithm 4. For +demonstration purposes, we make the following simplifications to our test model: 1) +17 + +Figure 1: Schematic overview of the Perth Basin section including (merged) geological layers, depths for +potential measurements, and configuration range for thermal conductivity θ on each subdomain. The bounds +are obtained from the reference values (c.f. [60, 61]) with a ±50% margin. Adapted from [61]. +We neglect radiogenic heat production; 2) we merge geological layers with similar +conductive behaviors; and 3) we scale the prior to emphasize the influence of different +sensor measurements on the posterior. All computations were performed in Python +3.7 on a computer with a 2.3 GHz Quad-Core Intel Core i5 processor and 16 GB of +RAM. The code will be available in a public GitHub repository for another geophysical +test problem.6 +5.1. Model Description +We model the temperature distribution xθ with the steady-state PDE +−∇ (θ∇xθ) = 0 +in Ω := (0, 0.2714) × (0, 0.9) × (0, 1) ⊂ R3, +(26) +where the domain Ω is a non-dimensionalized representation of the basin, and θ : Ω → +R>0 the local thermal conductivity. The section comprises three main geological layers +Ω = � +i=1,2,3 Ωi, each characterized by different rock properties, i.e. thermal conductiv- +ity θ|Ωi ≡ θi shown in Figure 1. We consider the position of the geological layers to be +fixed as these are often determined beforehand by geological and geophysical surveys +but allow the thermal conductivity to vary. In a slight abuse of notation, this lets us +identify the field θ with the vector +θ = (θ1, θ2, θ3) ∈ P := [0.453, 1.360] × [0.448, 1.343] × [0.360, 1.081]. +in the hyper-parameter domain P. +We impose zero-Dirichlet boundary conditions at the surface7, and zero-Neumann +(“no-flow”) boundary conditions at the lateral faces of the domain. The remaining +boundary ΓIn corresponds to an area spanning 63 km × 70 km area in the Perth basin +19 km below the surface. At this depth, local variations in the heat flux have mostly +stabilized which makes modeling possible, but since most boreholes – often originat- +ing from hydrocarbon exploration – are found in the uppermost 2 km we treat it as +6The Perth Basin Model is available upon request from the third author. +7Non-zero Dirichlet boundary conditions obtained from satellite data could be considered via a lifting +function and an affine transformation of the measurement data (see [62]). +18 + +Geology +Thermal Conductivity +Creteaous- +2 +with 1 E [0.453,1.360], [0refl1 = 0.9065 +-380 m +Yarragadee +Eneabba- +-760 m +0.2 +with 02 E [0.448,1.343], [0rerl2 = 0.9855 +Lesueur +D.1 +-1140 m +Permian +. +with 3 E [0.360,1.081], [0refl3 = 0.7205 +0.2 +Basement +-1520 m +0.4 +0.6 +-1900 m += 0.2443 +0.8 +X1uncertain. Specifically, we model it as a Neumann boundary condition +n · ∇xθ = u · p +a.e. on ΓIn := {0} × [0, 0.9] × [0, 1] +where n : ΓIn → R3 is the outward pointing unit normal on Ω, p : ΓIn → R5 is a vector +composed of quadratic, L2(ΓIn)-orthonormal polynomials on the basal boundary that +vary either in north-south or east-west direction, and u ∼ πpr = N(upr, Σpr) is a random +variable. The prior is chosen such that the largest uncertainty is attributed to a constant +entry in p, and the quadratic terms are treated as the most certain with prior zero. This +setup reflects typical geophysical boundary conditions, where it is most common to +assume a constant Neumann heat flux (e.g., [61]), and sometimes a linear one (e.g., +[60]). With the quadratic functions, we allow an additional degree of freedom than +typically considered. +The problem is discretized using a linear finite element (FE) basis of dimension +132,651. The underlying mesh was created with GemPy ([64]) and MOOSE ([65]). +Since the FE matrices decouple in θ, we precompute and store an affine decomposition +using DwarfElephant ([61]). Given a configuration θ and a coefficient vector u for the +heat flux at ΓIn, the computation of a full-order solution xθ(u) ∈ X then takes 2.96 s on +average. We then exploit the affine decomposition further to construct a reduced basis +(RB) surrogate model via a greedy algorithm (c.f. [49, 66]). Using the inner product8 +⟨x, φ⟩X := +� +Ω ∇x · ∇φdΩ and an a posteriori error bound ∆(θ), we prescribe the relative +target accuracy +max +u∈RM +∥xθ(u) − ˜xθ(u)∥X +∥˜xθ(u)∥X +≤ max +u∈RM +∆(θ) +∥˜xθ(u)∥X +< ε := 1e − 4 +(27) +to be reached for 511,000 consecutively drawn, uniformly distributed samples of θ. +The training phase and final computational performance of the RB surrogate model are +summarized in Figure 2. The speedup of the surrogate model (approximately a factor +of 3,000 without error bounds) justifies its offline training time, with computational +savings expected already after 152 approximations of βG(θ). +For taking measurements, we consider a 47 × 47 grid over the surface to represent +possible drilling sites. At each, a single point evaluation9 of the basin’s temperature +distribution may be made at any one of five possible depths as shown in Figure 1. In +total, we obtain a set L ⊂ Ω of 11, 045 admissible points for measurements. We model +the noise covariance between sensors ℓχ, ℓ˜χ ∈ L at points χ, ˜χ ∈ Ω via +cov(ℓχ, ℓ˜χ) := a + b − y(h) +with the exponential variogram model +y(h) := a + (b − a) +�3 +2 max{h +c, 1} − 1 +2 max{h +c, 1}3 +� +8Note that ⟨·, ·⟩X is indeed an inner product due to the Dirichlet boundary conditions. +9Point evaluations are standard for geophysical models because a borehole (diameter approximately 1 m) +is very small compared to the size of the model. +19 + +0 +10 +20 +30 +40 +50 +RB dimension +10 +3 +10 +2 +10 +1 +100 +norm of the error +max relative error bound +true relative error +Reduced-order model +RB dimension +83 +training time +37.58 min +training accuracy +1e-4 +RB solve +0.97 ms +�→ speedup +3,058 +RB error bound +4.78 ms +�→ speedup +515 +Figure 2: Training of the RB surrogate model for the Perth Basin section. On the left: Maximum relative +error bound (27) in the course of the greedy algorithm, computed over the training set Ξtrain together with the +true relative error at the corresponding configuration θ. On the right: Performance pointers for the obtained +RB model after (27) was reached; online computation times and speedups are averages computed over 1000 +randomly drawn configurations θ. +where h2 := (χ2 − ˜χ2)2 + (χ3 − ˜χ3)2 is the horizontal distance between the points and +a := 2.2054073480730403 +(sill) +b := 1.6850672040263555 +(nugget) +c := 20.606782733391228 +(range) +The covariance function was computed via kriging (c.f. [67]) from the existing mea- +surements [68]. With this covariance function, the noise between measurements at any +two sensor locations is increasingly correlated the closer they are on the horizontal +plane. Note that for any subset of sensor locations, the associated noise covariance ma- +trix remains regular as long as each sensor is placed at a distinct drilling location. We +choose this experimental setup because measurements in typical geothermal data sets +are often made at the bottom of a borehole (“bottom hole temperature measurements”) +within the first 2 km below the surface. +5.2. Restricted Library +To test the feasibility of the observability coefficient for sensor selection, we first +consider a small sensor library (denoted as L5×5 below) with 25 drilling locations po- +sitioned on a 5 × 5 grid. We consider the problem of choosing 8 pair-wise different, +unordered sensor locations out of the given 25 positions; this is a combinatorial prob- +lem with 1,081,575 possible combinations. +Sensor selection +We run Algorithm 4, using the RB surrogate model and a training set Ξtrain ⊂ P +with 512,000 configurations on an 80 × 80 × 80 regular grid on P. When new sensors +are chosen, the surrogate observability coefficient ˜βG(θ) increases monotonously with +a strong incline just after the initial M = 5 sensors, followed by a visible stagnation +(see Figure 3a) as is often observed for similar OMP-based sensor selection algorithms +20 + +5 +6 +7 +8 +number of sensors +10 +4 +10 +3 +10 +2 +10 +1 +observability coefficient +mean observability coefficient +min observability coefficient +improvement with next sensor +observability coefficient, fixed config. +(a) Observability during sensor selection +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +0.14 +observability coefficient +0.0 +2.5 +5.0 +7.5 +10.0 +12.5 +15.0 +17.5 +max. observability +A-OED +D-OED +E-OED +proposal +proposal, fixed +(b) Histogram of βG(θref) +Figure 3: Observability coefficient for different methods when choosing 8 out of 25 sensor locations. Left: +Minimum and mean over θ of ˜βG(θ) as well as βG(θref) obtained in the course of running Algorithm 4 once +for 512,000 configurations and once for the training set {θref}. Right: Distribution of βG(θref) over all possible +sensor combinations with indicators for the A-, D-, and E-optimal choices, the combination with maximum +observability, and the sensors chosen by the Algorithm 4 with Ξtrain-training (“proposal”, purple, marked +“x”) and θref-training (“proposal, fixed”, turquoise, marked “+”). Note that the height of the indicator line +was chosen solely for readability. +(e.g., [8, 69, 70, 7]). Algorithm 4 terminates in 7.93 min with a minimum reduced-order +observability of ˜βG(θ) = 7.3227e-2 and an average of 1.0995e-1. At the reference +configuration θref, the full-order observability coefficient is βG(θref) = 1.0985, slightly +below the reduced-order average. We call this training procedure “Ξtrain-training” here- +after and denote the chosen sensors as “Ξtrain-trained sensor set” in the subsequent text +and as “proposal” in the plots. +In order to get an accurate understanding of how the surrogate model ˜xθ(u) and the +large configuration training set Ξtrain influence the sensor selection, we run Algorithm 4 +again, this time restricted on the full-order FE model xθref(u) at only the reference con- +figuration θref. The increase in βG(θref) in the course of the algorithm is shown in Fig- +ure 3a. The curve starts significantly above the average for Ξtrain-training, presumably +because conflicting configurations cannot occur, e.g., when one sensor would signifi- +cantly increase the observability at one configuration but cause little change in another. +However, in the stagnation phase, the curve comes closer to the average achieved with +Ξtrain-training. The computation finishes within 12.53 s, showing that the long runtime +before can be attributed to the size of Ξtrain. The final observability coefficient with 8 +sensors is βG(θref) = 1.2647e-1, above the average over ˜βG(θ) achieved training on +Ξtrain. We call this training procedure “θref-training” hereafter, and the sensor configu- +ration “θref-trained” in the text or “proposal, fixed config.” in the plots. +Comparison at the reference configuration +For comparing the performance of the Ξtrain- and θref-trained sensor combinations, +we compute – at the reference configuration θref – all 1,081,575 posterior covariance +matrices Σθref,L +post for all unordered combinations L of 8 distinct sensors in the sensor li- +brary L5×5. For each matrix, we compute the trace (A-OED criterion), the determinant +(D-OED criterion), the maximum eigenvalue (E-OED criterion), and the observability +coefficient βG(θref). This lets us identify the A-, D-, and E-optimal sensor combina- +21 + +0 +0.05 +0.1 +0.15 +0.2 +Observability Coefficient +10 +0.25 +100 +100.25 +100.5 +100.75 +101.0 +101.25 +101.5 +101.75 +posterior trace +A-optimal sensor combination +proposal +proposal, fixed configuration +maximum observability coefficient +all 25 sensors +best +sensor selection +0.587 % +proposal +0.022 % +proposal, fixed config. +1.778 % +max. observability +Figure 4: Distribution of trace(ΣL,θ +post) for θ = θref over all 1,081,575 combinations for choosing 8 out of the +25 sensor locations. On the left: distribution of trace(ΣL,θ +post) against the observability coefficient βG(θref). +Note that the marginal distribution of the horizontal axis is provided in Figure 3b. On the right: histogram +of trace(ΣL,θ +post) (marginal distribution for the plot on the left) with for the different sensor combinations (in +percent out of 1,081,575 combinations). The plots include markers for the A-optimal sensor choice, the +sensors chosen by Algorithm 4 with Ξtrain-training (“proposal”) and with {θref}-training (“proposal, fixed +configuration”), the sensor combination with maximum observability βG(θref), and when all 25 sensors are +included. +tions. The total runtime for these computations is 4 min – well above the 12.53 s of +θref-training. The (almost) 8 min for Ξtrain-training remain reasonable considering it is +trained on |Ξtrain| = 512, 000 configurations and not only θref. +A histogram for the distribution of βG(θref) is given in Figure 3b with markers for +the values of the A-, D-, and E-optimal choices and the Ξtrain- and θref-trained ob- +servation operators. Out of these five, the D-optimal choice has the smallest value, +since the posterior determinant is influenced less by the maximum posterior eigenvalue +and hence the observability coefficient. In contrast, both the A- and E-optimal sen- +sor choices are among the 700 combinations with the largest βG(θref) (this corresponds +to the top 0.065%). The θref-trained sensors have similar observability and are even +among the top 500 combinations. For the Ξtrain- trained sensors, the observability co- +efficient is smaller, presumably because Ξtrain-training is not as optimized for θref. Still, +it ranks among the top 0.705 % of sensor combinations with the largest observability. +In order to visualize the connection between the observability coefficient βG(θref) +and the classic A-, D-, and E-OED criteria, we plot the distribution of the posterior +covariance matrix’s trace, determinant, and maximum eigenvalue over all sensor com- +binations against βG(θ) in Figures 4, 5, 6. Overall we observe a strong correlation +between the respective OED criteria and βG(θref): It is the most pronounced in Figure +6 for E-optimality, and the least pronounced for D-optimality in Figure 5. For all OED +22 + +0 +0.05 +0.1 +0.15 +0.2 +Observability Coefficient +10 +5 +10 +4 +10 +3 +10 +2 +10 +1 +100 +101 +Posterior determinant +D-optimal sensor combination +proposal +proposal, fixed configuration +maximum observability coefficient +all 25 sensors +best +sensor selection +0.252 % +proposal +0.081 % +proposal, fixed config. +12.923 % +max. observability +Figure 5: Distribution of the posterior determinant det(ΣL,θ +post) for θ = θref. See Figure 4 for details about the +plot structure. +0 +0.05 +0.1 +0.15 +0.2 +Observability Coefficient +10 +0.5 +100 +100.5 +101 +101.5 +maximum posterior eigenvalue +E-optimal sensor combination +proposal +proposal, fixed configuration +maximum observability coefficient +all 25 sensors +best +sensor selection +1.679 % +proposal +0.001 % +proposal, fixed config. +4.080 % +max. observability +Figure 6: Distribution of the maximum eigenvalue of the posterior covariance matrix ΣL,θ +post for θ = θref. See +Figure 4 for details about the plot structure. Note that the θref-trained sensor combination has the 101-st +smallest maximum posterior eigenvalue among all 1,081,575 possibilities. +23 + +design criterion +training +pctl +A-OED +D-OED +E-OED +θref +Ξtrain +99-th +3.5835 +81.8508 +1.1724 +2.2223 +9.2512 +95-th +2.2747 +26.8430 +0.3601 +0.7846 +4.0374 +75-th +0.5141 +3.8600 +0.0532 +0.1419 +0.8106 +50-th +0.1527 +1.4641 +0.0159 +0.0438 +0.2354 +25-th +0.0414 +0.3669 +0.0035 +0.0068 +0.0621 +Figure 7: Ranking in βG(θref) of the A-, D-, E- optimal and the θref- and Ξtrain-trained sensor choices for +all possible combinations of choosing 8 unordered sensors in the library. Left: Boxplots obtained over 200 +random sensor libraries. Right: worst-case ranking (in percent) of the corresponding percentiles (“pctl”). +criteria, the correlation becomes stronger for smaller scaling factors σ2 and weakens +for large σ2 when the prior is prioritized (plots not shown). This behavior aligns with +the discussion in Section 3.1 that βG(θ) primarily targets the largest posterior eigenvalue +and is most decisive for priors with higher uncertainty. +Comparison for different libraries +We finally evaluate the influence of the library L5×5 on our results. To this end, +we randomly select 200 sets of new measurement positions, each consisting of 25 +drilling locations with an associated drilling depth. For each library, we run Algorithm +4 to choose 8 sensors, once with Ξtrain-training on the surrogate model, and once with +the full-order model at θref only. For comparison, we then consider in each library +each possible combination of choosing 8 unordered sensor sets and compute the trace, +determinant, and maximum eigenvalue of the associated posterior covariance matrix at +the reference configuration θref together with its observability coefficient. This lets us +identify the A-, D-, and E-optimal sensor combinations. +Figure 7 shows how βG(θref) is distributed over the 200 libraries, with percentiles +provided in the adjacent table. For 75% of the libraries, the A- and E-optimal, and the +Ξtrain- and θref-trained sensor choices rank among the top 1% of combinations with the +largest observability. Due to its non-optimized training for θref, the Ξtrain-trained sensor +set performs slightly worse than what is achieved with θref-training, but still yields +a comparatively large value for βG(θref). In contrast, overall, the D-optimal sensor +choices have smaller observability coefficients, presumably because the minimization +of the posterior determinant is influenced less by the maximum posterior eigenvalue. +The ranking of the Ξtrain- and θref-trained sensor configurations in terms of the pos- +terior covariance matrix’s trace, determinant, and maximum eigenvalue over the 200 +libraries is given in Figure 8. Both perform well and lie for 75% of the libraries within +the top 1% of combinations. As the ranking is performed for the configuration param- +eter θref, the θref-trained sensor combination performs better, remaining in 95% of the +libraries within the top 5% of sensor combinations. +24 + +A-OED +D-OED +E-OED +obs. coef. +100 +101 +102 +103 +104 +105 +106 +ranking +proposal, flexible configuration +opt. +top 10 +0.01% +0.1% +1% +5% +25% +100% +percentile +pctl +A-OED +D-OED +E-OED +βG(θref) +99-th +3.9240 +6.2372 +10.8391 +9.2512 +95-th +1.9093 +3.1544 +4.5583 +4.0374 +75-th +0.3083 +0.7718 +0.9185 +0.8106 +50-th +0.0664 +0.2361 +0.2763 +0.2354 +25-th +0.0177 +0.0536 +0.0596 +0.0621 +A-OED +D-OED +E-OED +obs. coef. +100 +101 +102 +103 +104 +105 +106 +ranking +proposal, fixed configuration +opt. +top 10 +0.01% +0.1% +1% +5% +25% +100% +percentile +pctl +A-OED +D-OED +E-OED +βG(θref) +99-th +2.5261 +2.9752 +11.1534 +2.2223 +95-th +1.0134 +1.8324 +2.8458 +0.7846 +75-th +0.1155 +0.4698 +0.3549 +0.1419 +50-th +0.0224 +0.1212 +0.0687 +0.0438 +25-th +0.0041 +0.0181 +0.0138 +0.0068 +Figure 8: Ranking of the posterior covariance matrix Σθref,L +post +in terms of the A-, D-, E-OED criteria and +the observability coefficient βG(θref) when the observation operator GL,θ is chosen with Algorithm 4 and +Ξtrain-training (top) or θref-training (bottom). The ranking is obtained by comparing all possible unordered +combinations of 8 sensors in each sensor library. On the left: Boxplots of the ranking over 200 sensor +libraries; on the right: ranking (in percent) among different percentiles. +25 + +(a) upmost layer, Ξtrain-training +(b) upmost layer, θref-training +(c) lowest layer, Ξtrain-training +(d) lowest layer, θref-training +Figure 9: Sensor positions chosen by Algorithm 4 from a grid of 47 × 47 available horizontal positions +with available 5 depths each, though only the lowest (bottom) and upmost (top) layers were chosen. The +underlying plot shows cuts through the full-order solution xθ(u) at θ = θref. Left: Ξtrain-training with the +RB surrogate model on a training set Ξtrain ⊂ P with 10,000 random configurations; runtime 14.19 s for 10 +sensors. Right: θref-training with full-order model at reference parameter; runtime 15.85 s for 10 sensors. +5.3. Unrestricted Library +We next verify the scalability of Algorithm 4 to large sensor libraries by permitting +all 2,209 drilling locations, at each of which at most one measurement may be taken +at any of the 5 available measurement depths. Choosing 10 unordered sensors yields +approximately 7.29e+33 possible combinations. Using the RB surrogate model from +before, we run Algorithm 4 once on a training grid Ξtrain ⊂ P consisting of 10,000 +randomly chosen configurations using only the surrogate model (runtime 14.19 s), and +once on the reference configuration θref using the full-order model (runtime 15.85 s) for +comparison. We terminate the algorithm whenever 10 sensors are selected. Compared +to the training time on L5×5 before, the results confirm that the size of the library itself +has little influence on the overall runtime but that the full-order computations and the +size of Ξtrain relative to the surrogate compute dominate. +The sensors chosen by the two runs of Algorithm 4 are shown in Figure 9. They +share many structural similarities: +• Depth: Despite the availability of 5 measurement depths, sensors have only been +chosen on the lowest and the upmost layers with 5 sensors each. The lower sen- +sors were chosen first (with one exception, sensor 3 in θref-training), presumably +26 + +z +1 +0.8 +0.6 +0.4 +0.2 +0 +0.9 +0.9 +P +9 +0.8 +0.8 +Temperature +0.7 +0.7 +0.18 +0.6 +0.6 +10 +0.16 +0.5 +-0.5 +Y +Y +0.4 +-0.4 +E0.14 +0.3 +0.3 +E0.12 +0.2 +0.2 +4 +E0.10 +0.1 +0.1 +0 +0.8 +0.6 +0.4 +0.2 +1 +0 +zz +1 +0.8 +0.6 +0.4 +0.2 +0 +0.9 +0.9 +P +0.8 +0.8 +Temperature +0.7 +0.7 +0.18 +0.6 +0.6 +0.16 +0.5 +-0.5 +Y +5.2 +6 +Y +0.4 +-0.4 +E0.14 +0.3 +0.3 +E0.12 +0.2 +-0.2 +E0.10 +9 +0.1 +0.1 +8 +0 +0.8 +0.6 +0.4 +0.2 +1 +0 +zz +1 +0.8 +0.6 +0.4 +0.2 +0 +0.9 +0.9 +3 +0.8 +-0.8 +Temperature +0.7 +0.7 +0.91 +0.6 +0.6 +6 +0.81 +0.5 +-0.5 +Y +Y +0.4- +0.4 +E0.71 +0.3 +-0.3 +E0.61 +0.2 +0.2 +E0.50 +0.1 +0.1 +-0 +-0 +0.8 +0.6 +0.4 +0.2 +0 +1 +zz +1 +0.8 +0.6 +0.4 +0.2 +0 +0.9 +0.9 +0.8 +0.8 +Temperature +0.7 +0.7 +0.91 +0.6 +0.6 +0.81 +0.5 +-0.5 +5 +Y ++ +Y +0.4 +0.4 +E0.71 +0.3 +0.3 +E0.61 +0.2 +0.2 +2 +E0.50 +3 +0.1 +0.1 +-0 +-0 +0.8 +0.6 +0.4 +0.2 +0 +1 +zbecause the lower layer is closer to the uncertain Neumann boundary condition +and therefore yields larger measurement values. +• Pairing Each sensor on the lowest layer has a counterpart on the upmost layer +that has almost the same position on the horizontal plane. This pairing targets +noise sensitivity: With the prescribed error covariance function, the noise in two +measurements is increasingly correlated the closer the measurements lie horizon- +tally, independent of their depth coordinate. Choosing a reference measurement +near the zero-Dirichlet boundary at the surface helps filter out noise terms in the +lower measurement. +• Organization On each layer, the sensors are spread out evenly and approxi- +mately aligned in 3 rows and 3 columns. The alignment helps distinguish be- +tween the constant, linear, and quadratic parts of the uncertain Neumann flux +function in north-south and east-west directions. +Figure 10 (left side) shows the increase in the observability coefficients ˜βG(θ) (for +Ξtrain-training) and βG(θref) (for θref-training) over the number of chosen sensors. We +again observe a strong initial incline followed by stagnation for the Ξtrain-trained sen- +sors, whereas the curve for θref-training already starts at a large value to remain then +almost constant. The latter is explained by the positions of the first 5 sensors in Fig- +ure 9 (right), as they are already spaced apart in both directions for the identification of +quadratic polynomials. In contrast, for Ξtrain-training, the “3 rows, 3 columns” structure +is only completed after the sixth sensor (c.f. Figure 9, left). With 6 sensors, the observ- +ability coefficients in both training schemes have already surpassed the final observ- +ability coefficients with 8 sensors in the previous training on the smaller library L5×5. +The final observability coefficients at the reference parameter θref are βG(θref) = 0.4042 +for θref-training, and βG(θref) = 0.3595 for Ξtrain-training. +As a final experiment, we compare the eigenvalues of the posterior covariance ma- +trix ΣL,θref +post for the Ξtrain- and θref-trained sensors against 50,000 sets of 10 random sen- +sors each. We confirm that all 50,000 sensor combinations comply with the combina- +torial restrictions. Boxplots of the eigenvalues are provided in Figure 10 (right side). +The eigenvalues of the posterior covariance matrix with sensors chosen by Algorithm +4 are smaller10 than all posterior eigenvalues for the random sensor combinations. +6. Conclusion +In this work, we analyzed the connection between the observation operator and +the eigenvalues of the posterior covariance matrix in the inference of an uncertain pa- +rameter via Bayesian inversion for a linear, hyper-parameterized forward model. We +identified an observability coefficient whose maximization decreases the uncertainty in +the posterior probability distribution for all hyper-parameters. To this end, we proposed +10Here we compare the largest eigenvalue of one matrix to the largest eigenvalue of another, the second +largest to the second largest, and so on. +27 + +5 +6 +7 +8 +9 +10 +number of sensors +10 +3 +10 +2 +10 +1 +100 +observability coefficient +mean observability coefficient +min observability coefficient +improvement with next sensor +observability coefficient, fixed config. +1 +2 +3 +4 +5 +posterior eigenvalue (largest to smallest) +10 +3 +10 +2 +10 +1 +100 +101 +random combinations with 10 sensors each +proposal +proposal, fixed config +Figure 10: Left: Observabity coefficients during sensor selection with Ξtrain- and θref-training for a library +with 11,045 measurement positions and combinatorial restrictions. Shown are 1) the minimum and mean +surrogate observability coefficient ˜βG(θ) over a training set with 10,000 random configurations with final +values minθ ˜βG(θ) = 0.4160 and meanθ ˜βG(θ) = 0.6488, and 2) the full-order observability coefficient βG(θref) +when training on the reference parameter θref alone (final value βG(θref) = 0.4042). Right: Boxplots for the +5 eigenvalues of the posterior covariance matrix ΣL,θ +post over 50,000 sets of 10 sensors chosen uniformly from +a 5 × 47 × 47 grid with imposed combinatorial restrictions. The eigenvalues are compared according to their +order from largest to smallest. Indicated are also the eigenvalues for the Ξtrain-trained (purple, “x”-marker) +and θref-trained (turquoise, “+”-marker) sensors from Figure 9. +a sensor selection algorithm that expands an observation operator iteratively to guaran- +tee a uniformly large observability coefficient for all hyper-parameters. Computational +feasibility is retained through a reduced-order model in the greedy step and an OMP +search for the next sensor that only requires a single full-order model evaluation. The +validity of the approach was demonstrated on a large-scale heat conduction problem +over a section of the Perth Basin in Western Australia. Future extensions of this work +are planned to address 1) high-dimensional parameter spaces through parameter reduc- +tion techniques, 2) the combination with the PBDW inf-sup-criterion to inform sensors +by functionalanalytic means in addition to the noise covariance, and 3) the expansion +to non-linear models through a Laplace approximation. +Acknowledgments +We would like to thank Tan Bui-Thanh, Youssef Marzouk, Francesco Silva, An- +drew Stuart, Dariusz Ucinski, and Keyi Wu for very helpful discussions, and Florian +Wellmann at the Institute for Computational Geoscience, Geothermics and Reservoir +Geophysics at RWTH Aachen University for providing the Perth Basin Model. This +work was supported by the Excellence Initiative of the German federal and state gov- +ernments and the German Research Foundation through Grants GSC 111 and 33849990/GRK2379 +(IRTG Modern Inverse Problems). This project has also received funding from the +European Research Council (ERC) under the European Union’s Horizon 2020 re- +search and innovation programme (grant agreement n° 818473), the US Department +of Energy (grant DE-SC0021239), and the US Air Force Office of Scientific Research +(grant FA9550-21-1-0084). Peng Chen is partially supported by the NSF grant DMS +#2245674. +28 + +References +[1] A. M. Stuart, Inverse problems: a Bayesian perspective, Acta numerica 19 (2010) +451–559. +[2] I. Stober, K. Bucher, Geothermie, Springer, 2012. +[3] D. Ucinski, Optimal measurement methods for distributed parameter system iden- +tification, CRC press, 2004. +[4] V. B. Melas, Functional approach to optimal experimental design, volume 184, +Springer Science & Business Media, 2006. +[5] L. Pronzato, Optimal experimental design and some related control problems, +Automatica 44 (2008) 303–325. +[6] N. Aretz-Nellesen, P. Chen, M. A. Grepl, K. 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Taddei, Model order reduction methods for data assimilation: state estimation +and structural health monitoring, Ph.D. thesis, Massachusetts Institute of Tech- +nology, 2017. +34 + diff --git a/MtFLT4oBgHgl3EQfNS8Q/content/tmp_files/load_file.txt b/MtFLT4oBgHgl3EQfNS8Q/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..484fcad063af68a2c8d20937e732de632f0f04aa --- /dev/null +++ b/MtFLT4oBgHgl3EQfNS8Q/content/tmp_files/load_file.txt @@ -0,0 +1,1336 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf,len=1335 +page_content='A Greedy Sensor Selection Algorithm for Hyperparameterized Linear Bayesian Inverse Problems Nicole Aretza, Peng Chenb, Denise D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Degenc, Karen Veroyd aOden Institute for Computational Engineering and Sciences, University of Texas at Austin, 201 E 24th St, Austin, TX 78712, USA bSchool of Computational Science and Engineering, Georgia Institute of Technology, 756 W Peachtree St NW, Atlanta, GA 30308, USA cComputational Geoscience, Geothermics, and Reservoir Geophysics, RWTH Aachen University, Mathieustr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 30, 52074 Aachen, Germany dCenter for Analysis, Scientific Computing and Applications, Department of Mathematics and Computer Science, Eindhoven University of Technology, 5612 AZ Eindhoven, The Netherlands Abstract We consider optimal sensor placement for a family of linear Bayesian inverse prob- lems characterized by a deterministic hyper-parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The hyper-parameter describes distinct configurations in which measurements can be taken of the observed physical system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' To optimally reduce the uncertainty in the system’s model with a single set of sensors, the initial sensor placement needs to account for the non-linear state changes of all admissible configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We address this requirement through an observabil- ity coefficient which links the posteriors’ uncertainties directly to the choice of sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We propose a greedy sensor selection algorithm to iteratively improve the observability coefficient for all configurations through orthogonal matching pursuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The algorithm allows explicitly correlated noise models even for large sets of candidate sensors, and remains computationally efficient for high-dimensional forward models through model order reduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We demonstrate our approach on a large-scale geophysical model of the Perth Basin, and provide numerical studies regarding optimality and scalability with regard to classic optimal experimental design utility functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Introduction In the Bayesian approach to inverse problems (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' [1]), the uncertainty in a param- eter is described via a probability distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' With Bayes’ Theorem, the prior belief in a parameter is updated when new information is revealed such that the posterior distri- bution describes the parameter with improved certainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Bayes’ posterior is optimal in the sense that it is the unique minimizer of the sum of the relative entropy between the posterior and the prior, and the mean squared error between the model prediction and the experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The noise model drives, along with the measurements, how the posterior’s uncertainty is reduced in comparison to the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' A critical aspect – espe- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='12019v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='NA] 27 Jan 2023 cially for expensive experimental data1 – is how to select the measurements to improve the posterior’s credibility best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The selection of adequate sensors meeting individual applications’ needs is, therefore, a big goal of the optimal experimental design (OED) research field and its surrounding community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We refer to the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', [3, 4, 5]) for introductions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The analysis and algorithm presented in this work significantly extend our initial ideas presented in [6] in which we seek to generalize the 3D-VAR stability results from [7] to the probabilistic Bayesian setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Our proposed algorithm is directly re- lated to the orthogonal matching pursuit (OMP) algorithm [8, 9] for the parameterized- background data-weak (PBDW) method and the empirical interpolation method (EIM) ([10, 11]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Closely related OED methods for linear Bayesian inverse problems over partial differential equations (PDEs) include [12, 13, 14, 15, 16, 17], mostly for A- and D-OED and uncorrelated noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In recent years, these methods have also been extended to non-linear Bayesian inverse problems, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', [18, 19, 20, 21, 22], while an advance to correlated noise has been made in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In particular, [21, 22] use similar algorithmic approaches to this work by applying a greedy algorithm to maximize the expected in- formation gain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Common strategies for dealing with the high dimensions imposed by the PDE model use the framework in [24] for discretization, combined with parameter reduction methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', [25, 26, 27, 28, 29, 30, 31]) and model order reduction (MOR) methods for uncertainty quantification (UQ) problems (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', [32, 33, 34, 35, 36]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In this paper, we consider inverse problem settings, in which a deterministic hyper- parameter describes anticipated system configurations such as material properties or loading conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Each configuration changes the model non-linearly, so we obtain a family of possible posterior distributions for any measurement data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Supposing data can only be obtained with a single set of sensors regardless of the system’s configu- ration, the OED task becomes to reduce the uncertainty in each posterior uniformly over all hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' This task is challenging for high-dimensional models since 1) each configuration requires its own computationally expensive model solve, and 2) for large sets of admissible measurements, the comparison between sensors requires the inversion of the associated, possibly dense noise covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' By building upon [6], this paper addresses both challenges and proposes in detail a sensor selection algorithm that remains efficient even for correlated noise models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The main contributions are as follows: First, we identify an observability coeffi- cient as a link between the sensor choice and the maximum eigenvalue of each poste- rior distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We also provide an analysis of its sensitivity to model approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Second, we decompose the noise covariance matrix for any observation operator to al- low fast computation of the observability gain under expansion with additional sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Third, we propose a sensor selection algorithm that iteratively constructs an observa- tion operator from a large set of sensors to increase the observability coefficient over all hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The algorithm is applicable to correlated noise models, and re- quires, through the efficient use of MOR techniques, only a single full-order model 1For instance, for projects harvesting geothermal energy, the development costs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', drilling, stimula- tion, and tests) take up 50 − 70% of the total budget ([2]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' As each borehole can cost several million dollars, it is essential to plan their location carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 2 evaluation per selected sensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' While the main idea and derivation of the observability coefficient are similar to [6], this work additionally features 1) an analysis of the observability coefficient re- garding model approximations, 2) explicit computational details for treating correlated noise models, and 3) a comprehensive discussion of the individual steps in the sen- sor selection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Moreover, the proposed method is tested using a large-scale geophysical model of the Perth Basin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' This paper is structured as follows: In Section 2 we introduce the hyper-parameterized inverse problem setting, including all assumptions for the prior distribution, the noise model, and the forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In Section 3, we then establish and analyze the con- nection between the observability coefficient and the posterior uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We fi- nally propose our sensor selection algorithm in Section 4 which exploits the presented analysis to choose sensors that improve the observability coefficient even in a hyper- parameterized setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We demonstrate the applicability and scalability of our approach on a high-dimensional geophysical model in Section 5 before concluding in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Problem setting Let X be a Hilbert space with inner product ⟨·, ·⟩X and induced norm ∥x∥2 X := ⟨x, x⟩X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We consider the problem of identifying unknown states xtrue(θ) ∈ X of a single physical system under changeable configurations θ from noisy measurements d(θ) ≈ [ℓ1(xtrue(θ)), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ℓK(xtrue(θ))]T ∈ RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The measurements are obtained by a set of K unique sensors (or experiments) ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ℓK ∈ X′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Our goal is to choose these sensors from a large sensor library L ⊂ X′ of options in a way that optimizes how much information is gained from their measurements for any configurations θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Hyper-parameterized forward model We consider the unknown state xtrue to be uniquely characterized by two sources of information: an unknown parameter utrue ∈ RM describing uncertainties in the governing physical laws, and a hyper-parameter (or configuration2) θ ∈ P ⊂ Rp describing dependencies on controllable configurations under which the system may be observed (such as material properties or loading conditions) where P is a given compact set en- closing all possible configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For any given u ∈ RM and θ ∈ P, we let xθ(u) ∈ X be the solution of an abstract model equation Mθ(xθ(u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' u) = 0 and assume that the map u → xθ(u) is well-defined, linear, 2We call θ interchangeably hyper-parameter or configuration to either stress its role in the mathematical model or physical interpretation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 3 and uniformly continuous in u, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' ∃ ¯η > 0 : η(θ) := sup u∈RM ∥xθ(u)∥X ∥u∥Σ−1 pr < ¯η ∀ θ ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (1) Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Although we assumed that utrue lies in the Euclidean space RM, any other linear space can be considered via an affine transformation onto an appropriate basis (see [12, 37]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For infinite-dimensional spaces, we first discretize with appropriate treatment of the adjoint operator (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' [24]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' By keeping the model equation general, we stress the applicability of our approach to a wide range of problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For instance, time-dependent states can be treated by choosing X as a Bochner space or its discretization (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' [38]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We also do not formally restrict the dimension of X, though any implementation relies on the ability to compute xθ(u) with sufficient accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' To this end, we note that the analysis in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2 can be applied to determine how discretization errors affect the observ- ability criterion in the sensor selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Following a probabilistic approach to inverse problems, we express the initial un- certainty in utrue = utrue(θ) of any xtrue = xθ(utrue) in configuration θ through a random variable u with Gaussian prior µpr = N � upr, Σpr � , where upr ∈ RM is the prior mean and Σpr ∈ RM×M is a symmetric positive definite (s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=') covariance matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The latter defines the inner product ⟨·, ·⟩Σ−1 pr and its induced norm ∥ · ∥Σ−1 pr through ⟨u, v⟩Σ−1 pr := uTΣ−1 pr ˜u, ∥u∥2 Σ−1 pr := ⟨u, u⟩Σ−1 pr , ∀ u, v ∈ RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (2) With these definitions, the probability density function (pdf) for µpr is πpr(u) = 1 � (2π)M det Σpr exp � −1 2∥u − upr∥2 Σ−1 pr � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For simplicity, we assume {utrue(θ)}θ∈P to be independent realizations of u such that we may consider the same prior for all θ without accounting for a possible history of measurements at different configurations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Sensor library and noise model For taking measurements of the unknown states {xtrue(θ)}θ, we call any linear func- tional ℓ ∈ X′ a sensor, and its application to a state x ∈ X its measurement ℓ(x) ∈ R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We model experimental measurements dℓ ∈ R of the actual physical state xtrue as dℓ = ℓ(xtrue) + εℓ where εℓ ∼ N(0, cov(εℓ, εℓ)) is a Gaussian random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We permit noise in different sensor measurements to be correlated with a known covari- ance function cov.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In a slight overload of notation, we write cov : L × L → R, cov(ℓi, ℓj) := cov(εℓi, εℓj) as a symmetric bilinear form over the sensor library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Any ordered subset S = {ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ℓK} ⊂ L of sensors can then form a (linear and continuous) observation operator through L := [ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ℓK]T : X → RK, Lx := [ℓ1(x), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ℓK(x)]T .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 4 The experimental measurements of L have the form d = �ℓ1(xtrue) + εℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ℓK(xtrue) + εℓK �T = Lxtrue + ε with ε = �εℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , εℓK �T ∼ N(0, σ2ΣL), (3) where σ2ΣL is the noise covariance matrix defined through ΣL ∈ RK×K, such that � σ2ΣL � i, j := cov(ℓj, ℓi) = cov(εℓj, εℓi) (4) with an auxiliary scaling parameter3 σ2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We assume that the library L and the noise covariance function cov have been chosen such that ΣL is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' for any combination of sensors in L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' This assumption gives rise to the L-dependent inner product and its induced norm � d, ˜d � Σ−1 L := dTΣ−1 L ˜d, ∥d∥2 Σ−1 L := ⟨d, d⟩Σ−1 L , ∀ d, ˜d ∈ RK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (5) Measured with respect to this norm, the largest observation of any (normalized) state is thus γL := sup ∥x∥X=1 ∥Lx∥Σ−1 L = sup x∈X ∥Lx∥Σ−1 L ∥x∥X .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (6) We show in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1 that γL increases under expansion of L with additional sensors despite the change in norm, and is therefore bounded by γL ≤ γL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We also define the parameter-to-observable map GL,θ : RM → RK, such that GL,θ (u) := Lxθ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (7) With the assumptions above – in particular the linearity and uniform continuity (1) of x in u – the map GL,θ is linear and uniformly bounded in u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We let GL,θ ∈ RK×M denote its matrix representation with respect to the unit basis {em}M m=1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The likelihood of d ∈ RK obtained through the observation operator L for the parameter u ∈ RM and the system configuration θ is then ΦL � d ��� u, θ � := 1 � 2K det ΣL exp � − 1 2σ2 ���d − GL,θ (u) ���2 Σ−1 L � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Note that GL,θ and GL,θ may depend non-linearly on θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Posterior distribution Once noisy measurement data d ≈ Lxtrue(θ) is available, Bayes’ theorem yields the posterior pdf as πL,θ post(u | d) = 1 Z(θ) exp � − 1 2σ2 ���GL,θ (u) − d ���2 Σ−1 L − 1 2∥u − upr∥2 Σ−1 pr � ∝ πpr(u)·ΦL � d ��� u, θ � , (8) 3We introduce σ2 here as an additional variable to ease the discussion of scaling in Section 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' However, we can set σ2 = 1 without loss of generality (w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='o.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 5 with normalization constant Z(θ) := � Rp exp � − 1 2σ2 ���GL,θ (u) − d ���2 Σ−1 L � dµpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Due to the linearity of the parameter-to-observable map, the posterior measure µL,θ post is a Gaussian µL,θ post = N(uL,θ post(d), ΣL,θ post) with known (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' [1]) mean and covariance matrix uL,θ post(d) = ΣL,θ post � 1 σ2 GT L,θΣ−1 L d + Σ−1 pr upr � ∈ RM, (9) ΣL,θ post = � 1 σ2 GT L,θΣ−1 L GL,θ + Σ−1 pr �−1 ∈ RM×M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (10) The posterior µL,θ post thus depends not only on the choice of sensors, but also on the con- figuration θ under which their measurements were obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Therefore, to decrease the uncertainty in all possible posteriors with a single, θ-independent observation operator L, the construction of L should account for all admissible configurations θ ∈ P under which xtrue may be observed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Remark 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The linearity of xθ(u) in u is a strong assumption that dictates the Gaussian posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' However, in combination with the hyper-parameter θ, our setting here can be re-interpreted as the Laplace-approximation for a non-linear state map θ �→ x(θ) (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' [39, 21, 40]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The sensor selection presented here is then an intermediary step for OED over non-linear forward models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The Observability Coefficient In this section, we characterize how the choice of sensors in the observation op- erator L and its associated noise covariance matrix ΣL influence the uncertainty in the posteriors µL,θ post, θ ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We identify an observability coefficient that bounds the eigen- values of the posterior covariance matrices ΣL,θ post, θ ∈ P with respect to L, and facilitates the sensor selection algorithm presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Eigenvalues of the Posterior Covariance Matrix The uncertainty in the posterior πL,θ post for any configuration θ ∈ P is uniquely char- acterized by the posterior covariance matrix ΣL,θ post, which is in turn connected to the observation operator L through the parameter-to-observable map GL,θ and the noise co- variance matrix ΣL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' To measure the uncertainty in ΣL,θ post, the OED literature suggests a variety of different utility functions to be minimized over L in order to optimize the sen- sor choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Many of these utility functions can be expressed in terms of the eigenvalues 6 λθ,1 L ≥ · · · ≥ λθ,M L > 0 of ΣL,θ post, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', A-OED: trace(ΣL,θ post) = M � m=1 λθ,m L (mean variance) D-OED: det(ΣL,θ post) = M � m=1 λθ,m L (volume) E-OED: λmax(ΣL,θ post) = λθ,1 L (spectral radius).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In practice, the choice of the utility function is dictated by the application.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In E-optimal experimental design (E-OED), for instance, posteriors whose uncertainty ellipsoids stretch out into any one direction are avoided, whereas D-OED minimizes the overall volume of the uncertainty ellipsoid regardless of the uncertainty in any one parameter direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We refer to [3] for a detailed introduction and other OED criteria.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Considering the hyper-parameterized setting where each configuration θ influences the posterior uncertainty, we seek to choose a single observation operator L such that the selected utility function remains small for all configurations θ ∈ P, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', for E-OED, minimizing min ℓ1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=',ℓK∈L max θ∈P λmax(ΣL,θ post) such that L = [ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ℓK]T guarantees that the longest axis of each posterior covariance matrix ΣL,θ post for any θ ∈ P has the same guaranteed upper bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The difficulty here is that the minimization over P necessitates repeated, cost-intensive model evaluations to compute the utility func- tion for many different configurations θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In the following, we therefore introduce an upper bound to the posterior eigenvalues that can be optimized through an observabil- ity criterion with far fewer model solves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The bound’s optimization indirectly reduces the different utility functions through the posterior eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Recalling that ΣL,θ post is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', let {ψm}M m=1 be an orthonormal eigenvector basis of ΣL,θ post, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' ψT mψn = δm,n and ΣL,θ postψm = λθ,m L ψm m = 1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (11) Using the representation (10), any eigenvalue λθ,m L can be written in the form 1 λθ,m L = ψT m � ΣL,θ post �−1 ψm = ψT m � 1 σ2 GT L,θΣ−1 L GL,θ + Σ−1 pr � ψm = 1 σ2 ���GL,θ (ψm) ���2 Σ−1 L + ∥ψm∥2 Σ−1 pr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (12) Since ψm depends implicitly on L and θ through (11), we cannot use this representation directly to optimize over L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' To take out the dependency on ψm, we bound ∥ψm∥2 Σ−1 pr ≥ 1 λmax pr in terms of the maximum eigenvalue of the prior covariance matrix Σpr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Likewise, we define βG(θ) := inf u∈RM ���GL,θ (u) ���Σ−1 L ∥u∥Σ−1 pr = inf u∈RM ∥Lxθ(u)∥Σ−1 L ∥u∥Σ−1 pr , (13) 7 as the minimum ratio between an observation for a parameter u relative to the prior’s covariance norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' From (12) and (13) we obtain the upper bound λθ,m L = ����������� 1 σ2 ���GL,θ (ψm) ���2 Σ−1 L ∥ψm∥2 Σ−1 pr + 1 ����������� −1 ∥ψm∥−2 Σ−1 pr ≤ � 1 σ2 βG(θ)2 + 1 �−1 λmax pr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Geometrically, this bound means that the radius λθ,1 L of the outer ball around the pos- terior uncertainty ellipsoid is smaller than that of the prior uncertainty ellipsoid by at least the factor � 1 σ2 βG(θ)2 + 1 �−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' By choosing L to maximize minθ βG(θ), we therefore minimize this outer ball containing all uncertainty ellipsoids (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', for any θ ∈ P).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' As expected, the influence of L is strongest when the measurement noise is small such that data can be trusted (σ2 ≪ 1), and diminishes with increasing noise levels (σ2 ≫ 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Parameter Restriction An essential property of βG(θ) is that βG(θ) = 0 if K < M, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', the number of sen- sors in L is smaller than the number of parameter dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In this case, βG(θ) cannot distinguish between sensors during the first M − 1 steps of an iterative algorithm, or in general when less than a total of M sensors are supposed to be chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For medium- dimensional parameter spaces (M ∈ O(10)), we mitigate this issue by restricting u to the subspace span{ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ϕmin{K,M}} ⊂ RM spanned by the first min{K, M} eigenvec- tors of Σpr corresponding to its largest eigenvalues, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', the subspace with the largest prior uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For high-dimensional parameter spaces or when the model Mθ has a non-trivial null-space, we bound βG(θ) further βG(θ) = inf u∈RM ∥Lxθ(u)∥Σ−1 L ∥xθ(u)∥X ∥xθ(u)∥X ∥u∥Σ−1 pr ≥ inf x∈Wθ ∥Lx∥Σ−1 L ∥x∥X inf u∈RM ∥xθ(u)∥X ∥u∥Σ−1 pr = βL|W(θ) η(θ) (14) where we define the linear space Wθ of all achievable states Wθ := {xθ(u) ∈ X : u ∈ RM} and the coefficients βL|W(θ) := inf x∈Wθ ∥Lx∥Σ−1 L ∥x∥X , η(θ) := inf u∈RM ∥xθ(u)∥X ∥u∥Σ−1 pr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (15) The value of η(θ) describes the minimal state change that a parameter u can achieve relative to its prior-induced norm ∥u∥Σ−1 pr .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' It can filter out parameter directions that have little influence on the states xθ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In contrast, the observability coefficient βL|W(θ) depends on the prior only implicitly via Wθ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' it quantifies the minimum amount of information (measured with respect to the noise model) that can be obtained on any state in Wθ relative to its norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Future work will investigate how to optimally restrict the parameter space based on η(θ) before choosing sensors that maximize βL|W(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Existing parameter reduction approaches in a similar context include [28, 41, 42, 27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In this work, however, we solely focus on the maximization of βG(θ) and, by extension, βL|W(θ) and henceforth assume that M is sufficiently small and η := infθ∈P η(θ) > 0 is bounded away from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Observability under model approximations To optimize the observability coefficient βG(θ) or βL|W(θ), it must be computed for many different configurations θ ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The accumulating computational cost motivates the use of reduced-order surrogate models, which typically yield considerable com- putational savings versus the original full-order model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' However, this leads to errors in the state approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In the following, we thus quantify the influence of state approximation error on the observability coefficients βG(θ) and βL|W(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' An analysis of the change in posterior distributions when the entire model Mθ is substituted in the inverse problem can be found in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Suppose a reduced-order surrogate model ˜ Mθ(˜xθ(u);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' u) = 0 is available that yields for any configuration θ ∈ P and parameter u ∈ RM a unique solution ˜xθ(u) ∈ X such that ∥xθ(u) − ˜xθ(u)∥X ≤ εθ ∥xθ(u)∥X with accuracy 0 ≤ εθ ≤ ε < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (16) Analogously to (13) and (15), we define the reduced-order observability coefficients ˜βG(θ) := inf u∈RM ∥L˜xθ(u)∥Σ−1 L ∥u∥Σ−1 pr , ˜βL|W(θ) := inf u∈RM ∥L˜xθ(u)∥Σ−1 L ∥˜xθ(u)∥X (17) to quantify the smallest observations of the surrogate states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For many applications, it is possible to choose a reduced-order model whose solution can be computed at a significantly reduced cost such that ˜βG(θ) and ˜βL|W(θ) are much cheaper to compute than their full-order counterparts βG(θ) and βL|W(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Since the construction of such a surrogate model depends strongly on the application itself, we refer to the literature (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', [43, 44, 45, 46, 47]) for tangible approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Recalling the definition of γL in (6), we start by bounding how closely the surrogate observability coefficient ˜βL|W(θ) approximates the full-order βL|W(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Let η(θ) > 0 hold, and let ˜xθ(u) ∈ X be an approximation to xθ(u) such that (16) holds for all θ ∈ P, u ∈ RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Then (1 − εθ) ˜βL|W(θ) − γLεθ ≤ βL|W(θ) ≤ (1 + εθ) ˜βL|W(θ) + γLεθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (18) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Let u ∈ RM \\ {0} be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Using (16) and the (reversed) triangle inequality, we obtain the bound ∥˜xθ(u)∥X ∥xθ(u)∥X ≥ ∥xθ(u)∥X − ∥xθ(u) − ˜xθ(u)∥X ∥xθ(u)∥X ≥ 1 − εθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (19) Note here that η(θ) > 0 implies ∥xθ(u)∥X > 0 so the quotient is indeed well defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The ratio of observation to state can now be bounded from below by ∥Lxθ(u)∥Σ−1 L ∥xθ(u)∥X ≥ ∥L˜xθ(u)∥Σ−1 L ∥xθ(u)∥X − ∥L(xθ(u) − ˜xθ(u))∥Σ−1 L ∥xθ(u)∥X ≥ ∥˜xθ(u)∥X ∥xθ(u)∥X ∥L˜xθ(u)∥Σ−1 L ∥˜xθ(u)∥X − γL ∥xθ(u) − ˜xθ(u)∥X ∥xθ(u)∥X ≥ (1 − εθ) ∥L˜xθ(u)∥Σ−1 L ∥˜xθ(u)∥X − γLεθ ≥ (1 − εθ)˜βL|W(θ) − γLεθ, 9 where we have applied the reverse triangle inequality, definition (6), the bounds (16), (19), and definition (17) of ˜βL|W(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Since u is arbitrary, the lower bound in (18) follows from definition (13) of βL|W(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The upper bound in (18) follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For the observability of the parameter-to-observable map GL,θ and its approxima- tion u �→ L˜xθ(u), we obtain a similar bound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' It uses the norm η(θ) of xθ : u �→ xθ(u) as a map from the parameter to the state space, see (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Let ˜xθ(u) ∈ X be an approximation to xθ(u) such that (16) holds for all θ ∈ P, u ∈ RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Then ˜βG(θ) − γLη(θ)εθ ≤ βG(θ) ≤ ˜βG(θ) + γLη(θ)εθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (20) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Let u ∈ RM \\ {0} be arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Then ∥Lxθ(u)∥Σ−1 L ≥ ∥L˜xθ(u)∥Σ−1 L − ∥L(xθ(u) − ˜xθ(u))∥Σ−1 L ≥ ∥L˜xθ(u)∥Σ−1 L − γL ∥xθ(u) − ˜xθ(u)∥X ≥ ∥L˜xθ(u)∥Σ−1 L − γLεθ ∥xθ(u)∥X ≥ ∥L˜xθ(u)∥Σ−1 L − γLεθη(θ)∥u∥Σ−1 pr , where we have used the reverse triangle inequality, followed by (6), (16), and (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We divide by ∥u∥Σ−1 pr and take the infimum over u to obtain βG(θ) = inf u∈RM ∥Lxθ(u)∥Σ−1 L ∥u∥Σ−1 pr ≥ inf u∈RM ∥L˜xθ(u)∥Σ−1 L ∥u∥Σ−1 pr − γL η(θ) εθ = ˜βG(θ) − γL η(θ) εθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The upper bound in (20) follows analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' If εθ is sufficiently small, Propositions 1 and 2 justify employing the surrogates ˜βL|W(θ) and ˜βG(θ) instead of the original full-order observability coefficients βL|W(θ) and βG(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' This substitution becomes especially necessary when the computation of xθ(u) is too expensive to evaluate βL|W(θ) or βG(θ) repeatedly for different configura- tions θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Another approximation step in our sensor selection algorithm relies on the identifi- cation of a parameter direction v ∈ RM with comparatively small observability, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' ∥Lxθ(v)∥Σ−1 L ∥v∥Σ−1 pr ≈ inf u∈RM ∥Lxθ(u)∥Σ−1 L ∥u∥Σ−1 pr = βG(θ) or ∥Lxθ(v)∥Σ−1 L ∥xθ(v)∥X ≈ inf x∈Wθ ∥Lx∥Σ−1 L ∥x∥X = βL|W(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The ideal choice would be the infimizer of respectively βG(θ) or βL|W(θ), but its compu- tation involves M full-order model evaluations (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' To avoid these costly computations, we instead choose v as the infimizer of the respective reduced-order observability coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' This choice is justified for small εθ < 1 by the following proposition: 10 Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Let η(θ) > 0 hold, and let ˜xθ(u) ∈ X be an approximation to xθ(u) such that (16) holds for all θ ∈ P, u ∈ RM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Suppose v ∈ arg infu∈RM ∥u∥−1 Σ−1 pr ∥L˜xθ(u)∥Σ−1 L , then βG(θ) ≤ ∥Lxθ(v)∥Σ−1 L ∥v∥Σ−1 pr ≤ βG(θ) + 2γLη(θ)εθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (21) Likewise, if v ∈ arg infu∈RM ∥˜xθ(u)∥−1 X ∥L˜xθ(u)∥Σ−1 L , then βL|W(θ) ≤ ∥Lxθ(v)∥Σ−1 L ∥xθ(v)∥X ≤ 1 + εθ 1 − εθ �βL|W(θ) + γLεθ � + γLεθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (22) Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For both (21) and (22) the lower bound follows directly from definitions (13) and (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' To prove the upper bound in (21), let v ∈ arg infu∈RM ∥u∥−1 Σ−1 pr ∥L˜xθ(u)∥Σ−1 L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Following the same steps as in the proof of Proposition 2, we can then bound ∥Lxθ(v)∥Σ−1 L ∥v∥Σ−1 pr ≤ ∥L˜xθ(v)∥Σ−1 L ∥v∥Σ−1 pr + ∥L(xθ(v) − ˜xθ(v))∥Σ−1 L ∥v∥Σ−1 pr ≤ ˜βG(θ) + γLη(θ)εθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The upper bound in (21) then follows with Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' To prove the upper bound in (22), let v ∈ arg infu∈RM ∥˜xθ(u)∥−1 X ∥L˜xθ(u)∥Σ−1 L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Then ∥Lxθ(v)∥Σ−1 L ∥xθ(v)∥X ≤ ∥L˜xθ(v)∥Σ−1 L ∥˜xθ(v)∥X ∥˜xθ(v)∥X ∥xθ(v)∥X + ∥L(xθ(v) − ˜xθ(v))∥Σ−1 L ∥xθ(v)∥X ≤ (1 + ε) ˜βL|W(θ) + γLεθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The result then follows with Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Sensor selection In the following, we present a sensor selection algorithm that iteratively increases the minimal observability coefficient minθ∈P βG(θ) and thereby decreases the upper bound for the eigenvalues of the posterior covariance matrix for all admissible system configurations θ ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The iterative approach is relatively easy to implement, allows a simple way of dealing with combinatorial restrictions, and can deal with large4 sensor libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Cholesky decomposition The covariance function cov connects an observation operator L to its observabil- ity coefficients βG(θ) and βL|W(θ) through the noise covariance matrix ΣL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Its inverse enters the norm ∥·∥Σ−1 L and the posterior covariance matrix ΣL,θ post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The inversion poses a challenge when the noise is correlated, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', when ΣL is not diagonal, as even the expansion of L with a single sensor ℓ ∈ L changes each entry of Σ−1 L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In naive compu- tations of the observability coefficients and the posterior covariance matrix, this leads 4For instance, in Section 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='3 we apply the presented algorithm to a library with KL = 11, 045 available sensor positions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 11 Algorithm 1: CholeskyExpansion Input: observation operator L = [ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ℓK]T, noise covariance matrix ΣL, Cholesky matrix CL, new sensor ℓ ∈ X′ L ← [ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ℓK, ℓ]T // operator expansion if K = 0 then ΣL ← (cov(ℓ, ℓ)) , CL ← � √cov(ℓ, ℓ) � ∈ R1×1 // first sensor else v ← [cov(ℓ1, ℓ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , cov(ℓK, ℓ)]T ∈ RK // matrix expansion w ← C−1 L v ∈ RK, s ← cov(ℓ, ℓ), c ← s − wTw ∈ R ΣL ← � ΣL v vT s � , CL ← � CL 0 wT c � ∈ R(K+1)×(K+1) return L, ΣL, CL to M dense linear system solves of order O((K + 1)3) each time the observation oper- ator is expanded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In the following, we therefore expound on how Σ−1 L changes under expansion of L to exploit its structure when comparing potential sensor choices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Suppose L = [ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ℓK]T has already been chosen with sensors ℓk ∈ X′, but shall be expanded by another sensor ℓ to [L, ℓ] := [ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ℓK, ℓ]T : X → RK+1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Following definition (4), the noise covariance matrix Σ[L,ℓ] of the expanded operator [L, ℓ] has the form Σ[L,ℓ] = � ΣL vL,ℓ vT L,ℓ vℓ,ℓ � = � CL 0 cT L,ℓ cℓ,ℓ � � CT L cL,ℓ 0 cℓ,ℓ � , where CLCT L = ΣL ∈ RK×K is the Cholesky decomposition of the s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' noise covariance matrix ΣL for the original observation operator L, and vL,ℓ, cL,ℓ ∈ RK, vℓ,ℓ, cℓ,ℓ ∈ R are defined through �vL,ℓ � i := cov(ℓi, ℓ), cL,ℓ := C−1 L vL,ℓ, vℓ,ℓ := cov(ℓ, ℓ), cℓ,ℓ := � vℓ,ℓ − cT L,ℓcL,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Note that Σ[L,ℓ] is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' by the assumptions posed on cov in Section 2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' consequently, cℓ,ℓ is well-defined and strictly positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' With this factorization, the expanded Cholesky matrix C[L,ℓ] with C[L,ℓ]CT [L,ℓ] = Σ[L,ℓ] can be computed in O(K2), dominated by the linear system solve with the triangular CL for obtaining cL,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' It is summarized in Algo- rithm 1 for later use in the sensor selection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Using the Cholesky decomposition, the inverse of Σ[L,ℓ] factorizes to Σ−1 [L,ℓ] = � CT L cL,ℓ 0 cℓ,ℓ �−1 � CL 0 cT L,ℓ cℓ,ℓ �−1 = � C−T L rL,ℓ 0 1/cℓ,ℓ � � C−1 L 0 rT L,ℓ 1/cℓ,ℓ � , 12 Algorithm 2: ObservabilityGain Input: observation operator L = [ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ℓK]T, Cholesky matrix CL, sensor candidate ℓ ∈ X′, state x ∈ X d ← Lx, z ← C−1 L d // preparation if K = 0 then return ℓ(xK)2/cov(ℓ, ℓ) // one sensor only else v ← [cov(ℓ1, ℓ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , cov(ℓK, ℓ)]T ∈ RK // general case w ← C−1 L v ∈ RK return (ℓ(xK)−wT z) 2 cov(ℓ,ℓ)−wT w where rL,ℓ := − 1 cℓ,ℓ C−T L cL,ℓ = − 1 cℓ,ℓ C−T L C−1 L vL,ℓ = − 1 cℓ,ℓ Σ−1 L vL,ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For an arbitrary state x ∈ X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' the norm of the extended observation [L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' ℓ](x) = � LxT,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' ℓ(x) �T ∈ RK+1 in the corresponding norm ∥·∥Σ−1 [L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓ] is hence connected to the original observation Lx ∈ RK in the original norm ∥·∥Σ−1 L via ∥ [L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' ℓ](x) ∥2 Σ−1 [L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓ] = � Lx ℓ(x) �T � ΣL vL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓ vT L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓ vℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓ �−1 � Lx ℓ(x) � = � Lx ℓ(x) �T � C−T L rL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓ 0 1/cℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓ � � C−1 L 0 rT L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓ 1/cℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓ � � Lx ℓ(x) � = � C−1 L Lx rT L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓLx + ℓ(x)/cℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓ �T � C−1 L Lx rT L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓLx + ℓ(x)/cℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓ � = (Lx)TC−T L C−1 L Lx + (rT L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓLx + ℓ(x)/cℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓ)2 = ∥Lx∥2 Σ−1 L + (rT L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓLx + ℓK+1(x)/cℓ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='ℓ)2 ≥ ∥Lx∥2 Σ−1 L .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (23) We conclude from this result that the norm ∥Lx∥Σ−1 L of any observation, and therefore also the continuity coefficient γL defined in (6), is increasing under expansion of L despite the change in norms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For any configuration θ, the observability coefficients βG(θ) and βL|W(θ) are thus non-decreasing when sensors are selected iteratively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Given a state x ∈ X and an observation operator L, we can determine the sen- sor ℓK+1 ∈ L that increases the observation of x the most by comparing the increase (rT L,ℓLx + ℓ(x)/cℓ,ℓ)2 for all ℓ ∈ L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Algorithm 2 summarizes the computation of this observability gain for use in the sensor selection algorithm (see Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Its gen- eral runtime is determined by K + 1 sensor evaluations and two linear solves with the triangular Cholesky matrix CL in O(K2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' When called with the same L and the same 13 state x for different candidate sensors ℓ, the preparation step must only be performed once, which reduces the runtime to one sensor evaluation and one linear system solve in all subsequent calls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Compared to computing ∥ [L, ℓ](x) ∥2 Σ−1 [L,ℓ] for all KL candidate sensors in the library L, we save O(KLK2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Computation of the observability coefficient We next discuss the computation of the observability coefficient βG(θ) for a given configuration θ and observation operator L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Let Σpr = UTDprU be the eigenvalue decomposition of the s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' prior covariance matrix with U = �ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ϕM � ∈ RM×M, ϕ j ∈ RM orthonormal in the Euclidean inner product, and Dpr = diag(λ1 pr, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , λM pr) a diagonal matrix containing the eigenvalues λ1 pr ≥ · · · ≥ λM pr > 0 in decreasing order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Using the eigenvector basis {ϕm}M m=1, we define the matrix M(θ) := �Lxθ(ϕ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , Lxθ(ϕM)� ∈ RK×M (24) featuring all observations of the associated states xθ(ϕ j) for the configuration θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The observability coefficient βG(θ) can then be computed as the square root of the minimum eigenvalue λmin of the generalized eigenvalue problem M(θ)TC−T L C−1 L M(θ)umin = λminD−1 pr umin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (25) Note that (25) has M real, non-negative eigenvalues because the matrix on the left is symmetric positive semi-definite, and Dpr is s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' [48]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The eigenvector umin contains the basis coefficients in the eigenvector basis {ϕm}M m=1 of the “worst-case” pa- rameter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' the infimizer of βG(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For computing βL|W(θ), we exchange the right-hand side matrix D−1 pr in (25) with the X-inner-product matrix for the states xθ(ϕ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , xθ(ϕM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The solution of the eigenvalue problem can be computed in O(M3), with an addi- tional O(MK2 + M2K) for the computation of the left-hand side matrix in (25).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The dominating cost is hidden in M(θ) since it requires KM sensor observations and K full- order model solves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' To reduce the computational cost, we therefore approximate βG(θ) with ˜βG(θ) by exchanging the full-order states xθ(ϕ j) in (24) with their reduced-order approximations ˜xθ(ϕ j).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The procedure is summarized in Algorithm 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Remark 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' If K < M, Algorithm 3 restricts the parameter space, as discussed in Sec- tion 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2, to the span of the first K eigenvectors ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ϕK encoding the least certain directions in the prior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' A variation briefly discussed in [8] in the context of the PBDW method to prioritize the least certain parameters even further is to only expand the parameter space once the observability coefficient on the subspace surpasses a prede- termined threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Sensor selection In our sensor selection algorithm, we iteratively expand the observation operator L and thereby increase the observability coefficient L for all θ ∈ P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Although this 14 Algorithm 3: SurrogateObservability Input: configuration θ ∈ P, observation operator L = [ℓ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ℓK]T with K > 0, Cholesky matrix CL N ← min{M, K} // parameter restriction M ← �L˜xθ(ϕ1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , L˜xθ(ϕN)�, S ← �� xθ(ϕi), xθ(ϕ j) � X �N i, j=1 // matrix setup Find (λmin, umin) of � C−1 L M �T � C−1 L M � umin = λminSumin // eigenvalue problem return √ λmin, umin procedure cannot guarantee finding the maximum observability over all sensor com- binations, the underlying greedy searches are well-established in practice, and can be shown to perform with exponentially decreasing error rates in closely related settings, see [49, 8, 50, 51, 52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In each iteration, the algorithm performs two main steps: A greedy search over a training set Ξtrain ⊂ P to identify the configuration θ ∈ Ξtrain for which the observability coefficient βG(θ) is minimal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' A data-matching step to identify the sensor in the library that maximizes the observation of the “worst-case” parameter at the selected configuration θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The procedure is summarized in Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' It terminates when Kmax ≤ KL sensors have been selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='5 In the following, we explain its computational details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Preparations In order to increase βG(θ) uniformly over the hyper-parameter domain P, we con- sider a finite training set, Ξtrain ⊂ P, that is chosen to be fine enough to capture the θ- dependent variations in xθ(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We assume a reduced-order model is available such that we can compute approximations ˜xθ(ϕm) ≈ xθ(ϕm) for each θ ∈ Ξtrain and 1 ≤ m ≤ M within an acceptable computation time while guaranteeing the accuracy requirement (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' If necessary, the two criteria can be balanced via adaptive training domains (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', [53, 54]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' If storage allows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', with projection-based surrogate models), we only compute the surrogate states once and avoid unnecessary re-computations when up- dating the surrogate observability coefficients ˜βG(θ) in each iteration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' As a first “worst-case” parameter direction, u0, we choose the vector ϕ1 with the largest prior uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Likewise, we choose the “worst-case” configuration θK ∈ P as the one for which the corresponding state ˜xθ(ϕ1) is the largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 5This termination criterion can easily be adapted to prescribe a minimum value of the observability coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' This value should be chosen with respect to the observability βG(L) achieved with the entire sensor library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 15 Algorithm 4: SensorSelection Input: sensor library L ⊂ X′, training set Ξtrain ⊂ P, maximum number of sensors Kmax ≤ |KL|, surrogate model ˜ Mθ, covariance function cov : L × L → R Compute Σpr = �ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' , ϕM �T Dpr �ϕ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' ϕM � // eigenvalue decomposition For all θ ∈ Ξtrain,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 1 ≤ m ≤ M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' compute ˜xθ(ϕm) // preparation K ← 0,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' θ0 ← arg maxθ∈Ξtrain ∥˜xθ(ϕ1)∥X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' u0 ← ϕ1 // initialization while K < Kmax do Solve full-order equation MθK(xK,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' uK) for xK // "worst-case" state ℓK+1 ← arg maxℓ∈L ObservabilityGain(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' CL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' ℓ) // sensor selection L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' ΣL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' CL ← CholeskyExpansion(L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' ΣL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' CL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' ℓK+1) // expansion K ← K + 1 for θ ∈ Ξtrain do ˜βL|W(θ),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' umin(θ) ← SurrogateObservability(θ,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' CL) // update coefficients θK ← arg minθ∈Ξtrain ˜βL|W(θ) // greedy step uK ← �min{M,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='K} m=1 [umin(θK)]m ϕm return L,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' CL Data-matching step In each iteration,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' we first compute the full-order state xK = xθK(uK) at the “worst- case” parameter uK and configuration θK.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We then choose the sensor ℓK+1 which most improves the observation of the “worst-case” state xK under the expanded observation operator [LT, ℓK+1]T and its associated norm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We thereby iteratively approximate the information that would be obtained by measuring with all sensors in the library L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For fixed θK and in combination with selecting x to have the smallest observability in Wθ, we arrive at an algorithm similar to worst-case orthogonal matching pursuit (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' [8, 9]) but generalized to deal with the covariance function cov in the noise model (3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We use the full-order state xθK(uK) rather than its reduced-order approxi- mation in order to avoid training on local approximation inaccuracies in the reduced- order model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Here, by using the “worst-case” parameter direction uK, we only require a single full-order solve per iteration instead of the M required for setting up the entire posterior covariance matrix ΣL,θ post.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Greedy step We train the observation operator L on all configurations θ ∈ Ξtrain by varying for which θ the “worst-case” state is computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Specifically, we follow a greedy approach 16 where, in iteration K, we choose the minimizer θK of βG(θ) over the training domain Ξtrain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', the configuration for which the current observation operator L is the least advantageous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The corresponding “worst-case” parameter uK is the parameter direction for which the least significant observation is achieved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' By iteratively increasing the observability at the “worst-case” parameters and hyper-parameters, we increase the minimum of βG(θ) throughout the training domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Since the computation of ˜βG(θ) requires as many reduced-order model solves as needed for the posterior covariance matrix over the surrogate model, it is possible to directly target an (approximated) OED utility function in the greedy step in place of ˜βL|W(θ) without major concessions in the computational efficiency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The OMP step can then still be performed for the “worst-case” parameter with only one full-order model solve, though its benefit for the utility function should be evaluated carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Runtime Assuming the dominating computational restriction is the model evaluation to solve for xθ(u) – as is usually the case for PDE models – then the runtime of each iteration in Algorithm 4 is determined by one full-order model evaluation, and KL sensor measure- ments of the full-order state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Compared to computed the posterior covariance matrix for the chosen configuration, the OMP step saves N − 1 full-order model solves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The other main factor in the runtime of Algorithm 4 is the |Ξtrain|M reduced-order model evaluations with KL sensor evaluations each that need to be performed in each iteration (unless they can be pre-computed).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The parameter dimension M not only en- ters as a scaling factor, but also affects the cost of the reduced-order model itself since larger values of M generally require larger or more complicated reduced-order models to achieve the desired accuracy (16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In turn, the computational cost of the reduced- order model indicates how large Ξtrain may be chosen for a given computational budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' While some cost can be saved through adaptive training sets and models, overall, this connection to M stresses the need for an adequate initial parameter reduction as dis- cussed in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Numerical Results We numerically confirm the validity of our sensor selection approach using a geo- physical model of a section of the Perth Basin in Western Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The basin has raised interest in the geophysics community due to its high potential for geothermal energy, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', [55, 56, 57, 58, 59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We focus on a subsection that spans an area of 63 km × 70 km and reaches 19 km below the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The model was introduced in [60] and the presented section of the model was discussed extensively in the context of MOR in [61, 62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In particular, the subsurface temperature distribution is described through a steady-state heat conduction problem with different subdomains for the geo- logical layers, and local measurements may be obtained through boreholes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The bore- hole locations need to be chosen carefully due to their high costs (typically several million dollars, [63]), which in turn motivates our application of Algorithm 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For demonstration purposes, we make the following simplifications to our test model: 1) 17 Figure 1: Schematic overview of the Perth Basin section including (merged) geological layers, depths for potential measurements, and configuration range for thermal conductivity θ on each subdomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The bounds are obtained from the reference values (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' [60, 61]) with a ±50% margin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Adapted from [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We neglect radiogenic heat production;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 2) we merge geological layers with similar conductive behaviors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' and 3) we scale the prior to emphasize the influence of different sensor measurements on the posterior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' All computations were performed in Python 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='7 on a computer with a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='3 GHz Quad-Core Intel Core i5 processor and 16 GB of RAM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The code will be available in a public GitHub repository for another geophysical test problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='6 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Model Description We model the temperature distribution xθ with the steady-state PDE −∇ (θ∇xθ) = 0 in Ω := (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2714) × (0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='9) × (0, 1) ⊂ R3, (26) where the domain Ω is a non-dimensionalized representation of the basin, and θ : Ω → R>0 the local thermal conductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The section comprises three main geological layers Ω = � i=1,2,3 Ωi, each characterized by different rock properties, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' thermal conductiv- ity θ|Ωi ≡ θi shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We consider the position of the geological layers to be fixed as these are often determined beforehand by geological and geophysical surveys but allow the thermal conductivity to vary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In a slight abuse of notation, this lets us identify the field θ with the vector θ = (θ1, θ2, θ3) ∈ P := [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='453, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='360] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='448, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='343] × [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='360, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='081].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' in the hyper-parameter domain P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We impose zero-Dirichlet boundary conditions at the surface7, and zero-Neumann (“no-flow”) boundary conditions at the lateral faces of the domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The remaining boundary ΓIn corresponds to an area spanning 63 km × 70 km area in the Perth basin 19 km below the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' At this depth, local variations in the heat flux have mostly stabilized which makes modeling possible, but since most boreholes – often originat- ing from hydrocarbon exploration – are found in the uppermost 2 km we treat it as 6The Perth Basin Model is available upon request from the third author.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 7Non-zero Dirichlet boundary conditions obtained from satellite data could be considered via a lifting function and an affine transformation of the measurement data (see [62]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 18 Geology Thermal Conductivity Creteaous- 2 with 1 E [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='453,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='360], [0refl1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='9065 380 m Yarragadee Eneabba- 760 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2 with 02 E [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='448,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='343], [0rerl2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='9855 Lesueur D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1 1140 m Permian .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' with 3 E [0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='360,1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='081], [0refl3 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='7205 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2 Basement 1520 m 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='6 1900 m = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2443 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='8 X1uncertain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Specifically, we model it as a Neumann boundary condition n · ∇xθ = u · p a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' on ΓIn := {0} × [0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='9] × [0, 1] where n : ΓIn → R3 is the outward pointing unit normal on Ω, p : ΓIn → R5 is a vector composed of quadratic, L2(ΓIn)-orthonormal polynomials on the basal boundary that vary either in north-south or east-west direction, and u ∼ πpr = N(upr, Σpr) is a random variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The prior is chosen such that the largest uncertainty is attributed to a constant entry in p, and the quadratic terms are treated as the most certain with prior zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' This setup reflects typical geophysical boundary conditions, where it is most common to assume a constant Neumann heat flux (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', [61]), and sometimes a linear one (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', [60]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' With the quadratic functions, we allow an additional degree of freedom than typically considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The problem is discretized using a linear finite element (FE) basis of dimension 132,651.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The underlying mesh was created with GemPy ([64]) and MOOSE ([65]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Since the FE matrices decouple in θ, we precompute and store an affine decomposition using DwarfElephant ([61]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Given a configuration θ and a coefficient vector u for the heat flux at ΓIn, the computation of a full-order solution xθ(u) ∈ X then takes 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='96 s on average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We then exploit the affine decomposition further to construct a reduced basis (RB) surrogate model via a greedy algorithm (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' [49, 66]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Using the inner product8 ⟨x, φ⟩X := � Ω ∇x · ∇φdΩ and an a posteriori error bound ∆(θ), we prescribe the relative target accuracy max u∈RM ∥xθ(u) − ˜xθ(u)∥X ∥˜xθ(u)∥X ≤ max u∈RM ∆(θ) ∥˜xθ(u)∥X < ε := 1e − 4 (27) to be reached for 511,000 consecutively drawn, uniformly distributed samples of θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The training phase and final computational performance of the RB surrogate model are summarized in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The speedup of the surrogate model (approximately a factor of 3,000 without error bounds) justifies its offline training time, with computational savings expected already after 152 approximations of βG(θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For taking measurements, we consider a 47 × 47 grid over the surface to represent possible drilling sites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' At each, a single point evaluation9 of the basin’s temperature distribution may be made at any one of five possible depths as shown in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In total, we obtain a set L ⊂ Ω of 11, 045 admissible points for measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We model the noise covariance between sensors ℓχ, ℓ˜χ ∈ L at points χ, ˜χ ∈ Ω via cov(ℓχ, ℓ˜χ) := a + b − y(h) with the exponential variogram model y(h) := a + (b − a) �3 2 max{h c, 1} − 1 2 max{h c, 1}3 � 8Note that ⟨·, ·⟩X is indeed an inner product due to the Dirichlet boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 9Point evaluations are standard for geophysical models because a borehole (diameter approximately 1 m) is very small compared to the size of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 19 0 10 20 30 40 50 RB dimension 10 3 10 2 10 1 100 norm of the error max relative error bound true relative error Reduced-order model RB dimension 83 training time 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='58 min training accuracy 1e-4 RB solve 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='97 ms �→ speedup 3,058 RB error bound 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='78 ms �→ speedup 515 Figure 2: Training of the RB surrogate model for the Perth Basin section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' On the left: Maximum relative error bound (27) in the course of the greedy algorithm, computed over the training set Ξtrain together with the true relative error at the corresponding configuration θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' On the right: Performance pointers for the obtained RB model after (27) was reached;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' online computation times and speedups are averages computed over 1000 randomly drawn configurations θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' where h2 := (χ2 − ˜χ2)2 + (χ3 − ˜χ3)2 is the horizontal distance between the points and a := 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2054073480730403 (sill) b := 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='6850672040263555 (nugget) c := 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='606782733391228 (range) The covariance function was computed via kriging (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' [67]) from the existing mea- surements [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' With this covariance function, the noise between measurements at any two sensor locations is increasingly correlated the closer they are on the horizontal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Note that for any subset of sensor locations, the associated noise covariance ma- trix remains regular as long as each sensor is placed at a distinct drilling location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We choose this experimental setup because measurements in typical geothermal data sets are often made at the bottom of a borehole (“bottom hole temperature measurements”) within the first 2 km below the surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Restricted Library To test the feasibility of the observability coefficient for sensor selection, we first consider a small sensor library (denoted as L5×5 below) with 25 drilling locations po- sitioned on a 5 × 5 grid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We consider the problem of choosing 8 pair-wise different, unordered sensor locations out of the given 25 positions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' this is a combinatorial prob- lem with 1,081,575 possible combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Sensor selection We run Algorithm 4, using the RB surrogate model and a training set Ξtrain ⊂ P with 512,000 configurations on an 80 × 80 × 80 regular grid on P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' When new sensors are chosen,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' the surrogate observability coefficient ˜βG(θ) increases monotonously with a strong incline just after the initial M = 5 sensors,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' followed by a visible stagnation (see Figure 3a) as is often observed for similar OMP-based sensor selection algorithms 20 5 6 7 8 number of sensors 10 4 10 3 10 2 10 1 observability coefficient mean observability coefficient min observability coefficient improvement with next sensor observability coefficient,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' fixed config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (a) Observability during sensor selection 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='14 observability coefficient 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='5 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='5 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='5 max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' observability A-OED D-OED E-OED proposal proposal, fixed (b) Histogram of βG(θref) Figure 3: Observability coefficient for different methods when choosing 8 out of 25 sensor locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Left: Minimum and mean over θ of ˜βG(θ) as well as βG(θref) obtained in the course of running Algorithm 4 once for 512,000 configurations and once for the training set {θref}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Right: Distribution of βG(θref) over all possible sensor combinations with indicators for the A-, D-, and E-optimal choices, the combination with maximum observability, and the sensors chosen by the Algorithm 4 with Ξtrain-training (“proposal”, purple, marked “x”) and θref-training (“proposal, fixed”, turquoise, marked “+”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Note that the height of the indicator line was chosen solely for readability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', [8, 69, 70, 7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Algorithm 4 terminates in 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='93 min with a minimum reduced-order observability of ˜βG(θ) = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='3227e-2 and an average of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0995e-1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' At the reference configuration θref, the full-order observability coefficient is βG(θref) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0985, slightly below the reduced-order average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We call this training procedure “Ξtrain-training” here- after and denote the chosen sensors as “Ξtrain-trained sensor set” in the subsequent text and as “proposal” in the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In order to get an accurate understanding of how the surrogate model ˜xθ(u) and the large configuration training set Ξtrain influence the sensor selection, we run Algorithm 4 again, this time restricted on the full-order FE model xθref(u) at only the reference con- figuration θref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The increase in βG(θref) in the course of the algorithm is shown in Fig- ure 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The curve starts significantly above the average for Ξtrain-training, presumably because conflicting configurations cannot occur, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=', when one sensor would signifi- cantly increase the observability at one configuration but cause little change in another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' However, in the stagnation phase, the curve comes closer to the average achieved with Ξtrain-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The computation finishes within 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='53 s, showing that the long runtime before can be attributed to the size of Ξtrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The final observability coefficient with 8 sensors is βG(θref) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2647e-1, above the average over ˜βG(θ) achieved training on Ξtrain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We call this training procedure “θref-training” hereafter, and the sensor configu- ration “θref-trained” in the text or “proposal, fixed config.” in the plots.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Comparison at the reference configuration For comparing the performance of the Ξtrain- and θref-trained sensor combinations, we compute – at the reference configuration θref – all 1,081,575 posterior covariance matrices Σθref,L post for all unordered combinations L of 8 distinct sensors in the sensor li- brary L5×5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For each matrix, we compute the trace (A-OED criterion), the determinant (D-OED criterion), the maximum eigenvalue (E-OED criterion), and the observability coefficient βG(θref).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' This lets us identify the A-, D-, and E-optimal sensor combina- 21 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2 Observability Coefficient 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='25 100 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='25 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='5 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='75 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='25 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='5 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='75 posterior trace A-optimal sensor combination proposal proposal, fixed configuration maximum observability coefficient all 25 sensors best sensor selection 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='587 % proposal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='022 % proposal, fixed config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='778 % max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' observability Figure 4: Distribution of trace(ΣL,θ post) for θ = θref over all 1,081,575 combinations for choosing 8 out of the 25 sensor locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' On the left: distribution of trace(ΣL,θ post) against the observability coefficient βG(θref).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Note that the marginal distribution of the horizontal axis is provided in Figure 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' On the right: histogram of trace(ΣL,θ post) (marginal distribution for the plot on the left) with for the different sensor combinations (in percent out of 1,081,575 combinations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The plots include markers for the A-optimal sensor choice, the sensors chosen by Algorithm 4 with Ξtrain-training (“proposal”) and with {θref}-training (“proposal, fixed configuration”), the sensor combination with maximum observability βG(θref), and when all 25 sensors are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The total runtime for these computations is 4 min – well above the 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='53 s of θref-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The (almost) 8 min for Ξtrain-training remain reasonable considering it is trained on |Ξtrain| = 512, 000 configurations and not only θref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' A histogram for the distribution of βG(θref) is given in Figure 3b with markers for the values of the A-, D-, and E-optimal choices and the Ξtrain- and θref-trained ob- servation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Out of these five, the D-optimal choice has the smallest value, since the posterior determinant is influenced less by the maximum posterior eigenvalue and hence the observability coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In contrast, both the A- and E-optimal sen- sor choices are among the 700 combinations with the largest βG(θref) (this corresponds to the top 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='065%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The θref-trained sensors have similar observability and are even among the top 500 combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For the Ξtrain- trained sensors, the observability co- efficient is smaller, presumably because Ξtrain-training is not as optimized for θref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Still, it ranks among the top 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='705 % of sensor combinations with the largest observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In order to visualize the connection between the observability coefficient βG(θref) and the classic A-, D-, and E-OED criteria, we plot the distribution of the posterior covariance matrix’s trace, determinant, and maximum eigenvalue over all sensor com- binations against βG(θ) in Figures 4, 5, 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Overall we observe a strong correlation between the respective OED criteria and βG(θref): It is the most pronounced in Figure 6 for E-optimality, and the least pronounced for D-optimality in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For all OED 22 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2 Observability Coefficient 10 5 10 4 10 3 10 2 10 1 100 101 Posterior determinant D-optimal sensor combination proposal proposal, fixed configuration maximum observability coefficient all 25 sensors best sensor selection 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='252 % proposal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='081 % proposal, fixed config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='923 % max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' observability Figure 5: Distribution of the posterior determinant det(ΣL,θ post) for θ = θref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' See Figure 4 for details about the plot structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2 Observability Coefficient 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='5 100 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='5 101 101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='5 maximum posterior eigenvalue E-optimal sensor combination proposal proposal, fixed configuration maximum observability coefficient all 25 sensors best sensor selection 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='679 % proposal 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='001 % proposal, fixed config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='080 % max.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' observability Figure 6: Distribution of the maximum eigenvalue of the posterior covariance matrix ΣL,θ post for θ = θref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' See Figure 4 for details about the plot structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Note that the θref-trained sensor combination has the 101-st smallest maximum posterior eigenvalue among all 1,081,575 possibilities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 23 design criterion training pctl A-OED D-OED E-OED θref Ξtrain 99-th 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='5835 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='8508 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1724 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2223 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2512 95-th 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2747 26.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='8430 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='3601 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='7846 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0374 75-th 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='5141 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='8600 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0532 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1419 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='8106 50-th 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1527 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='4641 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0159 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0438 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2354 25-th 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0414 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='3669 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0035 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0068 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0621 Figure 7: Ranking in βG(θref) of the A-, D-, E- optimal and the θref- and Ξtrain-trained sensor choices for all possible combinations of choosing 8 unordered sensors in the library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Left: Boxplots obtained over 200 random sensor libraries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Right: worst-case ranking (in percent) of the corresponding percentiles (“pctl”).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' criteria, the correlation becomes stronger for smaller scaling factors σ2 and weakens for large σ2 when the prior is prioritized (plots not shown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' This behavior aligns with the discussion in Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1 that βG(θ) primarily targets the largest posterior eigenvalue and is most decisive for priors with higher uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Comparison for different libraries We finally evaluate the influence of the library L5×5 on our results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' To this end, we randomly select 200 sets of new measurement positions, each consisting of 25 drilling locations with an associated drilling depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For each library, we run Algorithm 4 to choose 8 sensors, once with Ξtrain-training on the surrogate model, and once with the full-order model at θref only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For comparison, we then consider in each library each possible combination of choosing 8 unordered sensor sets and compute the trace, determinant, and maximum eigenvalue of the associated posterior covariance matrix at the reference configuration θref together with its observability coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' This lets us identify the A-, D-, and E-optimal sensor combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Figure 7 shows how βG(θref) is distributed over the 200 libraries, with percentiles provided in the adjacent table.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' For 75% of the libraries, the A- and E-optimal, and the Ξtrain- and θref-trained sensor choices rank among the top 1% of combinations with the largest observability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Due to its non-optimized training for θref, the Ξtrain-trained sensor set performs slightly worse than what is achieved with θref-training, but still yields a comparatively large value for βG(θref).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In contrast, overall, the D-optimal sensor choices have smaller observability coefficients, presumably because the minimization of the posterior determinant is influenced less by the maximum posterior eigenvalue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The ranking of the Ξtrain- and θref-trained sensor configurations in terms of the pos- terior covariance matrix’s trace, determinant, and maximum eigenvalue over the 200 libraries is given in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Both perform well and lie for 75% of the libraries within the top 1% of combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' As the ranking is performed for the configuration param- eter θref, the θref-trained sensor combination performs better, remaining in 95% of the libraries within the top 5% of sensor combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 24 A-OED D-OED E-OED obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 100 101 102 103 104 105 106 ranking proposal, flexible configuration opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' top 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='01% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1% 1% 5% 25% 100% percentile pctl A-OED D-OED E-OED βG(θref) 99-th 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='9240 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2372 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='8391 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2512 95-th 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='9093 3.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0621 A-OED D-OED E-OED obs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' coef.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 100 101 102 103 104 105 106 ranking proposal, fixed configuration opt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' top 10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='01% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1% 1% 5% 25% 100% percentile pctl A-OED D-OED E-OED βG(θref) 99-th 2.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='7846 75-th 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1155 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='4698 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='3549 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1419 50-th 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0224 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='1212 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0687 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0438 25-th 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0041 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0181 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0138 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='0068 Figure 8: Ranking of the posterior covariance matrix Σθref,L post in terms of the A-, D-, E-OED criteria and the observability coefficient βG(θref) when the observation operator GL,θ is chosen with Algorithm 4 and Ξtrain-training (top) or θref-training (bottom).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The ranking is obtained by comparing all possible unordered combinations of 8 sensors in each sensor library.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' On the left: Boxplots of the ranking over 200 sensor libraries;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' on the right: ranking (in percent) among different percentiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 25 (a) upmost layer, Ξtrain-training (b) upmost layer, θref-training (c) lowest layer, Ξtrain-training (d) lowest layer, θref-training Figure 9: Sensor positions chosen by Algorithm 4 from a grid of 47 × 47 available horizontal positions with available 5 depths each, though only the lowest (bottom) and upmost (top) layers were chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The underlying plot shows cuts through the full-order solution xθ(u) at θ = θref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Left: Ξtrain-training with the RB surrogate model on a training set Ξtrain ⊂ P with 10,000 random configurations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' runtime 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='19 s for 10 sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Right: θref-training with full-order model at reference parameter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' runtime 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='85 s for 10 sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Unrestricted Library We next verify the scalability of Algorithm 4 to large sensor libraries by permitting all 2,209 drilling locations, at each of which at most one measurement may be taken at any of the 5 available measurement depths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Choosing 10 unordered sensors yields approximately 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='29e+33 possible combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Using the RB surrogate model from before, we run Algorithm 4 once on a training grid Ξtrain ⊂ P consisting of 10,000 randomly chosen configurations using only the surrogate model (runtime 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='19 s), and once on the reference configuration θref using the full-order model (runtime 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='85 s) for comparison.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We terminate the algorithm whenever 10 sensors are selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Compared to the training time on L5×5 before, the results confirm that the size of the library itself has little influence on the overall runtime but that the full-order computations and the size of Ξtrain relative to the surrogate compute dominate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The sensors chosen by the two runs of Algorithm 4 are shown in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' They share many structural similarities: Depth: Despite the availability of 5 measurement depths, sensors have only been chosen on the lowest and the upmost layers with 5 sensors each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The lower sen- sors were chosen first (with one exception, sensor 3 in θref-training), presumably 26 z 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='4 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='2 0 1 zbecause the lower layer is closer to the uncertain Neumann boundary condition and therefore yields larger measurement values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Pairing Each sensor on the lowest layer has a counterpart on the upmost layer that has almost the same position on the horizontal plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' This pairing targets noise sensitivity: With the prescribed error covariance function, the noise in two measurements is increasingly correlated the closer the measurements lie horizon- tally, independent of their depth coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Choosing a reference measurement near the zero-Dirichlet boundary at the surface helps filter out noise terms in the lower measurement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Organization On each layer, the sensors are spread out evenly and approxi- mately aligned in 3 rows and 3 columns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The alignment helps distinguish be- tween the constant, linear, and quadratic parts of the uncertain Neumann flux function in north-south and east-west directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Figure 10 (left side) shows the increase in the observability coefficients ˜βG(θ) (for Ξtrain-training) and βG(θref) (for θref-training) over the number of chosen sensors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We again observe a strong initial incline followed by stagnation for the Ξtrain-trained sen- sors, whereas the curve for θref-training already starts at a large value to remain then almost constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The latter is explained by the positions of the first 5 sensors in Fig- ure 9 (right), as they are already spaced apart in both directions for the identification of quadratic polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' In contrast, for Ξtrain-training, the “3 rows, 3 columns” structure is only completed after the sixth sensor (c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Figure 9, left).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' With 6 sensors, the observ- ability coefficients in both training schemes have already surpassed the final observ- ability coefficients with 8 sensors in the previous training on the smaller library L5×5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The final observability coefficients at the reference parameter θref are βG(θref) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='4042 for θref-training, and βG(θref) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='3595 for Ξtrain-training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' As a final experiment, we compare the eigenvalues of the posterior covariance ma- trix ΣL,θref post for the Ξtrain- and θref-trained sensors against 50,000 sets of 10 random sen- sors each.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We confirm that all 50,000 sensor combinations comply with the combina- torial restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Boxplots of the eigenvalues are provided in Figure 10 (right side).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The eigenvalues of the posterior covariance matrix with sensors chosen by Algorithm 4 are smaller10 than all posterior eigenvalues for the random sensor combinations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Conclusion In this work, we analyzed the connection between the observation operator and the eigenvalues of the posterior covariance matrix in the inference of an uncertain pa- rameter via Bayesian inversion for a linear, hyper-parameterized forward model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' We identified an observability coefficient whose maximization decreases the uncertainty in the posterior probability distribution for all hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' To this end, we proposed 10Here we compare the largest eigenvalue of one matrix to the largest eigenvalue of another, the second largest to the second largest, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 27 5 6 7 8 9 10 number of sensors 10 3 10 2 10 1 100 observability coefficient mean observability coefficient min observability coefficient improvement with next sensor observability coefficient, fixed config.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' 1 2 3 4 5 posterior eigenvalue (largest to smallest) 10 3 10 2 10 1 100 101 random combinations with 10 sensors each proposal proposal, fixed config Figure 10: Left: Observabity coefficients during sensor selection with Ξtrain- and θref-training for a library with 11,045 measurement positions and combinatorial restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Shown are 1) the minimum and mean surrogate observability coefficient ˜βG(θ) over a training set with 10,000 random configurations with final values minθ ˜βG(θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='4160 and meanθ ˜βG(θ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='6488, and 2) the full-order observability coefficient βG(θref) when training on the reference parameter θref alone (final value βG(θref) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content='4042).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Right: Boxplots for the 5 eigenvalues of the posterior covariance matrix ΣL,θ post over 50,000 sets of 10 sensors chosen uniformly from a 5 × 47 × 47 grid with imposed combinatorial restrictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The eigenvalues are compared according to their order from largest to smallest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Indicated are also the eigenvalues for the Ξtrain-trained (purple, “x”-marker) and θref-trained (turquoise, “+”-marker) sensors from Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' a sensor selection algorithm that expands an observation operator iteratively to guaran- tee a uniformly large observability coefficient for all hyper-parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Computational feasibility is retained through a reduced-order model in the greedy step and an OMP search for the next sensor that only requires a single full-order model evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' The validity of the approach was demonstrated on a large-scale heat conduction problem over a section of the Perth Basin in Western Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Future extensions of this work are planned to address 1) high-dimensional parameter spaces through parameter reduc- tion techniques, 2) the combination with the PBDW inf-sup-criterion to inform sensors by functionalanalytic means in addition to the noise covariance, and 3) the expansion to non-linear models through a Laplace approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' Acknowledgments We would like to thank Tan Bui-Thanh, Youssef Marzouk, Francesco Silva, An- drew Stuart, Dariusz Ucinski, and Keyi Wu for very helpful discussions, and Florian Wellmann at the Institute for Computational Geoscience, Geothermics and Reservoir Geophysics at RWTH Aachen University for providing the Perth Basin Model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' This work was supported by the Excellence Initiative of the German federal and state gov- ernments and the German Research Foundation through Grants GSC 111 and 33849990/GRK2379 (IRTG Modern Inverse Problems).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/MtFLT4oBgHgl3EQfNS8Q/content/2301.12019v1.pdf'} +page_content=' This project has also received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 re- search and innovation programme (grant agreement n° 818473), the US Department of Energy (grant DE-SC0021239), and the US Air Force Office of Scientific Research (grant FA9550-21-1-0084).' metadata={'source': 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+ +Zachary Hamblin-Frohman, Koralalage Don Raveen Amarasekera & Stefanie I. +Becker + +School of Psychology, The University of Queensland, Brisbane, Australia. + + + + +2 + +Abstract +It is well-known that visual attention can be tuned in a context-dependent manner to +elementary features, such as searching for all redder items or the reddest item, supporting a +relational theory of visual attention. However, in previous studies, the conditions were often +conducive for relational search, allowing successfully selecting the target relationally on 50% +of trials or more. Moreover, the search displays were often only sparsely populated and +presented repeatedly, rendering it possible that relational search was based on context +learning and not spontaneous. The present study tested the shape of the attentional tuning +function in 36-item search displays, when the target never had a maximal feature value (e.g., +was never the reddest or yellowest item), and when only the target colour but not the context +colour was known. The first fixations on a trial showed that these displays still reliably +evoked relational search, even when participants had no advance information about the +context and no on-task training. Context learning further strengthened relational tuning on +subsequent trials, but was not necessary for relational search. Analysing the progression of +visual search within a singe trial showed that attention is first guided to the relationally +maximal item (e.g., reddest), then the next-maximal (e.g., next-reddest) item, and so forth, +before attention can hone in on target-matching features. In sum, the results support two +tenets of the relational account, that information about the dominant feature in a display can +be rapidly extracted and used to guide attention to the relatively best-matching features. + + + +3 + +Introduction +It is well-known that we cannot consciously process all objects in a visual scene at once. +To address this, visual attention selects objects for in-depth processing, often guiding our +gaze to relevant parts in a scene (e.g., Deubel & Schneider, 1996). Much effort has been +devoted to determine which items in a scene will be attended first, and more generally, to +identify the processes involved in creating our rich mental representation of the visual +environment (for a review, see Carrasco, 2011; Wolfe, 2021). +To date, it is widely accepted that attention can be guided by both, bottom-up, stimulus- +driven processes and top-down, goal-driven processes (e.g., Wolfe, 2020). For example, +attention can be reflexively drawn to visually salient events such as a bright flash, a +movement, or the sudden appearance of an object (e.g., Theeuwes, 2004, 2013), or it can be +top-down tuned to select items with certain attributes (e.g., colours: red, green) to help goal- +related behaviours such as finding a friend in a crowd (e.g., Desimone & Duncan, 1995; +Wolfe, 1994, 2021) . Correspondingly, current models of visual attention typically include +both a bottom-up and a top-down component to predict which items in a visual scene will be +selected first (e.g., Wolfe, 1994, 2021). Top-down tuning is typically modelled as an increase +or decrease in the firing rate of sensory neurons in response to specific stimulus attributes +(e.g., red, green; Navalpakkam & Itti, 2007; Yu, Hanks & Geng, 2022). For example, when +looking for an orange in a fruit basket, we would tune attention to orange, which increases the +output of neurons that respond to orange and prioritises colour-matching items for selection. + +It is commonly assumed that attention is tuned to the feature value that a person is looking +for (e.g., particular shade of orange; e.g., Duncan & Humphreys, 1989; Navalpakkam & Itti, +2007). However, to date, there are also several accounts of non-veridical tuning. +Navalpakkam and Itti (2007) noted that tuning attention to the exact target feature value +would not be beneficial when the target is very similar to surrounding irrelevant non-target + +4 + +items, as tuning attention to, for example, orange, would also boost the response gain of red- +orange or yellow-orange, leading to a poor signal-to-noise ratio (SNR). They proposed that +attention would be tuned to a feature value that is slightly shifted away from similar +nontargets, to increase the SNR (e.g., to yellow-orange, when an orange target is presented +among red-orange items; Navalpakkam & Itti, 2007). According to their optimal tuning +account, attention is always tuned to the feature value that maximises the ability to +discriminate the target from the non-targets (i.e., maximise the SNR). Thus, attention would +only be tuned to the exact target feature value when the target is presented alone or among +dissimilar other items. In less discriminable cases, attention should be shifted to a slightly +exaggerated target feature value. In line with this idea, a perceptual probe task revealed that a +slightly shifted non-target colour was likely to be mistaken for the target when the target was +consistently embedded among similar featured non-targets in a prior visual search. For +example, if an orange target was always presented among similar, yellow-orange nontargets +in a visual search task, participants would pick a slightly redder (red-orange) colour as the +target colour in intermixed probe trials (Navalpakkam & Itti, 2007; see also Geng et al., 2017; +Scolari et al., 2012). +Another account of non-veridical tuning is the Relational Account, which proposes that +attention may not at all be tuned to a specific feature value. As noted by Becker (2010), +tuning attention to a particular feature value may not be beneficial in natural environments, as +specific feature values such as the size, shape and colour of objects vary a lot with differences +in distance, perspective and shading. To allow efficient selection of the targets in these noisy +environments, attention could be tuned in a context-dependent manner to objects, biasing +attention to a feature that the target has relative to the other items in the context (e.g., redder, +larger, darker; Becker, 2010). For example, when searching for an orange in a fruit basket, +the visual system would quickly assess the distribution of colours in the visual scene to + +5 + +determine the dominant colours, and tune attention to the relative colour that would best +discriminate the target from the dominant coloured non-target items. Thus, attention would be +tuned to all redder items or the reddest item when the fruit basket (or visual scene) contains +many yellow or green objects, and tune attention to all yellower items or the yellowest item +when the fruit basket (or visual scene) contains many red objects. As a consequence of this +very broad tuning, the item that maximally fulfils the relevant feature relationship will be +selected first (e.g., reddest or yellowest item). Hence, when attention is tuned to all redder +items, the reddest item in the visual field will be selected first, followed by the next-reddest, +and so forth. +In line with this prediction, several visual search studies showed that when an orange +target is presented among mostly yellow(er) items, a red irrelevant distractor was more likely +to be attended first than the target, even when it was quite dissimilar from the target, +suggesting that attention was biased to all redder items, or the reddest item (e.g., Becker, +2010; Becker et al., 2013; Hamblin-Frohman & Becker, 2021; York & Becker, 2020). +Selection of the distractor was reflected both in higher response times (RT) to the target, as +well as in a high proportion of first eye movements to the red distractor. Several studies also +included a visually salient distractor with a dissimilar colour (e.g., blue), but found no or only +very weak effects of saliency, ruling out that selection of the relatively matching (e.g., red) +distractor was mediated by bottom-up, stimulus-driven processes (e.g., Martin & Becker, +2018; York & Becker, 2020). + +Both optimal tuning and the relational account propose that attention can be tuned to +features in a non-veridical manner but propose very different underlying mechanisms. +Subsequent studies combined the two paradigms to determine whether attention is tuned to +all relatively matching features or an optimally shifted feature value (see Figure 1 for an +illustration of the different accounts). + +6 + + +Figure 1. Overview of different theoretical tuning functions to tune attention to the target in +the example display (left), which shows an orange target (T) among yellow-orange and +yellow non-targets (NT). The standard view (2nd from left) is that attention would be tuned to +the target colour (T). According to Optimal Tuning (3rd from left), attention would be tuned +to a slightly exaggerated, ‘optimal’ target feature value that is shifted away from the +nontargets, to increase the SNR (optimal colour; O). According to the Relational Account +(right), attention would be tuned to the relative colour of the target, that the target has relative +to the other items in the surround (here: redder), which can include vastly dissimilar colours +(relatively matching colour, R). + +The results of two studies showed that attention is biased to all relatively matching items, +indicating that early visual selection (i.e., which item is selected first) follows the predictions +of the relational account. However, target identification judgements were skewed towards the +slightly shifted feature value, showing that perceptual decision-making (i.e., decisions about +whether a selected item is the target or not) is best described by the optimal tuning account +(Hamblin-Frohman & Becker, 2021; Yu et al., 2022). Thus, visual attention can be tuned +rather broadly towards the target’s relative feature, leading to frequent selection of very +dissimilar colours that only share the target’s relative colour, whereas perceptual decision- +making is tuned rather sharply towards an exaggerated target feature value, so that only very +similar colours are mistaken for the target colour. +Despite the progress in describing the attentional tuning function, there are still significant +knowledge gaps. First, one important claim of the relational account is that the visual system +quickly assesses the dominant feature in a scene and biases attention to the relative feature +that discriminates the target from the (dominant feature in the) context. However, this fast, +automatic evaluation of the context has never been tested, as the target and non-targets were + +Standard Tuning +ExampleDisplay +Optimal Tuning +(and Optimal Tuning +Relational Tuning +(similar nontargets) +dissimilarnontargets) +NT +NT +R +NTNT +R +NT +NT +R7 + +typically repeated numerous times in previous studies, and the reported selection behaviour +was based on average trial performance (e.g., Becker, 2010). This leaves open the possibility +that the target’s discriminative relative feature must be learnt, or that the visual system adapts +weight settings over consecutive trials in a trial-and-error fashion to bias attention to the +relative feature – which is very different from the claim that attention can be biased to +relative features instantly, with the first glance. +Second and relatedly, the boundary conditions for tuning to relative features are currently +not clear. Previous work has shown that attention will be tuned to the relative target colour +when the target is the reddest (or greenest / yellowest / bluest) item in the display on ≥50% of +the trials, including when the target always has the same feature and could be found by tuning +attention to the exact feature value (e.g., Becker et al., 2013). Relational search was also +observed across different feature dimensions (e.g., colour, size, brightness, shape), and across +different search tasks such as pop-out search (e.g., Becker et al., 2014), feature search (e.g., +Becker et al., 2013) and conjunction search (e.g., Becker et al., 2017). However, it is +currently unknown if we would observe relational search when the target is never the reddest +(or greenest / yellowest / bluest) item in the display, but differs from the majority of non- +target items in one direction (e.g., redder). +Previous studies have shown that we can enforce tuning to a specific feature value; for +example, by presenting a target among equal numbers of flanking nontarget colours, such as +presenting an orange target among three yellow and three red nontargets (Becker et al., +2014). As the target is neither the reddest nor the yellowest item, attention cannot be tuned to +the relative feature but needs to be tuned to the target’s feature value in these displays (e.g., +Becker, Harris, Venini & Retell, 2014; Harris, Remington & Becker, 2013; Schoenhammer et +al., 2016; but see Becker et al., 2013). + +8 + +Given the ability to tune attention to specific feature values, it is unknown whether +participants would adopt a relational search or feature-specific search when the target is never +the reddest item in the visual field, yet differs from a majority of items in a single direction +(e.g., redder). With that, it is currently unclear whether relational search is indeed based on a +mechanism assessing the dominant feature in the visual field, or whether it is only adopted +when relational search allows localising the target successfully on a large proportion of trials +(i.e., through statistical learning). +Third, previous work on characterising the attentional tuning function is also limited in +that the studies typically used fairly sparse displays, of four to eight items (e.g., Becker, 2010; +Hamblin-Frohman & Becker, 2021, Navalpakkam & Itti, 2007; Yu et al., 2022). The results +typically showed that visually salient items (e.g., a red item among blue-green stimuli) did +not attract attention, or attracted attention only very weakly (e.g., Gaspelin et al., 2015; +Martin & Becker, 2018; York & Becker, 2020). These results were taken to show that strong +capture by relational items was not mediated by saliency, and that bottom-up saliency may +modulate attention to a lesser extent than assumed in current models (e.g., Theeuwes, 2013; +Wolfe, 1994). However, Wang and Theeuwes (2020) argued that item salience is drastically +weakened in sparse displays. If this is correct, previous studies on attentional tuning may +have overestimated the role of tuning to relative features and/or underestimated the role of +bottom-up saliency effects in guiding attention. Thus, it is still an open question whether +previous results would generalise to more realistic search settings, such as search displays +containing a larger number of search items. +Aims +The aim of the present study was to test some of the untested assumptions of the relational +account in more realistic search settings. To that end, the search displays always contained +thirty-six items, one of which was a saliently different distractor. This allowed assessing + +9 + +possible contributions of bottom-up saliency effects to visual selection in more realistic +search conditions with potentially stronger saliency effects. +To assess how attention is guided with versus without training on the task, we created four +blocked conditions in which the search colours varied (bluer, greener, yellower and redder +target; see Fig. 2). Participants completed ten trials in each block, and then switched to a +different search colour. Prior to each block, participants were only informed about the exact +target colour (teal or orange), but not about the colour of the non-target items or the target’s +relative colour (redder, yellower, greener, or bluer). Thus, analysing the first trial of each +block allows assessing how attention was tuned to the target under conditions of uncertainty +and in the absence of training effects. + +Example Display + + + + + + + + + Overview of Conditions + + +Figure 2. Left: Example of a visual search display. Search displays consisted of three target- +coloured items (here: orange) presented among 29 nontarget coloured items that could have +two different colours (here: yellow-orange or yellow). Participants were instructed to search +for a target-coloured item (here: orange) that contained the letter u or m, and to report +whether the target contained the u or m (whereas the other orange items contained the letters +w or m). To assess how attention was guided in search, we assessed eye movements to items +with a relationally matching colour (here: red), optimal colour (here: red-orange), and a +saliently different colour (here: violet). Right: Colours used to create the four blocked +conditions. The target could be teal or orange, and the teal target could be presented among +other green-ish non-targets (bluer target), blue-ish non-targets (greener target), while the +orange target could be among red-ish non-targets (yellower target) or yellow-ish non-targets +(redder target). Prior to a mini-block of 10 trials, participants were informed of the target +colour, but not of the colour of the non-targets, which determined whether the target would +be relationally redder/yellower or greener/bluer than the non-targets. + +mBluer +R +0 +T +NT +NT +S +NT +NT +T +0 +R +Greener +Yellower +R +0 +T +NT +NT +S +NT +NT +T +0 +R +Redder10 + +Deviating from previous studies, the search displays always contained all possible +distractor colours, allowing a more fine-grained measurement of distractor effects and an +assessment of possible boundary conditions for relational tuning. For example, in the redder +target condition, an orange target was presented among 29 yellow-orange and yellow +nontargets, one distractor with a relatively matching colour (red), two distractors with an +optimal colour; i.e., an exaggerated target colour that was slightly shifted away from the +nontarget colours (red-orange), two distractors with the target colour (orange), and a saliently +different distractor with an unrelated colour (e.g., a blue; see Figure 2). These displays allows +assessing how attention is tuned to the target when the target differs from the majority of non- +targets in a single direction (redder) but is never the relationally maximal item (e.g., reddest, +bluest, greenest or yellowest item) in the display. +To assess how attention was tuned to the target, we measured eye movements to each of +the different item types (target and distractors). Specifically, to tap into early processes of +visual selection, we analysed the first eye movements on a trial, which are not influenced by +prior fixations or search processes. +If attention is tuned to the exact target colour, we would expect a large proportion of first +eye movements on the target-coloured items (orange, teal), regardless of the context colour, +and only few eye movements on the differently coloured distractors (optimal, relational). On +the other hand, if attention is tuned to the optimal colour, we would expect most first eye +movements on the optimal colour and a decrease in selecting distractors with other-than- +optimal colours. According to the relational account, we would expect most first eye +movements directed towards the distractor that best matches the relative colour of the target, +viz., the distractor that is the reddest, yellowest, greenest, bluest in the visual field, followed +by the optimal distractor (as this is the next-maximal, e.g., next-reddest item), and the target- +coloured distractors. + +11 + +Moreover, if relational search is based on a fast, automatic assessment of the dominant +colour in the display, we would expect to see these results immediately in the first trial of +each block, prior to learning and knowledge of the relative values. +If the salient distractor automatically captures attention in these multiple-item displays +(e.g., Wang & Theeuwes, 2020), we would expect a large proportion of first eye movements +on the salient distractor. Moreover, if participants can learn to ignore or inhibit the distractor, +we would expect this saliency-effect to decrease with training (i.e., over the course of a +block; e.g., Gaspelin & Luck, 2018; Hamblin-Frohman et al., 2022). +Method +Participants. To estimate the required sample size, we examined the ability to detect +relational vs. feature-specific search in the first fixations on distractors in previous work +(Becker, Harris, Venini & Retell, 2014). The weakest effect was the feature-specific effect +(t(14) = 2.5, p = .024 in search for a redder target; t(14) = 2.5, p = .027 in search for a +yellower target; Becker et al., 2014; Exp. 3). The BUCSS tool suggested a target sample size +of N=32 for the present study, to achieve a power of 85% (with 50% assurance; Anderson et +al., 2017). +Thirty-four paid participants from the University of Queensland participated in the +experiment. Two participants were excluded for having a low search accuracy (< 70%), +leaving 32 in the final analysis (M age =23.1 years (SD = 1.9), 24 female). The study was +approved by the University of Queensland ethics board, and all procedures were in line with +the Declaration of Helsinki. +Apparatus. Stimuli were presented on a 21-inch CRT monitor with a refresh rate of 85Hz. +A chin and headrest were used to hold the participant’s heads 600mm from the screen. Gaze +location was measured by an SR-Research Eyelink-1000 eye tracker at 500Hz sampling rate. +The experiment was controlled by Python’s PsychoPy (Peirce, 2007). + +12 + +Stimuli. All stimuli were presented against a grey background. Each search array +contained thirty-six coloured circles (radius: 0.48°) arranged in a six-by-six grid format (see +Figure 2). Stimulus locations were initially selected to have a 4.96° horizontal separation and +4.30° vertical separation (centre-to-centre), which varied because the location of all non- +target stimuli (NT) was randomly jittered by ± 1.43° horizontally and vertically on each trial. +The relevant distractors (relational, optimal, target-similar and salient) and the target were not +jittered to retain precise eye tracking data and to ensure that these stimuli were never too +close to each other. The location of all items was randomly chosen on each trial, with the +restriction that the items of interest could never appear in the corner and corner-adjacent +positions of the search array (as these positions were too far from fixation), and never in the +central four (3.82° from fixation) positions (as these were too close to fixation). +Each coloured circle contained a letter, either ‘u’, ‘n’, ‘m’ or ‘w’ (height: 0.29°). The +target stimulus always contained a ‘u’ or ‘m’. The distractor items of interest never contained +the target response characters (u/m). Colours were selected from an equiluminant (30 ± +2cd/m2) RGB colour set (see Figure 2). There were two potential target colours, orange +(RGB: [227, 124,52]) and teal (RGB: [56,171,146]), that alternated between blocks. The +orange target could either appear among a set of redder non-targets, creating a yellower target +condition, or among yellower non-targets, creating a redder target condition. The teal target +could be presented either among greener or bluer non-targets, creating bluer or greener target +conditions, respectively. Each trial contained the same amount of distractor and non-target +items: One distractor with a relationally matching colour (R), two distractors with an optimal +colour (O), two target-matching distractors (T), and a salient distractor (S) that always had a +colour from the opposite side of colour space (e.g., a pink distractor for when the target was +teal). The stimuli were always presented among 29 non-target items that were selected from +two other colours (e.g., two blue-ish colours; see Fig. 2). + +13 + +Design. The colours of the target, distractors and non-targets were always repeated within +a mini-block of 10 trials, and mini-blocks alternated between orange and teal targets, with the +direction of search (redder/yellower or greener/bluer condition) determined randomly at the +start of each block. Participants completed 64 mini-blocks, for a total of 640 trials. The first +four blocks were treated as practice trials, leaving 600 trials for the final analysis. +Procedure. Prior to the experiment, participants were instructed to locate the target- +coloured stimuli containing the character u or m and respond with the corresponding +keyboard key as quickly and accurately as possible. Moreover, prior to each mini-block, +participants were shown the colour of the upcoming visual search target (either orange or +teal). Importantly, no information about the colour of the non-targets was provided; hence +participants did not know the relative colour of the target prior to the first trial (i.e., whether it +was redder/yellower or greener/bluer). +To ensure stable and accurate eye tracking, participants were calibrated with a randomised +9-point calibration at the beginning of the experiment and whenever the fixation control +failed. The fixation control was implemented prior to each trial, with participants maintaining +fixation for 650ms on the central fixation cross. If gaze remained within 2.0° from the centre, +the search array was presented until a manual keypress response (u, m) was recorded. If an +incorrect response was returned, error feedback was displayed. After each trial, a blank grey +screen was presented for 750ms, and the next trial commenced again with the fixation +control. +Results +Overall, accuracy on the letter identification task was high (>90%). Trials with incorrect +responses were excluded from all analyses (6.6% of trials). The average response time (RT) +was 2,207.7ms (SD = 1,919.4ms). Trials that were more than 2.5 standard deviations above +the mean RT (rounded to 7,000ms) were excluded from all analyses (1.9% of trials). + +14 + +Proportion of First Fixations on Each Item on Trials 1 - 10 + +Figure 3. The proportion of first fixations directed towards the target, or any of the distractors +of interest (target-matching, optimal, relational or salient), depicted as a function of trial +repetition. The relational and optimal distractors attracted the highest proportions of first +fixations and showed a linear increase over the ten trials. The target and the target-matching +distractor received more first fixations than the saliently different item, and also displayed a +linear increase in fixations. The salient item was selected least frequently and showed a linear +decrease in first fixations. + +First Fixations: Training Effects. To assess whether attentional priorities changed with +experience, we analysed the probability of fixating on each of the differently coloured +distractors as a function of trial repetition (see Fig. 3). The probability of fixating on each +differently coloured distractor was computed by dividing the proportion of trials in which a +given distractor was fixated (as the first item) by the number of distractors present on the trial +(e.g., as there were two optimal-coloured stimuli, the proportion of first fixations on an +optimal coloured stimulus was divided by two). The proportion of first fixations on the target +stimulus and the target-matching distractor did not differ from each other, F(1, 31) = 0.95, p += .337, and did not interact with trial repetition, F(9, 279) = 1.16, p = .321. This shows that + +14 +Target +TMatch +(%) suo!1e +Optimal +12 +Relational +Salient +10ortion of First Fix +8 +6 +42 +P +0 +T1 +T2 +T3 +T4 +Te +T7 +T: +Tg +T10 +Ts +Trial Repetition15 + +attention was not guided to the target item based on the contained response-related character, +and thus the data were collapsed across the target and target-coloured distractor. +A 4 (Item Type: Target-Matching, Optimal, Relational, Salient) x 10 (Trial Repetition: T1 +to T10) repeated measures analysis of variance (ANOVA) was computed over the +probabilities of fixating on each differently coloured item. The results showed a main effect +of item type, F(3, 93) = 53.5.47, p < .001, ƞ2p = 0.63, trial repetition, F(9, 279) = 4.42, p < +.001, ƞ2p = 0.13, as well as a significant interaction, F(27, 837) = 2.46, p < .001, ƞ2p = 0.07. +Pairwise t-tests showed that both the relational and optimal items attracted more first eye +movements than the target-coloured items (relational: t(31) = 5.33, p < .001, BF10 = 2500.17, +optimal: t(31) = 6.61, p < .001, BF10 = 7.24 * 104). In turn, the target-coloured items received +more first saccades than the saliently different item, t(31) = 3.96, p < .001, BF10 = 71.41. +Optimal and relational items did not differ, t(31) = 1.07, p = .294, BF10 = 0.32. +To investigate possible linear trends that may reflect learning or adaptation effects, we +next examined the slopes of first fixation locations separately for each of the differently +coloured items across the ten trials. The linear trends for all four item types differed +significantly from zero. The relational item had the steepest positive slope, β = 0.33, t(31) = +4.38, p < .001, followed by the optimal item, β = 0.18, t(31) = 2.87, p = .007, and the target- +matching item, β = 0.09, t(31) = 2.36, p = .025, reflecting that selection increased for all of +these items over the course of the block. Thus, the first fixations in a trial show relational +tuning, which increased slightly and remained dominant across all trials in the block. Finally, +the saliently different item showed a significant reduction in visual selection across blocks, β += -0.11, t(31) = 2.08, p = .046. +First Fixations: Initial Guidance on T1. We next analysed the eye movement behaviour +on the first trial of the block, where participants were unaware of the relative colour of the + +16 + +target (redder/yellower or greener/bluer), and only knew the exact target colour (orange or +teal; see Fig. 3, T1). +Paired-samples t-tests revealed that first saccade proportions on T1 were lower for the +target colour than both the optimal, t(31) = 4.34, p < .001, BF10 = 185.7, and relational +distractors, t(31) = 2.50, p = .018, BF10 = 2.71. Optimal and relational distractors did not +significantly differ from each other, t(31) = 1.59, p = .123, BF10 = 0.58. The salient item did +not differ significantly from the target-coloured item, t(31) = 0.88, p = .388, BF10 = 0.27. The +finding that relational search was conducted on trial 1 shows that experience or prior +knowledge is not necessary for relational search, and supports the hypothesis that the visual +system can quickly assess the dominant colour in the visual field and tune attention to the +relative colour of the target. +Fixation Progression. +The results above suggest that search was initially (on the first trial) relational and that this +relational guidance was maintained and increased across repeated trials. To assess whether +the same trends may be found within a single trial, we analysed the first five fixations in each +trial to reveal how attentional guidance developed within an individual trial (see Figure 4). +On average participants made 4.58 fixations per trial. We included all trials in this data set, +including those where the task was completed within the first five fixations (which resulted in +1.7% missing data for F2, F3: 7.4%, F4: 17.5%, and F5: 30.6%). +The first fixation (F1), now collapsed over trial repetition, followed the same results +pattern as previously described, with the relational and optimal stimuli attracting a higher +proportion of first fixations than the target and the target-matching distractor (all ps < .001). +For F2 results began to deviate from relational guidance. The target stimulus was now more +likely to be fixated than any other item (ps < .010). The target-matching distractors were now +equally likely to be fixated as the optimal distractors, t(31) = 0.14, p = .888, BF10 = 0.19, and + +17 + +the optimal distractors were now more likely to be fixated than the relational item, t(31) = +4.32, p < .001, BF10 = 178.09. From F3 onwards fixation patterns remained consistent: Now +the target-matching distractors were more likely to be fixated than the optimal distractor (all +ps < .001), and the optimal distractor was more likely to be fixated than the relational +distractor (all ps < .001). Finally, fixations on the salient item were highest on F1 than at any +other point in the trial (all ps < .001) and were significantly lower than all other stimuli of +interest across all fixations (ps < .004). + +First Five Fixations in a Trial +Figure 4. The first five fixations (F1 to F5) in a trial show how search progressed during a +trial. The first fixation (F1) displayed relational search, with the relational and optimal +distractors being selected more frequently than the target and target-matching distractor. The +second fixation (F2) showed a shift to more feature-specific guidance: The target was now the +most likely to be fixated, followed by the target-matching and optimal distractors, with fewer +fixations on the relational distractor. With the third fixation (F3), the target-matching +distractor was now more likely to be fixated than optimal or relational distractors. From this +point onwards, fixation proportions seem to be dictated by target-feature similarity. The +salient item was most frequently fixated on F1 then dropped off in subsequent fixations. + + +24 +Target +-T_Match +Optimal +(%) suo +20 +Relational +Salientoportion of Fixati +8P +4 +0 +F1 +F2 +F3 +F4 +Fs +Fixation18 + +In sum, the results are in line with the hypothesis that (in relational search) participants +will first select the relationally maximal item (e.g., reddest), followed by the optimal (e.g., +next-reddest) item, and the target-coloured items. Another way of describing these results is +that attention was always initially guided to the relatively matching items, with feature- +specific tuning developing only after the first (few) fixation(s), perhaps by inhibiting the +more extreme colours that were selected first. + +First Saccade Latencies. We next analysed the saccade latencies, that is, the time from +the onset of the search display to the onset of the first saccade in a trial, to assess the time- +course of attentional deployment using the first saccade in a trial. This analysis was collapsed +over trials, with the first trial being excluded (due to longer saccadic latencies).1 Figure 5 +shows the distributions of saccadic latencies separately for each item type (i.e., the proportion +of trials where saccadic latencies ranged from 125ms – 150ms, 150 – 175ms, etc.). +To analyse the data statistically, we fed the average saccade latencies of the first saccades in a +trial into a one-way (Item Type: Target, Optimal, Relational, Salient) repeated-measures +ANOVA. The results showed a significant effect of item type F(3,93) = 53.01, p < .001, ƞ2p = +0.63. Two-tailed comparisons revealed that latencies were shorter for the salient item (M = +217.6ms) than both the relational (M = 237.5ms) and optimal item (M = 240.9ms), all ts> +8.18, ps < .001, BF10 > 3.98 x 106. Saccades to target-matching items (M = 252.0ms) were +slower than to the relational and optimal items, ts(31) = 4.35, ps < .001, BF10 > 191.72, while +relational and optimal items did not differ, t(31) = 1.26, p = .218, BF10 = 0.39. Thus, the more +dissimilar the stimuli were from the target item, the shorter their saccadic latencies were. + + +1 Saccade latencies on T1 were longer than in all other trial repetitions (all ps < .001), but +there was no repetition by item type interaction (p = .650) or any linear trends for the +individual items after excluding the first trial (ps > .235). + +19 + + Distribution of First Saccade Latencies + +Figure 5. The distribution of saccade latencies, which was derived by sorting saccades to +each stimulus type into 25ms latency bins and depicting the proportions of trials within each +bin separately for each item type. The results showed a higher proportion of short-latency +saccades to the saliently different distractor than to the other distractors. Moreover, there was +a higher proportion of saccades with long latencies to target-matching items (pooled over +target and target-coloured distractor), producing differences between the target-coloured +items and the other distractors in the tail end of the distribution. + +Figure 5 shows that the distribution of saccadic latencies and reveals that differences in the +average saccade latencies were due to saccades to salient distractor being initiated +significantly earlier than to all other items. By contrast, the longer saccade latencies for the +target and target-matching distractors were due to a higher proportion of saccades with longer +latencies. +Distractor Dwell Times. As in previous studies, we also examined the mean dwell times +on each of the distractors, measured as the time spent fixating on each of the different item +types. The target-matching stimuli were examined independently from the actual target (M = +162.9ms), and results were collapsed over trial repetition. +0 +5 +10 +15 +20 +25 +30 +125ms 150ms 175ms 200ms 225ms 250ms 275ms 300ms 325ms 350ms +Proportions +Saccade Latency (25ms Brackets) +Target-Matching +Optimal +Relational +Salient + +20 + +The one-way (Distractor Type: Target-Matching, Optimal, Relational, Salient) repeated- +measures ANOVA revealed significant differences between the item types, F(3,93) = 40.50, +p < .001, ƞ2p = 0.57. Planned two-tailed comparisons showed that dwell times for the target- +matching (M = 173.0ms) and optimal distractors (M = 171.4ms) were longest and did not +differ from each other, t(31) = 0.92, p = .363, BF10 = 0.28. Dwell times were significantly +shorter for the relational distractor (M = 166.4ms) than the optimal distractor, t(31) = 5.74, p +< .001, BF10 = 7312.78, and target matching distractors, t(31) = 3.21, p = .003, BF10 = 11.99. +The relational distractor had longer dwell times than the salient item (M = 152.5ms), t(31) = +6.26, p < .001, BF10 = 2.90 * 104. In sum, dwell times increased with similarity to the target +colour, indicating that longer dwell times were needed to identify the items as distractors as +they became more similar to the target. +General Discussion + +The current study revealed that relational effects may be more pervasive than previously +assumed. This was the first study to examine attentional tuning functions in the absence of +target contextual knowledge and in displays containing numerous items. The relatively +matching distractors attracted the first saccade, including on the first trial in a mini-block, +when participants only knew the colour of the target but not its relative contextual colour. +These results show that relational/contextual information can be quickly extracted from a +visual scene, in line with the tenets of the relational account. Previous studies have already +shown that information about the statistical properties of a visual scene can be rapidly +extracted (e.g., feature averaging, Chong & Treisman, 2005a, b; Choo & Franconeri, 2010; +Joo et al., 2009; gist processing, Oliva & Torralba, 2006; see also Thorpe, Fize & Marlot, +1996). The present study extends on this research by showing that this information can be +used to bias attention to the relative features of the target, prior to executing the first eye +movement. In other words, relational guidance does not require learning of the context or + +21 + +knowledge of the relative target feature, but can be executed with the first glance at a novel +scene. + +The current results also revealed that relational guidance is more ubiquitous than +previously thought. In particular, we found relational guidance even when the target never +had a relationally maximal value – that is, when it was never the reddest, greenest, bluest or +yellowest item in the display. In every trial there were relationally better-matching distractors +(one relatively matching distractor and two optimal distractors), which frustrated selecting the +target with the first eye movement (in relational search). Yet, we reliably found that +participants searched relationally. This reveals that the visual system is indeed sensitive to the +dominant feature in the display, and initiates relational search as soon as the target differs +from most of the display items in a linear fashion. Furthermore, participants continued to +search relationally over as many as ten successive trials, even though relational guidance +reliably resulted in selecting one of the more extreme distractors first. This means that +relational guidance is more readily applied and more persistent than previously thought (e.g., +Becker et al., 2014). + +The results of the present study also provided evidence for selection proceeding in the way +as laid out in the relational account, with the most extreme, relatively matching item being +selected first (e.g., reddest), followed by the next most extreme, relatively matching item (i.e., +next reddest item), and so forth, until the target is selected. As shown in Figure 3, the first +saccade in a trial was most likely to select the most extreme, relatively matching item, +whereas later saccades were more likely to select target-matching items and the probability +for selecting extreme items declined after the second fixation. +The mechanism responsible for achieving this order of selection has never been specified +in the relational account. Other theories have proposed a memory-based, ‘inhibitory tagging’ +mechanism to explain how search proceeds to the next item (e.g., Klein, 2000). + +22 + +Correspondingly, it is possible that selection of target-matching items emerged during the +trial due to selection and subsequent inhibition of relatively matching, more extreme +distractors. Inhibitory tagging could be either location-based (i.e., inhibiting selected +nontarget locations during the trial), or feature-based (i.e., inhibiting a non-target or distractor +colour after selection; e.g., Bichot & Schall, 2002). If inhibitory tagging is responsible for the +selection sequence in the present study, it is perhaps more likely to be feature-based +inhibition rather than location-based inhibition: Selection of the target and target-matching +distractors commenced earlier than would be expected on the basis of location-based +inhibition. Specifically, as there were three relatively matching or optimal distractors in the +display, selection of these distractors should have declined after the 3rd fixation, but we +observed the decline after the 2nd fixation, which corresponds to the number of different +colours in these distractors. Also, selection of the target and target-matching items showed a +steep increase after only one fixation, which seems too early for location-based inhibition, +and indicates that the colour(s) rather than the locations of the more extreme distractors may +have been inhibited. +Alternatively, it is possible that, after the first fixation, attention was tuned to the exact +feature value of the target, swiftly switching from originally relational search to a more +feature-specific search (or target template; e.g., Duncan & Humphreys, 1989). We currently +consider this unlikely, as an attentional bias usually automatically carries over to the next trial +to influence selection (i.e., intertrial priming; Maljkovic & Nakayama, 1994), and the first +eye movements never showed evidence of a feature-specific bias. +Inhibition of a distractor feature can also carry over to the next trial to influence selection, +but the effects are weaker than effects of target guidance (e.g., Chang & Egeth, 2019; +Hamblin-Frohman et al., 2022). This could explain that relational guidance was somewhat +attenuated, as the relatively matching distractor mostly did not differ significantly from the + +23 + +optimal distractor, but was still clearly visible (as both the relatively matching and optimal +distractors were selected more frequently than the target-matching distractors; see Fig. 3). +However, the somewhat attenuated effect of the relatively matching distractor could also be +due to difficulties distinguishing the optimal and relatively matching distractor in the far +periphery (e.g., Noorlander et al., 1983), or to the fact that there was only one relatively +matching, extreme distractor and two optimal distractors. Further studies are necessary to +estimate the magnitude of the attenuation and clarify the mechanism that allows narrowing +search to target-matching items after the first fixation(s). +A fourth interesting finding relates to the salient item, which had a very dissimilar colour +compared to the other items and hence, had the largest bottom-up feature contrast. Wang and +Theeuwes (2020) proposed that experiments with sparse displays may underestimate bottom- +up saliency effects (see also Rangelov et al., 2017). The displays of the present study +contained the largest number of items tested for fine-grained attentional tuning (36 items), yet +we found only very weak effects for the salient item, which further declined over trials 1 – 10 +(see Fig. 3). Given the low proportion of first fixations on the salient item, we cannot claim +that the salient item significantly attracted attention.2 Even in the densely populated displays, +top-down tuning to the (relative) target feature led to significantly higher selection rates of +the corresponding distractors than bottom-up saliency, indicating that top-down processes +dominate attentional guidance over bottom-up saliency, contrary to bottom-up selection +views (e.g., Theeuwes, 2004; Wang & Theeuwes, 2020). +These results also rule out an alternative explanation of the results. Proponents of feature- +based theories (e.g., Guided Search 2.0; Wolfe, 1994) may argue that the results could be due + +2 Comparing the proportion of first fixations on the salient item to the proportion of fixations +on the non-salient non-target items (omitting the corner positions and central positions that +never contained a distractor) revealed that the salient item was actually selected less +frequently than the non-salient non-targets (p=.03). + +24 + +to a combination of top-down, feature-specific tuning to the target and bottom-up processes, +as the more extreme distractors (optimal and relatively matching distractors) may have been +similar enough to the target to be subject to (broad) top-down tuning and were simultaneously +more salient than the target, leading to higher selection rates. Contrary to this contention, we +found a very different results pattern for the salient distractor compared to the other +distractors, both in the trial analysis (which showed an increase in capture by extreme and +target-similar distractors over trials and a decrease for the salient distractor; see Fig. 3), and +in the time-course analysis (which showed shorter latencies of saccades to the salient +distractor than for all other distractors; see Fig. 5). These findings suggest that the more +extreme (and more salient) relatively matching and optimal distractors attracted attention and +the gaze in the same manner as the target-similar (less salient) distractor, viz., due to top- +down tuning and not bottom-up saliency, whereas the effects of the salient distractor were +due to bottom-up, stimulus-driven processes. +Admittedly, it is notoriously difficult to distinguish between combined top-down/bottom- +up theories, optimal tuning and the relational account, especially in standard visual search +experiments with a distractor, as the theories make quite similar predictions. However, the +relational account has been extensively tested against these other theories in previous +experiments using the spatial cueing paradigm (e.g., Becker et al., 2013) and visual search +with relatively matching distractors that were very dissimilar from the target (York & Becker, +2020) or whose similarity / saliency varied systematically (Hamblin-Frohman & Becker, +2021; Yu et al., 2022). The results unequivocally supported the relational account and +showed that early selection cannot be explained by optimal tuning or the combined saliency +and similarity to the target. +Previous studies testing the relational account against the standard, feature-based theories +moreover revealed that tuning to relations was a default search mode applied across a wide + +25 + +range of different tasks that would have allowed locating the target by tuning to its specific +feature value (e.g., when the target was always repeated). The present study extends on these +findings, by showing that attention is tuned to relative features even when the target is never +the relatively reddest, bluest, greenest or yellowest item in the display. In conditions such as +the present experiment, a feature-specific guidance mechanism may be the most efficient +search method for the current display, as it would have reduced competition for attention to +only target-similar distractors (i.e., two other items) by excluding relatively matching and +optimal distractors. Participants chose relational tuning over feature-specific tuning even +though, on the first trial of each block, they were only given information about the exact +feature-value of the target stimulus. The instructional prompts mentioning only the target +colour should have primed participants to initiate a guided search for the exact target colour. +Yet, participants consistently engaged in relational search and also did not learn to use +feature-specific tuning over the course of ten (identical) trials. As shown in Figure 3, linear +trends for first eye movements were stronger for the relational and the optimal distractors +compared to the target-matching items, suggesting that the relational effect became stronger +over the course of the block rather than weaker. + +Collectively, these results show that participants can be quite reluctant to tune attention to +the specific feature value of the target. We can only speculate why this may be the case. +Previous studies have shown that feature-specific tuning results in delays in selecting the +target, as well as longer dwell times on target-similar distractors, compared to tuning to +relations (in identical displays; Becker et al., 2014; Martin & Becker, 2018). This suggests +that tuning attention to exact feature values may cause delays in selection as well as decision- +making (about whether the selected item is the target), rendering search less efficient. +While this question warrants further research, the present study clearly showed that tuning +to relative features occurs in more densely populated, 36-item displays; in the absence of + +26 + +knowledge of the relative feature of the target, and in the absence of any training. 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Psychological Science, 33(1), +105–120. +https://doi.org/10.1177/09567976211032225/ASSET/IMAGES/LARGE/10.1177_095679762 +11032225-FIG2.JPEG + + + diff --git a/OtE1T4oBgHgl3EQfaASJ/content/tmp_files/load_file.txt b/OtE1T4oBgHgl3EQfaASJ/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..20c0832b4afaa8002a141cd92e1a0e25929a3301 --- /dev/null +++ b/OtE1T4oBgHgl3EQfaASJ/content/tmp_files/load_file.txt @@ -0,0 +1,1019 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf,len=1018 +page_content='1 The progression of visual search in multiple item displays: First relational, then feature-based.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Zachary Hamblin-Frohman, Koralalage Don Raveen Amarasekera & Stefanie I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Becker School of Psychology, The University of Queensland, Brisbane, Australia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 2 Abstract It is well-known that visual attention can be tuned in a context-dependent manner to elementary features, such as searching for all redder items or the reddest item, supporting a relational theory of visual attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' However, in previous studies, the conditions were often conducive for relational search, allowing successfully selecting the target relationally on 50% of trials or more.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Moreover, the search displays were often only sparsely populated and presented repeatedly, rendering it possible that relational search was based on context learning and not spontaneous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The present study tested the shape of the attentional tuning function in 36-item search displays, when the target never had a maximal feature value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', was never the reddest or yellowest item), and when only the target colour but not the context colour was known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The first fixations on a trial showed that these displays still reliably evoked relational search, even when participants had no advance information about the context and no on-task training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Context learning further strengthened relational tuning on subsequent trials, but was not necessary for relational search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Analysing the progression of visual search within a singe trial showed that attention is first guided to the relationally maximal item (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', reddest), then the next-maximal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', next-reddest) item, and so forth, before attention can hone in on target-matching features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' In sum, the results support two tenets of the relational account, that information about the dominant feature in a display can be rapidly extracted and used to guide attention to the relatively best-matching features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 3 Introduction It is well-known that we cannot consciously process all objects in a visual scene at once.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' To address this, visual attention selects objects for in-depth processing, often guiding our gaze to relevant parts in a scene (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Deubel & Schneider, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Much effort has been devoted to determine which items in a scene will be attended first, and more generally, to identify the processes involved in creating our rich mental representation of the visual environment (for a review, see Carrasco, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Wolfe, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' To date, it is widely accepted that attention can be guided by both, bottom-up, stimulus- driven processes and top-down, goal-driven processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Wolfe, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' For example, attention can be reflexively drawn to visually salient events such as a bright flash, a movement, or the sudden appearance of an object (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Theeuwes, 2004, 2013), or it can be top-down tuned to select items with certain attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', colours: red, green) to help goal- related behaviours such as finding a friend in a crowd (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Desimone & Duncan, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Wolfe, 1994, 2021) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Correspondingly, current models of visual attention typically include both a bottom-up and a top-down component to predict which items in a visual scene will be selected first (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Wolfe, 1994, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Top-down tuning is typically modelled as an increase or decrease in the firing rate of sensory neurons in response to specific stimulus attributes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', red, green;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Navalpakkam & Itti, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Yu, Hanks & Geng, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' For example, when looking for an orange in a fruit basket, we would tune attention to orange, which increases the output of neurons that respond to orange and prioritises colour-matching items for selection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' It is commonly assumed that attention is tuned to the feature value that a person is looking for (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', particular shade of orange;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Duncan & Humphreys, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Navalpakkam & Itti, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' However, to date, there are also several accounts of non-veridical tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Navalpakkam and Itti (2007) noted that tuning attention to the exact target feature value would not be beneficial when the target is very similar to surrounding irrelevant non-target 4 items, as tuning attention to, for example, orange, would also boost the response gain of red- orange or yellow-orange, leading to a poor signal-to-noise ratio (SNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' They proposed that attention would be tuned to a feature value that is slightly shifted away from similar nontargets, to increase the SNR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', to yellow-orange, when an orange target is presented among red-orange items;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Navalpakkam & Itti, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' According to their optimal tuning account, attention is always tuned to the feature value that maximises the ability to discriminate the target from the non-targets (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', maximise the SNR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Thus, attention would only be tuned to the exact target feature value when the target is presented alone or among dissimilar other items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' In less discriminable cases, attention should be shifted to a slightly exaggerated target feature value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' In line with this idea, a perceptual probe task revealed that a slightly shifted non-target colour was likely to be mistaken for the target when the target was consistently embedded among similar featured non-targets in a prior visual search.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' For example, if an orange target was always presented among similar, yellow-orange nontargets in a visual search task, participants would pick a slightly redder (red-orange) colour as the target colour in intermixed probe trials (Navalpakkam & Itti, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' see also Geng et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Scolari et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Another account of non-veridical tuning is the Relational Account, which proposes that attention may not at all be tuned to a specific feature value.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' As noted by Becker (2010), tuning attention to a particular feature value may not be beneficial in natural environments, as specific feature values such as the size, shape and colour of objects vary a lot with differences in distance, perspective and shading.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' To allow efficient selection of the targets in these noisy environments, attention could be tuned in a context-dependent manner to objects, biasing attention to a feature that the target has relative to the other items in the context (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', redder, larger, darker;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Becker, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' For example, when searching for an orange in a fruit basket, the visual system would quickly assess the distribution of colours in the visual scene to 5 determine the dominant colours, and tune attention to the relative colour that would best discriminate the target from the dominant coloured non-target items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Thus, attention would be tuned to all redder items or the reddest item when the fruit basket (or visual scene) contains many yellow or green objects, and tune attention to all yellower items or the yellowest item when the fruit basket (or visual scene) contains many red objects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' As a consequence of this very broad tuning, the item that maximally fulfils the relevant feature relationship will be selected first (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', reddest or yellowest item).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Hence, when attention is tuned to all redder items, the reddest item in the visual field will be selected first, followed by the next-reddest, and so forth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' In line with this prediction, several visual search studies showed that when an orange target is presented among mostly yellow(er) items, a red irrelevant distractor was more likely to be attended first than the target, even when it was quite dissimilar from the target, suggesting that attention was biased to all redder items, or the reddest item (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Becker, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Hamblin-Frohman & Becker, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' York & Becker, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Selection of the distractor was reflected both in higher response times (RT) to the target, as well as in a high proportion of first eye movements to the red distractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Several studies also included a visually salient distractor with a dissimilar colour (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', blue), but found no or only very weak effects of saliency, ruling out that selection of the relatively matching (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', red) distractor was mediated by bottom-up, stimulus-driven processes (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Martin & Becker, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' York & Becker, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Both optimal tuning and the relational account propose that attention can be tuned to features in a non-veridical manner but propose very different underlying mechanisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Subsequent studies combined the two paradigms to determine whether attention is tuned to all relatively matching features or an optimally shifted feature value (see Figure 1 for an illustration of the different accounts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 6 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Overview of different theoretical tuning functions to tune attention to the target in the example display (left), which shows an orange target (T) among yellow-orange and yellow non-targets (NT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The standard view (2nd from left) is that attention would be tuned to the target colour (T).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' According to Optimal Tuning (3rd from left), attention would be tuned to a slightly exaggerated, ‘optimal’ target feature value that is shifted away from the nontargets, to increase the SNR (optimal colour;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' O).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' According to the Relational Account (right), attention would be tuned to the relative colour of the target, that the target has relative to the other items in the surround (here: redder), which can include vastly dissimilar colours (relatively matching colour, R).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The results of two studies showed that attention is biased to all relatively matching items, indicating that early visual selection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', which item is selected first) follows the predictions of the relational account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' However, target identification judgements were skewed towards the slightly shifted feature value, showing that perceptual decision-making (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', decisions about whether a selected item is the target or not) is best described by the optimal tuning account (Hamblin-Frohman & Becker, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Thus, visual attention can be tuned rather broadly towards the target’s relative feature, leading to frequent selection of very dissimilar colours that only share the target’s relative colour, whereas perceptual decision- making is tuned rather sharply towards an exaggerated target feature value, so that only very similar colours are mistaken for the target colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Despite the progress in describing the attentional tuning function, there are still significant knowledge gaps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' First, one important claim of the relational account is that the visual system quickly assesses the dominant feature in a scene and biases attention to the relative feature that discriminates the target from the (dominant feature in the) context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' However, this fast, automatic evaluation of the context has never been tested, as the target and non-targets were Standard Tuning ExampleDisplay Optimal Tuning (and Optimal Tuning Relational Tuning (similar nontargets) dissimilarnontargets) NT NT R NTNT R NT NT R7 typically repeated numerous times in previous studies, and the reported selection behaviour was based on average trial performance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Becker, 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' This leaves open the possibility that the target’s discriminative relative feature must be learnt, or that the visual system adapts weight settings over consecutive trials in a trial-and-error fashion to bias attention to the relative feature – which is very different from the claim that attention can be biased to relative features instantly, with the first glance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Second and relatedly, the boundary conditions for tuning to relative features are currently not clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Previous work has shown that attention will be tuned to the relative target colour when the target is the reddest (or greenest / yellowest / bluest) item in the display on ≥50% of the trials, including when the target always has the same feature and could be found by tuning attention to the exact feature value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Relational search was also observed across different feature dimensions (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', colour, size, brightness, shape), and across different search tasks such as pop-out search (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2014), feature search (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2013) and conjunction search (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' However, it is currently unknown if we would observe relational search when the target is never the reddest (or greenest / yellowest / bluest) item in the display, but differs from the majority of non- target items in one direction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', redder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Previous studies have shown that we can enforce tuning to a specific feature value;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' for example, by presenting a target among equal numbers of flanking nontarget colours, such as presenting an orange target among three yellow and three red nontargets (Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' As the target is neither the reddest nor the yellowest item, attention cannot be tuned to the relative feature but needs to be tuned to the target’s feature value in these displays (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Becker, Harris, Venini & Retell, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Harris, Remington & Becker, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Schoenhammer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' but see Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 8 Given the ability to tune attention to specific feature values, it is unknown whether participants would adopt a relational search or feature-specific search when the target is never the reddest item in the visual field, yet differs from a majority of items in a single direction (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', redder).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' With that, it is currently unclear whether relational search is indeed based on a mechanism assessing the dominant feature in the visual field, or whether it is only adopted when relational search allows localising the target successfully on a large proportion of trials (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', through statistical learning).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Third, previous work on characterising the attentional tuning function is also limited in that the studies typically used fairly sparse displays, of four to eight items (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Becker, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Hamblin-Frohman & Becker, 2021, Navalpakkam & Itti, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The results typically showed that visually salient items (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', a red item among blue-green stimuli) did not attract attention, or attracted attention only very weakly (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Gaspelin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Martin & Becker, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' York & Becker, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' These results were taken to show that strong capture by relational items was not mediated by saliency, and that bottom-up saliency may modulate attention to a lesser extent than assumed in current models (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Theeuwes, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Wolfe, 1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' However, Wang and Theeuwes (2020) argued that item salience is drastically weakened in sparse displays.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' If this is correct, previous studies on attentional tuning may have overestimated the role of tuning to relative features and/or underestimated the role of bottom-up saliency effects in guiding attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Thus, it is still an open question whether previous results would generalise to more realistic search settings, such as search displays containing a larger number of search items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Aims The aim of the present study was to test some of the untested assumptions of the relational account in more realistic search settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' To that end, the search displays always contained thirty-six items, one of which was a saliently different distractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' This allowed assessing 9 possible contributions of bottom-up saliency effects to visual selection in more realistic search conditions with potentially stronger saliency effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' To assess how attention is guided with versus without training on the task, we created four blocked conditions in which the search colours varied (bluer, greener, yellower and redder target;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Participants completed ten trials in each block, and then switched to a different search colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Prior to each block, participants were only informed about the exact target colour (teal or orange), but not about the colour of the non-target items or the target’s relative colour (redder, yellower, greener, or bluer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Thus, analysing the first trial of each block allows assessing how attention was tuned to the target under conditions of uncertainty and in the absence of training effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Example Display Overview of Conditions Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Left: Example of a visual search display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Search displays consisted of three target- coloured items (here: orange) presented among 29 nontarget coloured items that could have two different colours (here: yellow-orange or yellow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Participants were instructed to search for a target-coloured item (here: orange) that contained the letter u or m, and to report whether the target contained the u or m (whereas the other orange items contained the letters w or m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' To assess how attention was guided in search, we assessed eye movements to items with a relationally matching colour (here: red), optimal colour (here: red-orange), and a saliently different colour (here: violet).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Right: Colours used to create the four blocked conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The target could be teal or orange, and the teal target could be presented among other green-ish non-targets (bluer target), blue-ish non-targets (greener target), while the orange target could be among red-ish non-targets (yellower target) or yellow-ish non-targets (redder target).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Prior to a mini-block of 10 trials, participants were informed of the target colour, but not of the colour of the non-targets, which determined whether the target would be relationally redder/yellower or greener/bluer than the non-targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' mBluer R 0 T NT NT S NT NT T 0 R Greener Yellower R 0 T NT NT S NT NT T 0 R Redder10 Deviating from previous studies, the search displays always contained all possible distractor colours, allowing a more fine-grained measurement of distractor effects and an assessment of possible boundary conditions for relational tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' For example, in the redder target condition, an orange target was presented among 29 yellow-orange and yellow nontargets, one distractor with a relatively matching colour (red), two distractors with an optimal colour;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', an exaggerated target colour that was slightly shifted away from the nontarget colours (red-orange), two distractors with the target colour (orange), and a saliently different distractor with an unrelated colour (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', a blue;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' These displays allows assessing how attention is tuned to the target when the target differs from the majority of non- targets in a single direction (redder) but is never the relationally maximal item (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', reddest, bluest, greenest or yellowest item) in the display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' To assess how attention was tuned to the target, we measured eye movements to each of the different item types (target and distractors).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Specifically, to tap into early processes of visual selection, we analysed the first eye movements on a trial, which are not influenced by prior fixations or search processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' If attention is tuned to the exact target colour, we would expect a large proportion of first eye movements on the target-coloured items (orange, teal), regardless of the context colour, and only few eye movements on the differently coloured distractors (optimal, relational).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' On the other hand, if attention is tuned to the optimal colour, we would expect most first eye movements on the optimal colour and a decrease in selecting distractors with other-than- optimal colours.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' According to the relational account, we would expect most first eye movements directed towards the distractor that best matches the relative colour of the target, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', the distractor that is the reddest, yellowest, greenest, bluest in the visual field, followed by the optimal distractor (as this is the next-maximal, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', next-reddest item), and the target- coloured distractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 11 Moreover, if relational search is based on a fast, automatic assessment of the dominant colour in the display, we would expect to see these results immediately in the first trial of each block, prior to learning and knowledge of the relative values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' If the salient distractor automatically captures attention in these multiple-item displays (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Wang & Theeuwes, 2020), we would expect a large proportion of first eye movements on the salient distractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Moreover, if participants can learn to ignore or inhibit the distractor, we would expect this saliency-effect to decrease with training (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', over the course of a block;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Gaspelin & Luck, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Hamblin-Frohman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Method Participants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' To estimate the required sample size, we examined the ability to detect relational vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' feature-specific search in the first fixations on distractors in previous work (Becker, Harris, Venini & Retell, 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The weakest effect was the feature-specific effect (t(14) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='5, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='024 in search for a redder target;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' t(14) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='5, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='027 in search for a yellower target;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Exp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The BUCSS tool suggested a target sample size of N=32 for the present study, to achieve a power of 85% (with 50% assurance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Anderson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Thirty-four paid participants from the University of Queensland participated in the experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Two participants were excluded for having a low search accuracy (< 70%), leaving 32 in the final analysis (M age =23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='1 years (SD = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='9), 24 female).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The study was approved by the University of Queensland ethics board, and all procedures were in line with the Declaration of Helsinki.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Apparatus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Stimuli were presented on a 21-inch CRT monitor with a refresh rate of 85Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' A chin and headrest were used to hold the participant’s heads 600mm from the screen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Gaze location was measured by an SR-Research Eyelink-1000 eye tracker at 500Hz sampling rate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The experiment was controlled by Python’s PsychoPy (Peirce, 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 12 Stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' All stimuli were presented against a grey background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Each search array contained thirty-six coloured circles (radius: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='48°) arranged in a six-by-six grid format (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Stimulus locations were initially selected to have a 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='96° horizontal separation and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='30° vertical separation (centre-to-centre), which varied because the location of all non- target stimuli (NT) was randomly jittered by ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='43° horizontally and vertically on each trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The relevant distractors (relational, optimal, target-similar and salient) and the target were not jittered to retain precise eye tracking data and to ensure that these stimuli were never too close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The location of all items was randomly chosen on each trial, with the restriction that the items of interest could never appear in the corner and corner-adjacent positions of the search array (as these positions were too far from fixation), and never in the central four (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='82° from fixation) positions (as these were too close to fixation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Each coloured circle contained a letter, either ‘u’, ‘n’, ‘m’ or ‘w’ (height: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='29°).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The target stimulus always contained a ‘u’ or ‘m’.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The distractor items of interest never contained the target response characters (u/m).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Colours were selected from an equiluminant (30 ± 2cd/m2) RGB colour set (see Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' There were two potential target colours, orange (RGB: [227, 124,52]) and teal (RGB: [56,171,146]), that alternated between blocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The orange target could either appear among a set of redder non-targets, creating a yellower target condition, or among yellower non-targets, creating a redder target condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The teal target could be presented either among greener or bluer non-targets, creating bluer or greener target conditions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Each trial contained the same amount of distractor and non-target items: One distractor with a relationally matching colour (R), two distractors with an optimal colour (O), two target-matching distractors (T), and a salient distractor (S) that always had a colour from the opposite side of colour space (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', a pink distractor for when the target was teal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The stimuli were always presented among 29 non-target items that were selected from two other colours (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', two blue-ish colours;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 13 Design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The colours of the target, distractors and non-targets were always repeated within a mini-block of 10 trials, and mini-blocks alternated between orange and teal targets, with the direction of search (redder/yellower or greener/bluer condition) determined randomly at the start of each block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Participants completed 64 mini-blocks, for a total of 640 trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The first four blocks were treated as practice trials, leaving 600 trials for the final analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Prior to the experiment, participants were instructed to locate the target- coloured stimuli containing the character u or m and respond with the corresponding keyboard key as quickly and accurately as possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Moreover, prior to each mini-block, participants were shown the colour of the upcoming visual search target (either orange or teal).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Importantly, no information about the colour of the non-targets was provided;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' hence participants did not know the relative colour of the target prior to the first trial (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', whether it was redder/yellower or greener/bluer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' To ensure stable and accurate eye tracking, participants were calibrated with a randomised 9-point calibration at the beginning of the experiment and whenever the fixation control failed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The fixation control was implemented prior to each trial, with participants maintaining fixation for 650ms on the central fixation cross.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' If gaze remained within 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='0° from the centre, the search array was presented until a manual keypress response (u, m) was recorded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' If an incorrect response was returned, error feedback was displayed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' After each trial, a blank grey screen was presented for 750ms, and the next trial commenced again with the fixation control.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Results Overall, accuracy on the letter identification task was high (>90%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Trials with incorrect responses were excluded from all analyses (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='6% of trials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The average response time (RT) was 2,207.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='7ms (SD = 1,919.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='4ms).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Trials that were more than 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='5 standard deviations above the mean RT (rounded to 7,000ms) were excluded from all analyses (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='9% of trials).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 14 Proportion of First Fixations on Each Item on Trials 1 - 10 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The proportion of first fixations directed towards the target, or any of the distractors of interest (target-matching, optimal, relational or salient), depicted as a function of trial repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The relational and optimal distractors attracted the highest proportions of first fixations and showed a linear increase over the ten trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The target and the target-matching distractor received more first fixations than the saliently different item, and also displayed a linear increase in fixations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The salient item was selected least frequently and showed a linear decrease in first fixations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' First Fixations: Training Effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' To assess whether attentional priorities changed with experience, we analysed the probability of fixating on each of the differently coloured distractors as a function of trial repetition (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The probability of fixating on each differently coloured distractor was computed by dividing the proportion of trials in which a given distractor was fixated (as the first item) by the number of distractors present on the trial (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', as there were two optimal-coloured stimuli, the proportion of first fixations on an optimal coloured stimulus was divided by two).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The proportion of first fixations on the target stimulus and the target-matching distractor did not differ from each other, F(1, 31) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='95, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='337, and did not interact with trial repetition, F(9, 279) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='16, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='321.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' This shows that 14 Target TMatch (%) suo!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='1e Optimal 12 Relational Salient 10ortion of First Fix 8 6 42 P 0 T1 T2 T3 T4 Te T7 T: Tg T10 Ts Trial Repetition15 attention was not guided to the target item based on the contained response-related character, and thus the data were collapsed across the target and target-coloured distractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' A 4 (Item Type: Target-Matching, Optimal, Relational, Salient) x 10 (Trial Repetition: T1 to T10) repeated measures analysis of variance (ANOVA) was computed over the probabilities of fixating on each differently coloured item.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The results showed a main effect of item type, F(3, 93) = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='47, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001, ƞ2p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='63, trial repetition, F(9, 279) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='42, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001, ƞ2p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='13, as well as a significant interaction, F(27, 837) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='46, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001, ƞ2p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Pairwise t-tests showed that both the relational and optimal items attracted more first eye movements than the target-coloured items (relational: t(31) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='33, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001, BF10 = 2500.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='17, optimal: t(31) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='61, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001, BF10 = 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='24 * 104).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' In turn, the target-coloured items received more first saccades than the saliently different item, t(31) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='96, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001, BF10 = 71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Optimal and relational items did not differ, t(31) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='07, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='294, BF10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' To investigate possible linear trends that may reflect learning or adaptation effects, we next examined the slopes of first fixation locations separately for each of the differently coloured items across the ten trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The linear trends for all four item types differed significantly from zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The relational item had the steepest positive slope, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='33, t(31) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='38, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001, followed by the optimal item, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='18, t(31) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='87, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='007, and the target- matching item, β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='09, t(31) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='36, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='025, reflecting that selection increased for all of these items over the course of the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Thus, the first fixations in a trial show relational tuning, which increased slightly and remained dominant across all trials in the block.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Finally, the saliently different item showed a significant reduction in visual selection across blocks, β = -0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='11, t(31) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='08, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='046.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' First Fixations: Initial Guidance on T1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' We next analysed the eye movement behaviour on the first trial of the block, where participants were unaware of the relative colour of the 16 target (redder/yellower or greener/bluer), and only knew the exact target colour (orange or teal;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 3, T1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Paired-samples t-tests revealed that first saccade proportions on T1 were lower for the target colour than both the optimal, t(31) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='34, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001, BF10 = 185.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='7, and relational distractors, t(31) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='50, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='018, BF10 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Optimal and relational distractors did not significantly differ from each other, t(31) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='59, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='123, BF10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='58.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The salient item did not differ significantly from the target-coloured item, t(31) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='88, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='388, BF10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The finding that relational search was conducted on trial 1 shows that experience or prior knowledge is not necessary for relational search, and supports the hypothesis that the visual system can quickly assess the dominant colour in the visual field and tune attention to the relative colour of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Fixation Progression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The results above suggest that search was initially (on the first trial) relational and that this relational guidance was maintained and increased across repeated trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' To assess whether the same trends may be found within a single trial, we analysed the first five fixations in each trial to reveal how attentional guidance developed within an individual trial (see Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' On average participants made 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='58 fixations per trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' We included all trials in this data set, including those where the task was completed within the first five fixations (which resulted in 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='7% missing data for F2, F3: 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='4%, F4: 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='5%, and F5: 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='6%).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The first fixation (F1), now collapsed over trial repetition, followed the same results pattern as previously described, with the relational and optimal stimuli attracting a higher proportion of first fixations than the target and the target-matching distractor (all ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' For F2 results began to deviate from relational guidance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The target stimulus was now more likely to be fixated than any other item (ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The target-matching distractors were now equally likely to be fixated as the optimal distractors, t(31) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='14, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='888, BF10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='19, and 17 the optimal distractors were now more likely to be fixated than the relational item, t(31) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='32, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001, BF10 = 178.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='09.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' From F3 onwards fixation patterns remained consistent: Now the target-matching distractors were more likely to be fixated than the optimal distractor (all ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001), and the optimal distractor was more likely to be fixated than the relational distractor (all ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Finally, fixations on the salient item were highest on F1 than at any other point in the trial (all ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001) and were significantly lower than all other stimuli of interest across all fixations (ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='004).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' First Five Fixations in a Trial Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The first five fixations (F1 to F5) in a trial show how search progressed during a trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The first fixation (F1) displayed relational search, with the relational and optimal distractors being selected more frequently than the target and target-matching distractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The second fixation (F2) showed a shift to more feature-specific guidance: The target was now the most likely to be fixated, followed by the target-matching and optimal distractors, with fewer fixations on the relational distractor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' With the third fixation (F3), the target-matching distractor was now more likely to be fixated than optimal or relational distractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' From this point onwards, fixation proportions seem to be dictated by target-feature similarity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The salient item was most frequently fixated on F1 then dropped off in subsequent fixations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 24 Target T_Match Optimal (%) suo 20 Relational Salientoportion of Fixati 8P 4 0 F1 F2 F3 F4 Fs Fixation18 In sum, the results are in line with the hypothesis that (in relational search) participants will first select the relationally maximal item (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', reddest), followed by the optimal (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', next-reddest) item, and the target-coloured items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Another way of describing these results is that attention was always initially guided to the relatively matching items, with feature- specific tuning developing only after the first (few) fixation(s), perhaps by inhibiting the more extreme colours that were selected first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' First Saccade Latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' We next analysed the saccade latencies, that is, the time from the onset of the search display to the onset of the first saccade in a trial, to assess the time- course of attentional deployment using the first saccade in a trial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' This analysis was collapsed over trials, with the first trial being excluded (due to longer saccadic latencies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='1 Figure 5 shows the distributions of saccadic latencies separately for each item type (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', the proportion of trials where saccadic latencies ranged from 125ms – 150ms, 150 – 175ms, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' To analyse the data statistically, we fed the average saccade latencies of the first saccades in a trial into a one-way (Item Type: Target, Optimal, Relational, Salient) repeated-measures ANOVA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The results showed a significant effect of item type F(3,93) = 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='01, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001, ƞ2p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='63.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Two-tailed comparisons revealed that latencies were shorter for the salient item (M = 217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='6ms) than both the relational (M = 237.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='5ms) and optimal item (M = 240.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='9ms), all ts> 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='18, ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001, BF10 > 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='98 x 106.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Saccades to target-matching items (M = 252.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='0ms) were slower than to the relational and optimal items, ts(31) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='35, ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001, BF10 > 191.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='72, while relational and optimal items did not differ, t(31) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='26, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='218, BF10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Thus, the more dissimilar the stimuli were from the target item, the shorter their saccadic latencies were.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 1 Saccade latencies on T1 were longer than in all other trial repetitions (all ps < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001), but there was no repetition by item type interaction (p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='650) or any linear trends for the individual items after excluding the first trial (ps > .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='235).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 19 Distribution of First Saccade Latencies Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The distribution of saccade latencies, which was derived by sorting saccades to each stimulus type into 25ms latency bins and depicting the proportions of trials within each bin separately for each item type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The results showed a higher proportion of short-latency saccades to the saliently different distractor than to the other distractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Moreover, there was a higher proportion of saccades with long latencies to target-matching items (pooled over target and target-coloured distractor), producing differences between the target-coloured items and the other distractors in the tail end of the distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Figure 5 shows that the distribution of saccadic latencies and reveals that differences in the average saccade latencies were due to saccades to salient distractor being initiated significantly earlier than to all other items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' By contrast, the longer saccade latencies for the target and target-matching distractors were due to a higher proportion of saccades with longer latencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Distractor Dwell Times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' As in previous studies, we also examined the mean dwell times on each of the distractors, measured as the time spent fixating on each of the different item types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The target-matching stimuli were examined independently from the actual target (M = 162.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='9ms), and results were collapsed over trial repetition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 0 5 10 15 20 25 30 125ms 150ms 175ms 200ms 225ms 250ms 275ms 300ms 325ms 350ms Proportions Saccade Latency (25ms Brackets) Target-Matching Optimal Relational Salient 20 The one-way (Distractor Type: Target-Matching, Optimal, Relational, Salient) repeated- measures ANOVA revealed significant differences between the item types, F(3,93) = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='50, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001, ƞ2p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='57.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Planned two-tailed comparisons showed that dwell times for the target- matching (M = 173.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='0ms) and optimal distractors (M = 171.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='4ms) were longest and did not differ from each other, t(31) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='92, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='363, BF10 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Dwell times were significantly shorter for the relational distractor (M = 166.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='4ms) than the optimal distractor, t(31) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='74, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001, BF10 = 7312.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='78, and target matching distractors, t(31) = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='21, p = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='003, BF10 = 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The relational distractor had longer dwell times than the salient item (M = 152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='5ms), t(31) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='26, p < .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='001, BF10 = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='90 * 104.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' In sum, dwell times increased with similarity to the target colour, indicating that longer dwell times were needed to identify the items as distractors as they became more similar to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' General Discussion The current study revealed that relational effects may be more pervasive than previously assumed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' This was the first study to examine attentional tuning functions in the absence of target contextual knowledge and in displays containing numerous items.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The relatively matching distractors attracted the first saccade, including on the first trial in a mini-block, when participants only knew the colour of the target but not its relative contextual colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' These results show that relational/contextual information can be quickly extracted from a visual scene, in line with the tenets of the relational account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Previous studies have already shown that information about the statistical properties of a visual scene can be rapidly extracted (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', feature averaging, Chong & Treisman, 2005a, b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Choo & Franconeri, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Joo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' gist processing, Oliva & Torralba, 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' see also Thorpe, Fize & Marlot, 1996).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The present study extends on this research by showing that this information can be used to bias attention to the relative features of the target, prior to executing the first eye movement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' In other words, relational guidance does not require learning of the context or 21 knowledge of the relative target feature, but can be executed with the first glance at a novel scene.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The current results also revealed that relational guidance is more ubiquitous than previously thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' In particular, we found relational guidance even when the target never had a relationally maximal value – that is, when it was never the reddest, greenest, bluest or yellowest item in the display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' In every trial there were relationally better-matching distractors (one relatively matching distractor and two optimal distractors), which frustrated selecting the target with the first eye movement (in relational search).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Yet, we reliably found that participants searched relationally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' This reveals that the visual system is indeed sensitive to the dominant feature in the display, and initiates relational search as soon as the target differs from most of the display items in a linear fashion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Furthermore, participants continued to search relationally over as many as ten successive trials, even though relational guidance reliably resulted in selecting one of the more extreme distractors first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' This means that relational guidance is more readily applied and more persistent than previously thought (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The results of the present study also provided evidence for selection proceeding in the way as laid out in the relational account, with the most extreme, relatively matching item being selected first (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', reddest), followed by the next most extreme, relatively matching item (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', next reddest item), and so forth, until the target is selected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' As shown in Figure 3, the first saccade in a trial was most likely to select the most extreme, relatively matching item, whereas later saccades were more likely to select target-matching items and the probability for selecting extreme items declined after the second fixation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The mechanism responsible for achieving this order of selection has never been specified in the relational account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Other theories have proposed a memory-based, ‘inhibitory tagging’ mechanism to explain how search proceeds to the next item (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Klein, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 22 Correspondingly, it is possible that selection of target-matching items emerged during the trial due to selection and subsequent inhibition of relatively matching, more extreme distractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Inhibitory tagging could be either location-based (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', inhibiting selected nontarget locations during the trial), or feature-based (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', inhibiting a non-target or distractor colour after selection;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Bichot & Schall, 2002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' If inhibitory tagging is responsible for the selection sequence in the present study, it is perhaps more likely to be feature-based inhibition rather than location-based inhibition: Selection of the target and target-matching distractors commenced earlier than would be expected on the basis of location-based inhibition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Specifically, as there were three relatively matching or optimal distractors in the display, selection of these distractors should have declined after the 3rd fixation, but we observed the decline after the 2nd fixation, which corresponds to the number of different colours in these distractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Also, selection of the target and target-matching items showed a steep increase after only one fixation, which seems too early for location-based inhibition, and indicates that the colour(s) rather than the locations of the more extreme distractors may have been inhibited.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Alternatively, it is possible that, after the first fixation, attention was tuned to the exact feature value of the target, swiftly switching from originally relational search to a more feature-specific search (or target template;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Duncan & Humphreys, 1989).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' We currently consider this unlikely, as an attentional bias usually automatically carries over to the next trial to influence selection (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', intertrial priming;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Maljkovic & Nakayama, 1994), and the first eye movements never showed evidence of a feature-specific bias.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Inhibition of a distractor feature can also carry over to the next trial to influence selection, but the effects are weaker than effects of target guidance (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Chang & Egeth, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Hamblin-Frohman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' This could explain that relational guidance was somewhat attenuated, as the relatively matching distractor mostly did not differ significantly from the 23 optimal distractor, but was still clearly visible (as both the relatively matching and optimal distractors were selected more frequently than the target-matching distractors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' However, the somewhat attenuated effect of the relatively matching distractor could also be due to difficulties distinguishing the optimal and relatively matching distractor in the far periphery (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Noorlander et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 1983), or to the fact that there was only one relatively matching, extreme distractor and two optimal distractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Further studies are necessary to estimate the magnitude of the attenuation and clarify the mechanism that allows narrowing search to target-matching items after the first fixation(s).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' A fourth interesting finding relates to the salient item, which had a very dissimilar colour compared to the other items and hence, had the largest bottom-up feature contrast.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Wang and Theeuwes (2020) proposed that experiments with sparse displays may underestimate bottom- up saliency effects (see also Rangelov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The displays of the present study contained the largest number of items tested for fine-grained attentional tuning (36 items), yet we found only very weak effects for the salient item, which further declined over trials 1 – 10 (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Given the low proportion of first fixations on the salient item, we cannot claim that the salient item significantly attracted attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='2 Even in the densely populated displays, top-down tuning to the (relative) target feature led to significantly higher selection rates of the corresponding distractors than bottom-up saliency, indicating that top-down processes dominate attentional guidance over bottom-up saliency, contrary to bottom-up selection views (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Theeuwes, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Wang & Theeuwes, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' These results also rule out an alternative explanation of the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Proponents of feature- based theories (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Guided Search 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Wolfe, 1994) may argue that the results could be due 2 Comparing the proportion of first fixations on the salient item to the proportion of fixations on the non-salient non-target items (omitting the corner positions and central positions that never contained a distractor) revealed that the salient item was actually selected less frequently than the non-salient non-targets (p=.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='03).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 24 to a combination of top-down, feature-specific tuning to the target and bottom-up processes, as the more extreme distractors (optimal and relatively matching distractors) may have been similar enough to the target to be subject to (broad) top-down tuning and were simultaneously more salient than the target, leading to higher selection rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Contrary to this contention, we found a very different results pattern for the salient distractor compared to the other distractors, both in the trial analysis (which showed an increase in capture by extreme and target-similar distractors over trials and a decrease for the salient distractor;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 3), and in the time-course analysis (which showed shorter latencies of saccades to the salient distractor than for all other distractors;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' These findings suggest that the more extreme (and more salient) relatively matching and optimal distractors attracted attention and the gaze in the same manner as the target-similar (less salient) distractor, viz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', due to top- down tuning and not bottom-up saliency, whereas the effects of the salient distractor were due to bottom-up, stimulus-driven processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Admittedly, it is notoriously difficult to distinguish between combined top-down/bottom- up theories, optimal tuning and the relational account, especially in standard visual search experiments with a distractor, as the theories make quite similar predictions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' However, the relational account has been extensively tested against these other theories in previous experiments using the spatial cueing paradigm (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2013) and visual search with relatively matching distractors that were very dissimilar from the target (York & Becker, 2020) or whose similarity / saliency varied systematically (Hamblin-Frohman & Becker, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Yu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The results unequivocally supported the relational account and showed that early selection cannot be explained by optimal tuning or the combined saliency and similarity to the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Previous studies testing the relational account against the standard, feature-based theories moreover revealed that tuning to relations was a default search mode applied across a wide 25 range of different tasks that would have allowed locating the target by tuning to its specific feature value (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', when the target was always repeated).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The present study extends on these findings, by showing that attention is tuned to relative features even when the target is never the relatively reddest, bluest, greenest or yellowest item in the display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' In conditions such as the present experiment, a feature-specific guidance mechanism may be the most efficient search method for the current display, as it would have reduced competition for attention to only target-similar distractors (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', two other items) by excluding relatively matching and optimal distractors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Participants chose relational tuning over feature-specific tuning even though, on the first trial of each block, they were only given information about the exact feature-value of the target stimulus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' The instructional prompts mentioning only the target colour should have primed participants to initiate a guided search for the exact target colour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Yet, participants consistently engaged in relational search and also did not learn to use feature-specific tuning over the course of ten (identical) trials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' As shown in Figure 3, linear trends for first eye movements were stronger for the relational and the optimal distractors compared to the target-matching items, suggesting that the relational effect became stronger over the course of the block rather than weaker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Collectively, these results show that participants can be quite reluctant to tune attention to the specific feature value of the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' We can only speculate why this may be the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Previous studies have shown that feature-specific tuning results in delays in selecting the target, as well as longer dwell times on target-similar distractors, compared to tuning to relations (in identical displays;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' Martin & Becker, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' This suggests that tuning attention to exact feature values may cause delays in selection as well as decision- making (about whether the selected item is the target), rendering search less efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' While this question warrants further research, the present study clearly showed that tuning to relative features occurs in more densely populated, 36-item displays;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' in the absence of 26 knowledge of the relative feature of the target, and in the absence of any training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' This demonstrates the existence of a mechanism capable of extracting the dominant feature in a visual scene and using that information to tune attention to the relative feature of a target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' This renders it likely that relational search is the default search mode in everyday situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' 27 Author note.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content=' This research was supported by Australian Research Council (ARC) grant DP210103430 to SIB.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='1177_095679762 11032225-FIG2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} +page_content='JPEG' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/OtE1T4oBgHgl3EQfaASJ/content/2301.03157v1.pdf'} diff --git a/P9E0T4oBgHgl3EQfTwDw/content/tmp_files/2301.02242v1.pdf.txt b/P9E0T4oBgHgl3EQfTwDw/content/tmp_files/2301.02242v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..44bbb873b7b7d7c30040d1d2a9cd9cf38db83578 --- /dev/null +++ b/P9E0T4oBgHgl3EQfTwDw/content/tmp_files/2301.02242v1.pdf.txt @@ -0,0 +1,590 @@ +GRAPH CONTRASTIVE LEARNING FOR MULTI-OMICS DATA +Nishant Rajadhyaksha, +K.J Somaiya College of Engineering +Mumbai +n.rajadhyaksha@somaiya.edu +Aarushi Chitkara +St Xaviers College +Mumbai +aarushi5812@gmail.com +ABSTRACT +Advancements in technologies related to working with omics data require novel computation methods +to fully leverage information and help develop a better understanding of human diseases. This paper +studies the effects of introducing graph contrastive learning to help leverage graph structure and infor- +mation to produce better representations for downstream classification tasks for multi-omics datasets. +We present a learnining framework named Multi-Omics Graph Contrastive Learner(MOGCL) which +outperforms several aproaches for integrating multi-omics data for supervised learning tasks. We +show that pre-training graph models with a contrastive methodology along with fine-tuning it in a +supervised manner is an efficient strategy for multi-omics data classification. +Keywords Graph Contrastive Learning · Graph Neural Networks · Multi-Omics +1 +Introduction +Omics, referring to a field of study in biological sciences that ends with -omics, aims at the collective characterization +and quantification of pools of biological molecules that translate into the structure, function, and dynamics of an +organism or organisms. The use of high throughput biomedical research methods has increased substantially over the +past few years such as Whole Genome Sequencing(WGS), RNA sequencing(RNA-seq), chromosome conformation +capture (Hi-C) and liquid chromatography coupled with mass spectrometry [1,2]. It is particularly helpful to integrate +data from different molecular sources such as microRNA(miRNA),mRNA and DNA methylation to provide insight into +the classification and processes of different diseases. Integration of multi-omics data requires efficient computational +methods which are capable of correctly modelling and interpreting these relationships. Whilst each omics technology +can present only a part of the true biological complexity integrating data from multiple omics technologies can help +provide a more universal picture which can help improve results for classification tasks [3–9]. +Several different methodologies have been proposed to help integrate multi-omics data for classification tasks. Generally, +different omics data have often been concatenated together or have been subject to ensemble learning where prediction +for each omics type is ensembled together [10,11]. The recent emergence of Graph Neural Networks(GNNs) as an +efficient deep learning methodology to model complex systems has prompted researchers to study its effects when +paired with multi-omics data [12, 13]. Graph contrastive learning is a learning paradigm where data-data pairs are +utilised to perform discrimination between positive and negative pairs of graph data. Contrastive learning can be used +as an effective pre-training strategy for training graphical models on supervised learning tasks. This paper describes +a framework named MOGCL which constructs graphs from multi-omics data and utilises contrastive learning as a +pre-training strategy to aid downstream classification tasks. We compare our results on benchmark datasets with several +different machine-learning methodologies. MOGCL performs better on several metrics such as Area Under the ROC +Curve(AUC), Accuracy and F1 score etc. +arXiv:2301.02242v1 [q-bio.GN] 3 Jan 2023 + +2 +Literature Review +2.1 +Machine learning for multi-omics data +A significant number of methods have been studied to integrate multi-omics data for various tasks. A large number of +methods focus on the semi-supervised integration of multi-omics data without utilising the information from ground +truth labels present in the dataset [14–17]. Most self-supervised methods focus on assigning clusters to different omics +groups present in the dataset. With the rapid advancements of biotechnology and the detailed curation of datasets, +more labelled samples of phenotypes or traits are being made available for research. This has led to the development +of supervised learning models which perform better on multi-omics datasets [18–22]. Kernel methods are powerful +machine learning models which operate on a higher dimensional space where linear interpolations can be found between +the given data points. Kernel methods are often used as classical machine learning models for analysing and classifying +multi-omics. Support Vector Machines(SVM) [23] and partial least squares [24] are examples of classical machine +learning approaches for multi-omics data. Currently, deep learning approaches are commonly adopted for multi-omics +integration for various tasks. Islam et al. [25] propose an approach which utilises a convolutional neural network to +learn important features for multiple omics types and finally concatenate them to predict breast cancer subtypes. An +approach using stacked Sparse Autoencoders (SAE) was proposed by Xu et al. [26] where representations are produced +for each type of omics data using autoencoders which are then fed to a deep flexible neural forest for predicting cancer +subtypes. +2.2 +Graph based learning approaches +Graphs are complex data structures which are used to model many complex phenomena in the real world. Graph +Neural Networks (GNN) deal with applying deep learning to graphical data structures. GNNs have several applications +such as combinatorial optimizations, neural machine translation, protein-protein interactions, drug discovery [27–34]. +Recently graph-based approaches have been used for multi-omics integration. Wang et al. [35] proposed a methodology +to convert multi-omics data to graphs and a model named MOGONET consisting of convolutional network (GCN) +layers [36] to produce refined representations for a downstream classification task on multiple multi-omics datasets +whilst also identifying important biomarkers. Xiao et al. [37] proposed a model named MOGCN for different cancer +sub-type classification tasks based on multi-omics data. Graph contrastive learning is an example of a self-supervised +training methodology where different graph augmentations are utilised to exploit both structural information and +information about features of the dataset to produce better representations. Some common training strategies are +pre-training followed by fine-tuning [38] and joint learning [39]. Zhu et al. [40] proposed a framework titled deep +GRAph Contrastive rEpresentation learning (GRACE) which specifically generates two graph views by corruption and +attempts to learn node representations by maximizing the agreement of node representations in these two views. +3 +Methodology +3.1 +Datasets +We demonstrate the effectiveness of MOGCL by applying our model on two benchmark datasets namely ROSMAP [41] +which describes information for patients with Alzheimer’s Disease (AD) concerning a normal control group (NC) and +BRCA which contains data for breast invasive carcinoma (BRCA) PAM50 subtype classification. The omics data +used were namely were DNA methylation data (meth), miRNA expression data (miRNA) and mRNA expression data +(mRNA). Details about the datasets are further described in table 1. +Dataset Name +Labels +Number of features for +mRNA, meth, miRNA +Number of features for training +mRNA, meth, miRNA +ROSMAP +NC: 169, AD: 182 +55,889, 23,788, 309 +200, 200, 200 +BRCA +Normal-like: 115, +Basal-like: 131, +HER2-enriched: 46, +Luminal A: 436, +Luminal B: 147 +20,531, 20,106, 503 +1000, 1000, 503 +Table 1: Summary of datasets +2 + +3.2 +Contrusting graphs from multi-omics data +In this section, we describe the methodology of converting multi-omics data to a graphical structure which can then +be leveraged by powerful graph neural network models for further processing. Our task can be defined as defining +graphs G = (V, E) where V, E represent vertices and edges of the graph respectively. We utilise the feature matrices we +obtain after preprocessing each type of omics data. The feature matrix for each omics type is represented as X ∈ Rn×d +where for the ROSMAP dataset d is 200 for each of the omics types and n is 351. Similarly for the BRCA dataset n +is 875 and d ranges from 1000 for mRNA and meth data to 503 for miRNA data. The nodes V of graph G represent +the users from which the omics data is collected. We construct an adjacency matrix A ∈ Rn×n to represent G with +each element in the adjacency matrix representing a node. We denote a weighted edge from node i to node j of the +graph as the element present at the ith row and jth column of A. Such an adjacency matrix is constructed for each type +of omics data respectively. A pairwise distance matrix is constructed for data for the points of the particular omics +dataset using cosine similarity [42] as the distance metric. The distance between node i and node j is denoted by tij. A +parameter k is introduced which represents the number of edges to be formed per node. An adjacency parameter is then +chosen by selecting the n × kth value from a sorted array of pairwise distances between all data points. Edges E is then +selected on the criteria of the distance between data points being smaller than the adjacency parameter. This ensures +that the number of edges per node is k. A weight of 1 − tij is assigned to the edge from node i to node j if belongs to +the set of selected edges. An adjacency matrix is prepared for each of the omics types present in the respective dataset +by following the methodology described above. +3.3 +Graph constrative learning +In this section, we describe our training methodology which utilises graph contrastive learning. We use GRACE [40] +which serves as our self-supervision model. Our contrastive learning methodology consists of two stages namely i) +data augmentation and ii) contrastive learning. Augmentation functions such as removing random edges and feature +masking are used to create augmented views of a graph. For augmenting edges, we randomly remove a portion of +edges from the original graph. We sample a random masking matrix ˜R ∈ {0, 1}N×N. Where the elements of R +are drawn from a Bernoulli distribution ˜R ∼ Bern(1 − pr) where pr is the probability of each edge is removed. +We choose pr to be 0.4 for our study. The resulting adjacency matrix can be given as ˜A = A ◦ ˜R. For augmenting +features we randomly mask the dimensions of a feature vector by replacing them with zeros. We sample random feature +vectors to construct a matrix ˜ +M ∈ {0, 1} according to a Bernoulli distribution having a similar size as feature matrix +X. The augmented feature matrix can then be represented by ˜X = X ◦ ˜ +M. We use a GCN [36] model as an encoder +model which helps represent the augmented views of a given graph and denote it with f. Let U = f( ˜ +X1, ˜ +A1) and +V = f( ˜ +X2, ˜ +A2) be the representations generated after processing two graphs with our shared encoder model. We aim +to maximise the agreement between similar nodes in the latent space and minimise the agreement between the rest +of the contrasting nodes. To achieve this we make use of the Normalized Temperature-scaled Cross Entropy Loss +(NT-Xent) [43]. NT-Xent loss is given by eq 1. +ℓ(ui, vi) = log +eθ(ui,vi)/τ +eθ(ui,vi)/τ + +� +k̸=i +eθ(ui,vk)/τ + +� +k̸=i +eθ(ui,uk)/τ , +(1) +where ui and vi represent the ith feature vector from the feature matrix U and V respectively. τ represents a temperature +parameter. θ is a similarity function given in equation 2. +θ(u, v) = c(n(u), n(v)) +(2) +where c(.,.) is the cosine similarity function and n(.) represents any non-linear function such as ReLU [44] or +LeakyReLU [45] etc. We finally optimise the weights of the shared encoder model on the NT-Xent loss. +The GCN encoder is further trained in a supervised manner using labels from the given dataset. The encoder model +was trained for a downstream classification task using pre-training followed by fine-tuning. In pre-training, we first +fully train an encoder model for each omics type in an unsupervised manner. We later fine-tune the models using label +information from the given dataset. Let ˜f be the pre-trained GCN encoder. We utilise linear layers in conjunction +with concatenated features produced from the encoder models to produce predicted label ˜Y = ˜f(X, A). We use the +Cross-Entropy Loss to calculate the loss for predicted labels ˜Y and true labels Y and finally optimise our encoder +model on this loss. +3 + +Figure 1: Contrastive Learning for GNN Encoder +Figure 2: Downstream Supervised Training of GNN Encoder +3.4 +Experiments +In this section we describe the experiments we perform to evaluate our MOGCL framework. We first produce graphs +for each omics type in our datasets and train a separate encoder model for each one respectively. We finally concatenate +the features produced by each encoder model and train the encoder model in a pre-training followed by fine-tuning +methodology. We compare our classification results to the ones described in [35] to evaluate the efficiency of introducing +a contrastive learning methodology for the given classification task. Performance of all permutations of encoder models +is calculated by conducting r = 5 runs with random weight initialisations for each permutation. We measure the +performance of our model on metrics such as accuracy, f1-score and AUC for the ROSMAP dataset and use accuracy,f1- +weighted score and f1-macro score to evaluate the BRCA dataset. We use PyTorch-geometric [46], PyGCL [47] and +pytorch-lightning [48] for conducting our experiments. Adam [49] optimizer with a learning rate of 0.0001 is utilised +for all our experiments. We use a two-layered GCN as our encoder model which is used in graph contrastive learning. +We further use two linear layers in conjunction with our encoder model to perform fine-tuning with the given true labels. +We compress all feature vectors to a 100-dimensional latent space for all our experiments. We try to visualise the effects +of our pretraining strategy by visualising the feature vectors before and after processing them with our encoder models +for each omics type. t-SNE [50] was utilised to compress feature vectors to a two-dimensional space in order to produce +visualisations. +4 +Results and Discussion +The results for the classification task for ROSMAP and BRCA datasets are displayed in table 2 and table 3 respectively. +4 + +000 +0000 +t ~T +GNN +Intrarview +ui +0000 +contrast +000 +000 +U= f(G) +0000 +000 +000 +000 +G = t(9) = (X1, A1) +000 +G1 +0000 +Shared +000 +000 +=(X,A) +000 +000 +Y1 +GNN +0000C0元 +t +000 +000 +000 +V= f(G2) +0000 + T +G2 = t(G) = (X2, A2) +000 +92 +0000Linear Layers +DNA Methylation +samples +Pre-trainedGCN +(meth) +Concatenated +Features +Final Logits +mRNA expression +samples +Pre-trained GCN +(mRNA) +miRNA expression +samples +Pre-trained GCN +(miRNA)Table 2: Results for classification task on ROSMAP +Method +Accuracy +F1 +AUC +KNN +0.657 ± 0.036 +0.671 ± 0.044 +0.709 ± 0.045 +SVM +0.770 ± 0.024 +0.778 ± 0.016 +0.770 ± 0.026 +Lasso +0.694 ± 0.037 +0.730 ± 0.033 +0.770 ± 0.035 +RF +0.726 ± 0.029 +0.734 ± 0.021 +0.811 ± 0.019 +XGBoost +0.760 ± 0.046 +0.772 ± 0.045 +0.837 ± 0.030 +NN +0.755 ± 0.021 +0.764 ± 0.021 +0.827 ± 0.025 +GRridge +0.760 ± 0.034 +0.769 ± 0.029 +0.841 ± 0.023 +block PLSDA +0.742 ± 0.024 +0.755 ± 0.023 +0.830 ± 0.025 +block sPLSDA +0.753 ± 0.033 +0.764 ± 0.035 +0.838 ± 0.021 +NN_NN +0.766 ± 0.023 +0.777 ± 0.019 +0.819 ± 0.017 +NN_VCDN +0.775 ± 0.026 +0.790 ± 0.018 +0.843 ± 0.021 +MOGONET_NN +0.804 ± 0.016 +0.808 ± 0.010 +0.858 ± 0.024 +MOGCL (ours) +0.818 ± 0.014 +0.818 ± 0.014 +0.866 ± 0.021 +Table 3: Results of classification task on BRCA. +Method +Accuracy +F1-Weighted +F1-Macro +KNN +0.742 ± 0.024 +0.730 ± 0.023 +0.682 ± 0.025 +SVM +0.729 ± 0.018 +0.702 ± 0.015 +0.640 ± 0.017 +Lasso +0.732 ± 0.012 +0.698 ± 0.015 +0.642 ± 0.026 +RF +0.754 ± 0.009 +0.733 ± 0.010 +0.649 ± 0.013 +XGBoost +0.781 ± 0.008 +0.764 ± 0.010 +0.701 ± 0.017 +NN +0.754 ± 0.028 +0.740 ± 0.034 +0.668 ± 0.047 +GRridge +0.745 ± 0.016 +0.726 ± 0.019 +0.656 ± 0.025 +block PLSDA +0.642 ± 0.009 +0.534 ± 0.014 +0.369 ± 0.017 +block sPLSDA +0.639 ± 0.008 +0.522 ± 0.016 +0.351 ± 0.022 +NN_NN +0.796 ± 0.012 +0.784 ± 0.014 +0.723 ± 0.018 +NN_VCDN +0.792 ± 0.010 +0.781 ± 0.006 +0.721 ± 0.018 +MOGONET_NN +0.805 ± 0.017 +0.782 ± 0.030 +0.737 ± 0.038 +MOGCL (ours) +0.853 ± 0.005 +0.851 ± 0.010 +0.823 ± 0.006 +The performance of MOGCL is compared with the following classification algorithms 1) K-nearest neighbour classifier +(KNN). K-nearest neighbours are chosen from the training data to make label predictions during evaluation. 2) Support +Vector Machine classifier (SVM). 3) Lasso which is L1-regularised linear regression. A unique model was trained +to forecast each class’s probability in Lasso, and the class with the greatest foretasted probability was chosen as the +final prediction of the model’s overall class label 4) Random Forest classifier (RF). 5) Extreme Gradient Boosting +(XGBoost) is a distributed, scalable gradient-boosted decision tree (GBDT) machine learning framework. 6) Fully +connected Neural Network (NN) classifier. loss for the fully connected NN was calculated by the cross-entropy loss. 7) +Adaptive group-regularized ridge regression (GRridge). 8) block PLSDA mentioned in DIABLO [6]. block PLSDA +performs latent Discriminant Analysis (LDA) to project multi-omics data to a latent space. To categorise a discrete +outcome, block PLSDA integrates various omics data types measured on the same set of samples. 9) block sPLSDA. +10) MOGONET_NN. MOGONET_NN is architecturally similar to MOGCL but does not use a pre-training strategy. +We achieve significant results by following our pre-training methodology as it performs better than the other models +on all metrics used to measure the results. For the ROSMAP dataset MOGCL achieves an average accuracy of 0.818 +in comparison to 0.804 achieved by MOGONET_NN. following this trend MOGCL achieves an F1-score and AUC +of 0.818 and 0.866 as compared to 0.808 and 0.856 achieved by MOGONET_NN. For the BRCA dataset MOGCL +achieves an accuracy of 0.853 as compared to 0.805 for MOGONET_NN. MOGCL receives an F1-weighted score +of 0.851 and an F1-macro score of 0.823 as compared to 0.782 and 0.737 respectively for MOGONET_NN. This +demonstrates that adopting a graph based semi-supervised learning strategy in addition to fine-tuning for a downstream +task is an effective training strategy for training models on multi-omics datasets. +We demonstrate the effects of adopting a semi-supervised methodology of training by analysing 3 and 4. We visualise +the feature matrices X by projecting data points into a two-dimensional plane by utilising t-SNE. Similarly, we map the +feature vectors produced by the GCN encoders to a 2-dimensional space and compare the results. MOGCL tries to +cluster embeddings in the absence of labels to create more structured representations during the pre-training phase. +Better representation help during the fine-tuning phase which in turn helps produce better classification scores. +5 + +Figure 3: BRCA Embeddings +Figure 4: ROSMAP Embeddings +6 + +DNA-Methylation-BRCA +DNA-Methylation-BRCA-Embeddings +mRNA-BRCA +miRNA-BRCA +miRNA-BRCA-Embeddings +mRNA-BRCA-EmbeddingsDNA-Methylation-ROSMAP +DNA-Methylation-ROSMAP-Embeddings +mRNA-ROSMAP +miRNA-ROSMAP +miRNA-ROSMAP-Embeddings +mRNA-ROSMAP-EmbeddingsFigure 5 represents the performance of permutation of different omics types when processed by MOGCL. We pre-train +three encoder models for all omics types in the study respectively. To calculate performance we select a permutation of +these encoder models and train them using true labels in a supervised manner. MOGCL performs its best when fed +information by concatenating all the omics types together for both the ROSMAP and BRCA datasets. For the BRCA +dataset, a combination of mRNA and DNA-Methylation data provides the next best results however for the ROSMAP +dataset a combination of mRNA and miRNA provides the next best set of results. For both the ROSMAP and BRCA +datasets using only a single omics type provides the worst results. Using only DNA-Methylation data is the least useful +option followed by miRNA and mRNA data across both BRCA and ROSMAP datasets. +Figure 5: Performance of Omics-Types +5 +Conclusion +This paper introduces a novel framework named MOGCL which introduces a graph contrastive learning methodology +for multi-omics data classification. We first provide a comprehensive literature survey regarding work done in the field +of machine learning relating to graph-based learning methods and multi-omics data. A method for constructing graphs +from multi-omics data is discussed. We then describe our framework MOGCL which uses GRACE as a pre-training +method followed by fine-tuning with true labels in a supervised setting. We discuss our results for the BRCA and +ROSMAP datasets and show that our framework performs better than other baselines used for this study. The use +of permutations of different omics types is discussed by analysing performance across different metrics. We discuss +the effects of adopting a semi-supervised pre-training strategy by visualising the embeddings produced by our graph +encoders. We finally conclude that adopting a pre-training methodology is an efficient way to train graphical models +for classification problems involving multi-omics datasets. Future works could include experimenting with different +contrastive learning methodologies to determine which one is the most efficient. 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Visualizing data using t-SNE. Journal of Machine Learning +Research, 9:2579–2605, 2008. +10 + diff --git a/P9E0T4oBgHgl3EQfTwDw/content/tmp_files/load_file.txt b/P9E0T4oBgHgl3EQfTwDw/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..d8bce18ade5976d3947c36a8a140f10060f46fc5 --- /dev/null +++ b/P9E0T4oBgHgl3EQfTwDw/content/tmp_files/load_file.txt @@ -0,0 +1,560 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf,len=559 +page_content='GRAPH CONTRASTIVE LEARNING FOR MULTI-OMICS DATA Nishant Rajadhyaksha, K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='J Somaiya College of Engineering Mumbai n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='rajadhyaksha@somaiya.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='edu Aarushi Chitkara St Xaviers College Mumbai aarushi5812@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='com ABSTRACT Advancements in technologies related to working with omics data require novel computation methods to fully leverage information and help develop a better understanding of human diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' This paper studies the effects of introducing graph contrastive learning to help leverage graph structure and infor- mation to produce better representations for downstream classification tasks for multi-omics datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We present a learnining framework named Multi-Omics Graph Contrastive Learner(MOGCL) which outperforms several aproaches for integrating multi-omics data for supervised learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We show that pre-training graph models with a contrastive methodology along with fine-tuning it in a supervised manner is an efficient strategy for multi-omics data classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Keywords Graph Contrastive Learning · Graph Neural Networks · Multi-Omics 1 Introduction Omics, referring to a field of study in biological sciences that ends with -omics, aims at the collective characterization and quantification of pools of biological molecules that translate into the structure, function, and dynamics of an organism or organisms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' The use of high throughput biomedical research methods has increased substantially over the past few years such as Whole Genome Sequencing(WGS), RNA sequencing(RNA-seq), chromosome conformation capture (Hi-C) and liquid chromatography coupled with mass spectrometry [1,2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' It is particularly helpful to integrate data from different molecular sources such as microRNA(miRNA),mRNA and DNA methylation to provide insight into the classification and processes of different diseases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Integration of multi-omics data requires efficient computational methods which are capable of correctly modelling and interpreting these relationships.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Whilst each omics technology can present only a part of the true biological complexity integrating data from multiple omics technologies can help provide a more universal picture which can help improve results for classification tasks [3–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Several different methodologies have been proposed to help integrate multi-omics data for classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Generally, different omics data have often been concatenated together or have been subject to ensemble learning where prediction for each omics type is ensembled together [10,11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' The recent emergence of Graph Neural Networks(GNNs) as an efficient deep learning methodology to model complex systems has prompted researchers to study its effects when paired with multi-omics data [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Graph contrastive learning is a learning paradigm where data-data pairs are utilised to perform discrimination between positive and negative pairs of graph data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Contrastive learning can be used as an effective pre-training strategy for training graphical models on supervised learning tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' This paper describes a framework named MOGCL which constructs graphs from multi-omics data and utilises contrastive learning as a pre-training strategy to aid downstream classification tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We compare our results on benchmark datasets with several different machine-learning methodologies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' MOGCL performs better on several metrics such as Area Under the ROC Curve(AUC), Accuracy and F1 score etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='02242v1 [q-bio.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='GN] 3 Jan 2023 2 Literature Review 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='1 Machine learning for multi-omics data A significant number of methods have been studied to integrate multi-omics data for various tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' A large number of methods focus on the semi-supervised integration of multi-omics data without utilising the information from ground truth labels present in the dataset [14–17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Most self-supervised methods focus on assigning clusters to different omics groups present in the dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' With the rapid advancements of biotechnology and the detailed curation of datasets, more labelled samples of phenotypes or traits are being made available for research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' This has led to the development of supervised learning models which perform better on multi-omics datasets [18–22].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Kernel methods are powerful machine learning models which operate on a higher dimensional space where linear interpolations can be found between the given data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Kernel methods are often used as classical machine learning models for analysing and classifying multi-omics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Support Vector Machines(SVM) [23] and partial least squares [24] are examples of classical machine learning approaches for multi-omics data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Currently, deep learning approaches are commonly adopted for multi-omics integration for various tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Islam et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' [25] propose an approach which utilises a convolutional neural network to learn important features for multiple omics types and finally concatenate them to predict breast cancer subtypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' An approach using stacked Sparse Autoencoders (SAE) was proposed by Xu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' [26] where representations are produced for each type of omics data using autoencoders which are then fed to a deep flexible neural forest for predicting cancer subtypes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='2 Graph based learning approaches Graphs are complex data structures which are used to model many complex phenomena in the real world.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Graph Neural Networks (GNN) deal with applying deep learning to graphical data structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' GNNs have several applications such as combinatorial optimizations, neural machine translation, protein-protein interactions, drug discovery [27–34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Recently graph-based approaches have been used for multi-omics integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Wang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' [35] proposed a methodology to convert multi-omics data to graphs and a model named MOGONET consisting of convolutional network (GCN) layers [36] to produce refined representations for a downstream classification task on multiple multi-omics datasets whilst also identifying important biomarkers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' [37] proposed a model named MOGCN for different cancer sub-type classification tasks based on multi-omics data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Graph contrastive learning is an example of a self-supervised training methodology where different graph augmentations are utilised to exploit both structural information and information about features of the dataset to produce better representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Some common training strategies are pre-training followed by fine-tuning [38] and joint learning [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Zhu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' [40] proposed a framework titled deep GRAph Contrastive rEpresentation learning (GRACE) which specifically generates two graph views by corruption and attempts to learn node representations by maximizing the agreement of node representations in these two views.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' 3 Methodology 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='1 Datasets We demonstrate the effectiveness of MOGCL by applying our model on two benchmark datasets namely ROSMAP [41] which describes information for patients with Alzheimer’s Disease (AD) concerning a normal control group (NC) and BRCA which contains data for breast invasive carcinoma (BRCA) PAM50 subtype classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' The omics data used were namely were DNA methylation data (meth), miRNA expression data (miRNA) and mRNA expression data (mRNA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Details about the datasets are further described in table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Dataset Name Labels Number of features for mRNA, meth, miRNA Number of features for training mRNA, meth, miRNA ROSMAP NC: 169, AD: 182 55,889, 23,788, 309 200, 200, 200 BRCA Normal-like: 115, Basal-like: 131, HER2-enriched: 46, Luminal A: 436, Luminal B: 147 20,531, 20,106, 503 1000, 1000, 503 Table 1: Summary of datasets 2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='2 Contrusting graphs from multi-omics data In this section, we describe the methodology of converting multi-omics data to a graphical structure which can then be leveraged by powerful graph neural network models for further processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Our task can be defined as defining graphs G = (V, E) where V, E represent vertices and edges of the graph respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We utilise the feature matrices we obtain after preprocessing each type of omics data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' The feature matrix for each omics type is represented as X ∈ Rn×d where for the ROSMAP dataset d is 200 for each of the omics types and n is 351.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Similarly for the BRCA dataset n is 875 and d ranges from 1000 for mRNA and meth data to 503 for miRNA data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' The nodes V of graph G represent the users from which the omics data is collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We construct an adjacency matrix A ∈ Rn×n to represent G with each element in the adjacency matrix representing a node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We denote a weighted edge from node i to node j of the graph as the element present at the ith row and jth column of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Such an adjacency matrix is constructed for each type of omics data respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' A pairwise distance matrix is constructed for data for the points of the particular omics dataset using cosine similarity [42] as the distance metric.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' The distance between node i and node j is denoted by tij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' A parameter k is introduced which represents the number of edges to be formed per node.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' An adjacency parameter is then chosen by selecting the n × kth value from a sorted array of pairwise distances between all data points.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Edges E is then selected on the criteria of the distance between data points being smaller than the adjacency parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' This ensures that the number of edges per node is k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' A weight of 1 − tij is assigned to the edge from node i to node j if belongs to the set of selected edges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' An adjacency matrix is prepared for each of the omics types present in the respective dataset by following the methodology described above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='3 Graph constrative learning In this section, we describe our training methodology which utilises graph contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We use GRACE [40] which serves as our self-supervision model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Our contrastive learning methodology consists of two stages namely i) data augmentation and ii) contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Augmentation functions such as removing random edges and feature masking are used to create augmented views of a graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' For augmenting edges, we randomly remove a portion of edges from the original graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We sample a random masking matrix ˜R ∈ {0, 1}N×N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Where the elements of R are drawn from a Bernoulli distribution ˜R ∼ Bern(1 − pr) where pr is the probability of each edge is removed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We choose pr to be 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='4 for our study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' The resulting adjacency matrix can be given as ˜A = A ◦ ˜R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' For augmenting features we randomly mask the dimensions of a feature vector by replacing them with zeros.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We sample random feature vectors to construct a matrix ˜ M ∈ {0, 1} according to a Bernoulli distribution having a similar size as feature matrix X.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' The augmented feature matrix can then be represented by ˜X = X ◦ ˜ M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We use a GCN [36] model as an encoder model which helps represent the augmented views of a given graph and denote it with f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Let U = f( ˜ X1, ˜ A1) and V = f( ˜ X2, ˜ A2) be the representations generated after processing two graphs with our shared encoder model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We aim to maximise the agreement between similar nodes in the latent space and minimise the agreement between the rest of the contrasting nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' To achieve this we make use of the Normalized Temperature-scaled Cross Entropy Loss (NT-Xent) [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' NT-Xent loss is given by eq 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' ℓ(ui, vi) = log eθ(ui,vi)/τ eθ(ui,vi)/τ + � k̸=i eθ(ui,vk)/τ + � k̸=i eθ(ui,uk)/τ , (1) where ui and vi represent the ith feature vector from the feature matrix U and V respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' τ represents a temperature parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' θ is a similarity function given in equation 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' θ(u, v) = c(n(u), n(v)) (2) where c(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=',.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=') is the cosine similarity function and n(.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=') represents any non-linear function such as ReLU [44] or LeakyReLU [45] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We finally optimise the weights of the shared encoder model on the NT-Xent loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' The GCN encoder is further trained in a supervised manner using labels from the given dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' The encoder model was trained for a downstream classification task using pre-training followed by fine-tuning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' In pre-training, we first fully train an encoder model for each omics type in an unsupervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We later fine-tune the models using label information from the given dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Let ˜f be the pre-trained GCN encoder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We utilise linear layers in conjunction with concatenated features produced from the encoder models to produce predicted label ˜Y = ˜f(X, A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We use the Cross-Entropy Loss to calculate the loss for predicted labels ˜Y and true labels Y and finally optimise our encoder model on this loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' 3 Figure 1: Contrastive Learning for GNN Encoder Figure 2: Downstream Supervised Training of GNN Encoder 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='4 Experiments In this section we describe the experiments we perform to evaluate our MOGCL framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We first produce graphs for each omics type in our datasets and train a separate encoder model for each one respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We finally concatenate the features produced by each encoder model and train the encoder model in a pre-training followed by fine-tuning methodology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We compare our classification results to the ones described in [35] to evaluate the efficiency of introducing a contrastive learning methodology for the given classification task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Performance of all permutations of encoder models is calculated by conducting r = 5 runs with random weight initialisations for each permutation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We measure the performance of our model on metrics such as accuracy, f1-score and AUC for the ROSMAP dataset and use accuracy,f1- weighted score and f1-macro score to evaluate the BRCA dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We use PyTorch-geometric [46], PyGCL [47] and pytorch-lightning [48] for conducting our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Adam [49] optimizer with a learning rate of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='0001 is utilised for all our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We use a two-layered GCN as our encoder model which is used in graph contrastive learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We further use two linear layers in conjunction with our encoder model to perform fine-tuning with the given true labels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We compress all feature vectors to a 100-dimensional latent space for all our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We try to visualise the effects of our pretraining strategy by visualising the feature vectors before and after processing them with our encoder models for each omics type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' t-SNE [50] was utilised to compress feature vectors to a two-dimensional space in order to produce visualisations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' 4 Results and Discussion The results for the classification task for ROSMAP and BRCA datasets are displayed in table 2 and table 3 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' 4 000 0000 t ~T GNN Intrarview ui 0000 contrast 000 000 U= f(G) 0000 000 000 000 G = t(9) = (X1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' A1) 000 G1 0000 Shared 000 000 =(X,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='A) 000 000 Y1 GNN 0000C0元 t 000 000 000 V= f(G2) 0000 T G2 = t(G) = (X2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' A2) 000 92 0000Linear Layers DNA Methylation samples Pre-trainedGCN (meth) Concatenated Features Final Logits mRNA expression samples Pre-trained GCN (mRNA) miRNA expression samples Pre-trained GCN (miRNA)Table 2: Results for classification task on ROSMAP Method Accuracy F1 AUC KNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='657 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='036 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='671 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='044 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='021 NN_NN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='766 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='777 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='819 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='017 NN_VCDN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='775 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='026 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='790 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='843 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='021 MOGONET_NN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='804 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='808 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='858 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='024 MOGCL (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='818 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='818 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='866 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='021 Table 3: Results of classification task on BRCA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Method Accuracy F1-Weighted F1-Macro KNN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='742 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='024 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='730 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='023 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='682 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='025 SVM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='729 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='018 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='702 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='640 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='017 Lasso 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='732 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='698 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='642 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='026 RF 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='754 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='733 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='649 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='013 XGBoost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='781 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='764 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='701 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='017 NN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='754 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='740 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='034 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='668 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='047 GRridge 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='745 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='726 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='019 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='656 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='025 block PLSDA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='642 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='009 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='534 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='369 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='017 block sPLSDA 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='639 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='522 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='016 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='351 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='022 NN_NN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='796 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='012 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='784 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='014 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='723 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='018 NN_VCDN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='792 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='781 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='721 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='018 MOGONET_NN 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='805 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='017 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='782 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='737 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='038 MOGCL (ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='853 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='851 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='823 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='006 The performance of MOGCL is compared with the following classification algorithms 1) K-nearest neighbour classifier (KNN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' K-nearest neighbours are chosen from the training data to make label predictions during evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' 2) Support Vector Machine classifier (SVM).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' 3) Lasso which is L1-regularised linear regression.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' A unique model was trained to forecast each class’s probability in Lasso, and the class with the greatest foretasted probability was chosen as the final prediction of the model’s overall class label 4) Random Forest classifier (RF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' 5) Extreme Gradient Boosting (XGBoost) is a distributed, scalable gradient-boosted decision tree (GBDT) machine learning framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' 6) Fully connected Neural Network (NN) classifier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' loss for the fully connected NN was calculated by the cross-entropy loss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' 7) Adaptive group-regularized ridge regression (GRridge).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' 8) block PLSDA mentioned in DIABLO [6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' block PLSDA performs latent Discriminant Analysis (LDA) to project multi-omics data to a latent space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' To categorise a discrete outcome, block PLSDA integrates various omics data types measured on the same set of samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' 9) block sPLSDA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' 10) MOGONET_NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' MOGONET_NN is architecturally similar to MOGCL but does not use a pre-training strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We achieve significant results by following our pre-training methodology as it performs better than the other models on all metrics used to measure the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' For the ROSMAP dataset MOGCL achieves an average accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='818 in comparison to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='804 achieved by MOGONET_NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' following this trend MOGCL achieves an F1-score and AUC of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='818 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='866 as compared to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='808 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='856 achieved by MOGONET_NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' For the BRCA dataset MOGCL achieves an accuracy of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='853 as compared to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='805 for MOGONET_NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' MOGCL receives an F1-weighted score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='851 and an F1-macro score of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='823 as compared to 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='782 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='737 respectively for MOGONET_NN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' This demonstrates that adopting a graph based semi-supervised learning strategy in addition to fine-tuning for a downstream task is an effective training strategy for training models on multi-omics datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We demonstrate the effects of adopting a semi-supervised methodology of training by analysing 3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We visualise the feature matrices X by projecting data points into a two-dimensional plane by utilising t-SNE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Similarly, we map the feature vectors produced by the GCN encoders to a 2-dimensional space and compare the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' MOGCL tries to cluster embeddings in the absence of labels to create more structured representations during the pre-training phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Better representation help during the fine-tuning phase which in turn helps produce better classification scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' 5 Figure 3: BRCA Embeddings Figure 4: ROSMAP Embeddings 6 DNA-Methylation-BRCA DNA-Methylation-BRCA-Embeddings mRNA-BRCA miRNA-BRCA miRNA-BRCA-Embeddings mRNA-BRCA-EmbeddingsDNA-Methylation-ROSMAP DNA-Methylation-ROSMAP-Embeddings mRNA-ROSMAP miRNA-ROSMAP miRNA-ROSMAP-Embeddings mRNA-ROSMAP-EmbeddingsFigure 5 represents the performance of permutation of different omics types when processed by MOGCL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We pre-train three encoder models for all omics types in the study respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' To calculate performance we select a permutation of these encoder models and train them using true labels in a supervised manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' MOGCL performs its best when fed information by concatenating all the omics types together for both the ROSMAP and BRCA datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' For the BRCA dataset, a combination of mRNA and DNA-Methylation data provides the next best results however for the ROSMAP dataset a combination of mRNA and miRNA provides the next best set of results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' For both the ROSMAP and BRCA datasets using only a single omics type provides the worst results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Using only DNA-Methylation data is the least useful option followed by miRNA and mRNA data across both BRCA and ROSMAP datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Figure 5: Performance of Omics-Types 5 Conclusion This paper introduces a novel framework named MOGCL which introduces a graph contrastive learning methodology for multi-omics data classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We first provide a comprehensive literature survey regarding work done in the field of machine learning relating to graph-based learning methods and multi-omics data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' A method for constructing graphs from multi-omics data is discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We then describe our framework MOGCL which uses GRACE as a pre-training method followed by fine-tuning with true labels in a supervised setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We discuss our results for the BRCA and ROSMAP datasets and show that our framework performs better than other baselines used for this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' The use of permutations of different omics types is discussed by analysing performance across different metrics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We discuss the effects of adopting a semi-supervised pre-training strategy by visualising the embeddings produced by our graph encoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' We finally conclude that adopting a pre-training methodology is an efficient way to train graphical models for classification problems involving multi-omics datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Future works could include experimenting with different contrastive learning methodologies to determine which one is the most efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' Experiments can be conducted for different GNNs such as Graph Attention Networks (GAT) or Graph Isomorphism Networks (GIN) etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' to determine which one can serve as the best encoder for supervised learning on multi-omics datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content=' References [1] 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0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='6 Performance Performance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content="5 E'0 0." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='2 mRNA+meth+miRNA mRNA+meth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/P9E0T4oBgHgl3EQfTwDw/content/2301.02242v1.pdf'} +page_content='2 mRNA+meth mRNA+miRNA mRNA+miRNA meth+miRNA mRNA meth+miRNA 0.' metadata={'source': 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Comerford,1 Francisco M¨uller S´anchez,2 Mitchell Revalski,3 Rogemar A. Riffel,4, 5 +Kevin Bundy,6 Rebecca Nevin,7 and Sandro B. Rembold4, 5 +1The University of Colorado Boulder, 2000 Colorado Avenue, Boulder, CO 80309, USA +2The University of Memphis, 3720 Alumni Avenue, Memphis, TN 38152, USA +3Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA +4Departamento de F´ısica, CCNE, Universidade Federal de Santa Maria, 97105-900, Santa Maria, RS, Brazil +5Laborat´orio Interinstitucional de e-Astronomia - LIneA, Rua Gal. Jos´e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil +6UC Santa Cruz, 1156 High Street, Santa Cruz, CA 95064 +7Fermi National Accelerator Laboratory, Batavia, IL 60510, USA +ABSTRACT +Despite the importance of AGN in galaxy evolution, accurate AGN identification is often challenging, +as common AGN diagnostics can be confused by contributions from star formation and other effects +(e.g., Baldwin-Phillips-Terlevich diagrams). However, one promising avenue for identifying AGNs are +“coronal emission lines” (“CLs”), which are highly ionized species of gas with ionization potentials +≥ 100 eV. These CLs may serve as excellent signatures for the strong ionizing continuum of AGN. +To determine if CLs are in fact strong AGN tracers, we assemble and analyze the largest catalog of +optical CL galaxies using the Sloan Digital Sky Survey’s Mapping Nearby Galaxies at Apache Point +Observatory (MaNGA) catalog. We detect CL emission in 71 MaNGA galaxies, out of the 10,010 +unique galaxies from the final MaNGA catalog, with ≥ 5σ confidence. In our sample, we measure +[NeV]λ3347, λ3427, [FeVII]λ3586, λ3760, λ6086, and [FeX]λ6374 emission and crossmatch the CL +galaxies with a catalog of AGNs that were confirmed with broad line, X-ray, IR, and radio observations. +We find that [NeV] emission, compared to [FeVII] and [FeX] emission, is best at identifying high +luminosity AGN. Moreover, we find that the CL galaxies with the least dust extinction yield the most +iron CL detections. We posit that the bulk of the iron CLs are destroyed by dust grains in the galaxies +with the highest [OIII] luminosities in our sample, and that AGN in the galaxies with low [OIII] +luminosities are possibly too weak to be detected using traditional techniques. +1. INTRODUCTION +Active Galactic Nucleus (AGN) feedback, the process +by which an active accretion disk converts gravitational +energy into radiative or mechanical energy (e.g., AGN- +induced photoionization, outflows, shocks, winds, and +jets), has been shown to dynamically influence the evo- +lution of a host galaxy (e.g., the tight correlation be- +tween stellar velocity dispersion and black hole mass and +the quenching of star formation; e.g., Ferrarese & Mer- +ritt 2000; Gebhardt et al. 2000; Hopkins et al. 2005; Di +Matteo et al. 2005; Fabian 2012; Kormendy & Ho 2013; +Heckman & Best 2014). However, the full spatial extent, +ionization properties, and impact of AGN feedback on +the host galaxy have yet to be fully unraveled. +The Unified Model of AGN (Antonucci 1993; Urry & +Padovani 1995) provides a fundamental architecture for +⋆ james.negus@colorado.edu +understanding the evolution of AGN feedback. In this +model, an AGN is either Type I or Type II. Type I are +viewed pole-on and are observed to have broad (FWHM +⪆ 1,000 km s−1) and narrow (FWHM ⪅ 1,000 km s−1) +emission lines, whereas Type II are viewed edge-on and +are observed to only have narrow emission lines. These +regions are termed the broad-line region (BLR) and the +narrow-line region (NLR), respectively. +In Negus et al. 2021, we considered the Unified Model +before investigating the “coronal line region” (CLR), +an area surrounding a supermassive black hole (SMBH; +MBH > 106 M⊙) that produces highly ionized species of +gas with ionization potentials (IPs) ≳ 100 eV (termed +“coronal lines” (CLs) since they were first observed in +the solar corona). CLs are suspected to primarily orig- +inate from the strong ionizing continuum of an AGN; +in particular, nuclear CLs are produced in the inner +edge of the dusty torus and extended CLs are tied +to the presence of a jet or AGN-driven outflows (due +arXiv:2301.13322v1 [astro-ph.GA] 30 Jan 2023 + +2 +Negus et al. +to the highly energetic nature of these processes; e.g., +Rodr´ıguez-Ardila et al. 2002; Prieto et al. 2005; Gel- +bord et al. 2009; Mullaney et al. 2009; Mazzalay et al. +2010; Rodr´ıguez-Ardila et al. 2011; M¨uller-S´anchez et +al. 2011; Glidden et al. 2016; Riffel et al. 2021; Trindade +Falc˜ao et al. 2022). +Further, CLs in the mid-infrared have been extensively +used to probe for AGNs, and to subsequently analyze +their physical environments, within dusty galaxies (e.g., +Genzel et al. 1998, Sturm et al. 2002, Armus et al. 2004 +Lutz et al. 2005, Weedman et al. 2006, Dasyra et al. +2008, Armus et al. 2022). In fact, several studies have +shown that AGNs, even those missed by optical surveys +(due to obscuration, for example), are uncovered by ob- +servations of infrared CLs (e.g., Satyapal et al. 2008, +2021; Sajina et al. 2022). Additionally, since CL emis- +sion from Type II supernovae is infrequent, weak, and +short lived, CL infrared observations have been partic- +ularly useful for accurately identifying CL emission ex- +clusively from AGNs (e.g., Smith et al. 2009). +In regard to optical studies, Baldwin-Phillips-Terlevich +diagnostics diagrams (Baldwin et al. 1981; Veilleux & +Osterbrock 1987; Kewley et al. 2001, 2006) are pre- +dominantly used to differentiate emission-line sources as +star-forming, AGN, or a composite of the two. However, +diffuse ionized gas, extraplanar gas, photoionization by +hot stars, metallicity, and shocks can elevate sources be- +yond the star formation threshold and potentially lead +to AGN misclassification (e.g., Wylezalek et al. 2018). +Moreover, while the NLR is the largest observable +structure directly affected by an AGN’s ionizing radi- +ation (out to several kpcs; e.g., M¨uller-S´anchez et al. +2011), star formation can also produce some of the nar- +row lines usually associated with AGN (e.g., [OIII] 5007; +“[OIII]” hereafter). +Further, while the BLR provides +definitive evidence of AGN activity, due to the elevated +cloud velocities, its compact radial extent (≈ 0.1 kpc; +e.g., Laor 2004) is spatially unresolved in most spectro- +scopic surveys. On the other hand, CLs require energies +well above the limit of stellar emission (55 eV; Haehnelt +et al. 2001) and are typically spatially resolved beyond +the BLR and well into the NLR (e.g., Negus et al. 2021). +If CLs can provide accurate AGN identification in opti- +cal spectroscopic surveys of galaxies, as they have been +shown to do in infrared surveys, then detecting them +may be a critical step in constraining the complexities +of AGN feedback (e.g., Molina et al. 2021). +The Sloan Digital Sky Survey’s (SDSS) Mapping +Nearby Galaxies at Apache Point Observatory cata- +log (MaNGA; Bundy et al. 2015) has provided an un- +precedented lens into the dynamic environments that +surround the SMBHs of nearly 10,010 nearby (0.01 < +z < 0.15; average z ≈ 0.03) galaxies. +Using integral +field spectroscopy (IFS), MaNGA provides a 1 - 2 kpc +spatial sampling across the field of view of each observed +galaxy, which offers direct insight into the spatial extent, +ionization properties, and the environmental impact of +AGN feedback. For reference, previous SDSS surveys +(e.g., SDSS-I to SDSS-III; York et al. 2000; Eisenstein +et al. 2011) observed galaxies with small (3” diame- +ter) optical fibers. +The resulting spectra only traced +a small region close to the galactic center, potentially +missing nuclear activity outside of this region. Yan et +al. (2016) further report that 80% of SDSS galaxies ob- +served with a single fiber have less than 36% of their +light covered. Moreover, long-slit spectroscopic surveys +of galaxies also reveal limited spatial information, since +only narrow elongated regions of each galaxy are ob- +served (e.g., Newman et al. 2013). In contrast, MaNGA +offers the ability to capture spatially extended galac- +tic features, which can reveal off-nuclear activity and +large-scale emission line regions. +In Negus et al. 2021, we scanned for [NeV]λ3347, +λ3427, [FeVII]λ3586, λ3760, λ6086, and [FeX]λ6374 +emission at ≥ 5σ above the background continuum in +the 6,623 galaxies from MaNGA’s eighth data release +(MPL-8). We identified 10 CL galaxies in MPL-8, the +largest such catalog at the time; seven of which were con- +firmed to host an AGN, which suggests that CL emission +can be useful for tracing AGN activity. The remaining +three visually appear to be undergoing galactic mergers. +We also found that the average spatial extent of the CLR +from the nuclear center is 6.6 kpc - well into the NLR. +Further, we measured the average electron number den- +sity of the CLRs in our sample to be on the order of ≈ +102 cm−3, also consistent with the CLR occupying the +traditional NLR, beyond the BLR (typical NLR den- +sities range from 101 - 107 cm−3; e.g., Peterson 1997; +Revalski et al. 2022). +However, we also reported a range of power-law in- +dices (α) above the threshold expected for pure AGN +photoionization (α = −2; we measured -1.8 ± 0.3 +≤ α ≤ 0.2 ± 0.1), and electron temperature values +slightly above the threshold for pure AGN photoioniza- +tion (Te = 20, 000 K; Osterbrock 1981). We found that +the average CLR electron temperatures varied between +12,331 K - 22,530 K. These results suggest that shock- +induced compression and heating may also play a role +in the production of CLs. +Comparatively, Mazzalay et al. (2010) investigated the +CLR for 10 pre-selected AGNs. They used the Hubble +Space Telescope/Space Telescope Imaging Spectrograph +to study [NeV] λ3427, [FeVII] λ3586, λ3760, λ6086, +[FeX] λ6374, [FeXIV] λ5303, [FeXI] λ7892, and [SXII] + +Coronal Lines in MaNGA +3 +λ7611 emission in their sample. The authors deduced +that AGN photoionization is the main driving mecha- +nism for the CLs. Moreover, Gelbord et al. (2009) used +the sixth SDSS data release (Adelman-McCarthy et al. +2008) to analyze the CLR in 63 AGNs with [FeX] λ6374 +(IP = 233.60 eV), [FeXI] λ7892 (IP = 262.10 eV), and +[FeVII] λ6086 (IP = 99.10 eV) emission. They used X- +ray observations from Rosat (Voges et al. 1999, 2000) +to similarly posit that AGN photoionization is the main +ionization source of the CLs. Finally, Reefe et al. (2022) +executed the first systematic survey of twenty optical +CLs in the spectra of nearly 1 million galaxies from the +eighth SDSS data release (Aihara et al. 2011). The au- +thors found that CL emission is extremely rare (≈ 0.03% +of the sample show at least one CL), and that the highest +ionization potential CLs tend to be found in lower mass +galaxies. They reasoned that this finding is consistent +with theory that hotter accretion disks are produced by +lower mass black holes, which typically reside in lower +mass galaxies. +Here, we use MaNGA’s eleventh, and final, data re- +lease (MPL-11; 10,010 unique galaxies) to further re- +solve the physics of the CLR, and to better understand +the relationship between the production of CLs and +AGN activity. With our custom pipeline, we identify 71 +unique galaxies with emission from either [NeV]λ3347, +λ3427, [FeVII]λ3586, λ3760, λ6086, or [FeX]λ6374 de- +tected at ≥ 5σ above the continuum, which makes it the +most extensive such catalog of MaNGA CL galaxies to +date. +This paper is outlined as follows: Section 2 details the +technical components of the SDSS-IV MaNGA survey +and its data pipeline, Section 3 describes the methodol- +ogy we use to build the CL catalog and to analyze the +physical properties of the CLR, Section 4 reviews our re- +sults, Section 5 provides interpretations of our findings, +and Section 6 includes our conclusions and intended fu- +ture work. All wavelengths are provided in vacuum and +we assume a ΛCDM cosmology with the following val- +ues: ΩM = 0.287, ΩΛ = 0.713 and H0 = 69.3 km s−1 +Mpc−1. +2. OBSERVATIONS +2.1. Sample of Galaxies +We assemble our sample from the SDSS-IV MaNGA +catalog (Bundy et al. 2015; Drory et al. 2015; Law et +al. 2016; Yan et al. 2016; Blanton et al. 2017; Wake et +al. 2017). MaNGA observations occurred between 2014 +to 2020, using the SDSS 2.5 m telescope (Gunn et al. +2006). The IFS survey contains data for 10,010 nearby +galaxies (0.01 < z < 0.15; average z ≈ 0.03) with stellar +mass distributions between 109 M⊙ and 1012 M⊙. The +spectra were taken at wavelengths between 3622 ˚A - +10354 ˚A, with a typical spectral resolving power of ≈ +2000, corresponding to a velocity resolution of ≈ 60 km +s−1 (see Bundy et al. 2015). +MaNGA contains spectroscopic maps out to at least +1.5 times the effective radius; the typical galaxy is +mapped out to a radius of 15 kpc. Each MaNGA spa- +tial pixel, or spaxel, covers 0.′′5 × 0.′′5, and the average +full-width half maximum (FWHM) of the on-sky point +spread function (PSF) is 2.′′5, which corresponds to a +typical spatial resolution of 1 -2 kpc (Drory et al. 2015). +2.2. MaNGA Data Analysis Pipeline +The MaNGA Data Analysis Pipeline (DAP; Westfall +et al. 2019) offers publicly available high-level data prod- +ucts. The MaNGA DAP algorithms have been in de- +velopment since 2014 and its main outputs are stellar +kinematics, fluxes and kinematics of prominent emis- +sion lines, and continuum spectral indices. To measure +each parameter, the DAP relies on spectral fitting with +pPXF (Cappellari 2012, 2017), where each fit features a +blend of stellar templates with a multiplicative polyno- +mial component to the stellar continuum. In particular, +the DAP incorporates the MILESHC stellar templates +library (Westfall et al. 2019) to fit the stellar kinematics. +The inputs for the DAP are data reduced by the +MaNGA Data Reduction Pipeline (DRP). The DRP is +fed spectra from the MaNGA fiber-feed system, which +consists of 17 IFUs: two 19-fiber IFUs, four 37-fiber +IFUs, four 61-fiber IFUs, two 91-fiber IFUs, and five +127-fiber IFUs (see Drory et al. 2015 for a more detailed +description). The DRP subsequently wavelength, flux, +and astrometrically calibrates the spectra. +3. ANALYSIS +3.1. CL Continuum Subtraction and Emission Line +Fitting +We scan for [NeV]λ3347, λ3427, [FeVII]λ3586, λ3760, +λ6086, and [FeX]λ6374 emission to better understand +their effectiveness as AGN indicators. These CLs are +selected because MaNGA’s DAP does not provide emis- +sion line measurements for them. As a result, we expand +upon the custom pipeline detailed in Negus et al. 2021 +to measure these CLs in MPL-11. Note, all 10 CL galax- +ies reported in Negus et al. 2021 are recovered using the +new MPL-11 pipeline. +3.1.1. CL Stellar Continuum Subtraction +To measure the stellar kinematics, and subsequently +subtract the stellar continuum for each CL galaxy’s ob- +served spectra, we use pPXF (Cappellari 2012, 2017). +pPXF performs a polynomial fit on each galaxy’s spec- +trum while masking gas emission lines. For each fit, we + +4 +Negus et al. +use the MILES1 stellar templates library to represent +the stellar population synthesis model. This library con- +tains ≈ 1,000 stars, with spectra obtained by the Isaac +Newton Telescope. These spectra cover the wavelength +range of 3525 ˚A - 7500 ˚A at a 2.5 ˚A FWHM resolution. +We first access the DRP to extract the necessary data +cubes for each MaNGA galaxy before performing the +pPXF stellar continuum subtraction. The data cubes +provide a spectrum for each individual spaxel across the +FoV of each galaxy. We then use the spectroscopic red- +shifts of each galaxy, adopted from the NASA Sloan At- +las catalogs (Blanton et al. 2011), to adjust the spectra +to rest vacuum wavelengths. We also use a minimum +redshift threshold (z min) for CLs near the lower wave- +length limit of MaNGA (3622 ˚A; Table 1) to ensure CLs +of interest are not shifted out of MaNGA’s spectral cov- +erage. For [NeV]λ3347, λ3427 and [FeVII]λ3586, ≈ 93% +(9,152), ≈ 83% (8,096), and ≈ 3% (229) of the MPL-11 +galaxies, respectively, feature redshifts that place each +CL out of MaNGA’s spectral range; as a result, we are +unable to scan for [NeV]λ3347, λ3427 and [FeVII]λ3586 +in these respective galaxies. +We then apply a mask to each datacube, such that the +imported wavelength range for each spectrum matches +the wavelength range of the stellar templates library +(3525 ˚A- 7500 ˚A). Next, we normalize each spectrum by +dividing fluxes in this wavelength range by each spec- +trum’s median flux value (to avoid numerical issues; see +Cappellari 2017 for a more detailed discussion). Sub- +sequently, we define a typical instrument resolution of +≈ 2.5 ˚A, construct a set of Gaussian emission line tem- +plates (to mask emission lines; provided by pPXF), and +fit the stellar templates. Note, for the CLs near the lower +limit of the mask (3525 ˚A; e.g., [NeV]λ3347, λ3427 and +[FeVII]λ3586), we perform a custom stellar continuum +fit and subtraction before measuring the target emis- +sion line. In these instances, we execute a polynomial +fit on a narrow spectral region, ≈ 300 ˚A wide, of con- +tinuum (free of prominent absorption or emission lines) +near the rest wavelength of the target CL to model the +background stellar continuum and subtract it from the +spectrum. +3.1.2. [NeV] and [FeVII] Emission Line Measurements +Once the spectra are stellar continuum subtracted, we +attempt a single Gaussian fit on a ≈ 30 ˚A region cen- +tered on the rest wavelengths of the CLs ([FeX]λ6374 +being the exception; see Section 3.1.3). We found that +this wavelength range is adequate for capturing the full +extent of CL emission in our preliminary scans. +We +1 http://miles.iac.es/ +Table 1. Target CLs +Emission Line +Wavelength +IP +z min +(˚A) +(eV) +[NeV] +3347 +126.2 +0.088 +[NeV] +3427 +126.2 +0.061 +[FeVII] +3586 +125.0 +0.016 +[FeVII] +3760 +125.0 +- +[FeVII] +6086 +125.0 +- +[FeX] +6374 +262.1 +- +Note: Columns are (1) emission line, (2) rest wavelength, +(3) ionization potential, and (4) minimum redshift value +required for MaNGA detection. +then determine the root mean square (RMS) flux of two +continuum regions (≈ 60 ˚A wide) that neighbor each +target CL, free of absorption or emission lines, and re- +quire that CL amplitudes are detected at ≥ 5σ above the +mean RMS flux values in these continuum regions. We +consider the spectral resolution of MaNGA (R = λ/∆λ +≈ 1400 at 3600 ˚A; R ≈ 2000 at 6000 ˚A; Smee et al. +2013) to eliminate fits with ∆λ ⪅ 2.4 ˚A(for [NeV]λ3347, +λ3427), ⪅ 2.6 ˚A(for [FeVII]λ3586, λ3760), and ⪅ 3 ˚A(for +[FeVII]λ6086 and [FeX]λ6374). We provide an example +of a single Gaussian fit for the [FeVII]λ3586 line in Fig- +ure 1. +3.1.3. [FeX] Emission Line Measurements +For [FeX]λ6374, the broad blue wing of this line is +often blended with [OI]λ6364 due to their close prox- +imity. +Consequently, we attempt a double Gaussian +fit to isolate the [FeX]λ6374 line. If this routine does +not successfully fit both lines with ≥ 5σ confidence, +then we attempt a single Gaussian fit and apply the +method used in Gelbord et al. (2009) and Rose et al. +(2015), whereby the emission line ratio [OI]λ6300/λ6364 +is used to determine if the [OI]λ6364 and [FeX]λ6374 +lines are blended. Specifically, from atomic physics, if +[OI]λ6300/λ6364 = 3, then the [OI]λ6364 line is free +from contamination (see also Elmhamdi 2011 for a full +review). If [FeX]λ6374 emission is present and blended +with [OI]λ6364, it will reduce the [OI]λ6300/λ6364 ra- +tio below three. The MaNGA DAP provides flux values +for both [OI]λ6364 and [OI]λ6300 lines. We adopt this +method and require this ratio to be below three when +fitting for [FeX]λ6374 with a single Gaussian fit to avoid +confusing [OI]λ6364 and [FeX]λ6374 emission. Once we +isolate the [FeX]λ6374 emission, we impose the same +thresholds used to identify the [NeV] and [FeVII] emis- +sion lines (e.g., amplitudes ≥ 5σ; Section 3.1.2). +3.2. Coronal Line Flux Maps + +Coronal Lines in MaNGA +5 +Similar to Negus et al. 2021, we create custom CL +flux maps to analyze the strength and distribution of +the CLs in the CLR. We create these maps using the +integrated CL flux value from each spaxel for each CL +galaxy (Figure 2). +The center of each MaNGA observation corresponds +to the galactic center (Yan et al. 2016). We use this posi- +tion and the galaxy’s inclination angle to determine the +de-projected galactocentric distance of each CL spaxel. +We do acknowledge that the CL gas may not be re- +stricted to the galactic disk; i.e., the CL emission may +associated with an ionization “cone” and therefore, in +these instances, the de-projected distances are approxi- +mations. +The MaNGA DAP provides the ratio of the semi- +minor to semi-major axes (b/a) for each galaxy, and we +use this value to determine the cosine of each galaxy’s +inclination angle (i): cos(i)= b/a. The de-projected dis- +tance of each CL spaxel to the center of the galaxy is +then measured by: +CLD = +� +(x − xcenter)2 + +� +(y − ycenter) ∗ cos(i) +�2 +(1) +where x is the projected distance between the spaxel and +the galaxy center measured along the galaxy’s major +axis, and y for the minor axis. +We then convert spaxel distances to a physical unit +(kpc) using the astropy.cosmology Python package. +The resulting value corresponds to the coronal line dis- +tance (CLD) of each CL emitting spaxel from the galac- +tic center. Further, the minimum coronal line distance +(CLDmin) corresponds to the distance of each galaxy’s +closest CL-emitting spaxel from the galactic center. Fi- +nally, the maximum coronal line distance (CLDmax) cor- +responds to the distance of each galaxy’s most distant +CL-emitting spaxel from the galactic center. +3.3. Galaxy Morphology +To uncover the correlation, if any, between CL emis- +sion and galaxy morphology (e.g., spiral and elliptical), +we use the MaNGA Morphologies Galaxy Zoo value- +added catalog to classify the morphologies of the galax- +ies in our sample. +This catalog features data from +Galaxy Zoo 2, a “citizen science” catalog with more +than 16 million visual morphological classifications for +> 304,000 galaxies in SDSS (GZ2; Willett et al. 2013), +The weighted vote fraction (discussed in Willett et al. +2013) accounts for voter consistency when participants +select morphological classifications, and we require this +fraction to be ≥ 50% before assigning a morphological +classification (e.g., “E” for elliptical, or “S” for spiral). +We also use the weighted vote fraction to determine if a +Figure 1. +A sample spectrum from an individual spaxel +showing the [FeVII]λ3586 line detected at ≥ 5σ above the +continuum in J0906. The dotted black line is the continuum +subtracted spectrum, the shaded gray region is the uncer- +tainty, the solid red line represents the best fit, the red dotted +vertical lines mark the fitting window, the blue dotted line +signifies the rest wavelength of the [FeVII]λ3586 line, and +the two sets of black dotted vertical lines correspond to the +neighboring continuum windows where the RMS flux values +of the continuum are calculated. +CL galaxy features a bar, and/or is categorized as odd +(“b” and“o”, respectively). +In addition, to determine the fraction of CL galaxies +undergoing a merger, we consider the analysis being per- +formed by Nevin et al., in prep (“Nevin catalog” here- +after). The authors determine the merger probability for +each of the 1.3 million galaxies in the SDSS DR16 pho- +tometric sample, using a statistical learning tool that is +built on a linear discriminant analysis framework, which +is trained to separate mock images of simulated merg- +ing and non-merging galaxies using imaging predictors +(see Nevin et al. 2019 for a full review). +We investi- +gate the MPL-11 galaxies from the broader SDSS DR16 +Nevin catalog, and classify a CL galaxy as a merger if +the Nevin catalog gives it a merger value (pmerg) > 0.5. +3.4. AGN Bolometric and [OIII] Luminosities +The AGN bolometric luminosity effectively traces the +energetic output of an AGN (across the entire electro- +magnetic spectrum). To compare the luminosity of the +CL AGN candidates with other known AGN candidates, +we thus consider the bolometric luminosity parameter. +We determine the AGN bolometric luminosity for +each CL galaxy using the summed [OIII] flux val- +ues (F[OIII]) across the entire galaxy (provided by the +MaNGA DAP), and the procedure outlined in Pennell +et al. (2017), which assumes [OIII] emission comes from + +[FeVIl] 35866 +Negus et al. +Figure 2. A sample CL flux map showing [NeV]λ3427 emis- +sion detected ≥ 5σ above the continuum in J1714. For this +galaxy, the strongest [NeV]λ3427 emission is located near +the center of the galaxy. The gray region is outside of the +MaNGA FoV and the black region are spaxels with no CL +emission. North is up, south is down, east is to the left, and +west is to the right. +an AGN: +log +� Lbol +ergs−1 +� += (0.5617 ± 0.0978) log +� L[OIII] +ergs−1 +� ++(21.186 ± 4.164) +(2) +where L[OIII] = F[OIII](4πR2) and R is the DAP pro- +vided luminosity distance based on redshift and a stan- +dard cosmology of ΩM = 0.3 and ΩΛ = 0.7 (redshift is +also measured by the DAP). +We then measure the total [OIII] luminosity (using the +summed [OIII] fluxes across the entire galaxy) for each +CL galaxy in our sample. Next, we compare the [OIII] +luminosities of the CLs in our pipeline (Section 4.3) to +determine the relative strength of [OIII] for each CL. We +do so to assess if specific CLs are preferentially found +in higher or lower luminosity [OIII]-emitting galaxies, +which is useful to determine if CLs uniformly trace all +AGN, or if there may be an [OIII] luminosity depen- +dence. +3.5. Narrow-Line BPT Diagnostics Diagrams +Baldwin-Phillips-Terlevich optical emission-line diag- +nostic diagrams (BPT diagrams; Baldwin et al. 1981; +Veilleux & Osterbrock 1987; Kewley et al. 2001, 2006) +are widely accepted to be effective tools for categorizing +gas ionization sources as star-forming, Seyfert (AGN), +low-ionization nuclear emission-line region (LINER), or +a composite of multiple ionization sources. They serve +as the traditional AGN selection tool for most spectro- +scopic surveys. Specifically, these diagrams compare line +ratios between high and low ionization species, most +commonly [OIII]λ5007/Hβ vs. [NII]λ6583/Hα (“[NII]/ +Hα diagram” hereafter). +In this paper, we construct spatially resolved narrow- +line BPT diagnostic diagrams for the CL galaxies to +better constrain the ionization sources of the CLs. To +do so, we require emission line measurements for the +[NII]λ6583, [OIII]λ5007, Hα, and Hβ emission lines. +The DAP measures the continuum subtracted flux for +each of these emission lines. Note, these fluxes account +for galactic reddening using the E(B-V) values deter- +mined by the DRP, which assumes an O’Donnell (1994) +reddening law. +Once we determine the necessary emission line flux +measurements, we compute the ratios for the [NII]/ Hα +diagram, for each CL-emitting spaxel. We then use these +values to create custom spatially-resolved BPT maps, +whereby we present the BPT-classification for each CL- +emitting spaxel within the MaNGA FoV, for each CL +galaxy. Figure 3 shows an example BPT map. +3.6. Dust Attenuation +Mullaney et al. (2009) investigated the [FeVII]λ6086, +[FeX]λ6374, and [FeXI]λ7892 emission lines in the +Seyfert 1 galaxy Ark 564. The authors used the pho- +toionization code CLOUDY (Ferland et al. 1998) to +determine the location and kinematics of these lines. +They found that the CLs are launched from a dusty +torus near the SMBH, where the gas is quickly ac- +celerated. Moreover, using the CLOUDY models, they +determined that some iron carrying grains are destroyed +during the initial acceleration of the gas. +To follow up on the analysis performed by Mullaney +et al. (2009), and to better understand the role of dust +grains on the potential depletion of the iron CLs, we +use the E(B - V) color excess index. This index traces +the degree of interstellar reddening caused by photons +that are scattered off of dust; in essence, it measures +the difference between an object’s observed color index +and its intrinsic color index. E(B - V) values for each +CL galaxy are provided by the MaNGA DRP (using +Schlegel et al. 1998 maps), and assume the extinction +law provided by O’Donnell (1994). +3.7. Shock Diagnostics +We explore the role of shocks (e.g., supernova rem- +nant (SNR) and [OI]λ6300 (“[OI]” hereafter) shocks) in +our analysis to elucidate the role of collisional excita- +tion in the production CLs (e.g., Penston et al. 1984). +To do so, we consider the strength of the [SII]λ6717, + +J171411.63+575834.0 +14 +10 +8 +6 +4 +2 +5 kpc +0Coronal Lines in MaNGA +7 +Figure 3. A sample BPT map showing AGN spaxels in red +and composite spaxels in green, for CL-emitting spaxels in +J2051 (a [NeV]λ3427 galaxy). The gray region is outside of +the MaNGA FoV and the black region are spaxels with no +CL emission. North is up, south is down, east is to the left, +and west is to the right. +λ6731 doublet with respect to the Hα line, which has +traditionally been used to differentiate SNR shocks from +photoionized regions. Specifically, Dodorico (1978) and +Dodorico et al. (1980) first determined that regions with +[SII] (λ6717 + λ6731)/Hα > 0.4 can be used to identify +SNR shocks. Additionally, the [OI] emission line is gen- +erally a strong tracer of shock excitation, and line flux +ratios with [OI]/Hα> 0.1 indicate that shocks with ve- +locities 160-300 km s −1 are the main excitation source +of [OI] (e.g., Dopita 1976; Allen et al. 2008; Farage et al. +2010; Rich et al. 2010, 2011; Riffel et al. 2021; Comerford +et al. 2022). The MaNGA DAP provides flux measure- +ments for the [SII]λ6717, λ6731, [OI], and Hα emission +lines. +4. RESULTS +In this section, we report the main findings for the CL +galaxies in our sample. First, we present the fraction of +confirmed AGN in the CL galaxies. Then, we analyze +the spatial distribution and extent of the CLs. Next, we +inspect the CL galaxy bolometric and [OIII] luminosi- +ties to deduce the effectiveness of using each species for +accurate AGN identification. After, we assess the BPT +classification of the CL-emitting spaxels. Finally, we in- +vestigate the role of dust extinction and shocks in the +CLR to determine the impact of dust grains on CL emis- +sion, and to further constrain the ionization source(s) of +the CLs. +In total, we find 71 galaxies with CL emission at ≥ 5σ +above the background continuum in MaNGA’s MPL- +11 (33 feature [NeV] emission, 39 feature [FeVII] emis- +sion, and 4 feature [FeX] emission). Note, in our sample, +40 unique CL galaxies with either [NeV]λ3427, [FeVII], +or [FeX] emission, or a combination of the three, fea- +ture redshifts below the z min threshold for [NeV]λ3347 +(z min = 0.088); further, 24 unique CL galaxies with +either [FeVII] or [FeX] emission feature redshifts be- +low the z min threshold for [NeV]λ3427 (z min = 0.061). +Therefore, we are unable to scan for [NeV]λ3347 or +[NeV]λ3427 in these respective galaxies. +In general, +most of the MPL-11 galaxies feature redshifts that place +[NeV]λ3347, λ3427 out of MaNGA’s spectral range (see +Section 3.1.1). +Moreover, in light of the extensive work to detect +AGNs using infrared CLs, we crossmatched our catalog +of 71 unique CL galaxies with the infrared CL catalogs +presented in Genzel et al. 1998, Sturm et al. 2002, Ar- +mus et al. 2004 Lutz et al. 2005, Weedman et al. 2006, +Dasyra et al. 2008, Goulding & Alexander 2009, Armus +et al. 2022. We do not identify any of the MaNGA CL +galaxies in these samples. +For 63/71 CL galaxies with GZ2 classifications (89%), +we determine a nearly even fraction of spirals and el- +lipticals (48% and 52%, respectively). In addition, we +measure the average size of the CLR (from the galac- +tic center) for [NeV]λ3347, λ3427, [FeVII]λ3586, λ3760, +λ6086, and [FeX]λ6374 to be 1.9 kpc, 2.3 kpc, 3.7 kpc, +5.3 kpc, 4.1 kpc, and 2.5 kpc, respectively (Table 2). +Further, we find that the vast majority of [NeV] galax- +ies feature at least one CL-emitting spaxel in their nu- +clear regions (98.5%; 2.′′5 x 2.′′5 FoV surrounding the +central spaxel), whereas [FeVII] and [FeX] galaxies gen- +erally feature a smaller fraction (73% and 75%, respec- +tively). The corresponding fraction of confirmed AGN +in these galaxies (determined by comparing our sample +to the largest catalog of confirmed MaNGA AGN; Sec- +tion 4.1) is 94%, 14%, and 25%, respectively. +4.1. MaNGA AGN Comparison +Comerford et al. (in prep) provide the most complete +sample of AGN in MaNGA’s MPL-11 (see Comerford et +al. 2020 for a full review of their MPL-8 MaNGA AGN +catalog). +The authors compile a catalog of MaNGA +AGN that were detected using SDSS broad emission +lines, NVSS/ FIRST 1.4 GHz radio observations, WISE +mid-infrared color cuts, and Swift/BAT hard X-ray ob- +servations. + +5 kpc +0205141.54±005135.48 +Negus et al. +Broad Balmer emission lines (FWHM > 1,000 km s−1) +are strong tracers of the rapidly rotating, high density +gas, near the SMBH. They serve as reliable tracers for +AGN activity. Oh et al. (2015) assembled a catalog of +nearby (z ≤ 2) Type I AGN in SDSS’s seventh data +release using the broad Hα emission line, and Comerford +et al. (in prep) identify 78 broad line AGN from this +catalog in MPL-11. +Powerful AGN radio jets can expand several kpcs from +the SMBH, and can thus serve as strong signatures for +AGN activity. +As a result, detecting the radio emis- +sion from these sources is a great tool for accurate AGN +identification. +Best & Heckman (2012) used observa- +tions from the 1.4 GHz NRAO Very Large Array Sky +Survey (NVSS; Condon et al. 1998) and the Faint Im- +ages of the Radio Sky at Twenty Centimeters (FIRST; +Becker et al. 1995) to detect AGN in the SDSS’s seventh +data release (DR7). They differentiated AGN activity +from star formation emission using the correlation be- +tween the 4000 ˚A break strength and radio luminosity +per stellar mass, emission line diagnostics, and the re- +lation between Hα and radio luminosity (Becker et al. +1995). Comerford et al. (in prep) find 221 radio AGN +from this catalog in MPL-11. +Heated dust that surrounds an AGN can produce mid- +infrared emission, which can expose obscured and un- +obscured AGN activity. +Comerford et al. +(in prep) +thus rely upon observations from the Wide-Field In- +frared Survey Explorer (WISE; Wright et al. 2010) to +help identify AGN. They consider the four bands ob- +served with WISE (3.4 µm (W1), 4.6 µm (W2), 12 µm +(W3), and 22 µm (W4)) and apply a 75% reliability +criteria of W1 - W2 > 0.486 exp{0.092(W2 - 13.07)2} +and W2 > 13.07, or W1 - W2 > 0.486 and W2 ≤ 13.07 +(Assef et al. 2018) to select AGN. Comerford et al. (in +prep) detect 130 WISE AGN in MPL-11. +X-ray emission produced by AGN generally result +from inverse Compton scattering of low energy UV pho- +tons by energetic electrons from the accretion disk (e.g., +Antonucci 1993; Haardt & Maraschi 1991; Hinkle & +Mushotzky 2021). +Therefore, X-rays can be a useful +indicator of AGN activity. Accordingly, the authors use +the X-ray catalog assembled by Oh et al. (2018), which +consists of ≈ 1,000 AGN observed by the Swift Obser- +vatory’s Burst Alert Telescope (BAT) in the ultra hard +X-ray (14 - 195 keV), to detect AGN. Comerford et al. +(in prep) uncover 30 AGN from this catalog in MPL-11. +We compare our CL sample to the AGN catalog re- +ported by Comerford et al. (in prep; “Comerford sam- +ple” hereafter) and crossmatch 35 CL galaxies in it (52% +of our sample). Further, we consider the fraction of CL +galaxies with confirmed AGN by specific CL species. +We determine that 94% (31/33) of the [NeV] galaxies +host an AGN; 14% (5/36) of the [FeVII] galaxies and +25% (1/4) of the [FeX] galaxies. +Overall, 35 unique +CL galaxies host a confirmed AGN; 80% (28/35) are +confirmed with WISE observations, 63% (22/35) with +broad Balmer emission lines, 14% (5/35) with NVSS +observations, and 11% (4/35) with BAT AGN. +All of the [NeV]λ3347 galaxies feature an AGN and +[NeV]λ3427 emission, and of the five [FeVII] galaxies +with a confirmed AGN, two (J0736 and J1714) also fea- +ture both [NeV]λ3347, λ3427 emission. +Further, two +of the remaining three [FeVII] galaxies with a con- +firmed AGN (J0807 and J1157) feature emission from +more than one [FeVII] emission line (J0807 features +[FeVII]λ3586, λ3760, λ6086 emission; J1157 features +[FeVII]λ3586, λ3760 emission). The final [FeVII] galaxy +with a confirmed AGN exclusively features [FeVII]λ6086 +emission, and the sole [FeX] galaxy with a confirmed +AGN (J1628) exclusively features [FeX] emission. +Provided that 80% of the CL galaxies in our sample +are confirmed to host an AGN via WISE diagnostics, we +consider the fact that these WISE diagnostics are likely +to miss low luminosity AGN (e.g., Assef et al. 2018). +Perhaps, one possible explanation for the discrepancy in +AGN across the CLs in our sample is that [NeV] traces +high-luminosity AGN, while [FeVII] and [FeX] may pos- +sibly trace low-luminosity AGN. We explore this further +in Sections 4.3 and 4.5. +4.2. Spatial Distribution and Extent of the CLs +To better constrain the ionization source(s) of the +CLs, we first map the measured fluxes of the CLs within +the MaNGA FoV for each CL galaxy (Figure 2). These +flux maps provide a snapshot of the orientation, extent, +and intensity of CL emission for the galaxies in our sam- +ple. +Then, we compute the de-projected distance of +each CL spaxel from the nuclear center of each galaxy +(i.e. the photometric center; Section 3.2) to determine +the distance of each CL-emitting spaxel from the galaxy +center. Finally, we define the nuclear region of each CL +galaxy to be a 2.′′5 x 2.′′5 aperture (5 x 5 spaxel grid; +where each spaxel covers a 0.′′5 x 0.′′5 FoV) surrounding +the central spaxel. +If AGN photoionization is the primary mechanism +producing the CLs, it is likely that CL emission is pre- +dominantly within the nuclear region of each galaxy, +close to the SMBH and the accretion disk (e.g., Gel- +bord et al. 2009; Mazzalay et al. 2010). On the other +hand, if shocks, AGN outflows, or stellar processes play +an active role in generating CLs, we anticipate that CL +emission will not be found exclusively in the nuclear +region. +Rather, we would expect to find emission in + +Coronal Lines in MaNGA +9 +Table 2. Spatial Properties for the CL Galaxies +Detected +Wavelength +Confirmed +Nuclear Emission +CLDmin +CLDmax +CLDavg +CL +(˚A) +Galaxies +(%) +kpc +kpc +kpc +(1) +(2) +(3) +(4) +(5) +(6) +(7) +[NeV] +3347 +8 +100 +0.52 +5.3 +1.9 +[NeV] +3427 +33 +97 +0.34 +19 +2.3 +[FeVII] +3586 +4 +100 +0.10 +9.6 +3.7 +[FeVII] +3760 +16 +56 +0.11 +36 +5.3 +[FeVII] +6086 +19 +63 +0.10 +21 +4.1 +[FeX] +6374 +4 +75 +0.60 +4.9 +2.5 +Note: Columns are (1) detected CL, (2) rest wavelength, (3) number of galaxies with CL +emission detected, (4) percentage of CL galaxies with at least one CL-emitting spaxel in +a nuclear 2.′′5 FoV, (5) the average CLD (distance of CL-emitting spaxel from the galaxy +center), (6) the distance of the furthest CL-emitting spaxel from the galaxy center, and (7) +the distance of the closest CL-emitting spaxel from the galaxy center. +regions off-center or off-axis from the SMBH and the +galaxy’s rotational plane (see Negus et al. 2021 for more +discussion). +To analyze the CL distribution within the nuclear re- +gion of the CL galaxies, we measure the fraction of CL +galaxies with at least one CL emitting spaxel in their +center, for each CL (Table 2). We find that the vast +majority of [NeV] λ3347, λ3427 galaxies feature at least +one [NeV]-emitting spaxel in their nuclear regions (100% +and 97%, respectively). This finding is consistent with +our results in Section 4.1, that [NeV] is a strong tracer +of AGN activity (i.e. CL emission is likely dominated by +AGN photoionization near the SMBH). Comparatively, +the fraction of CL galaxies with nuclear emission from +[FeVII] λ3586, λ3760, or λ6086 varies significantly more +(100%, 56%, and 63%, respectively). +In Figure 4, we present a sample of CL flux maps for +six representative CL galaxies. In three of the galaxies +(J1104, J1349, and J2152), it is apparent that the source +of the CLs is within the nuclear region, as the CL flux +is concentrated here. +However, for the three remain- +ing galaxies (J0023, J1613, and J0920), the CL-emitting +spaxels are highly offset from the nuclear region. Based +on the orientation of the CL flux in J0023 and J0920, +it is possible that AGN outflows are generating the CL +since the CL emission is generally perpendicular to the +orbital plane of each galaxy. For the J1613 observation, +we determine that several optical emission lines (e.g., +[OIII] and Hα) measured in the secondary galaxy (with +the featured “CL emission”; southwest of J1613 in the +MaNGA FoV) have large velocity shifts (> 2,000 km +s−1) compared to the center of J1613, which suggests +this may not be a companion galaxy (i.e., this is not +a merging system; J1613 is not in the Nevin catalog). +As a result, the “CL emission” in this galaxy is likely +from a separate emission line, from a background galaxy +with a different redshift than the primary galaxy. For +J1349, we acknowledge that there appears to be a visual +companion galaxy near the nuclear region; it is possible +that both merger induced shocks and AGN photoioniza- +tion could be producing the CL emission. In addition, +we suspect that dust grains may also have a significant +impact on the presence of [FeVII] and [FeX] emission in +our sample. In Section 4.5, we review the likelihood of +iron depletion by dust grains more thoroughly. +We also compute the CLDs for the CL galaxies. The +CLD, which is the distance of each CL-emitting spaxel +from the galactic center, reveals the physical scale of the +CLR for each CL galaxy. We measure the average CLDs +for [NeV]λ3347, λ3427 to be 1.9 kpc and 2.3 kpc, respec- +tively; 3.7 kpc, 5.3 kpc, and 4.1 kpc for [FeVII]λ3586, +λ3760, λ6086, respectively; 2.5 kpc for [FeX]λ6374. We +find no correlation between IPs and CLDs (IP = 262.1 +eV for [FeX], IP = 126.2 eV for [NeV], and IP = 125 +eV for [FeVII]). Moreover, for the [NeV] galaxies, the +minimum and maximum distances of each CL-emitting +spaxel from the nuclear center (labeled CLDmin and +CLDmax in Table 2) ranges between 340 pc to 19 kpc, +100 pc to 36 kpc for the [FeVII] galaxies, and 600 pc +to 4.9 kpc for the [FeX] galaxies. These large variances +in CL distance suggest that the CLR extends from just +beyond the BLR (≈ 0.1 kpc) and well into the NLR +(several kpcs). +Finally, to confirm that CL emission is indeed resolved +for each CL galaxy, we consider the instrument PSF (≈ +2.′′5 for MaNGA), and find that 60/71 CL galaxies show +resolved and continuous emission in excess of the typical +instrument PSF. The remaining 11 CL galaxies (J0205, +J1010, J1117, J1317, J1344, J1416, J1604, J1626, J1628, +J1658, and J1649) lack CL emission in excess of the typ- +ical instrument PSF. We reason that these CLRs are be- +low the instrument PSF, and not spatially resolved. As + +10 +Negus et al. +discussed in Negus et al. (2021), these CLRs may still be +spatially resolved by other instruments (e.g., Mazzalay +et al. 2010 and their use of STIS/HST optical spectra), +and it is also posible that CL emission may be oriented +along an ionization cone; however here we consider the +CLDs of these galaxies to be upper limits. +4.3. AGN Bolometric and [OIII] Luminosities +AGN bolometric luminosity, which scales with [OIII] +luminosity, is effectively the “power” of an AGN. As +outlined in Pennell et al. (2017), [OIII] emission is the +most utilized line for measuring bolometric luminosity, +due its strength in most AGN spectra and the relatively +weak blending of emission from photoionized gas in star +forming regions with the line (e.g., Heckman et al. 2004; +Heckman & Best 2014). +Therefore, to help resolve the discrepancy between the +differing fractions of confirmed AGN in our sample (Sec- +tion 4.1; 94% of the [NeV] galaxies feature a confirmed +AGN, 14% for the [FeVII] galaxies, and 25% of the [FeX] +galaxies; for CL galaxies with multiple CLs, we measure +this fraction independently for each CL), and to eval- +uate the overall effectiveness of using CL detections to +identify AGN, we consider the bolometric and [OIII] lu- +minosities of the CL galaxies, and further inspect the +Comerford sample of MaNGA AGN. In particular, we +compare the mean bolometric luminosities of the CL +galaxies (Lbol; using the summed [OIII] flux across the +entire galaxy; Section 3.4) with the total population of +MPL-11 AGN in the Comerford sample (Section 4.1; +Figure 5). +We find that the mean bolometric luminosity for the +[NeV] galaxies (mean z = 0.10; median z = 0.11), +log(Lbol) = 44.5 erg s−1, is consistent with the mean +value of Comerford’s population of MaNGA galaxies +that host an AGN (log(Lbol) = 44.6 erg s−1). On the +other hand, we measure the mean bolometric luminosi- +ties of the [FeVII] galaxies (mean z = 0.06; median z = +0.05) and the [FeX] galaxies (mean z = 0.07; median z = +0.06) to be an order of magnitude lower than the mean +log(Lbol) value of the Comerford sample (log(Lbol) = +43.7 erg s−1 and log(Lbol) = 43.5 erg s−1 for the [FeVII] +and [FeX] galaxies, respectively). Note, we also present +the [OIII] luminosity distribution for the CL galaxies +in Figure 6 (the mean [OIII] luminosity for the [NeV] +galaxies is 41.5 erg s−1; 40.1 erg s−1 and 39.8 erg s−1 for +the [FeVII] and [FeX] galaxies, respectively). We reason +that the [FeVII] and [FeX] galaxies may be preferen- +tially tracing lower luminosity AGN in MaNGA, which +are generally more difficult to detect in multi-wavelength +observations. +However, we find that the five [FeVII] galaxies with +a confirmed AGN (J0736, J0807, J1157, J1535, and +J1714) all feature relatively high [OIII] luminosities of +log(L[OIII]) ⪆ 41 erg s−1. Additionally, the three remain- +ing [FeVII] galaxies with [OIII] luminosities at or above +this limit (without confirmed AGN) are J0906, J1349, +and J2152. Both J0906 and J1349 visually appear to +be actively undergoing a merger; J2152 shows no appar- +ent companion galaxy. We reason that, for the [FeVII] +galaxies in our sample, the log(L[OIII]) cutoff of ≈ 41 erg +s−1 is a useful threshold for identifying confirmed AGN +(from the Comerford sample) and may also be helpful +for detecting mergers. Further, for the [NeV] galaxies, +J1344 features the lowest [OIII] luminosity (log(L[OIII]) += 40.6 erg s−1) and in fact hosts a confirmed AGN. We +consider the [OIII] luminosity threshold for the [FeVII] +galaxies (log(L[OIII]) ≈ 41 erg s−1) to be similar for the +[NeV] galaxies. +We also determine that the two [NeV] galaxies (J1658 +and J1104) that do not feature a confirmed AGN (out +of 33 total [NeV] galaxies), feature [OIII] luminosities +of log(L[OIII]) = 41.2 erg s−1 and log(L[OIII]) = 41.4 erg +s−1, respectively. +Considering these high [OIII] lumi- +nosities, and the high [NeV] AGN detection rate (94%), +we propose that these two galaxies are strong AGN can- +didates. +On the other hand, the one [FeX] galaxy with a con- +firmed AGN, J1628, features an [OIII] luminosity of +log(L[OIII]) = 39.8 erg s−1 (the remaining three [FeX] +galaxies, which do not host a confirmed AGN, also have +log([OIII]) luminosities < 40 erg s−1). +Consequently, +while we consider the log(L[OIII]) ≈ 41 erg s−1 threshold +useful for identifying CL galaxies with a confirmed AGN, +it is important to acknowledge that CL galaxies with a +confirmed AGN can have [OIII] luminosities below this +limit. +4.4. BPT Analysis +The BPT diagram has long served as the standard +tool for identifying ionization mechanisms in emission +line sources (e.g, Baldwin et al. 1981; Veilleux & Oster- +brock 1987; Kewley et al. 2001, 2006). While effects such +as stellar shocks and emission from post-AGB stars are +liable to elevate SF sources beyond the AGN threshold +(see Yan & Blanton 2012; Belfiore et al. 2016; Agostino +& Salim 2019 for a further discussion), we nonethe- +less explore the BPT classification for each CL-emitting +spaxel to help pin down the source of CL emission in our +sample. To do so, we compute the log([OIII]/Hβ) and +log([NII]/Hα) ratios required for the [NII]/Hα diagram +(Section 3.5). +Using the thresholds outlined in Kew- +ley et al. (2006), we categorize each CL spaxel as either + +Coronal Lines in MaNGA +11 +Figure 4. +CL flux maps for 6/71 CL galaxies in our sample. +From top to bottom and left to right: J1104 ([NeV]λ3427 +map), J0023 ([FeVII]λ3760 map), J1349 ([FeVII]λ3760 map), J1613 ([FeVII]λ3760 map), J2152 ([FeVII]λ6086 map), and J0940 +([FeVII]λ6086 map). For J0023 and J0940, the maps display CL emission spatially offset from the galaxy center. For each +galaxy, the emission is offset perpendicular to the rotational plane of the galaxy, suggestive of the source of the CLs being +AGN outflows. For J1349 we observe a possible companion galaxy and consider the possibility that these two galaxies to be +undergoing a merger. For the J1613 observation, we determine that several optical emission lines (e.g., [OIII] and Hα) measured +in the secondary galaxy (with the featured “CL emission”; southwest of J1613 in the MaNGA FoV) have large velocity shifts (> +2,000 km s−1) compared to the center of J1613, which suggests this may not be a companion galaxy (i.e., this is not a merging +system; J1613 is not in the Nevin catalog). As a result, the “CL emission” in this galaxy is likely from a separate emission line, +from a background galaxy with a different redshift than the primary galaxy. For J1104 and J2152, CL emission is concentrated +towards the galaxy center, likely produced by AGN photoionization. +[HII] (i.e. star forming), AGN, or a composite of the +two. Note, for some CL-spaxels, the DAP reports nega- +tive values for the necessary emission line fluxes, likely +because the emission lines of interest yield low flux lev- +els and the DAP’s subtraction of the stellar continuum +results in a net absorption at the expected wavelength +of the emission line. As such, we exclude these spaxels +from our analysis. +For the [NeV], [FeVII], and [FeX] emission lines, +we determine that, on average, the majority of CL- +emitting spaxels are AGN or composite (Table 3). For +[NeV]λ3347, λ3427 we find that 87.5% and 90% of these +spaxels are classified as AGN, respectively; 12.5% and +10% composite, respectively; 0% SF for both. Moreover, +we measure the BPT ratios for the [FeVII]λ3586, λ3760, +λ6086 spaxels, and find that 78.5%, 80.3%, and 67.9% of +these spaxels are classified as AGN, respectively; 19%, +5.8%, and 19.7% composite, respectively; 2.5%, 13.9%, +and 12.5% SF, respectively. For [FeX]λ6374, 88.3% of +the CL-emitting spaxels are classified as AGN; 0% com- +posite; 11.7% SF. In total, 100% of the [NeV] spaxels in +our sample are either BPT AGN or BPT composite, 91% +of the [FeVII] spaxels, and 88.3% of the [FeX] spaxels. +These results suggest that the CLs are perhaps useful +tracers of AGN, and that the lack of confirmed AGN +in our [FeVII] and [FeX] galaxies may trace back to the +nearly bimodal log([OIII]) and bolometric luminosity +distributions presented in Section 4.3 (i.e. [FeVII] and + +400 +1.2 +300 +cm +erg s +200m +(10- +150 +Flux +2 +0.2 +50 +J110431.08+423721.2 +J002343.86+141824.2 +5kpc +5.0 +0 +12 +60 +6 +(10- +10 +J134918.20+240544.9 +J161358.56+393150.2 +5kpc +1.75 +1.0 +-1) +1.50% +0.75 +0.2 +0.25 +J215259.07-000903.4 +5 kpc +J094036.39+033436.9 +5kpo +0.00 +0.012 +Negus et al. +Table 3. BPT Classifications for the CL Galaxies +. +Detected +Rest +Confirmed +Confirmed +Confirmed +[NII] +[NII] +[NII] +CL +Wavelength +CL Galaxies +AGN +AGN Fraction +AGN Fraction +Composite Fraction +SF Fraction +˚A +% +% +% +% +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +[NeV] +λ3347 +8 +8 +100 +87.5 +12.5 +0 +λ3427 +33 +31 +94 +90 +10 +0 +[FeVII] +λ3586 +4 +3 +75 +78.5 +19 +2.5 +λ3760 +16 +2 +13 +80.3 +5.8 +13.9 +λ6086 +19 +3 +16 +67.9 +19.7 +12.5 +[FeX] +λ6374 +4 +1 +25 +88.3 +0 +11.7 +Note: Columns are (1) detected CL, (2) rest wavelength, (3) the number of galaxies that feature emission from the +respective line, (4) the fraction of galaxies that host a confirmed AGN, (5) the average fraction of [NII] AGN BPT +spaxels, (6) the average fraction of [NII] Composite BPT spaxels, and (7) the average fraction of [NII] SF BPT spaxels. +Figure 5. +Average mean bolometric luminosities for the +71 CL galaxies in our sample (analyzed by each CL species; +[NeV], [FeVII], and [FeX]), compared to the MaNGA galax- +ies confirmed to feature an AGN in the Comerford sample. +The AGN in the Comerford sample were verified using SDSS +broad emission lines, NVSS/ FIRST 1.4 GHz radio observa- +tions, WISE mid-infrared color cuts, and Swift/BAT hard +X-ray observations. The mean bolometric luminosity of the +[NeV] galaxies, log(Lbol) = 44.5 erg s−1, is consistent with +the AGN reported in the Comerford sample (mean log(Lbol) += 44.6 erg s−1 for the Comerford sample). +However, the +[FeVII] and [FeX] galaxies feature mean bolometric lumi- +nosities an order of magnitude lower (log(Lbol) ≤ 43.7 erg +s−1) than the Comerford sample. We suspect that the [Fe- +VII] and [FeX] emission lines may primarily be detecting low +luminosity AGN in MaNGA. +[FeX] may generally be found in low luminosity AGN +that are potentially missed by traditional AGN detec- +tion techniques; though, it is also possible that [FeVII] +Figure 6. The log([OIII]) luminosity distribution for the 71 +CL galaxies. The blue, orange, and green histograms repre- +sent the [NeV], [FeVII], and [FeX] galaxies in our sample, re- +spectively. [NeV] emitting galaxies tend to have higher [OIII] +luminosities than [FeVII] or [FeX], which suggests that these +galaxies may host higher luminosity AGN. The mean of the +[NeV] log([OIII]) luminosity distribution is 41.5 erg s−1; 40.1 +erg s−1 and 39.8 erg s−1 for [FeVII] and [FeX], respectively. +and [FeX] may not host an AGN at all). We explore +this possibility, and the corresponding impact of dust +extinction on iron CL emission in Section 4.5. +4.5. The Impact of Dust on CL Emission +The role of dust extinction (i.e. the impact of dust +grains) on CL emission has yet to be fully unraveled. +Mullaney et al. (2009) suggest that dust grains can po- +tentially deplete heavier CL species (e.g., iron). Further, +Ferguson et al. (1997) posit that there are three pri- + +45.5 +) (erg/s) +45.0 +44.5 +44.0 +T +43.5 +Nev +FeVIl +FeX +BAT +WISE +RADIO +BROAD +Source12 +[NeV] +[FeVII] +[FeX] +10 +Number of CL Galaxies +2 +39.0 +39.5 40.0 40.5 41.0 41.5 42.0 +42.5 +log [Oll] Luminosity (erg/s)Coronal Lines in MaNGA +13 +mary effects of dust on line formation: 1) emission lines +weaken due to the absorption of the incident continuum +by dust, 2) grains photoelectrially heat the gas, and 3) +some of the gas-phase elements (e.g., iron) are depleted +(see also Seab & Shull 1983; Snow & Witt 1996; Collins +et al. 2009; Kraemer et al. 2009). Comparatively, Fergu- +son et al. (1997) contend that neon (a noble gas; i.e. a +species of gas with a full outer shell of valence electrons, +and thus less chemical reactivity) is significantly less de- +pleted by dust grains, and therefore [NeV] is emitted al- +most fully outside the grain sublimation radii. Here, we +consider the likelihood that a significant population of +iron CL photons are destroyed by dust in our sample. +To explore the role of dust extinction on CL emission +in our sample, and to determine its relevance for the dis- +crepancy between the fraction of confirmed AGN in the +[NeV] galaxies (94%) vs. the [FeVII] and [FeX] galaxies +(14% and 25% respectively; Section 4.1), we use the E(B +- V) color excess index, which traces interstellar redden- +ing (Section 3.6). The MaNGA DRP provides this index +for each galaxy in MPL-11, and we use it to determine +if there is a correlation between the dust content of each +CL galaxy and its CL emission. We present our findings +in Table 4. +In particular, we find the mean E(B - V) values +for the [FeVII]λ3760, λ6086 galaxies to be the low- +est (i.e. +feature less dust grains) across our sam- +ple (0.029 and 0.039, respectively). +Because iron is +susceptible to destruction by dust grains, particularly +in the nuclear region where the presence of dust is +greater (also due to dust in the NLR), these relatively +low values provide a viable explanation for the pres- +ence of [FeVII]λ3760, λ6086 emission in these galax- +ies; for reference, our sample contains 16 [FeVII]λ3760 +galaxies and 19 [FeVII]λ6086 galaxies. Comparatively, +the [NeV]λ3347, λ3427, [FeVII]λ3586, and [FeX]λ6374 +galaxies feature higher mean E(B- V) values; 0.057, +0.045, 0.045, and 0.049, respectively. The correspond- +ing number of iron CL-emitting galaxies found in these +galaxies is only nine in total (J0736 features emission +from both [NeV] lines, as well as [FeVII]λ3586 emis- +sion; J1714 features emission from both [NeV] lines, as +well as [FeVII]λ6086 emission; J0807 features emission +from [FeVII]λ3586, λ3760, λ6086; J1157 features emis- +sion from [FeVII]λ3586, λ3760; J0906 features emission +from [FeVII]λ3586; J1628, J2311, J1649, and J1720 ex- +clusively feature emission from [FeX]λ6374). We suspect +that emission from the iron CL species is being dimin- +ished within these relatively dusty galaxies, which pro- +vides a physical explanation for the low number of iron +CL galaxies in the high E(B - V) value galaxies (E(B - +Table 4. CLR Dust Attenuation +Detected +Wavelength +Average E(B-V) Value +CL +(˚A) +(1) +(2) +(3) +[NeV] +3347 +0.057 +3427 +0.045 +[FeVII] +3586 +0.045 +3760 +0.029 +6086 +0.039 +[FeX] +6374 +0.049 +Note: Columns are (1) detected CL, (2) +rest wavelength, and (3) average E(B-V) +values, for each CL, reported by MaNGA’s +DRP. +V) ≥ 0.045; nine iron CL galaxies) vs. the low E(B - V) +galaxies (E(B - V) < 0.039; 34 iron CL galaxies). +Furthermore, Elitzur & Shlosman 2006 considered the +correlation between the AGN dusty torus and AGN +bolometric luminosity. They proposed that the dusty +torus diminishes at log(Lbol) ⪅ 42 erg s−1, due to mass +accretion no longer being able to sustain the necessary +cloud outflow rate, which effectively results in a de- +crease in column density (see also Chiaberge et al. 1999; +Whysong & Antonucci 2004; Maoz et al. 2005). While +the cloud component of the AGN is not immediately +extinguished below this threshold, the authors contend +that the cloud outflow rate at log(Lbol) ⪅ 42 erg s−1 is +less than the necessary “standard” observed in higher +luminosity AGN. As a result, we consider the Lbol val- +ues for the CL species (Figure 5, mean log(Lbol) ≥ 44.3 +erg s−1 for [NeV] galaxies; mean log(Lbol) ≤ 43.7 erg +s−1 for the [FeVII] and [FeX] galaxies) to conclude that +the lower Lbol values correlate with a diminishing dusty +torus, which results in less destruction of iron by dust +grains. Accordingly, we detect more iron CLs in these +low luminosity sources. On the other hand, the [NeV] +galaxies feature higher Lbol values, which likely cor- +respond to their elevated E(B - V) values. +Likewise, +since Lbol scales with LOIII, this reasoning elucidates +the nearly bimodal log(LOIII) distribution of the [NeV] +vs. the [FeVII] and [FeX] galaxies (Figure 6; the mean +of the [NeV] log([OIII]) luminosity distribution is 41.5 +erg s−1; 40.1 erg s−1 and 39.8 erg s−1 for [FeVII] and +[FeX], respectively). +4.6. SNR, [OI], and Merger-Induced Shocks in the +CLR +Astrophysical shocks can result from a variety of +mechanisms, which include, but are not limited to, +galaxy collisions, SNRs, cloud-cloud collisions, expand- +ing HII regions, and outflows from young stellar objects + +14 +Negus et al. +Table 5. Morphological and Merger Classifications of the CL Galaxies +SDSS Name +Detected CL(s) +Redshift +Morphology +Merger +(1) +(2) +(3) +(4) +(5) +J001938.78+144201.1 +[FeVII]λ6086 +0.116 +E +N +J002343.86+141824.2 +[FeVII]λ3760 +0.018 +S(b) +N +J020557.03+004623.9 +[FeVII]λ6086 +0.042 +E +N +J021257.59+140610.2 +[NeV]λ3427 +0.062 +- +N +J030639.57+000343.1 +[NeV]λ3427 +0.107 +E +Y +J072656.07+410136.0 +[NeV]λ3427 +0.129 +S(b) +Y +J073623.13+392617.7 +[NeV]λ3347, [NeV]λ3427, [FeVII]λ3586 +0.118 +- +Y +J074128.48+442431.6 +[NeV]λ3427 +0.132 +E +N +J075217.84+193542.2 +[NeV]λ3427 +0.117 +- +- +J075756.71+395936.1 +[NeV]λ3427 +0.066 +E(o) +Y +J080018.53+461112.3 +[FeVII]λ3760 +0.061 +E +N +J080403.40+404809.3 +[NeV]λ3427 +0.126 +S +Y +J080543.32+252710.9 +[FeVII]λ3760 +0.072 +E +Y +J080707.18+361400.5 +[FeVII]λ3586, [FeVII]λ3760, [FeVII]λ6086 +0.032 +S +N +J080859.19+364112.9 +[FeVII]λ6086 +0.03 +E +Y +J084002.36+294902.6 +[NeV]λ3427 +0.065 +E +N +J085208.48+511845.8 +[FeVII]λ3760 +0.115 +S +N +J085601.94+572327.4 +[FeVII]λ6086 +0.041 +S(bo) +N +J085835.98+013149.5 +[NeV]λ3427 +0.107 +S(b) +Y +J090659.46+204810.0 +[FeVII]λ3586 +0.109 +S(o) +N +J092002.85+054407.7 +[FeVII]λ6086 +0.038 +S(b) +- +J092739.77+050312.5 +[NeV]λ3427 +0.126 +S +N +J094036.39+033436.9 +[FeVII]λ6086 +0.016 +- +Y +J101042.59+061157.0 +[NeV]λ3427 +0.098 +E(o) +Y +J103825.16-002331.1 +[NeV]λ3347, [NeV]λ3427 +0.096 +S(o) +Y +J105439.31+475144.2 +[NeV]λ3427 +0.073 +S(b) +N +J105759.31+404940.6 +[FeVII]λ6086 +0.024 +E +Y +J110431.08+423721.2 +[NeV]λ3427 +0.126 +E(o) +Y +J111403.52+472653.4 +[FeVII]λ3760 +0.113 +E +Y +J111711.79+465134.0 +[FeVII]λ3760 +0.061 +E +N +J111724.94+443347.8 +[FeVII]λ3760 +0.066 +E +N +J111803.22+450646.8 +[NeV]λ3347, [NeV]λ3427 +0.107 +E(o) +Y +J112043.79+534337.4 +[FeVII]λ6086 +0.107 +E +N +J115710.68+221746.2 +[FeVII]λ3586, [FeVII]λ3760 +0.052 +S(b) +N +J122443.43+442438.8 +[NeV]λ3427 +0.126 +E +Y +J123521.03+422002.6 +[FeVII]λ6086 +0.039 +E +N +J130626.65+451720.4 +[FeVII]λ6086 +0.051 +E +- +J131730.11+474659.3 +[FeVII]λ3760 +0.027 +E +N +J134401.90+255628.3 +[NeV]λ3427 +0.062 +S(b) +N +J134918.20+240544.9 +[FeVII]λ3760 +0.021 +- +Y +J141623.14+381127.4 +[NeV]λ3427 +0.135 +- +Y +J142004.29+470716.8 +[NeV]λ3427 +0.07 +S(b) +N +J144454.24+522648.5 +[FeVII]λ3760 +0.146 +E +N +Note: Columns are (1) SDSS Name, (2) detected CL(s), (3) redshift, (4) GZ2 morphological classifications; +“E” is for elliptical, “S” is for spiral, “b” is for bar, “o” is for odd, and “-” indicates no morphological +classification was assigned, and (5) the merger classification from the Nevin catalog; “Y” marks galaxies +with pmerg > 0.5, “N” identifies galaxies with pmerg ≤ 0.5, and “-” represents galaxies that are not in the +Nevin catalog. + +Coronal Lines in MaNGA +15 +Table 6. Morphological and Merger Classifications of the CL Galaxies (Continued) +SDSS Name +CL +Redshift +Morphology +Merger +(1) +(2) +(3) +(4) +(5) +J145420.10+470022.3 +[FeVII]λ3760 +0.126 +E(o) +Y +J151600.58+342119.1 +[NeV]λ3427 +0.125 +S(o) +Y +J151856.39+332152.2 +[FeVII]λ6086 +0.069 +E +N +J153552.40+575409.4 +[FeVII]λ6086 +0.03 +E(o) +N +J160455.20+280956.9 +[NeV]λ3427 +0.077 +S +Y +J161301.62+371714.9 +[NeV]λ3427 +0.069 +S(b) +Y +J161358.56+393150.2 +[FeVII]λ3760 +0.038 +S +- +J161413.20+260416.3 +[NeV]λ3347, [NeV]λ3427 +0.131 +-(o) +Y +J162428.39+483548.0 +[FeVII]λ6086 +0.057 +E(o) +Y +J162621.91+405442.7 +[FeVII]λ3760 +0.03 +S(o) +Y +J162845.89+252938.0 +[FeX]λ6374 +0.04 +E +N +J162908.95+383256.6 +[FeVII]λ6086 +0.033 +E +N +J163014.63+261223.3 +[NeV]λ3347, [NeV]λ3427 +0.131 +S(b) +N +J163053.84+243343.5 +[FeVII]λ6086 +0.063 +E +Y +J163430.87+374143.6 +[NeV]λ3427 +0.099 +E +N +J164956.39+351243.5 +[FeX]λ6374 +0.1 +E +N +J165810.10+622456.3 +[NeV]λ3427 +0.119 +S(b) +N +J171411.63+575834.0 +[NeV]λ3347, [NeV]λ3427, [FeVII]λ6086 +0.093 +E +Y +J172032.02+280602.9 +[FeX]λ6374 +0.083 +- +Y +J205141.54+005135.4 +[NeV]λ3347, [NeV]λ3427 +0.106 +S +Y +J211646.34+110237.4 +[NeV]λ3427 +0.081 +S +Y +J212401.89-002158.6 +[NeV]λ3427 +0.062 +S(b) +Y +J212900.75+001057.3 +[FeVII]λ3760 +0.133 +E(o) +N +J213227.90+100816.9 +[NeV]λ3427 +0.063 +S(o) +- +J215259.07-000903.4 +[FeVII]λ6086 +0.028 +S +N +J223338.41+131243.6 +[NeV]λ3347, [NeV]λ3427 +0.093 +S +Y +J231142.05+150638.2 +[FeX]λ6374 +0.04 +S(o) +N +J232538.54+152115.8 +[FeVII]λ6086 +0.041 +S +N +Note: Columns are (1) SDSS Name, (2) detected CL(s), (3) redshift, (4) GZ2 morphological classifications; +“E” is for elliptical, “S” is for spiral, “b” is for bar, “o” is for odd, and “-” indicates no morphological +classification was assigned, and (5) the merger classification from the Nevin catalog; “Y” marks galaxies +with pmerg > 0.5, “N” identifies galaxies with pmerg ≤ 0.5, and “-” represents galaxies that are not in the +Nevin catalog. + +16 +Negus et al. +Table 7. SNR Shocks +Detected +Wavelength +CL AGN +CL Non-AGN +CL Mergers +CL Non-Mergers +CL Nuclear +CL Non-Nuclear +CL +(˚A) +SNR Shocks +SNR Shocks +SNR Shocks +SNR Shocks +SNR Shocks +SNR Shocks +(1) +(2) +(3) +(4) +(5) +(6) +(7) +(8) +[NeV] +3347 +13% +- +14% +0% +13% +- +[NeV] +3427 +33% +100% +30% +50% +39% +0% +[FeVII] +3586 +3% +38% +0% +8% +12% +- +[FeVII] +3760 +6% +55% +49% +49% +52% +45% +[FeVII] +6086 +95% +42% +64% +41% +61% +25% +[FeX] +6374 +100% +45% +36% +67% +67% +36% +Note: Columns are (1) detected CL, (2) rest wavelength, (3) percentage of CL-emitting spaxels in the CL Galaxies +with a confirmed AGN that feature SNR shocks, (4) percentage of CL-emitting spaxels in the CL Galaxies without a +confirmed AGN that feature SNR shocks, (5) percentage of CL-emitting spaxels in the CL Galaxies undergoing a merger +that feature SNR shocks, (6) percentage of CL-emitting spaxels in the CL Galaxies not undergoing a merger that feature +SNR shocks, (7) percentage of CL-emitting spaxels in the CL Galaxies with nuclear CL emission (See Section 4.2) that +feature SNR shocks, and (8) percentage of CL-emitting spaxels in the CL Galaxies without nuclear CL emission that +feature SNR shocks. “-” indicates an empty sample set. +Table 8. [OI] Shocks +Detected +Wavelength +CL AGN +CL Non-AGN +CL Mergers +CL Non-Mergers +CL Nuclear +CL Non-Nuclear +CL +(˚A) +SNR Shocks +[OI] Shocks +[OI] Shocks +[OI] Shocks +[OI] Shocks +[OI] Shocks +(1) +(2) +(3) +(4) +(5) +(6) +(7) +[NeV] +3347 +25% +- +29% +0% +25% +- +[NeV] +3427 +29% +100% +33% +31% +35% +0% +[FeVII] +3586 +4% +12% +0% +6% +6% +- +[FeVII] +3760 +3% +28% +20% +26% +25% +25% +[FeVII] +6086 +50% +36% +45% +33% +31% +49% +[FeX] +6374 +100% +67% +100% +67% +67% +100% +Note: Columns are (1) detected CL, (2) rest wavelength, (3) percentage of CL-emitting spaxels in the CL Galaxies +with a confirmed AGN that feature [OI] shocks, (4) percentage of CL-emitting spaxels in the CL Galaxies without a +confirmed AGN that feature [OI] shocks, (5) percentage of CL-emitting spaxels in the CL Galaxies undergoing a merger +that feature [OI] shocks, (6) percentage of CL-emitting spaxels in the CL Galaxies not undergoing a merger that feature +[OI] shocks, (7) percentage of CL-emitting spaxels in the CL Galaxies with nuclear CL emission (See Section 4.2) that +feature [OI] shocks, and (8) percentage of CL-emitting spaxels in the CL Galaxies without nuclear CL emission that +feature [OI] shocks. “-” indicates an empty sample set. +(see Allen et al. 2008 for a further review). To deduce +the role of shocks in the CLR, we consider the [SII] +(λ6717 + λ6731)/Hα and [OI]λ6300/Hα ratios for each +CL-emitting spaxel in our sample (values > 0.4 indicate +SNR shocks and values > 0.1 trace [OI] shocks, respec- +tively; Section 3.7). +We also investigate the fraction of CL galaxies actively +undergoing a merger using the Nevin et al., catalog (Ta- +ble 5; Table 6; Section 3.3). In Negus et al. 2021, we +found that the 3/10 CL galaxies without a confirmed +AGN were all strong merger candidates (J0906, J1349, +and J1454). Therefore, we consider the possibility that +companion galaxies can drive gas inflows towards the +galactic centers, resulting in merger-induced shock ex- +citation (e.g., Farage et al. 2010) that may also pro- +duce CLs. Using the Nevin catalog, here we determine +that 32/66 of the CL galaxies (48%; 5 CL galaxies are +not reported in the Nevin catalog: J0752, J0920, J1306, +J1613, and J2132) have pmerg values > 0.5 - indicative +of an ongoing merger. +Further, we present our SNR and [OI] shocks results +in Tables 7 and 8. Overall, we find that the fraction of +SNR and [OI] shocks do not vary significantly for the +CL galaxies. In particular, on average and across all CL +species, 42% of the CL-emitting spaxels in the CL galax- +ies with a confirmed AGN feature SNR shocks (35% fea- +ture [OI] shocks), whereas 56% of the CL-emitting spax- +els in the CL galaxies without a confirmed AGN feature +SNR shocks (49% feature [OI] shocks). Further, on av- +erage and across all CL species, 32% of the CL-emitting + +Coronal Lines in MaNGA +17 +spaxels in the CL galaxies undergoing a merger feature +SNR shocks (38% feature [OI] shocks). On the other +hand, 36% of the CL-emitting spaxels in the CL galaxies +not undergoing a merger feature SNR shocks (27% fea- +ture [OI] shocks). Finally, on average and across all CL +species, 41% of the CL-emitting spaxels in the CL galax- +ies with nuclear CL emission feature SNR shocks (32% +feature [OI] shocks). +Comparatively, 27% of the CL- +emitting spaxels in the CL galaxies without nuclear CL +emission feature SNR shocks (44% feature [OI] shocks). +We find clear evidence of SNR and [OI] shocks in the +CL-emitting spaxels of each CL species in our sample. +However, the fraction of these shocks does not strongly +trace CL galaxies with or without: a confirmed AGN, a +companion galaxy, or nuclear CL emission. We reason +that SNR and [OI] shocks may be viable CL-emission +mechanisms; however, they are not likely dominant, and +we find little evidence that they produce CLs away from +the nuclear region, in the absence of a confirmed AGN, +or when a merging companion galaxy is present. +5. DISCUSSION +Based on our findings, we reason that the efficacy of +using CLs to detect AGN varies by species of CL. While +the ionization potential of each CL is ≥ 100 eV (Table +1; well above the 55 eV threshold for pure star forma- +tion; consistent with the strong continuum of an AGN +being the ionization source), we find that certain CLs +are better at identifying higher luminosity AGN than +others (log(L[OIII]) ⪆ 41 erg s−1). In particular, [NeV] +emission is predominately present in higher [OIII] lu- +minosity galaxies that feature a confirmed AGN (mean +log(L[OIII]) = 41.5 erg s−1 for the [NeV] galaxies; 94% +of the [NeV] galaxies host a confirmed AGN). On the +other hand, we detect [FeVII] and [FeX] emission in +lower [OIII] luminosity galaxies with fewer confirmed +AGNs (mean log(L[OIII]) ≤ 40.1 erg s−1 for both CLs; +14% and 25% confirmed AGN for the [FeVII] and [FeX] +galaxies, respectively). +We reason that the destruction of iron CLs by dust +grains, which we find is inversely proportional to AGN +bolometric luminosity (the dusty torus diminishes at +log(Lbol) ⪅ 42 erg s−1; e.g., Elitzur & Shlosman 2006), +may be directly impacting [FeVII] and [FeX] emission; +the CL galaxies with the lowest E(B-V) values yield the +most iron CL detections (nine iron CL galaxies with +E(B - V) ≥ 0.045; 34 iron CL galaxies with (E(B - +V) < 0.039). +We posit that if the [FeVII] and [FeX] +galaxies host AGNs, that they may be lower luminosity +AGNs, which are potentially too weak to be detected +via SDSS broad emission lines, NVSS/ FIRST 1.4 GHz +radio observations, WISE mid-infrared color cuts, and +Swift/BAT hard X-ray observations. +We determine that there are primarily two distinct +populations of CL galaxies in our sample: 1) a subset of +CL galaxies that emit [NeV] (33/71 CL galaxies), with +relatively high [OIII] and bolometric luminosities, and a +high fraction of confirmed AGN (94%), and 2) a group +of CL galaxies that emit [FeVII] and [FeX] (40/71 CL +galaxies), with relatively low [OIII] and bolometric lu- +minosities, and a low fraction of confirmed AGN (14% +and 25%, respectively). +Overall, we consider the similar IPs of [NeV] and [Fe- +VII] (126.2 eV and 125 eV, respectively), the high IP +of [FeX] (262.21 eV), and our BPT analysis (100% of +the [NeV] spaxels in our sample are either BPT AGN or +NPT composite, 91% of the [FeVII] spaxels, and 88.3% +of the [FeX] spaxels; Table 3), to deduce that each CL +in our sample is likely linked to AGN activity, but that +[FeVII] and [FeX] emission may preferentially be found +in less luminous AGN (we also consider the possibility +that some of the iron CL emission may not exclusively +be produced by AGN; e.g., shocks may also play a role). +We conclude that the BPT diagram is generally effective +at tracing large populations of AGN; however, [NeV], in +particular, can also be used as an additional resource +to help trace AGN, specifically for instances where one +or more of the optical BPT line ratios is unable to be +determined. +Moreover, Ferguson et al. (1997) reported CL criti- +cal densities between 107 - 1010 cm−3, which suggest +that the CLR is a region between the classical NLR and +the BLR. The authors also indicate that lower ioniza- +tion CLs (e.g., [NeV] and [FeVII]; IPs ≈ 125 eV) are +more likely to form in lower density gas that should be +spatially resolved. +In contrast, higher ionization CLs +(e.g., [FeX]; IP = 262.1 eV) form in a region closer +to the nucleus where the ionizing flux, and ionization +parameters, are higher (i.e., these CLs form in denser, +more efficiently emitting regions). Here we determine +that the average size of the CLR for [NeV]λ3347, λ3427, +[FeVII]λ3586, λ3760, λ6086, and [FeX]λ6374 is 1.9 kpc, +2.3 kpc, 3.7 kpc, 5.3 kpc, 4.1 kpc, and 2.5 kpc, respec- +tively (Table 2) - well into the NLR for all CL species. +With the enhanced capabilities of IFS, which enables us +to spatially resolve the CLR, we find that the CLR for +the galaxies in MaNGA is larger than reported in pre- +vious works (tens to hundreds of pcs; e.g. Prieto et al. +2005; Mazzalay et al. 2010; M¨uller-S´anchez et al. 2011; +Rodr´ıguez-Ardila et al. 2011). +Finally, while the bulk of the CL galaxies in our sam- +ple feature CL emission in their nuclear regions (within +a central 2.′′5 x 2.′′5 FoV; Section 4.2; Table 2), a sig- + +18 +Negus et al. +nificant fraction do not, which is inconsistent with pure +AGN photoionization. In Section 4.6, we show that [OI] +and SNR shocks are present in the CL-emitting spax- +els of each CL species. However, the fraction of SNR +and [OI] shocks, across all CL species, do not signifi- +cantly vary for the CL galaxies with or without nuclear +CL emission. We reason, instead, that AGN radio jets +or outflows may be interacting with gas clouds away +from the nuclear region, ionizing them, and producing +the non-centric CL emission we uncover in our sample +(Figure 4; e.g., Tadhunter et al. 1988; M¨uller-S´anchez +et al. 2011). It is also possible that this CL emission is +tracing a different species of gas, that is not a CL, from +a non-companion galaxy within the MaNGA FoV (at a +different redshift than the target galaxy; e.g., J1613 - +Figure 4). +6. SUMMARY AND FUTURE WORK +We construct the most extensive sample of MaNGA +CL galaxies to date. +With our custom pipeline, +we measure emission from [NeV]λ3347, [NeV]λ3427, +[FeVII]λ3586, +[FeVII]λ3760, +[FeVII]λ6086, +and/or +[FeX]λ6374 at ≥ 5σ above the background continuum +in 71 galaxies in MaNGA’s MPL-11 catalog of 10,010 +unique galaxies. +Our main findings are: +1. The average size of the CLR for [NeV], [FeVII], +and [FeX] is 1.9 kpc, 3.8 kpc, and 2.5 kpc, respec- +tively - beyond the BLR and into the traditional +NLR. +2. The fraction of [NeV], [FeVII], and [FeX] galax- +ies with at least one CL-emitting spaxel in their +nuclear 2.′′5 region is 98.5%, 73%, and 75%, re- +spectively. Nuclear CL emission is preferentially +found in [NeV] galaxies. +3. We identify two main populations of CL galax- +ies: 1) galaxies that mostly feature [NeV] emis- +sion (33/71 CL galaxies), with relatively high +[OIII] and bolometric luminosities (mean [NeV] +log([OIII]) luminosity = 41.5 erg s−1; mean [NeV] +log(Lbol) = 44.5 erg s−1), and a high fraction +of confirmed AGN (94%), and 2) galaxies that +predominately emit [FeVII] and [FeX] (40/71 CL +galaxies), with relatively low [OIII] and bolometric +luminosities (mean [FeVII] and [FeX] log([OIII]) +luminosities are 40.1 erg s−1 and 39.8 erg s−1, re- +spectively; mean [FeVII] and [FeX] log(Lbol) val- +ues are 43.7 erg s−1 and 43.5 erg s−1, respectively), +and a low fraction of confirmed AGN (14% and +25%, respectively). +4. 100% of the [NeV] spaxels in our sample are either +BPT AGN or BPT composite, 91% of the [FeVII] +spaxels, and 88.3% of the [FeX] spaxels. The CLs +are strong tracers of BPT AGN and BPT compos- +ite sources. +5. We detect a low number of iron CL galaxies in high +E(B - V) value galaxies (E(B - V) ≥ 0.045; nine +iron CL galaxies) vs. low E(B - V) galaxies (E(B +- V) < 0.039; 34 iron CL galaxies). +We reason +that the destruction of iron CLs by dust grains, +which is inversely proportional to AGN bolomet- +ric luminosity, may likely be depleting [FeVII] and +[FeX] emission, particularly in the nuclear region +where the presence of dust is greater. The [FeVII] +and [FeX] galaxies may be tracing lower luminosity +AGN, which are possibly too weak to be confirmed +by traditional AGN detection techniques. +6. SNR and [OI] shock excitation are viable CL pro- +duction mechanisms; however, they are not likely +primary, as the abundance of SNR and [OI] shocks +does not vary significantly across our sample for +galaxies with or without: nuclear CL emission, an +AGN, or a merging companion. +We will explore the CLR kinematics in a future publi- +cation to better comprehend the role of outflows on CL +production. In particular, we will use [OIII] flux maps to +evaluate the likelihood that AGN outflows produce CL +emission, provided the strong correlation between [OIII] +emission and AGN outflows (e.g., Sun et al. 2017; Com- +erford et al. 2018). Further, we will measure the rotation +and cloud velocities of the gas for each CL galaxy (to +determine how the bulk motion of gas in the CL galaxies +correlates with CL emission), and also analyze the emis- +sion line profiles of the CLs to determine if, for example, +the CLs feature any blue shifted emission - indicative of +outflows. +Moreover, additional multi-wavelength observations of +the CLs would help deduce their nature. X-ray obser- +vations from Chandra, for example, would allow us to +better confirm the population of low luminosity AGN in +the CL galaxies. This will help to determine the effec- +tiveness of using CL emission as an unambiguous tracer +of AGN in large-scale spectroscopic surveys of galaxies. +Finally, our work here is also relevant for motivating +near IR measurements of additional CLs that are observ- +able by the James Webb Space Telescope, particularly in +cases where optical CLs may be obscured. + +Coronal Lines in MaNGA +19 +ACKNOWLEDGMENTS +J.N. and J.M.C. acknowledge support from NSF +AST1714503 and NSF AST1847938. +Funding for the Sloan Digital Sky Survey IV has been +provided by the Alfred P. Sloan Foundation, the U.S. +Department of Energy Office of Science, and the Partici- +pating Institutions. SDSS-IV acknowledges support and +resources from the Center for High-Performance Com- +puting at the University of Utah. The SDSS web site is +www.sdss.org. +SDSS-IV is managed by the Astrophysical Research +Consortium for the Participating Institutions of the +SDSS Collaboration including the Brazilian Participa- +tion Group, the Carnegie Institution for Science, +Carnegie Mellon University, the Chilean Participa- +tion Group, the French Participation Group, Harvard +Smithsonian Center for Astrophysics, Instituto de As- +trof´ısica de Canarias, The Johns Hopkins University, +Kavli Institute for the Physics and Mathematics of +the Universe (IPMU) / University of Tokyo, the Ko- +rean Participation Group, Lawrence Berkeley National +Laboratory, Leibniz Institut f¨ur Astrophysik Potsdam +(AIP), Max-Planck-Institut f¨ur Astronomie (MPIA Hei- +delberg), Max-Planck-Institut f¨ur Astrophysik (MPA +Garching), Max-Planck-Institut f¨ur Extraterrestrische +Physik (MPE), National Astronomical Observatories of +China, New Mexico State University, New York Uni- +versity, University of Notre Dame, Observat´ario Na- +cional / MCTI, The Ohio State University, Pennsylva- +nia State University, Shanghai Astronomical Observa- +tory, United Kingdom Participation Group, Universidad +Nacional Aut´onoma de M´exico, University of Arizona, +University of Colorado Boulder, University of Oxford, +University of Portsmouth, University of Utah, Univer- +sity of Virginia, University of Washington, University of +Wisconsin, Vanderbilt University, and Yale University. +This publication makes use of data products from the +Wide-field Infrared Survey Explorer, which is a joint +project of the University of California, Los Angeles, and +the Jet Propulsion Laboratory/California Institute of +Technology, funded by the National Aeronautics and +Space Administration. +This research has made use of data supplied by the UK +Swift Science Data Centre at the University of Leicester. +This work utilized the Summit supercomputer, which +is +supported +by +the +National +Science +Foundation +(awards ACI-1532235 and ACI-1532236), the Univer- +sity of Colorado Boulder, and Colorado State Univer- +sity. +The Summit supercomputer is a joint effort of +the University of Colorado Boulder and Colorado State +University. +Software: +This work made use of Astropy2: +a +community-developed core Python package and an +ecosystem of tools and resources for astronomy (As- +tropy Collaboration et al. 2013, 2018, 2022). +REFERENCES +Adelman-McCarthy, J. 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E., et al. 2000, AJ, +120, 1579. doi:10.1086/301513 et al 21020 + diff --git a/R9FQT4oBgHgl3EQfbDai/content/tmp_files/load_file.txt b/R9FQT4oBgHgl3EQfbDai/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..7fde2d4d3be1d6f21562dee8f9f66e237b6ba999 --- /dev/null +++ b/R9FQT4oBgHgl3EQfbDai/content/tmp_files/load_file.txt @@ -0,0 +1,2298 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf,len=2297 +page_content='Draft version February 1, 2023 Typeset using LATEX twocolumn style in AASTeX62 A Catalog of 71 Coronal Line Galaxies in MaNGA: [NeV] is an Effective AGN Tracer James Negus,1 Julia M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Comerford,1 Francisco M¨uller S´anchez,2 Mitchell Revalski,3 Rogemar A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Riffel,4, 5 Kevin Bundy,6 Rebecca Nevin,7 and Sandro B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Rembold4, 5 1The University of Colorado Boulder, 2000 Colorado Avenue, Boulder, CO 80309, USA 2The University of Memphis, 3720 Alumni Avenue, Memphis, TN 38152, USA 3Space Telescope Science Institute, 3700 San Martin Drive, Baltimore, MD 21218, USA 4Departamento de F´ısica, CCNE, Universidade Federal de Santa Maria, 97105-900, Santa Maria, RS, Brazil 5Laborat´orio Interinstitucional de e-Astronomia - LIneA, Rua Gal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Jos´e Cristino 77, Rio de Janeiro, RJ - 20921-400, Brazil 6UC Santa Cruz, 1156 High Street, Santa Cruz, CA 95064 7Fermi National Accelerator Laboratory, Batavia, IL 60510, USA ABSTRACT Despite the importance of AGN in galaxy evolution, accurate AGN identification is often challenging, as common AGN diagnostics can be confused by contributions from star formation and other effects (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Baldwin-Phillips-Terlevich diagrams).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' However, one promising avenue for identifying AGNs are “coronal emission lines” (“CLs”), which are highly ionized species of gas with ionization potentials ≥ 100 eV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' These CLs may serve as excellent signatures for the strong ionizing continuum of AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' To determine if CLs are in fact strong AGN tracers, we assemble and analyze the largest catalog of optical CL galaxies using the Sloan Digital Sky Survey’s Mapping Nearby Galaxies at Apache Point Observatory (MaNGA) catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We detect CL emission in 71 MaNGA galaxies, out of the 10,010 unique galaxies from the final MaNGA catalog, with ≥ 5σ confidence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In our sample, we measure [NeV]λ3347, λ3427, [FeVII]λ3586, λ3760, λ6086, and [FeX]λ6374 emission and crossmatch the CL galaxies with a catalog of AGNs that were confirmed with broad line, X-ray, IR, and radio observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We find that [NeV] emission, compared to [FeVII] and [FeX] emission, is best at identifying high luminosity AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Moreover, we find that the CL galaxies with the least dust extinction yield the most iron CL detections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We posit that the bulk of the iron CLs are destroyed by dust grains in the galaxies with the highest [OIII] luminosities in our sample, and that AGN in the galaxies with low [OIII] luminosities are possibly too weak to be detected using traditional techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' INTRODUCTION Active Galactic Nucleus (AGN) feedback, the process by which an active accretion disk converts gravitational energy into radiative or mechanical energy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', AGN- induced photoionization, outflows, shocks, winds, and jets), has been shown to dynamically influence the evo- lution of a host galaxy (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', the tight correlation be- tween stellar velocity dispersion and black hole mass and the quenching of star formation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Ferrarese & Mer- ritt 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Gebhardt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Hopkins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Di Matteo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Fabian 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Kormendy & Ho 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Heckman & Best 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' However, the full spatial extent, ionization properties, and impact of AGN feedback on the host galaxy have yet to be fully unraveled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The Unified Model of AGN (Antonucci 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Urry & Padovani 1995) provides a fundamental architecture for ⋆ james.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='negus@colorado.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='edu understanding the evolution of AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In this model, an AGN is either Type I or Type II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Type I are viewed pole-on and are observed to have broad (FWHM ⪆ 1,000 km s−1) and narrow (FWHM ⪅ 1,000 km s−1) emission lines, whereas Type II are viewed edge-on and are observed to only have narrow emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' These regions are termed the broad-line region (BLR) and the narrow-line region (NLR), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2021, we considered the Unified Model before investigating the “coronal line region” (CLR), an area surrounding a supermassive black hole (SMBH;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' MBH > 106 M⊙) that produces highly ionized species of gas with ionization potentials (IPs) ≳ 100 eV (termed “coronal lines” (CLs) since they were first observed in the solar corona).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' CLs are suspected to primarily orig- inate from the strong ionizing continuum of an AGN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' in particular, nuclear CLs are produced in the inner edge of the dusty torus and extended CLs are tied to the presence of a jet or AGN-driven outflows (due arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='13322v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='GA] 30 Jan 2023 2 Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' to the highly energetic nature of these processes;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Rodr´ıguez-Ardila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2002;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Prieto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Gel- bord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Mullaney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Mazzalay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Rodr´ıguez-Ardila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' M¨uller-S´anchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Glidden et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Riffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Trindade Falc˜ao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Further, CLs in the mid-infrared have been extensively used to probe for AGNs, and to subsequently analyze their physical environments, within dusty galaxies (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Genzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1998, Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2002, Armus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2004 Lutz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2005, Weedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2006, Dasyra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2008, Armus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In fact, several studies have shown that AGNs, even those missed by optical surveys (due to obscuration, for example), are uncovered by ob- servations of infrared CLs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Satyapal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2008, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Sajina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Additionally, since CL emis- sion from Type II supernovae is infrequent, weak, and short lived, CL infrared observations have been partic- ularly useful for accurately identifying CL emission ex- clusively from AGNs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Smith et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In regard to optical studies, Baldwin-Phillips-Terlevich diagnostics diagrams (Baldwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Veilleux & Osterbrock 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2001, 2006) are pre- dominantly used to differentiate emission-line sources as star-forming, AGN, or a composite of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' However, diffuse ionized gas, extraplanar gas, photoionization by hot stars, metallicity, and shocks can elevate sources be- yond the star formation threshold and potentially lead to AGN misclassification (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Wylezalek et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Moreover, while the NLR is the largest observable structure directly affected by an AGN’s ionizing radi- ation (out to several kpcs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', M¨uller-S´anchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2011), star formation can also produce some of the nar- row lines usually associated with AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', [OIII] 5007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' “[OIII]” hereafter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Further, while the BLR provides definitive evidence of AGN activity, due to the elevated cloud velocities, its compact radial extent (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 kpc;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Laor 2004) is spatially unresolved in most spectro- scopic surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' On the other hand, CLs require energies well above the limit of stellar emission (55 eV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Haehnelt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2001) and are typically spatially resolved beyond the BLR and well into the NLR (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' If CLs can provide accurate AGN identification in opti- cal spectroscopic surveys of galaxies, as they have been shown to do in infrared surveys, then detecting them may be a critical step in constraining the complexities of AGN feedback (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Molina et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The Sloan Digital Sky Survey’s (SDSS) Mapping Nearby Galaxies at Apache Point Observatory cata- log (MaNGA;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Bundy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2015) has provided an un- precedented lens into the dynamic environments that surround the SMBHs of nearly 10,010 nearby (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='01 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' average z ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='03) galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Using integral field spectroscopy (IFS), MaNGA provides a 1 - 2 kpc spatial sampling across the field of view of each observed galaxy, which offers direct insight into the spatial extent, ionization properties, and the environmental impact of AGN feedback.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' For reference, previous SDSS surveys (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', SDSS-I to SDSS-III;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' York et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Eisenstein et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2011) observed galaxies with small (3” diame- ter) optical fibers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The resulting spectra only traced a small region close to the galactic center, potentially missing nuclear activity outside of this region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2016) further report that 80% of SDSS galaxies ob- served with a single fiber have less than 36% of their light covered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Moreover, long-slit spectroscopic surveys of galaxies also reveal limited spatial information, since only narrow elongated regions of each galaxy are ob- served (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Newman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In contrast, MaNGA offers the ability to capture spatially extended galac- tic features, which can reveal off-nuclear activity and large-scale emission line regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2021, we scanned for [NeV]λ3347, λ3427, [FeVII]λ3586, λ3760, λ6086, and [FeX]λ6374 emission at ≥ 5σ above the background continuum in the 6,623 galaxies from MaNGA’s eighth data release (MPL-8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We identified 10 CL galaxies in MPL-8, the largest such catalog at the time;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' seven of which were con- firmed to host an AGN, which suggests that CL emission can be useful for tracing AGN activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The remaining three visually appear to be undergoing galactic mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We also found that the average spatial extent of the CLR from the nuclear center is 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='6 kpc - well into the NLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Further, we measured the average electron number den- sity of the CLRs in our sample to be on the order of ≈ 102 cm−3, also consistent with the CLR occupying the traditional NLR, beyond the BLR (typical NLR den- sities range from 101 - 107 cm−3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Peterson 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Revalski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' However, we also reported a range of power-law in- dices (α) above the threshold expected for pure AGN photoionization (α = −2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' we measured -1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 ≤ α ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1), and electron temperature values slightly above the threshold for pure AGN photoioniza- tion (Te = 20, 000 K;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Osterbrock 1981).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We found that the average CLR electron temperatures varied between 12,331 K - 22,530 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' These results suggest that shock- induced compression and heating may also play a role in the production of CLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Comparatively, Mazzalay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2010) investigated the CLR for 10 pre-selected AGNs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' They used the Hubble Space Telescope/Space Telescope Imaging Spectrograph to study [NeV] λ3427, [FeVII] λ3586, λ3760, λ6086, [FeX] λ6374, [FeXIV] λ5303, [FeXI] λ7892, and [SXII] Coronal Lines in MaNGA 3 λ7611 emission in their sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The authors deduced that AGN photoionization is the main driving mecha- nism for the CLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Moreover, Gelbord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2009) used the sixth SDSS data release (Adelman-McCarthy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2008) to analyze the CLR in 63 AGNs with [FeX] λ6374 (IP = 233.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='60 eV), [FeXI] λ7892 (IP = 262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='10 eV), and [FeVII] λ6086 (IP = 99.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='10 eV) emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' They used X- ray observations from Rosat (Voges et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1999, 2000) to similarly posit that AGN photoionization is the main ionization source of the CLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Finally, Reefe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2022) executed the first systematic survey of twenty optical CLs in the spectra of nearly 1 million galaxies from the eighth SDSS data release (Aihara et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The au- thors found that CL emission is extremely rare (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='03% of the sample show at least one CL), and that the highest ionization potential CLs tend to be found in lower mass galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' They reasoned that this finding is consistent with theory that hotter accretion disks are produced by lower mass black holes, which typically reside in lower mass galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Here, we use MaNGA’s eleventh, and final, data re- lease (MPL-11;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 10,010 unique galaxies) to further re- solve the physics of the CLR, and to better understand the relationship between the production of CLs and AGN activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' With our custom pipeline, we identify 71 unique galaxies with emission from either [NeV]λ3347, λ3427, [FeVII]λ3586, λ3760, λ6086, or [FeX]λ6374 de- tected at ≥ 5σ above the continuum, which makes it the most extensive such catalog of MaNGA CL galaxies to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' This paper is outlined as follows: Section 2 details the technical components of the SDSS-IV MaNGA survey and its data pipeline, Section 3 describes the methodol- ogy we use to build the CL catalog and to analyze the physical properties of the CLR, Section 4 reviews our re- sults, Section 5 provides interpretations of our findings, and Section 6 includes our conclusions and intended fu- ture work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' All wavelengths are provided in vacuum and we assume a ΛCDM cosmology with the following val- ues: ΩM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='287, ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='713 and H0 = 69.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 km s−1 Mpc−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' OBSERVATIONS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Sample of Galaxies We assemble our sample from the SDSS-IV MaNGA catalog (Bundy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Drory et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Law et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Blanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Wake et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' MaNGA observations occurred between 2014 to 2020, using the SDSS 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 m telescope (Gunn et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The IFS survey contains data for 10,010 nearby galaxies (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='01 < z < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='15;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' average z ≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='03) with stellar mass distributions between 109 M⊙ and 1012 M⊙.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The spectra were taken at wavelengths between 3622 ˚A - 10354 ˚A, with a typical spectral resolving power of ≈ 2000, corresponding to a velocity resolution of ≈ 60 km s−1 (see Bundy et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' MaNGA contains spectroscopic maps out to at least 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 times the effective radius;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' the typical galaxy is mapped out to a radius of 15 kpc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Each MaNGA spa- tial pixel, or spaxel, covers 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='′′5 × 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='′′5, and the average full-width half maximum (FWHM) of the on-sky point spread function (PSF) is 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='′′5, which corresponds to a typical spatial resolution of 1 -2 kpc (Drory et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' MaNGA Data Analysis Pipeline The MaNGA Data Analysis Pipeline (DAP;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Westfall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2019) offers publicly available high-level data prod- ucts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The MaNGA DAP algorithms have been in de- velopment since 2014 and its main outputs are stellar kinematics, fluxes and kinematics of prominent emis- sion lines, and continuum spectral indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' To measure each parameter, the DAP relies on spectral fitting with pPXF (Cappellari 2012, 2017), where each fit features a blend of stellar templates with a multiplicative polyno- mial component to the stellar continuum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In particular, the DAP incorporates the MILESHC stellar templates library (Westfall et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2019) to fit the stellar kinematics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The inputs for the DAP are data reduced by the MaNGA Data Reduction Pipeline (DRP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The DRP is fed spectra from the MaNGA fiber-feed system, which consists of 17 IFUs: two 19-fiber IFUs, four 37-fiber IFUs, four 61-fiber IFUs, two 91-fiber IFUs, and five 127-fiber IFUs (see Drory et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2015 for a more detailed description).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The DRP subsequently wavelength, flux, and astrometrically calibrates the spectra.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' ANALYSIS 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' CL Continuum Subtraction and Emission Line Fitting We scan for [NeV]λ3347, λ3427, [FeVII]λ3586, λ3760, λ6086, and [FeX]λ6374 emission to better understand their effectiveness as AGN indicators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' These CLs are selected because MaNGA’s DAP does not provide emis- sion line measurements for them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' As a result, we expand upon the custom pipeline detailed in Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2021 to measure these CLs in MPL-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Note, all 10 CL galax- ies reported in Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2021 are recovered using the new MPL-11 pipeline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' CL Stellar Continuum Subtraction To measure the stellar kinematics, and subsequently subtract the stellar continuum for each CL galaxy’s ob- served spectra, we use pPXF (Cappellari 2012, 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' pPXF performs a polynomial fit on each galaxy’s spec- trum while masking gas emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' For each fit, we 4 Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' use the MILES1 stellar templates library to represent the stellar population synthesis model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' This library con- tains ≈ 1,000 stars, with spectra obtained by the Isaac Newton Telescope.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' These spectra cover the wavelength range of 3525 ˚A - 7500 ˚A at a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 ˚A FWHM resolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We first access the DRP to extract the necessary data cubes for each MaNGA galaxy before performing the pPXF stellar continuum subtraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The data cubes provide a spectrum for each individual spaxel across the FoV of each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We then use the spectroscopic red- shifts of each galaxy, adopted from the NASA Sloan At- las catalogs (Blanton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2011), to adjust the spectra to rest vacuum wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We also use a minimum redshift threshold (z min) for CLs near the lower wave- length limit of MaNGA (3622 ˚A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Table 1) to ensure CLs of interest are not shifted out of MaNGA’s spectral cov- erage.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' For [NeV]λ3347, λ3427 and [FeVII]λ3586, ≈ 93% (9,152), ≈ 83% (8,096), and ≈ 3% (229) of the MPL-11 galaxies, respectively, feature redshifts that place each CL out of MaNGA’s spectral range;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' as a result, we are unable to scan for [NeV]λ3347, λ3427 and [FeVII]λ3586 in these respective galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We then apply a mask to each datacube, such that the imported wavelength range for each spectrum matches the wavelength range of the stellar templates library (3525 ˚A- 7500 ˚A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Next, we normalize each spectrum by dividing fluxes in this wavelength range by each spec- trum’s median flux value (to avoid numerical issues;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' see Cappellari 2017 for a more detailed discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Sub- sequently, we define a typical instrument resolution of ≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 ˚A, construct a set of Gaussian emission line tem- plates (to mask emission lines;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' provided by pPXF), and fit the stellar templates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Note, for the CLs near the lower limit of the mask (3525 ˚A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', [NeV]λ3347, λ3427 and [FeVII]λ3586), we perform a custom stellar continuum fit and subtraction before measuring the target emis- sion line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In these instances, we execute a polynomial fit on a narrow spectral region, ≈ 300 ˚A wide, of con- tinuum (free of prominent absorption or emission lines) near the rest wavelength of the target CL to model the background stellar continuum and subtract it from the spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' [NeV] and [FeVII] Emission Line Measurements Once the spectra are stellar continuum subtracted, we attempt a single Gaussian fit on a ≈ 30 ˚A region cen- tered on the rest wavelengths of the CLs ([FeX]λ6374 being the exception;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We found that this wavelength range is adequate for capturing the full extent of CL emission in our preliminary scans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We 1 http://miles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='iac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='es/ Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Target CLs Emission Line Wavelength IP z min (˚A) (eV) [NeV] 3347 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='088 [NeV] 3427 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='061 [FeVII] 3586 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='016 [FeVII] 3760 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='0 [FeVII] 6086 125.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='0 [FeX] 6374 262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 Note: Columns are (1) emission line, (2) rest wavelength, (3) ionization potential, and (4) minimum redshift value required for MaNGA detection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' then determine the root mean square (RMS) flux of two continuum regions (≈ 60 ˚A wide) that neighbor each target CL, free of absorption or emission lines, and re- quire that CL amplitudes are detected at ≥ 5σ above the mean RMS flux values in these continuum regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We consider the spectral resolution of MaNGA (R = λ/∆λ ≈ 1400 at 3600 ˚A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' R ≈ 2000 at 6000 ˚A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Smee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2013) to eliminate fits with ∆λ ⪅ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='4 ˚A(for [NeV]λ3347, λ3427), ⪅ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='6 ˚A(for [FeVII]λ3586, λ3760), and ⪅ 3 ˚A(for [FeVII]λ6086 and [FeX]λ6374).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We provide an example of a single Gaussian fit for the [FeVII]λ3586 line in Fig- ure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' [FeX] Emission Line Measurements For [FeX]λ6374, the broad blue wing of this line is often blended with [OI]λ6364 due to their close prox- imity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Consequently, we attempt a double Gaussian fit to isolate the [FeX]λ6374 line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' If this routine does not successfully fit both lines with ≥ 5σ confidence, then we attempt a single Gaussian fit and apply the method used in Gelbord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2009) and Rose et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2015), whereby the emission line ratio [OI]λ6300/λ6364 is used to determine if the [OI]λ6364 and [FeX]λ6374 lines are blended.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Specifically, from atomic physics, if [OI]λ6300/λ6364 = 3, then the [OI]λ6364 line is free from contamination (see also Elmhamdi 2011 for a full review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' If [FeX]λ6374 emission is present and blended with [OI]λ6364, it will reduce the [OI]λ6300/λ6364 ra- tio below three.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The MaNGA DAP provides flux values for both [OI]λ6364 and [OI]λ6300 lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We adopt this method and require this ratio to be below three when fitting for [FeX]λ6374 with a single Gaussian fit to avoid confusing [OI]λ6364 and [FeX]λ6374 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Once we isolate the [FeX]λ6374 emission, we impose the same thresholds used to identify the [NeV] and [FeVII] emis- sion lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', amplitudes ≥ 5σ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Coronal Line Flux Maps Coronal Lines in MaNGA 5 Similar to Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2021, we create custom CL flux maps to analyze the strength and distribution of the CLs in the CLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We create these maps using the integrated CL flux value from each spaxel for each CL galaxy (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The center of each MaNGA observation corresponds to the galactic center (Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We use this posi- tion and the galaxy’s inclination angle to determine the de-projected galactocentric distance of each CL spaxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We do acknowledge that the CL gas may not be re- stricted to the galactic disk;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', the CL emission may associated with an ionization “cone” and therefore, in these instances, the de-projected distances are approxi- mations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The MaNGA DAP provides the ratio of the semi- minor to semi-major axes (b/a) for each galaxy, and we use this value to determine the cosine of each galaxy’s inclination angle (i): cos(i)= b/a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The de-projected dis- tance of each CL spaxel to the center of the galaxy is then measured by: CLD = � (x − xcenter)2 + � (y − ycenter) ∗ cos(i) �2 (1) where x is the projected distance between the spaxel and the galaxy center measured along the galaxy’s major axis, and y for the minor axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We then convert spaxel distances to a physical unit (kpc) using the astropy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='cosmology Python package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The resulting value corresponds to the coronal line dis- tance (CLD) of each CL emitting spaxel from the galac- tic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Further, the minimum coronal line distance (CLDmin) corresponds to the distance of each galaxy’s closest CL-emitting spaxel from the galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Fi- nally, the maximum coronal line distance (CLDmax) cor- responds to the distance of each galaxy’s most distant CL-emitting spaxel from the galactic center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Galaxy Morphology To uncover the correlation, if any, between CL emis- sion and galaxy morphology (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', spiral and elliptical), we use the MaNGA Morphologies Galaxy Zoo value- added catalog to classify the morphologies of the galax- ies in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' This catalog features data from Galaxy Zoo 2, a “citizen science” catalog with more than 16 million visual morphological classifications for > 304,000 galaxies in SDSS (GZ2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Willett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2013), The weighted vote fraction (discussed in Willett et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2013) accounts for voter consistency when participants select morphological classifications, and we require this fraction to be ≥ 50% before assigning a morphological classification (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', “E” for elliptical, or “S” for spiral).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We also use the weighted vote fraction to determine if a Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' A sample spectrum from an individual spaxel showing the [FeVII]λ3586 line detected at ≥ 5σ above the continuum in J0906.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The dotted black line is the continuum subtracted spectrum, the shaded gray region is the uncer- tainty, the solid red line represents the best fit, the red dotted vertical lines mark the fitting window, the blue dotted line signifies the rest wavelength of the [FeVII]λ3586 line, and the two sets of black dotted vertical lines correspond to the neighboring continuum windows where the RMS flux values of the continuum are calculated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' CL galaxy features a bar, and/or is categorized as odd (“b” and“o”, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In addition, to determine the fraction of CL galaxies undergoing a merger, we consider the analysis being per- formed by Nevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', in prep (“Nevin catalog” here- after).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The authors determine the merger probability for each of the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 million galaxies in the SDSS DR16 pho- tometric sample, using a statistical learning tool that is built on a linear discriminant analysis framework, which is trained to separate mock images of simulated merg- ing and non-merging galaxies using imaging predictors (see Nevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2019 for a full review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We investi- gate the MPL-11 galaxies from the broader SDSS DR16 Nevin catalog, and classify a CL galaxy as a merger if the Nevin catalog gives it a merger value (pmerg) > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' AGN Bolometric and [OIII] Luminosities The AGN bolometric luminosity effectively traces the energetic output of an AGN (across the entire electro- magnetic spectrum).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' To compare the luminosity of the CL AGN candidates with other known AGN candidates, we thus consider the bolometric luminosity parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We determine the AGN bolometric luminosity for each CL galaxy using the summed [OIII] flux val- ues (F[OIII]) across the entire galaxy (provided by the MaNGA DAP), and the procedure outlined in Pennell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2017), which assumes [OIII] emission comes from [FeVIl] 35866 Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' A sample CL flux map showing [NeV]λ3427 emis- sion detected ≥ 5σ above the continuum in J1714.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' For this galaxy, the strongest [NeV]λ3427 emission is located near the center of the galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The gray region is outside of the MaNGA FoV and the black region are spaxels with no CL emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' North is up, south is down, east is to the left, and west is to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' an AGN: log � Lbol ergs−1 � = (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5617 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='0978) log � L[OIII] ergs−1 � +(21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='186 ± 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='164) (2) where L[OIII] = F[OIII](4πR2) and R is the DAP pro- vided luminosity distance based on redshift and a stan- dard cosmology of ΩM = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 and ΩΛ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7 (redshift is also measured by the DAP).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We then measure the total [OIII] luminosity (using the summed [OIII] fluxes across the entire galaxy) for each CL galaxy in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Next, we compare the [OIII] luminosities of the CLs in our pipeline (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3) to determine the relative strength of [OIII] for each CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We do so to assess if specific CLs are preferentially found in higher or lower luminosity [OIII]-emitting galaxies, which is useful to determine if CLs uniformly trace all AGN, or if there may be an [OIII] luminosity depen- dence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Narrow-Line BPT Diagnostics Diagrams Baldwin-Phillips-Terlevich optical emission-line diag- nostic diagrams (BPT diagrams;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Baldwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Veilleux & Osterbrock 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2001, 2006) are widely accepted to be effective tools for categorizing gas ionization sources as star-forming, Seyfert (AGN), low-ionization nuclear emission-line region (LINER), or a composite of multiple ionization sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' They serve as the traditional AGN selection tool for most spectro- scopic surveys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Specifically, these diagrams compare line ratios between high and low ionization species, most commonly [OIII]λ5007/Hβ vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' [NII]λ6583/Hα (“[NII]/ Hα diagram” hereafter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In this paper, we construct spatially resolved narrow- line BPT diagnostic diagrams for the CL galaxies to better constrain the ionization sources of the CLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' To do so, we require emission line measurements for the [NII]λ6583, [OIII]λ5007, Hα, and Hβ emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The DAP measures the continuum subtracted flux for each of these emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Note, these fluxes account for galactic reddening using the E(B-V) values deter- mined by the DRP, which assumes an O’Donnell (1994) reddening law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Once we determine the necessary emission line flux measurements, we compute the ratios for the [NII]/ Hα diagram, for each CL-emitting spaxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We then use these values to create custom spatially-resolved BPT maps, whereby we present the BPT-classification for each CL- emitting spaxel within the MaNGA FoV, for each CL galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Figure 3 shows an example BPT map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Dust Attenuation Mullaney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2009) investigated the [FeVII]λ6086, [FeX]λ6374, and [FeXI]λ7892 emission lines in the Seyfert 1 galaxy Ark 564.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The authors used the pho- toionization code CLOUDY (Ferland et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1998) to determine the location and kinematics of these lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' They found that the CLs are launched from a dusty torus near the SMBH, where the gas is quickly ac- celerated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Moreover, using the CLOUDY models, they determined that some iron carrying grains are destroyed during the initial acceleration of the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' To follow up on the analysis performed by Mullaney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2009), and to better understand the role of dust grains on the potential depletion of the iron CLs, we use the E(B - V) color excess index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' This index traces the degree of interstellar reddening caused by photons that are scattered off of dust;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' in essence, it measures the difference between an object’s observed color index and its intrinsic color index.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' E(B - V) values for each CL galaxy are provided by the MaNGA DRP (using Schlegel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1998 maps), and assume the extinction law provided by O’Donnell (1994).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Shock Diagnostics We explore the role of shocks (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', supernova rem- nant (SNR) and [OI]λ6300 (“[OI]” hereafter) shocks) in our analysis to elucidate the role of collisional excita- tion in the production CLs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Penston et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1984).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' To do so, we consider the strength of the [SII]λ6717, J171411.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='63+575834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='0 14 10 8 6 4 2 5 kpc 0Coronal Lines in MaNGA 7 Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' A sample BPT map showing AGN spaxels in red and composite spaxels in green, for CL-emitting spaxels in J2051 (a [NeV]λ3427 galaxy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The gray region is outside of the MaNGA FoV and the black region are spaxels with no CL emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' North is up, south is down, east is to the left, and west is to the right.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' λ6731 doublet with respect to the Hα line, which has traditionally been used to differentiate SNR shocks from photoionized regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Specifically, Dodorico (1978) and Dodorico et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (1980) first determined that regions with [SII] (λ6717 + λ6731)/Hα > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='4 can be used to identify SNR shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Additionally, the [OI] emission line is gen- erally a strong tracer of shock excitation, and line flux ratios with [OI]/Hα> 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 indicate that shocks with ve- locities 160-300 km s −1 are the main excitation source of [OI] (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Dopita 1976;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Farage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Rich et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2010, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Riffel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Comerford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The MaNGA DAP provides flux measure- ments for the [SII]λ6717, λ6731, [OI], and Hα emission lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' RESULTS In this section, we report the main findings for the CL galaxies in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' First, we present the fraction of confirmed AGN in the CL galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Then, we analyze the spatial distribution and extent of the CLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Next, we inspect the CL galaxy bolometric and [OIII] luminosi- ties to deduce the effectiveness of using each species for accurate AGN identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' After, we assess the BPT classification of the CL-emitting spaxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Finally, we in- vestigate the role of dust extinction and shocks in the CLR to determine the impact of dust grains on CL emis- sion, and to further constrain the ionization source(s) of the CLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In total, we find 71 galaxies with CL emission at ≥ 5σ above the background continuum in MaNGA’s MPL- 11 (33 feature [NeV] emission, 39 feature [FeVII] emis- sion, and 4 feature [FeX] emission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Note, in our sample, 40 unique CL galaxies with either [NeV]λ3427, [FeVII], or [FeX] emission, or a combination of the three, fea- ture redshifts below the z min threshold for [NeV]λ3347 (z min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='088);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' further, 24 unique CL galaxies with either [FeVII] or [FeX] emission feature redshifts be- low the z min threshold for [NeV]λ3427 (z min = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='061).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Therefore, we are unable to scan for [NeV]λ3347 or [NeV]λ3427 in these respective galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In general, most of the MPL-11 galaxies feature redshifts that place [NeV]λ3347, λ3427 out of MaNGA’s spectral range (see Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Moreover, in light of the extensive work to detect AGNs using infrared CLs, we crossmatched our catalog of 71 unique CL galaxies with the infrared CL catalogs presented in Genzel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1998, Sturm et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2002, Ar- mus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2004 Lutz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2005, Weedman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2006, Dasyra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2008, Goulding & Alexander 2009, Armus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We do not identify any of the MaNGA CL galaxies in these samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' For 63/71 CL galaxies with GZ2 classifications (89%), we determine a nearly even fraction of spirals and el- lipticals (48% and 52%, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In addition, we measure the average size of the CLR (from the galac- tic center) for [NeV]λ3347, λ3427, [FeVII]λ3586, λ3760, λ6086, and [FeX]λ6374 to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='9 kpc, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 kpc, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7 kpc, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 kpc, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 kpc, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 kpc, respectively (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Further, we find that the vast majority of [NeV] galax- ies feature at least one CL-emitting spaxel in their nu- clear regions (98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='′′5 x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='′′5 FoV surrounding the central spaxel), whereas [FeVII] and [FeX] galaxies gen- erally feature a smaller fraction (73% and 75%, respec- tively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The corresponding fraction of confirmed AGN in these galaxies (determined by comparing our sample to the largest catalog of confirmed MaNGA AGN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1) is 94%, 14%, and 25%, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' MaNGA AGN Comparison Comerford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (in prep) provide the most complete sample of AGN in MaNGA’s MPL-11 (see Comerford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2020 for a full review of their MPL-8 MaNGA AGN catalog).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The authors compile a catalog of MaNGA AGN that were detected using SDSS broad emission lines, NVSS/ FIRST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='4 GHz radio observations, WISE mid-infrared color cuts, and Swift/BAT hard X-ray ob- servations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 5 kpc 0205141.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='54±005135.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='48 Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Broad Balmer emission lines (FWHM > 1,000 km s−1) are strong tracers of the rapidly rotating, high density gas, near the SMBH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' They serve as reliable tracers for AGN activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2015) assembled a catalog of nearby (z ≤ 2) Type I AGN in SDSS’s seventh data release using the broad Hα emission line, and Comerford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (in prep) identify 78 broad line AGN from this catalog in MPL-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Powerful AGN radio jets can expand several kpcs from the SMBH, and can thus serve as strong signatures for AGN activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' As a result, detecting the radio emis- sion from these sources is a great tool for accurate AGN identification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Best & Heckman (2012) used observa- tions from the 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='4 GHz NRAO Very Large Array Sky Survey (NVSS;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Condon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1998) and the Faint Im- ages of the Radio Sky at Twenty Centimeters (FIRST;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1995) to detect AGN in the SDSS’s seventh data release (DR7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' They differentiated AGN activity from star formation emission using the correlation be- tween the 4000 ˚A break strength and radio luminosity per stellar mass, emission line diagnostics, and the re- lation between Hα and radio luminosity (Becker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1995).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Comerford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (in prep) find 221 radio AGN from this catalog in MPL-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Heated dust that surrounds an AGN can produce mid- infrared emission, which can expose obscured and un- obscured AGN activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Comerford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (in prep) thus rely upon observations from the Wide-Field In- frared Survey Explorer (WISE;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Wright et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2010) to help identify AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' They consider the four bands ob- served with WISE (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='4 µm (W1), 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='6 µm (W2), 12 µm (W3), and 22 µm (W4)) and apply a 75% reliability criteria of W1 - W2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='486 exp{0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='092(W2 - 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='07)2} and W2 > 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='07, or W1 - W2 > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='486 and W2 ≤ 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='07 (Assef et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2018) to select AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Comerford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (in prep) detect 130 WISE AGN in MPL-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' X-ray emission produced by AGN generally result from inverse Compton scattering of low energy UV pho- tons by energetic electrons from the accretion disk (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Antonucci 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Haardt & Maraschi 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Hinkle & Mushotzky 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Therefore, X-rays can be a useful indicator of AGN activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Accordingly, the authors use the X-ray catalog assembled by Oh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2018), which consists of ≈ 1,000 AGN observed by the Swift Obser- vatory’s Burst Alert Telescope (BAT) in the ultra hard X-ray (14 - 195 keV), to detect AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Comerford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (in prep) uncover 30 AGN from this catalog in MPL-11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We compare our CL sample to the AGN catalog re- ported by Comerford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (in prep;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' “Comerford sam- ple” hereafter) and crossmatch 35 CL galaxies in it (52% of our sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Further, we consider the fraction of CL galaxies with confirmed AGN by specific CL species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We determine that 94% (31/33) of the [NeV] galaxies host an AGN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 14% (5/36) of the [FeVII] galaxies and 25% (1/4) of the [FeX] galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Overall, 35 unique CL galaxies host a confirmed AGN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 80% (28/35) are confirmed with WISE observations, 63% (22/35) with broad Balmer emission lines, 14% (5/35) with NVSS observations, and 11% (4/35) with BAT AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' All of the [NeV]λ3347 galaxies feature an AGN and [NeV]λ3427 emission, and of the five [FeVII] galaxies with a confirmed AGN, two (J0736 and J1714) also fea- ture both [NeV]λ3347, λ3427 emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Further, two of the remaining three [FeVII] galaxies with a con- firmed AGN (J0807 and J1157) feature emission from more than one [FeVII] emission line (J0807 features [FeVII]λ3586, λ3760, λ6086 emission;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' J1157 features [FeVII]λ3586, λ3760 emission).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The final [FeVII] galaxy with a confirmed AGN exclusively features [FeVII]λ6086 emission, and the sole [FeX] galaxy with a confirmed AGN (J1628) exclusively features [FeX] emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Provided that 80% of the CL galaxies in our sample are confirmed to host an AGN via WISE diagnostics, we consider the fact that these WISE diagnostics are likely to miss low luminosity AGN (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Assef et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Perhaps, one possible explanation for the discrepancy in AGN across the CLs in our sample is that [NeV] traces high-luminosity AGN, while [FeVII] and [FeX] may pos- sibly trace low-luminosity AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We explore this further in Sections 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Spatial Distribution and Extent of the CLs To better constrain the ionization source(s) of the CLs, we first map the measured fluxes of the CLs within the MaNGA FoV for each CL galaxy (Figure 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' These flux maps provide a snapshot of the orientation, extent, and intensity of CL emission for the galaxies in our sam- ple.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Then, we compute the de-projected distance of each CL spaxel from the nuclear center of each galaxy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' the photometric center;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2) to determine the distance of each CL-emitting spaxel from the galaxy center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Finally, we define the nuclear region of each CL galaxy to be a 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='′′5 x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='′′5 aperture (5 x 5 spaxel grid;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' where each spaxel covers a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='′′5 x 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='′′5 FoV) surrounding the central spaxel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' If AGN photoionization is the primary mechanism producing the CLs, it is likely that CL emission is pre- dominantly within the nuclear region of each galaxy, close to the SMBH and the accretion disk (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Gel- bord et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Mazzalay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' On the other hand, if shocks, AGN outflows, or stellar processes play an active role in generating CLs, we anticipate that CL emission will not be found exclusively in the nuclear region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Rather, we would expect to find emission in Coronal Lines in MaNGA 9 Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Spatial Properties for the CL Galaxies Detected Wavelength Confirmed Nuclear Emission CLDmin CLDmax CLDavg CL (˚A) Galaxies (%) kpc kpc kpc (1) (2) (3) (4) (5) (6) (7) [NeV] 3347 8 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='52 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='9 [NeV] 3427 33 97 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='34 19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 [FeVII] 3586 4 100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='10 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7 [FeVII] 3760 16 56 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='11 36 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 [FeVII] 6086 19 63 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='10 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 [FeX] 6374 4 75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='60 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 Note: Columns are (1) detected CL, (2) rest wavelength, (3) number of galaxies with CL emission detected, (4) percentage of CL galaxies with at least one CL-emitting spaxel in a nuclear 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='′′5 FoV, (5) the average CLD (distance of CL-emitting spaxel from the galaxy center), (6) the distance of the furthest CL-emitting spaxel from the galaxy center, and (7) the distance of the closest CL-emitting spaxel from the galaxy center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' regions off-center or off-axis from the SMBH and the galaxy’s rotational plane (see Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2021 for more discussion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' To analyze the CL distribution within the nuclear re- gion of the CL galaxies, we measure the fraction of CL galaxies with at least one CL emitting spaxel in their center, for each CL (Table 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We find that the vast majority of [NeV] λ3347, λ3427 galaxies feature at least one [NeV]-emitting spaxel in their nuclear regions (100% and 97%, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' This finding is consistent with our results in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1, that [NeV] is a strong tracer of AGN activity (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' CL emission is likely dominated by AGN photoionization near the SMBH).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Comparatively, the fraction of CL galaxies with nuclear emission from [FeVII] λ3586, λ3760, or λ6086 varies significantly more (100%, 56%, and 63%, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In Figure 4, we present a sample of CL flux maps for six representative CL galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In three of the galaxies (J1104, J1349, and J2152), it is apparent that the source of the CLs is within the nuclear region, as the CL flux is concentrated here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' However, for the three remain- ing galaxies (J0023, J1613, and J0920), the CL-emitting spaxels are highly offset from the nuclear region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Based on the orientation of the CL flux in J0023 and J0920, it is possible that AGN outflows are generating the CL since the CL emission is generally perpendicular to the orbital plane of each galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' For the J1613 observation, we determine that several optical emission lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', [OIII] and Hα) measured in the secondary galaxy (with the featured “CL emission”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' southwest of J1613 in the MaNGA FoV) have large velocity shifts (> 2,000 km s−1) compared to the center of J1613, which suggests this may not be a companion galaxy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', this is not a merging system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' J1613 is not in the Nevin catalog).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' As a result, the “CL emission” in this galaxy is likely from a separate emission line, from a background galaxy with a different redshift than the primary galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' For J1349, we acknowledge that there appears to be a visual companion galaxy near the nuclear region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' it is possible that both merger induced shocks and AGN photoioniza- tion could be producing the CL emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In addition, we suspect that dust grains may also have a significant impact on the presence of [FeVII] and [FeX] emission in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5, we review the likelihood of iron depletion by dust grains more thoroughly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We also compute the CLDs for the CL galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The CLD, which is the distance of each CL-emitting spaxel from the galactic center, reveals the physical scale of the CLR for each CL galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We measure the average CLDs for [NeV]λ3347, λ3427 to be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='9 kpc and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 kpc, respec- tively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7 kpc, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 kpc, and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 kpc for [FeVII]λ3586, λ3760, λ6086, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 kpc for [FeX]λ6374.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We find no correlation between IPs and CLDs (IP = 262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 eV for [FeX], IP = 126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2 eV for [NeV], and IP = 125 eV for [FeVII]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Moreover, for the [NeV] galaxies, the minimum and maximum distances of each CL-emitting spaxel from the nuclear center (labeled CLDmin and CLDmax in Table 2) ranges between 340 pc to 19 kpc, 100 pc to 36 kpc for the [FeVII] galaxies, and 600 pc to 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='9 kpc for the [FeX] galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' These large variances in CL distance suggest that the CLR extends from just beyond the BLR (≈ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 kpc) and well into the NLR (several kpcs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Finally, to confirm that CL emission is indeed resolved for each CL galaxy, we consider the instrument PSF (≈ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='′′5 for MaNGA), and find that 60/71 CL galaxies show resolved and continuous emission in excess of the typical instrument PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The remaining 11 CL galaxies (J0205, J1010, J1117, J1317, J1344, J1416, J1604, J1626, J1628, J1658, and J1649) lack CL emission in excess of the typ- ical instrument PSF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We reason that these CLRs are be- low the instrument PSF, and not spatially resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' As 10 Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' discussed in Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2021), these CLRs may still be spatially resolved by other instruments (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Mazzalay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2010 and their use of STIS/HST optical spectra), and it is also posible that CL emission may be oriented along an ionization cone;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' however here we consider the CLDs of these galaxies to be upper limits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' AGN Bolometric and [OIII] Luminosities AGN bolometric luminosity, which scales with [OIII] luminosity, is effectively the “power” of an AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' As outlined in Pennell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2017), [OIII] emission is the most utilized line for measuring bolometric luminosity, due its strength in most AGN spectra and the relatively weak blending of emission from photoionized gas in star forming regions with the line (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Heckman et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Heckman & Best 2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Therefore, to help resolve the discrepancy between the differing fractions of confirmed AGN in our sample (Sec- tion 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 94% of the [NeV] galaxies feature a confirmed AGN, 14% for the [FeVII] galaxies, and 25% of the [FeX] galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' for CL galaxies with multiple CLs, we measure this fraction independently for each CL), and to eval- uate the overall effectiveness of using CL detections to identify AGN, we consider the bolometric and [OIII] lu- minosities of the CL galaxies, and further inspect the Comerford sample of MaNGA AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In particular, we compare the mean bolometric luminosities of the CL galaxies (Lbol;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' using the summed [OIII] flux across the entire galaxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='4) with the total population of MPL-11 AGN in the Comerford sample (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Figure 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We find that the mean bolometric luminosity for the [NeV] galaxies (mean z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='10;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' median z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='11), log(Lbol) = 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 erg s−1, is consistent with the mean value of Comerford’s population of MaNGA galaxies that host an AGN (log(Lbol) = 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='6 erg s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' On the other hand, we measure the mean bolometric luminosi- ties of the [FeVII] galaxies (mean z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='06;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' median z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='05) and the [FeX] galaxies (mean z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='07;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' median z = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='06) to be an order of magnitude lower than the mean log(Lbol) value of the Comerford sample (log(Lbol) = 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7 erg s−1 and log(Lbol) = 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 erg s−1 for the [FeVII] and [FeX] galaxies, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Note, we also present the [OIII] luminosity distribution for the CL galaxies in Figure 6 (the mean [OIII] luminosity for the [NeV] galaxies is 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 erg s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 erg s−1 and 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='8 erg s−1 for the [FeVII] and [FeX] galaxies, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We reason that the [FeVII] and [FeX] galaxies may be preferen- tially tracing lower luminosity AGN in MaNGA, which are generally more difficult to detect in multi-wavelength observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' However, we find that the five [FeVII] galaxies with a confirmed AGN (J0736, J0807, J1157, J1535, and J1714) all feature relatively high [OIII] luminosities of log(L[OIII]) ⪆ 41 erg s−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Additionally, the three remain- ing [FeVII] galaxies with [OIII] luminosities at or above this limit (without confirmed AGN) are J0906, J1349, and J2152.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Both J0906 and J1349 visually appear to be actively undergoing a merger;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' J2152 shows no appar- ent companion galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We reason that, for the [FeVII] galaxies in our sample, the log(L[OIII]) cutoff of ≈ 41 erg s−1 is a useful threshold for identifying confirmed AGN (from the Comerford sample) and may also be helpful for detecting mergers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Further, for the [NeV] galaxies, J1344 features the lowest [OIII] luminosity (log(L[OIII]) = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='6 erg s−1) and in fact hosts a confirmed AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We consider the [OIII] luminosity threshold for the [FeVII] galaxies (log(L[OIII]) ≈ 41 erg s−1) to be similar for the [NeV] galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We also determine that the two [NeV] galaxies (J1658 and J1104) that do not feature a confirmed AGN (out of 33 total [NeV] galaxies), feature [OIII] luminosities of log(L[OIII]) = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2 erg s−1 and log(L[OIII]) = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='4 erg s−1, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Considering these high [OIII] lumi- nosities, and the high [NeV] AGN detection rate (94%), we propose that these two galaxies are strong AGN can- didates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' On the other hand, the one [FeX] galaxy with a con- firmed AGN, J1628, features an [OIII] luminosity of log(L[OIII]) = 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='8 erg s−1 (the remaining three [FeX] galaxies, which do not host a confirmed AGN, also have log([OIII]) luminosities < 40 erg s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Consequently, while we consider the log(L[OIII]) ≈ 41 erg s−1 threshold useful for identifying CL galaxies with a confirmed AGN, it is important to acknowledge that CL galaxies with a confirmed AGN can have [OIII] luminosities below this limit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' BPT Analysis The BPT diagram has long served as the standard tool for identifying ionization mechanisms in emission line sources (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g, Baldwin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Veilleux & Oster- brock 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Kewley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2001, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' While effects such as stellar shocks and emission from post-AGB stars are liable to elevate SF sources beyond the AGN threshold (see Yan & Blanton 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Belfiore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Agostino & Salim 2019 for a further discussion), we nonethe- less explore the BPT classification for each CL-emitting spaxel to help pin down the source of CL emission in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' To do so, we compute the log([OIII]/Hβ) and log([NII]/Hα) ratios required for the [NII]/Hα diagram (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Using the thresholds outlined in Kew- ley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2006), we categorize each CL spaxel as either Coronal Lines in MaNGA 11 Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' CL flux maps for 6/71 CL galaxies in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' From top to bottom and left to right: J1104 ([NeV]λ3427 map), J0023 ([FeVII]λ3760 map), J1349 ([FeVII]λ3760 map), J1613 ([FeVII]λ3760 map), J2152 ([FeVII]λ6086 map), and J0940 ([FeVII]λ6086 map).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' For J0023 and J0940, the maps display CL emission spatially offset from the galaxy center.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' For each galaxy, the emission is offset perpendicular to the rotational plane of the galaxy, suggestive of the source of the CLs being AGN outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' For J1349 we observe a possible companion galaxy and consider the possibility that these two galaxies to be undergoing a merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' For the J1613 observation, we determine that several optical emission lines (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', [OIII] and Hα) measured in the secondary galaxy (with the featured “CL emission”;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' southwest of J1613 in the MaNGA FoV) have large velocity shifts (> 2,000 km s−1) compared to the center of J1613, which suggests this may not be a companion galaxy (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', this is not a merging system;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' J1613 is not in the Nevin catalog).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' As a result, the “CL emission” in this galaxy is likely from a separate emission line, from a background galaxy with a different redshift than the primary galaxy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' For J1104 and J2152, CL emission is concentrated towards the galaxy center, likely produced by AGN photoionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' [HII] (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' star forming), AGN, or a composite of the two.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Note, for some CL-spaxels, the DAP reports nega- tive values for the necessary emission line fluxes, likely because the emission lines of interest yield low flux lev- els and the DAP’s subtraction of the stellar continuum results in a net absorption at the expected wavelength of the emission line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' As such, we exclude these spaxels from our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' For the [NeV], [FeVII], and [FeX] emission lines, we determine that, on average, the majority of CL- emitting spaxels are AGN or composite (Table 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' For [NeV]λ3347, λ3427 we find that 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5% and 90% of these spaxels are classified as AGN, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5% and 10% composite, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 0% SF for both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Moreover, we measure the BPT ratios for the [FeVII]λ3586, λ3760, λ6086 spaxels, and find that 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5%, 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3%, and 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='9% of these spaxels are classified as AGN, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 19%, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='8%, and 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7% composite, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5%, 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='9%, and 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5% SF, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' For [FeX]λ6374, 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3% of the CL-emitting spaxels are classified as AGN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 0% com- posite;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7% SF.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In total, 100% of the [NeV] spaxels in our sample are either BPT AGN or BPT composite, 91% of the [FeVII] spaxels, and 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3% of the [FeX] spaxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' These results suggest that the CLs are perhaps useful tracers of AGN, and that the lack of confirmed AGN in our [FeVII] and [FeX] galaxies may trace back to the nearly bimodal log([OIII]) and bolometric luminosity distributions presented in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' [FeVII] and 400 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2 300 cm erg s 200m (10- 150 Flux 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2 50 J110431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='08+423721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2 J002343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='86+141824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2 5kpc 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='0 0 12 60 6 (10- 10 J134918.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='20+240544.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='9 J161358.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='56+393150.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2 5kpc 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='0 1) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='50% 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='75 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='25 J215259.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='07-000903.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='4 5 kpc J094036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='39+033436.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='9 5kpo 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='012 Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' BPT Classifications for the CL Galaxies .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Detected Rest Confirmed Confirmed Confirmed [NII] [NII] [NII] CL Wavelength CL Galaxies AGN AGN Fraction AGN Fraction Composite Fraction SF Fraction ˚A % % % % (1) (2) (3) (4) (5) (6) (7) (8) [NeV] λ3347 8 8 100 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 0 λ3427 33 31 94 90 10 0 [FeVII] λ3586 4 3 75 78.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 19 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 λ3760 16 2 13 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='8 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='9 λ6086 19 3 16 67.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='9 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 [FeX] λ6374 4 1 25 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 0 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7 Note: Columns are (1) detected CL, (2) rest wavelength, (3) the number of galaxies that feature emission from the respective line, (4) the fraction of galaxies that host a confirmed AGN, (5) the average fraction of [NII] AGN BPT spaxels, (6) the average fraction of [NII] Composite BPT spaxels, and (7) the average fraction of [NII] SF BPT spaxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Average mean bolometric luminosities for the 71 CL galaxies in our sample (analyzed by each CL species;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' [NeV], [FeVII], and [FeX]), compared to the MaNGA galax- ies confirmed to feature an AGN in the Comerford sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The AGN in the Comerford sample were verified using SDSS broad emission lines, NVSS/ FIRST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='4 GHz radio observa- tions, WISE mid-infrared color cuts, and Swift/BAT hard X-ray observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The mean bolometric luminosity of the [NeV] galaxies, log(Lbol) = 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 erg s−1, is consistent with the AGN reported in the Comerford sample (mean log(Lbol) = 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='6 erg s−1 for the Comerford sample).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' However, the [FeVII] and [FeX] galaxies feature mean bolometric lumi- nosities an order of magnitude lower (log(Lbol) ≤ 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7 erg s−1) than the Comerford sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We suspect that the [Fe- VII] and [FeX] emission lines may primarily be detecting low luminosity AGN in MaNGA.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' [FeX] may generally be found in low luminosity AGN that are potentially missed by traditional AGN detec- tion techniques;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' though, it is also possible that [FeVII] Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The log([OIII]) luminosity distribution for the 71 CL galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The blue, orange, and green histograms repre- sent the [NeV], [FeVII], and [FeX] galaxies in our sample, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' [NeV] emitting galaxies tend to have higher [OIII] luminosities than [FeVII] or [FeX], which suggests that these galaxies may host higher luminosity AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The mean of the [NeV] log([OIII]) luminosity distribution is 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 erg s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 erg s−1 and 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='8 erg s−1 for [FeVII] and [FeX], respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' and [FeX] may not host an AGN at all).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We explore this possibility, and the corresponding impact of dust extinction on iron CL emission in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The Impact of Dust on CL Emission The role of dust extinction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' the impact of dust grains) on CL emission has yet to be fully unraveled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Mullaney et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2009) suggest that dust grains can po- tentially deplete heavier CL species (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', iron).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Further, Ferguson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (1997) posit that there are three pri- 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 ) (erg/s) 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='0 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='0 T 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 Nev FeVIl FeX BAT WISE RADIO BROAD Source12 [NeV] [FeVII] [FeX] 10 Number of CL Galaxies 2 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='0 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='0 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='0 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='0 42.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 log [Oll] Luminosity (erg/s)Coronal Lines in MaNGA 13 mary effects of dust on line formation: 1) emission lines weaken due to the absorption of the incident continuum by dust, 2) grains photoelectrially heat the gas, and 3) some of the gas-phase elements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', iron) are depleted (see also Seab & Shull 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Snow & Witt 1996;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Collins et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Kraemer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Comparatively, Fergu- son et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (1997) contend that neon (a noble gas;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' a species of gas with a full outer shell of valence electrons, and thus less chemical reactivity) is significantly less de- pleted by dust grains, and therefore [NeV] is emitted al- most fully outside the grain sublimation radii.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Here, we consider the likelihood that a significant population of iron CL photons are destroyed by dust in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' To explore the role of dust extinction on CL emission in our sample, and to determine its relevance for the dis- crepancy between the fraction of confirmed AGN in the [NeV] galaxies (94%) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' the [FeVII] and [FeX] galaxies (14% and 25% respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1), we use the E(B V) color excess index, which traces interstellar redden- ing (Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The MaNGA DRP provides this index for each galaxy in MPL-11, and we use it to determine if there is a correlation between the dust content of each CL galaxy and its CL emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We present our findings in Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In particular, we find the mean E(B - V) values for the [FeVII]λ3760, λ6086 galaxies to be the low- est (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' feature less dust grains) across our sam- ple (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='029 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='039, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Because iron is susceptible to destruction by dust grains, particularly in the nuclear region where the presence of dust is greater (also due to dust in the NLR), these relatively low values provide a viable explanation for the pres- ence of [FeVII]λ3760, λ6086 emission in these galax- ies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' for reference, our sample contains 16 [FeVII]λ3760 galaxies and 19 [FeVII]λ6086 galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Comparatively, the [NeV]λ3347, λ3427, [FeVII]λ3586, and [FeX]λ6374 galaxies feature higher mean E(B- V) values;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='057, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='045, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='045, and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='049, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The correspond- ing number of iron CL-emitting galaxies found in these galaxies is only nine in total (J0736 features emission from both [NeV] lines, as well as [FeVII]λ3586 emis- sion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' J1714 features emission from both [NeV] lines, as well as [FeVII]λ6086 emission;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' J0807 features emission from [FeVII]λ3586, λ3760, λ6086;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' J1157 features emis- sion from [FeVII]λ3586, λ3760;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' J0906 features emission from [FeVII]λ3586;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' J1628, J2311, J1649, and J1720 ex- clusively feature emission from [FeX]λ6374).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We suspect that emission from the iron CL species is being dimin- ished within these relatively dusty galaxies, which pro- vides a physical explanation for the low number of iron CL galaxies in the high E(B - V) value galaxies (E(B - Table 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' CLR Dust Attenuation Detected Wavelength Average E(B-V) Value CL (˚A) (1) (2) (3) [NeV] 3347 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='057 3427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='045 [FeVII] 3586 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='045 3760 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='029 6086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='039 [FeX] 6374 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='049 Note: Columns are (1) detected CL, (2) rest wavelength, and (3) average E(B-V) values, for each CL, reported by MaNGA’s DRP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' V) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='045;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' nine iron CL galaxies) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' the low E(B - V) galaxies (E(B - V) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='039;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 34 iron CL galaxies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Furthermore, Elitzur & Shlosman 2006 considered the correlation between the AGN dusty torus and AGN bolometric luminosity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' They proposed that the dusty torus diminishes at log(Lbol) ⪅ 42 erg s−1, due to mass accretion no longer being able to sustain the necessary cloud outflow rate, which effectively results in a de- crease in column density (see also Chiaberge et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Whysong & Antonucci 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Maoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' While the cloud component of the AGN is not immediately extinguished below this threshold, the authors contend that the cloud outflow rate at log(Lbol) ⪅ 42 erg s−1 is less than the necessary “standard” observed in higher luminosity AGN.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' As a result, we consider the Lbol val- ues for the CL species (Figure 5, mean log(Lbol) ≥ 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 erg s−1 for [NeV] galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' mean log(Lbol) ≤ 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7 erg s−1 for the [FeVII] and [FeX] galaxies) to conclude that the lower Lbol values correlate with a diminishing dusty torus, which results in less destruction of iron by dust grains.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Accordingly, we detect more iron CLs in these low luminosity sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' On the other hand, the [NeV] galaxies feature higher Lbol values, which likely cor- respond to their elevated E(B - V) values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Likewise, since Lbol scales with LOIII, this reasoning elucidates the nearly bimodal log(LOIII) distribution of the [NeV] vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' the [FeVII] and [FeX] galaxies (Figure 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' the mean of the [NeV] log([OIII]) luminosity distribution is 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 erg s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 erg s−1 and 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='8 erg s−1 for [FeVII] and [FeX], respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' SNR, [OI], and Merger-Induced Shocks in the CLR Astrophysical shocks can result from a variety of mechanisms, which include, but are not limited to, galaxy collisions, SNRs, cloud-cloud collisions, expand- ing HII regions, and outflows from young stellar objects 14 Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Table 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Morphological and Merger Classifications of the CL Galaxies SDSS Name Detected CL(s) Redshift Morphology Merger (1) (2) (3) (4) (5) J001938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='78+144201.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 [FeVII]λ6086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='116 E N J002343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='86+141824.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2 [FeVII]λ3760 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='018 S(b) N J020557.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='03+004623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='9 [FeVII]λ6086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='042 E N J021257.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='59+140610.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2 [NeV]λ3427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='062 N J030639.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='57+000343.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 [NeV]λ3427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='107 E Y J072656.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='07+410136.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='0 [NeV]λ3427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='129 S(b) Y J073623.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='13+392617.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7 [NeV]λ3347, [NeV]λ3427, [FeVII]λ3586 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='118 Y J074128.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='48+442431.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='6 [NeV]λ3427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='132 E N J075217.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='84+193542.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2 [NeV]λ3427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='117 J075756.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='71+395936.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 [NeV]λ3427 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='066 E(o) Y J080018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='53+461112.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 [FeVII]λ3760 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='061 E N J080403.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='8 [FeVII]λ6086 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='041 S N Note: Columns are (1) SDSS Name, (2) detected CL(s), (3) redshift, (4) GZ2 morphological classifications;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' “E” is for elliptical, “S” is for spiral, “b” is for bar, “o” is for odd, and “-” indicates no morphological classification was assigned, and (5) the merger classification from the Nevin catalog;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' “Y” marks galaxies with pmerg > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5, “N” identifies galaxies with pmerg ≤ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5, and “-” represents galaxies that are not in the Nevin catalog.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 16 Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Table 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' SNR Shocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='Detected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='Wavelength ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='CL AGN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='CL Non-AGN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='CL Mergers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='CL Non-Mergers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='CL Nuclear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='CL Non-Nuclear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='CL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='(˚A) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='SNR Shocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='SNR Shocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='SNR Shocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='SNR Shocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='SNR Shocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='36% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='Note: Columns are (1) detected CL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2) rest wavelength,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (3) percentage of CL-emitting spaxels in the CL Galaxies with a confirmed AGN that feature SNR shocks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (4) percentage of CL-emitting spaxels in the CL Galaxies without a confirmed AGN that feature SNR shocks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (5) percentage of CL-emitting spaxels in the CL Galaxies undergoing a merger that feature SNR shocks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (6) percentage of CL-emitting spaxels in the CL Galaxies not undergoing a merger that feature SNR shocks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (7) percentage of CL-emitting spaxels in the CL Galaxies with nuclear CL emission (See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2) that feature SNR shocks, and (8) percentage of CL-emitting spaxels in the CL Galaxies without nuclear CL emission that feature SNR shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' “-” indicates an empty sample set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Table 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' [OI] Shocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='Detected ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='Wavelength ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='CL AGN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='CL Non-AGN ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='CL Mergers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='CL Non-Mergers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='CL Nuclear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='CL Non-Nuclear ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='CL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='(˚A) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='SNR Shocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='[OI] Shocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='[OI] Shocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='[OI] Shocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='[OI] Shocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='[OI] Shocks ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='(1) ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='67% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='67% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='67% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='100% ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='Note: Columns are (1) detected CL,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (2) rest wavelength,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (3) percentage of CL-emitting spaxels in the CL Galaxies with a confirmed AGN that feature [OI] shocks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (4) percentage of CL-emitting spaxels in the CL Galaxies without a confirmed AGN that feature [OI] shocks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (5) percentage of CL-emitting spaxels in the CL Galaxies undergoing a merger that feature [OI] shocks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (6) percentage of CL-emitting spaxels in the CL Galaxies not undergoing a merger that feature [OI] shocks,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (7) percentage of CL-emitting spaxels in the CL Galaxies with nuclear CL emission (See Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2) that feature [OI] shocks, and (8) percentage of CL-emitting spaxels in the CL Galaxies without nuclear CL emission that feature [OI] shocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' “-” indicates an empty sample set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (see Allen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2008 for a further review).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' To deduce the role of shocks in the CLR, we consider the [SII] (λ6717 + λ6731)/Hα and [OI]λ6300/Hα ratios for each CL-emitting spaxel in our sample (values > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='4 indicate SNR shocks and values > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 trace [OI] shocks, respec- tively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We also investigate the fraction of CL galaxies actively undergoing a merger using the Nevin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', catalog (Ta- ble 5;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Table 6;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2021, we found that the 3/10 CL galaxies without a confirmed AGN were all strong merger candidates (J0906, J1349, and J1454).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Therefore, we consider the possibility that companion galaxies can drive gas inflows towards the galactic centers, resulting in merger-induced shock ex- citation (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Farage et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2010) that may also pro- duce CLs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Using the Nevin catalog, here we determine that 32/66 of the CL galaxies (48%;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 5 CL galaxies are not reported in the Nevin catalog: J0752, J0920, J1306, J1613, and J2132) have pmerg values > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 - indicative of an ongoing merger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Further, we present our SNR and [OI] shocks results in Tables 7 and 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Overall, we find that the fraction of SNR and [OI] shocks do not vary significantly for the CL galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In particular, on average and across all CL species, 42% of the CL-emitting spaxels in the CL galax- ies with a confirmed AGN feature SNR shocks (35% fea- ture [OI] shocks), whereas 56% of the CL-emitting spax- els in the CL galaxies without a confirmed AGN feature SNR shocks (49% feature [OI] shocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Further, on av- erage and across all CL species, 32% of the CL-emitting Coronal Lines in MaNGA 17 spaxels in the CL galaxies undergoing a merger feature SNR shocks (38% feature [OI] shocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' On the other hand, 36% of the CL-emitting spaxels in the CL galaxies not undergoing a merger feature SNR shocks (27% fea- ture [OI] shocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Finally, on average and across all CL species, 41% of the CL-emitting spaxels in the CL galax- ies with nuclear CL emission feature SNR shocks (32% feature [OI] shocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Comparatively, 27% of the CL- emitting spaxels in the CL galaxies without nuclear CL emission feature SNR shocks (44% feature [OI] shocks).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We find clear evidence of SNR and [OI] shocks in the CL-emitting spaxels of each CL species in our sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' However, the fraction of these shocks does not strongly trace CL galaxies with or without: a confirmed AGN, a companion galaxy, or nuclear CL emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We reason that SNR and [OI] shocks may be viable CL-emission mechanisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' however, they are not likely dominant, and we find little evidence that they produce CLs away from the nuclear region, in the absence of a confirmed AGN, or when a merging companion galaxy is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' DISCUSSION Based on our findings, we reason that the efficacy of using CLs to detect AGN varies by species of CL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' While the ionization potential of each CL is ≥ 100 eV (Table 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' well above the 55 eV threshold for pure star forma- tion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' consistent with the strong continuum of an AGN being the ionization source), we find that certain CLs are better at identifying higher luminosity AGN than others (log(L[OIII]) ⪆ 41 erg s−1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In particular, [NeV] emission is predominately present in higher [OIII] lu- minosity galaxies that feature a confirmed AGN (mean log(L[OIII]) = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 erg s−1 for the [NeV] galaxies;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 94% of the [NeV] galaxies host a confirmed AGN).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' On the other hand, we detect [FeVII] and [FeX] emission in lower [OIII] luminosity galaxies with fewer confirmed AGNs (mean log(L[OIII]) ≤ 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 erg s−1 for both CLs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 14% and 25% confirmed AGN for the [FeVII] and [FeX] galaxies, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We reason that the destruction of iron CLs by dust grains, which we find is inversely proportional to AGN bolometric luminosity (the dusty torus diminishes at log(Lbol) ⪅ 42 erg s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Elitzur & Shlosman 2006), may be directly impacting [FeVII] and [FeX] emission;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' the CL galaxies with the lowest E(B-V) values yield the most iron CL detections (nine iron CL galaxies with E(B - V) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='045;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 34 iron CL galaxies with (E(B - V) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='039).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We posit that if the [FeVII] and [FeX] galaxies host AGNs, that they may be lower luminosity AGNs, which are potentially too weak to be detected via SDSS broad emission lines, NVSS/ FIRST 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='4 GHz radio observations, WISE mid-infrared color cuts, and Swift/BAT hard X-ray observations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We determine that there are primarily two distinct populations of CL galaxies in our sample: 1) a subset of CL galaxies that emit [NeV] (33/71 CL galaxies), with relatively high [OIII] and bolometric luminosities, and a high fraction of confirmed AGN (94%), and 2) a group of CL galaxies that emit [FeVII] and [FeX] (40/71 CL galaxies), with relatively low [OIII] and bolometric lu- minosities, and a low fraction of confirmed AGN (14% and 25%, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Overall, we consider the similar IPs of [NeV] and [Fe- VII] (126.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2 eV and 125 eV, respectively), the high IP of [FeX] (262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='21 eV), and our BPT analysis (100% of the [NeV] spaxels in our sample are either BPT AGN or NPT composite, 91% of the [FeVII] spaxels, and 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3% of the [FeX] spaxels;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Table 3), to deduce that each CL in our sample is likely linked to AGN activity, but that [FeVII] and [FeX] emission may preferentially be found in less luminous AGN (we also consider the possibility that some of the iron CL emission may not exclusively be produced by AGN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', shocks may also play a role).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We conclude that the BPT diagram is generally effective at tracing large populations of AGN;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' however, [NeV], in particular, can also be used as an additional resource to help trace AGN, specifically for instances where one or more of the optical BPT line ratios is unable to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Moreover, Ferguson et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' (1997) reported CL criti- cal densities between 107 - 1010 cm−3, which suggest that the CLR is a region between the classical NLR and the BLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The authors also indicate that lower ioniza- tion CLs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', [NeV] and [FeVII];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' IPs ≈ 125 eV) are more likely to form in lower density gas that should be spatially resolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In contrast, higher ionization CLs (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', [FeX];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' IP = 262.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 eV) form in a region closer to the nucleus where the ionizing flux, and ionization parameters, are higher (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', these CLs form in denser, more efficiently emitting regions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Here we determine that the average size of the CLR for [NeV]λ3347, λ3427, [FeVII]λ3586, λ3760, λ6086, and [FeX]λ6374 is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='9 kpc, 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 kpc, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7 kpc, 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3 kpc, 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 kpc, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 kpc, respec- tively (Table 2) - well into the NLR for all CL species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' With the enhanced capabilities of IFS, which enables us to spatially resolve the CLR, we find that the CLR for the galaxies in MaNGA is larger than reported in pre- vious works (tens to hundreds of pcs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Prieto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Mazzalay et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' M¨uller-S´anchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Rodr´ıguez-Ardila et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Finally, while the bulk of the CL galaxies in our sam- ple feature CL emission in their nuclear regions (within a central 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='′′5 x 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='′′5 FoV;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Table 2), a sig- 18 Negus et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' nificant fraction do not, which is inconsistent with pure AGN photoionization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='6, we show that [OI] and SNR shocks are present in the CL-emitting spax- els of each CL species.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' However, the fraction of SNR and [OI] shocks, across all CL species, do not signifi- cantly vary for the CL galaxies with or without nuclear CL emission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We reason, instead, that AGN radio jets or outflows may be interacting with gas clouds away from the nuclear region, ionizing them, and producing the non-centric CL emission we uncover in our sample (Figure 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Tadhunter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' M¨uller-S´anchez et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2011).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' It is also possible that this CL emission is tracing a different species of gas, that is not a CL, from a non-companion galaxy within the MaNGA FoV (at a different redshift than the target galaxy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', J1613 - Figure 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' SUMMARY AND FUTURE WORK We construct the most extensive sample of MaNGA CL galaxies to date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' With our custom pipeline, we measure emission from [NeV]λ3347, [NeV]λ3427, [FeVII]λ3586, [FeVII]λ3760, [FeVII]λ6086, and/or [FeX]λ6374 at ≥ 5σ above the background continuum in 71 galaxies in MaNGA’s MPL-11 catalog of 10,010 unique galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Our main findings are: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The average size of the CLR for [NeV], [FeVII], and [FeX] is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='9 kpc, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='8 kpc, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 kpc, respec- tively - beyond the BLR and into the traditional NLR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The fraction of [NeV], [FeVII], and [FeX] galax- ies with at least one CL-emitting spaxel in their nuclear 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='′′5 region is 98.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5%, 73%, and 75%, re- spectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Nuclear CL emission is preferentially found in [NeV] galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We identify two main populations of CL galax- ies: 1) galaxies that mostly feature [NeV] emis- sion (33/71 CL galaxies), with relatively high [OIII] and bolometric luminosities (mean [NeV] log([OIII]) luminosity = 41.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 erg s−1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' mean [NeV] log(Lbol) = 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 erg s−1), and a high fraction of confirmed AGN (94%), and 2) galaxies that predominately emit [FeVII] and [FeX] (40/71 CL galaxies), with relatively low [OIII] and bolometric luminosities (mean [FeVII] and [FeX] log([OIII]) luminosities are 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1 erg s−1 and 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='8 erg s−1, re- spectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' mean [FeVII] and [FeX] log(Lbol) val- ues are 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='7 erg s−1 and 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='5 erg s−1, respectively), and a low fraction of confirmed AGN (14% and 25%, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 100% of the [NeV] spaxels in our sample are either BPT AGN or BPT composite, 91% of the [FeVII] spaxels, and 88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3% of the [FeX] spaxels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The CLs are strong tracers of BPT AGN and BPT compos- ite sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We detect a low number of iron CL galaxies in high E(B - V) value galaxies (E(B - V) ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='045;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' nine iron CL galaxies) vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' low E(B - V) galaxies (E(B V) < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='039;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 34 iron CL galaxies).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We reason that the destruction of iron CLs by dust grains, which is inversely proportional to AGN bolomet- ric luminosity, may likely be depleting [FeVII] and [FeX] emission, particularly in the nuclear region where the presence of dust is greater.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The [FeVII] and [FeX] galaxies may be tracing lower luminosity AGN, which are possibly too weak to be confirmed by traditional AGN detection techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' SNR and [OI] shock excitation are viable CL pro- duction mechanisms;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' however, they are not likely primary, as the abundance of SNR and [OI] shocks does not vary significantly across our sample for galaxies with or without: nuclear CL emission, an AGN, or a merging companion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' We will explore the CLR kinematics in a future publi- cation to better comprehend the role of outflows on CL production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' In particular, we will use [OIII] flux maps to evaluate the likelihood that AGN outflows produce CL emission, provided the strong correlation between [OIII] emission and AGN outflows (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Sun et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Com- erford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Further, we will measure the rotation and cloud velocities of the gas for each CL galaxy (to determine how the bulk motion of gas in the CL galaxies correlates with CL emission), and also analyze the emis- sion line profiles of the CLs to determine if, for example, the CLs feature any blue shifted emission - indicative of outflows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Moreover, additional multi-wavelength observations of the CLs would help deduce their nature.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' X-ray obser- vations from Chandra, for example, would allow us to better confirm the population of low luminosity AGN in the CL galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' This will help to determine the effec- tiveness of using CL emission as an unambiguous tracer of AGN in large-scale spectroscopic surveys of galaxies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Finally, our work here is also relevant for motivating near IR measurements of additional CLs that are observ- able by the James Webb Space Telescope, particularly in cases where optical CLs may be obscured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Coronal Lines in MaNGA 19 ACKNOWLEDGMENTS J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' and J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' acknowledge support from NSF AST1714503 and NSF AST1847938.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Funding for the Sloan Digital Sky Survey IV has been provided by the Alfred P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Sloan Foundation, the U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Department of Energy Office of Science, and the Partici- pating Institutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' SDSS-IV acknowledges support and resources from the Center for High-Performance Com- puting at the University of Utah.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The SDSS web site is www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='sdss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='org.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' SDSS-IV is managed by the Astrophysical Research Consortium for the Participating Institutions of the SDSS Collaboration including the Brazilian Participa- tion Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' the Carnegie Institution for Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Carnegie Mellon University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' the Chilean Participa- tion Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' the French Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Harvard Smithsonian Center for Astrophysics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Instituto de As- trof´ısica de Canarias,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The Johns Hopkins University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Kavli Institute for the Physics and Mathematics of the Universe (IPMU) / University of Tokyo,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' the Ko- rean Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Lawrence Berkeley National Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Leibniz Institut f¨ur Astrophysik Potsdam (AIP),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Max-Planck-Institut f¨ur Astronomie (MPIA Hei- delberg),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Max-Planck-Institut f¨ur Astrophysik (MPA Garching),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Max-Planck-Institut f¨ur Extraterrestrische Physik (MPE),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' National Astronomical Observatories of China,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' New Mexico State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' New York Uni- versity,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' University of Notre Dame,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Observat´ario Na- cional / MCTI,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The Ohio State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Pennsylva- nia State University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Shanghai Astronomical Observa- tory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' United Kingdom Participation Group,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Universidad Nacional Aut´onoma de M´exico,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' University of Arizona,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' University of Colorado Boulder,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' University of Oxford,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' University of Portsmouth,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' University of Utah,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Univer- sity of Virginia,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' University of Washington,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' University of Wisconsin,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Vanderbilt University,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' and Yale University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' This publication makes use of data products from the Wide-field Infrared Survey Explorer, which is a joint project of the University of California, Los Angeles, and the Jet Propulsion Laboratory/California Institute of Technology, funded by the National Aeronautics and Space Administration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' This research has made use of data supplied by the UK Swift Science Data Centre at the University of Leicester.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' This work utilized the Summit supercomputer, which is supported by the National Science Foundation (awards ACI-1532235 and ACI-1532236), the Univer- sity of Colorado Boulder, and Colorado State Univer- sity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' The Summit supercomputer is a joint effort of the University of Colorado Boulder and Colorado State University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' Software: This work made use of Astropy2: a community-developed core Python package and an ecosystem of tools and resources for astronomy (As- tropy Collaboration et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2013, 2018, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3847/1538-3881/aabc4f Astropy Collaboration, Price-Whelan, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Lim, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2022, ApJ, 935, 167.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1088/0004-637X/694/2/765 Comerford, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Nevin, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Stemo, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' & Mushotzky, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2021, MNRAS, 506, 4960.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1093/mnras/stab1976 doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1146/annurev-astro-081913-035722 Hopkins, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' F.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1111/j.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1365-2966.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2010.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='16533.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='x Molina, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Reines, A.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='x Reefe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Satyapal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Sexton, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2022, ApJ, 936, 140.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3847/1538-4357/ac8981 Revalski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Crenshaw, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Rafelski, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2022, ApJ, 930, 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3847/1538-4357/ac5f3d Rich, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Dopita, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Kewley, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2010, ApJ, 721, 505.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1088/0004-637X/721/1/505 Rich, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Kewley, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', & Dopita, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2011, ApJ, 734, 87.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1088/0004-637X/734/2/87 Riffel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Bianchin, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Riffel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2021, MNRAS, 503, 5161.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1093/mnras/stab788 Riffel, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Dors, O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Armah, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2021, MNRAS, 501, L54.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1093/mnrasl/slaa194 Rodr´ıguez-Ardila, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Viegas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Pastoriza, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2002, ApJ, 579, 214.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1086/342840 Rodr´ıguez-Ardila, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Prieto, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Portilla, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2015, MNRAS, 448, 2900.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1093/mnras/stv113 Sajina, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Lacy, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', & Pope, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2022, Universe, 8, 356.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3390/universe8070356 Satyapal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Vega, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Dudik, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2008, ApJ, 677, 926.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1086/529014 Satyapal, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Kamal, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Cann, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2021, ApJ, 906, 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3847/1538-4357/abbfaf Schlegel, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Finkbeiner, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' P.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1983, ApJ, 275, 652.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1086/161563 Smee, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Gunn, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Chornock, R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2009, ApJ, 695, 1334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1088/0004-637X/695/2/1334 Snow, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2002, A&A, 393, 821.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1051/0004-6361:20021043 Sun, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='-L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Greene, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1093/mnras/235.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='403 Trindade Falc˜ao, A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Kraemer, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', Crenshaw, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2017, AJ, 154, 86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='3847/1538-3881/aa7ecc Ward, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' & Morris, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 1984, MNRAS, 207, 867.' metadata={'source': 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+page_content=', Anderson, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=', et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' 2000, AJ, 120, 1579.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} +page_content='1086/301513 et al 21020' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/R9FQT4oBgHgl3EQfbDai/content/2301.13322v1.pdf'} diff --git a/SNAyT4oBgHgl3EQfVPeU/content/tmp_files/load_file.txt b/SNAyT4oBgHgl3EQfVPeU/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0b7e84f0caff9f1b30f217b02e4ebf2c2ee21ab0 --- /dev/null +++ b/SNAyT4oBgHgl3EQfVPeU/content/tmp_files/load_file.txt @@ -0,0 +1,2368 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf,len=2367 +page_content='Published at 1st Conference on Lifelong Learning Agents, 2022 SELF-ACTIVATING NEURAL ENSEMBLES FOR CONTINUAL REINFORCEMENT LEARNING Sam Powers Carnegie Mellon University snpowers@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='edu Eliot Xing Georgia Institute of Technology exing@gatech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='edu Abhinav Gupta Carnegie Mellon University gabhinav@cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='cmu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='edu ABSTRACT The ability for an agent to continuously learn new skills without catastrophically forgetting existing knowledge is of critical importance for the development of generally intelligent agents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Most meth- ods devised to address this problem depend heavily on well-defined task boundaries, and thus depend on human supervision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Our task-agnostic method, Self-Activating Neural Ensembles (SANE), uses a modular architecture designed to avoid catastrophic forgetting without making any such assump- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' At the beginning of each trajectory, a module in the SANE ensemble is activated to determine the agent’s next policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' During training, new modules are created as needed and only activated modules are updated to ensure that unused modules remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' This system enables our method to retain and leverage old skills, while growing and learning new ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We demonstrate our approach on visually rich procedurally generated environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 1 INTRODUCTION Lifelong learning (Thrun & Mitchell, 1995) is of critical importance for the field of robotics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' An agent that interacts with the world should continuously learn from it and act intelligently in a wide variety of situations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In contrast to this ideal, most standard deep reinforcement learning methods are centered around a single task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' First, a task is defined, then a policy is learned to maximize the rewards the agent receives in that setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' If the task is changed, a new model is learned from scratch, discarding the previous model and previous interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Task specification thus plays a central role in current end-to-end deep reinforcement learning frameworks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In contrast, humans do not require concrete task boundaries to be able to effectively learn separate tasks—instead, we perform continual (lifelong) learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We learn new skills efficiently by leveraging prior knowledge, without forgetting old behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' However, when placed into continual learning settings, current deep reinforcement learning approaches do neither: the forward transfer properties of these systems are negligible, and they suffer from catastrophic forgetting (McCloskey & Cohen, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' French, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The core issue of catastrophic forgetting is that a neural network trained on one task starts to forget what it knows when trained on a second task, and this issue only becomes exacerbated as more tasks are added.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The problem ultimately stems from sequentially training a single network in an end-to-end manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The shared nature of the weights and the use of backpropagation to update them mean that later tasks overwrite earlier ones (McCloskey & Cohen, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Ratcliff, 1990).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' To handle this, past approaches have proposed a wide variety of ideas: from task-based regularization (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017), to learning different sub-modules for different tasks (Rusu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2016), and dual-system slow/fast learners inspired by the human hippocampus (Schwarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The fundamental problem of continual learning, which few methods address, is that the agent should autonomously determine how and when to adapt to changing environments, or stabilize existing knowledge, without explicit task specification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' It is infeasible for a human to indefinitely provide agents with task-boundary supervision, and doing so side-steps the core problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' There are a few existing task-agnostic (Zeno et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018) methods, though most have only been demonstrated on classification or behavior cloning: for example Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' (2019b) addresses the problem by detecting plateaus and using those as boundaries, Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' (2019) adaptively creates new clusters using Dirichlet processes, and Veness et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' (2021) replaces backpropagation completely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Methods that have been demonstrated on reinforcement learning are rarer;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' exceptions include Rolnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' (2019), which utilizes a large replay buffer, and Lomonaco et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' (2020) which uses the error in the value estimate to determine when to consolidate modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 1 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='00141v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='LG] 31 Dec 2022 Published at 1st Conference on Lifelong Learning Agents, 2022 We approach the problem by introducing a system that continuously, dynamically adapts to changing environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Our ensemble-based method, Self-Activating Neural Ensembles1 (SANE), depicted in Figure 1, is the core of our proposal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Each module in the ensemble is a separate, task-agnostic network.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Periodically, a single module from the ensemble is activated to determine which policy to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Only activated modules are updated, leaving unused modules unchanged and therefore protected from catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Crucially, our ensemble is dynamic: new modules are created when existing modules are found to be insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In this way, modules are created when novel scenarios are encountered, preventing destructive updates to other modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Additionally, SANE is simple;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' modules control their own relevance, activating when the situation to which they are specialized is encountered, and remaining untouched the rest of the time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' SANE provides the following desirable properties for continual reinforcement learning: (a) It mitigates catastrophic forgetting by only updating relevant modules;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' (b) Because of its task-agnostic nature, unlike previous approaches, it does not require explicit supervision with task IDs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' (c) It achieves these targets with bounded resources and computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We demonstrate SANE on three visually rich, challenging level sequences based on Procgen (Cobbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2020) environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Additionally, we analyze the behavior of SANE at a more fine-grained level on 2 individual runs, to gain more understanding of the dynamics of training SANE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Figure 1: The overall structure of the SANE system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Each module contains an actor and a critic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Upon activation, collection occurs from several environments in parallel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 2 RELATED WORK Continual learning Any continual learning system must balance stability (the extent to which existing knowledge is retained) and plasticity (how readily new knowledge is acquired) (Grossberg, 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Abraham & Robins, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Mermillod et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2013).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Stability has posed a substantial challenge due to catastrophic forgetting, by which neural networks trained by backpropagation abruptly forget learned behavior for solving old tasks when presented with new tasks (Kemker et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' McCloskey & Cohen, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Ratcliff, 1990;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Lewandowsky & Li, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' French, 1999).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Broadly, methods for continual learning can be categorized under Regularization, Rehearsal, or Architectural ap- proaches, as well as combinations of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We refer the reader to the survey papers by Parisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' (2019);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Lesort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' (2020);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Mundt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' (2020) for general discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Here we review methods for continual learning relevant to our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Recent strategies for mitigating catastrophic forgetting such as Elastic Weight Consolidation (EWC) (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017), among other Regularization approaches (Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Li & Hoiem, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Zenke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Ritter et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Chaudhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Jaeger, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' He & Jaeger, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Serra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Park et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2019), constrain updates to network parameters important for past tasks when learning new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' However, these methods fundamentally run into the stability-plasticity dilemma, as over-constraining updates can hinder the learning of new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' To improve plasticity, dynamic architectures (Ring, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Zhou et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Terekhov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Rusu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2016) incorporate additional network parameters to help learn new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Furthermore, to prevent model size from growing unbounded, such approaches (Xiao et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Cortes et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Yoon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Mallya et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Mallya & Lazebnik, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Xu & Zhu, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Schwarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Kaplanis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Traor´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2019) use distillation (Buciluˇa et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2006;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Hinton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Rusu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Teh et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017), pruning, and related techniques to consolidating learned behavior while reducing parameter count.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Similarly, Rehearsal and (generative) memory-based approaches (Robins, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' French, 1997;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Gepperth & Karaoguz, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Furlanello et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Rebuffi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Shin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Draelos et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Kamra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Lopez-Paz & Ranzato, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Chaudhry et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Wu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Isele & Cosgun, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Parisi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Riemer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Soltoggio et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Kemker & Kanan, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Van de Ven & Tolias, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Xiang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2019a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Lesort et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 1Code available: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='com/AGI-Labs/continual_rl 2 State s Activate SANE Train Collect P P P Env Env EnvPublished at 1st Conference on Lifelong Learning Agents, 2022 Caccia et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Caselles-Dupr´e et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2021) must also balance data storage and memory network constraints when determining which examples are needed to preserve previously learned behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We build our ensemble approach off of CLEAR (Rolnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2019), a state-of-the-art asynchronous continual RL method which uses Rehearsal, by maintaining a replay buffer that uniformly preserves past experience via reservoir sampling (Isele & Cosgun, 2018), along with Regularization, via behavioral cloning and a KL penalty to preserve prior learned behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Ensemble methods Falling under Architectural approaches, aggregation ensembles (Cheung et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Wen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Veness et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2021) combine predictions from multiple models to produce a final output.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' These types of ensem- bles are also commonly used for uncertainty estimation (Lakshminarayanan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017), exploration (Pathak et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2019), or reducing overestimation bias such as in double Q-learning (Van Hasselt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Fujimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In contrast, modular ensembles (Aljundi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Fernando et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Parascandolo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Kessler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2021) use a subset of the entire ensemble’s parameters to select an appropriate expert model for the task presented.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Selectively updating a subset of parameters or specific modules instead of the entire ensemble can circumvent catastrophic forgetting while bounding compute costs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' this is a feature we utilize in SANE, which is a type of modular ensemble rather than the former, aggregation ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Our method is similar to Multiple Choice Learning (Guzman-Rivera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Lee et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 2017a;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Seo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2020), which chooses and updates only the best expert from an ensemble, encouraging specialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' However, Multiple Choice Learning uses fixed-size static ensembles, while SANE is a dynamic ensemble that merges similar modules and works with a given resource budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' For supervised continual learning, LMC (Ostapenko et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2021) also proposes a modular ensemble approach, although LMC assumes access to task IDs at training time and can only add modules, meaning that its computational footprint is linear relative to the number of tasks learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In contrast, SANE is completely task-agnostic at train and test time, while also creating and merging modules to meet a given compute budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Hierarchical RL can be seen as a hierarchy of meta-policies that control access to an ensemble of (often hand-designed) sub-policies that act at differing temporal resolutions (Sutton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 1999;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Brunskill & Li, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Tessler et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Analogous to our own value-based activation score, some hierarchical RL methods use predicted Q-values to select amongst their ensemble, as in (Dayan & Hinton, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Dietterich, 2000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' (2019) demonstrates the utility of avoiding meta-policies, instead relying on primitives that independently determine their own relevance, similar to self-activation in our approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' However, their primitives distinguish themselves by factorizing a state space, placing strong assumptions on the learnable policies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Additionally, their primitives are not created over time, so the method relies on regularization to ensure primitives in their ensemble are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 3 BACKGROUND We review background on the continual RL setting we study in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Traditional neural networks suffer from catastrophic forgetting because weights in the network are changed by backpropagation every update (McCloskey & Cohen, 1989), causing information learned in a new scenario to overwrite prior behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Instead of learning and updating a single neural network for policy π across multiple tasks, we propose using an dynamic ensemble of self- activating modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Our approach partitions, allocates, and manages parameters for separate modules, so that each module may handle different situations without interfering with others.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' If a module is relevant to the current situation, it activates during inference and is updated during training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' If a module is irrelevant, it is unused and remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' One way of viewing these modules is as latent behaviors, each specialized to a particular circumstance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' For example, if in one context an agent must carefully wait to allow an enemy to pass, we don’t want this to disrupt a behavior where moving quickly to dodge an enemy is the best action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' How may we know when to use which module, when task boundaries are ambiguous and not given by human super- vision?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Each module in our ensemble predicts an activation score, which estimates the relevancy of a given module’s behavior to the current situation, and the module with the highest activation score is selected from the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' An appropriate activation score will protect modules against catastrophic forgetting, and can also enable forward transfer, by activating modules with prior learned behaviors that are advantageous in new settings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' How should such an ensemble be structured?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Pre-defining a static fixed-size ensemble is ineffective for module- based behavior specialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In such a static ensemble, one module will tend to perform well at a task, leading to that module being chosen as the starting point for future tasks which results in catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Regularizing with additional losses would be necessary to distribute activation across the ensemble’s experts, as in (Jacobs et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 1991;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Shazeer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Goyal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Instead, we design SANE as a dynamic ensemble, in which modules are created and merged together as necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Intuitively, modules are created when existing latent behaviors fail to perform as expected, and the ensemble determines that a new latent behavior is needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Modules may also be merged to conserve resource consumption and meet a given compute budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 3 Published at 1st Conference on Lifelong Learning Agents, 2022 Bringing self-activating modules and a dynamic ensemble together, we present Self-Activating Neural Ensembles (SANE).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' To summarize, our approach differs from traditionally-used ensembles in two ways: (i) We do not aggregate results across modules, in order to keep modules isolated from one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' This circumvents catastrophic forgetting, by not backpropogating through the entire ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' (ii).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The ensemble itself is dynamic, in that modules are being created and merged throughout training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 4 SELF-ACTIVATING NEURAL ENSEMBLES FOR CONTINUAL RL We now proceed to formally describe SANE in full detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' SANE is a dynamic collection of modules {M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' , Mk} where, based on the context, one module Mt activates and is used for inference.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Subsequently, given transitions from collected episodes, only the selected module Mt is updated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We describe an individual SANE module in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1, including how activation scores are computed to determine which module to use.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We present the learning process to manage a dynamic ensemble in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Pseudocode is provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 SELF-ACTIVATING MODULE Every module Mi is an actor-critic algorithm represented by: a policy πi(a|s), a critic Vi(v, u|s), as well as a replay buffer Bi that holds experience transitions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We modify the critic Vi from the standard formulation in the following way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Given a state s at timestep t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' the critic Vi predicts two scalars: vi(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' the value estimate of the return Rt received if module Mi is activated,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' and ui(s),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' an uncertainty estimate of the absolute error: ui(s) ≈ |Rt − vi(s)| (1) We proceed by defining an optimistic estimate vUCB i (upper confidence bound) and a pessimistic estimate vLCB i (lower confidence bound) for the return that the module M⟩ can achieve from state s: vUCB i (s) = vi(s) + αu ∗ ui(s) (2) vLCB i (s) = vi(s) − αl ∗ ui(s) (3) where αu,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' αl > 0 are hyperparameters which represent how wide a margin around the expected value to allow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We use these margins to: (i) choose which module to activate during inference;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' (ii) decide when to create a new module during Structure Update (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In all, each module Mi can be considered to be a tuple ⟨πi, Vi, Bi, V i, Ai⟩, where V i and Ai are two other versions of the critic Vi, which we proceed to describe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The target network V i is used for the confidence bounds estimates (Equation 2 and 3) instead of Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Target networks are commonly used in Q-learning (Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Lillicrap et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Anschel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Fujimoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018) to stabilize training by reducing variance from approximation error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Similarly, we update V i with an exponential moving average (Ruppert, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Polyak & Juditsky, 1992;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Izmailov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We denote Vi’s parameters by θi, V i’s parameters by θ′ i, and the update rate by τV ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' we use the update: θ′ i ← τV θi + (1 − τV )θ′ i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The anchor Ai is a frozen instance of the critic Vi from when the module Mi was created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We describe how we use the anchor Ai to measure drift in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Module update SANE can be applied to any actor-critic algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' we describe the specifics of our implementation in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Let LMi denote the loss function of the active SANE module and Lrl be the loss of the actor- critic RL algorithm, with components associated with module Mi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We perform a module update by optimizing LMi = Lrl + µLue, where Lue is MSE loss to estimate uncertainty from Equation 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Inference (Self-Activation) SANE consists of several modules where each module represents the behavior for a particular situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Activating the right module for the right situation is key to the success of the SANE method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In an RL setting, the critic predicts a value estimate, which can serve as an effective proxy for how successful a module Mi may be in obtaining high return.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' At the beginning of the episode, we compute vUCB i for each module in the ensemble {M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' , Mk} using the target network critic V i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Then, we greedily select the module whose critic predicts the highest such value, and use that module for the whole episode.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 DYNAMIC ENSEMBLE We propose a process to dynamically update the structure of the ensemble in SANE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' If the current set of modules behave in an expected manner (returns are within the expected range) then the current set is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' However 4 Published at 1st Conference on Lifelong Learning Agents, 2022 at some point in training, if the returns are outside the expected range, then we know the current set of modules is insufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We create new modules to handle the new situation, and merge modules together to stay within a given compute budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 MEASURING DRIFT The key to successfully updating the SANE structure lies in our ability to detect that we have moved outside this expected range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Our main assumption here is that the change in rewards received is sufficient for distinguishing relevant changes in setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Therefore, we detect change in setting by measuring drift in rewards.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Drift describes when an environment is non-stationary, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' when the reward distribution or the state transition distribution is changing over time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Often where drift occurs, catastrophic forgetting follows because networks update to the new setting, forgetting the old.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' To recognize drift with SANE, at the time of their creation modules have their critic cloned and frozen, creating a static critic called the anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We compare the prediction of a module’s critic to the prediction of its anchor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We say that sufficient change has occurred when the bounds of the expected return, as defined in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1, predicted by a module’s critic do not include the value predicted by its anchor, which serves as a static baseline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Let vAi denote the value estimate of the anchor Ai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Formally, we say that sufficient change has occurred when for a given state s, critic Vi, and anchor Ai, either of the following inequalities hold: vUCB i (s) < vAi(s) (4) vLCB i (s) > vAi(s) (5) In practice, we use the target network critic V i to predict vUCB i (s) and vLCB i (s), instead of Vi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 CREATING A NEW MODULE When drift occurs such that the returns are better than anticipated, we expect that this corresponds to the case that the policy has simply improved in the current setting, as intended by standard module policy training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In this setup, we just update the anchor to improve expectation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' However, the case of negative drift, where the UCB falls below the estimate of the anchor, requires a different strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' This situation occurs when the behavior (policy) starts under-performing expectation, which can occur when the task has been changed and the old policy is no longer as effective as it had been.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' What we do in this case is create a new module that is a clone of the one that was activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We empty the replay buffer and update the anchor at the time of creation of the new module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The goal is for the new module to be activated by the new setting while the old one continues to be activated by the old setting, splitting the input space to more effectively handle the two desired behaviors that are not well handled by a single policy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='3 MERGING MODULES To prevent unlimited memory consumption, we limit the the total number of modules in our ensemble by merging modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' To execute a merge, we start by finding the two modules in the ensemble that are closest by the L2 distance between frames averaged from a sample of trajectories from the replay buffers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We then keep the more frequently used module and drop the less frequent module from the ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Before dropping, we combine the replay buffer of the two modules and run a module update (Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1) on the combined module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Note that in combining the replay buffers of the two modules we use the reservoir sampling technique from CLEAR (Rolnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We maintain a reservoir value for each trajectory, defined as a random value be- tween 0 and 1, that allows every trajectory to have an equal chance of being stored in the buffer, regardless of when it was collected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The trajectories from the module being dropped are added to the replay buffer of the module being kept using the reservoir values that were originally generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='3 IMPLEMENTATION DETAILS Leveraging CLEAR and IMPALA We have chosen to base our modules on IMPALA-based CLEAR as implemented by CORA (Powers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2021), as it allows us to get several useful features for free: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Learning is done efficiently, in a highly parallel manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The expected return, used for training both the critic and the policy (for SANE as well as all baselines), is computed using vtrace, an effective credit assignment method, as described in (Espeholt et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The CLEAR replay buffers are maintained using reservoir sampling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' CLEAR provides auxiliary losses that maintain consistency of both the policy and critic with the replay buffer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 5 Published at 1st Conference on Lifelong Learning Agents, 2022 Model architecture The base implementation uses the Nature CNN model from Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We augment the baseline network with 2 hidden linear layers of dimension 32 with ReLU nonlinearities to increase its representational capacity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' All other hyperparameters for the experiments are provided in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Our code is provided in additional materials, and will be open sourced upon publication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Parallelism By collecting data from multiple environments in parallel, training is considerably faster, but it requires us to make one key assumption: the activated module must be guaranteed to be applicable to all actors being run at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' This requires that all actors be running the same task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 5 EXPERIMENTS Task sequences We choose three procedurally-generated game environments (Climber, Miner, Fruitbot) from Proc- gen (Cobbe et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We construct three task sequences using each of these game environments, by isolating sequences of levels that are likely to cause catastrophic forgetting and where approaches like CLEAR would perform poorly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We selected four levels for Climber and Miner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' For Fruitbot, we added an easier fifth level at the start as a simple curriculum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We run each set of levels for three cycles, to see how learning evolves as the levels are seen again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The first frame of the selected levels are visualized in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' (a) Climber Levels (b) Miner Levels (c) Fruitbot Levels Figure 2: The first frame of each sequence of levels used in our experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Baselines We compare our approach to three baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We also perform an ablation showing the importance of the dynamic ensemble compared to a static set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The baselines we selected are: CLEAR We compare to CLEAR (Rolnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2019), a state-of-the-art continual RL method (Powers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In addition to comparing to CLEAR with the default number of parameters (the same as each module in the SANE ensemble), we also compare to a version of CLEAR with as many total parameters as we use in our SANE ensemble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We refer to this as “CLEAR 8x”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Note that while SANE takes around 14 hours to run our Fruitbot sequence and standard CLEAR takes around 10 hours, these larger models take longer to run: CLEAR 8x took 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We would have liked to compare to a CLEAR 32x as well, but such an experiment was on track to take more than 2 weeks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' This exemplifies another benefit of SANE: the effective usage of more parameters without such a dramatic increase in runtime.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Elastic Weight Consolidation (EWC) We compare to EWC (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017), which uses the diag- onal of the Fisher matrix to estimate the importance of parameters for past tasks, and slows updates to those parameters when learning new tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Progress & Compress (P&C) We additionally compare to P&C (Schwarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018), which uses an online variant of EWC to consolidate learned behavior between dual networks, after each task is learned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Static SANE Ensemble To validate the utility of our dynamic SANE ensemble, we compare to a SANE ensemble that is static: all modules are initialized upfront, and no creation or merging occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' SANE Oracle We also compare against an Oracle version of SANE, where each task has its own pre-specified module, which is looked up by task ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Experimental setup & metrics All hyperparameters for the methods used are given in the Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' For fairness of comparison we hold constant the number of replay frames each method has access to in total, at 400k frames.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' All implementations for baselines are based on those provided by (Powers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We use Continual Evaluation to generate plots for each task in the task sequence, which show how well each task was learned and how well each task was remembered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Every method was run for 5 seeds, and the mean and standard error of the mean are shown in the graphs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Gray shaded rectangles show when the agent trains on each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We also report the Forgetting metric introduced in (Powers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We reproduce the definition of their Forgetting metric here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' ri,j,end expected return achieved on task i after training on task j (6) ri,all,max maximum expected return achieved on task i after training on all tasks (7) 6 Published at 1st Conference on Lifelong Learning Agents, 2022 SANE Static SANE CLEAR CLEAR 8x EWC 8x P&C 8x SANE Oracle Climber 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 Miner 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='3 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='9 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 Fruitbot 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='7 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='3 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='7 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='7 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='3 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 Table 1: Forgetting (F) summary statistics for all experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' EWC and P&C exhibit little forgetting because they also exhibit little learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Of the methods that learned the tasks, we see SANE performs best.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Forgetting compares the maximum final expected return achieved for a task i at any prior point to the expected return while training on task j, where j > i: Fi,j = 10 s � s �ri,j,end − ri,j−1,end |ri,all,max| � We compute the Forgetting statistic for only the first cycle for each seed and take the average across tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We report the average and standard error of the mean across seeds for the Forgetting summary statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 RESULTS We present the Forgetting summary statistics (Powers et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2021) for all methods in Table 1 and the Continual Eval- uation graphs, which present the average and standard error of the mean of the returns received from the environment versus steps taken in the environment, in each section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We also present the final average performance and standard error of the mean for all benchmarks in the Appendix, Tables 3-5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Climber: First we demonstrate SANE on Climber, a side-view task where the agent must ascend a series of platforms while collecting coins and dodging bats.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The selected levels are particularly challenging because avoiding the bats requires relatively precise timing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' a slight decay in policy performance results in significantly reduced reward.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We start by analyzing the Continual Evaluation results in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' SANE and Static SANE both learn the tasks, but we can see that our dynamic model consistently learns and remembers, while Static SANE overall shows more inconsistent performance, doing particularly poorly on Envs 0 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Both versions of CLEAR learn the tasks but readily forget them, indicating that SANE is not improving by merely adding more parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' EWC has mixed results;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' it does worse than SANE uniformly on all cycles of Env 0 and the first cycle of the other Envs, but approximately ties it on the other cycles of Envs 2 and 3, and exceeds it on the other cycles of Env 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' P&C largely fails to learn the tasks at all, with some exception on Env 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' These results are further validated by looking at the Forgetting summary statistics presented in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' By this metric EWC does the best, likely aided by poorer learning during the first cycle of Envs 0 and 1 and the particularly good later performance on Env 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Of the four methods that learned all tasks immediately (SANE, Static SANE, CLEAR, and CLEAR 8x), SANE exhibits the least forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE Oracle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Static SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='CLEAR 8x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='CLEAR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 8x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='P&C 8x ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 8x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='P&C 8x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Figure 3: Results for Continual Evaluation on the Climber sequence of tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We observe that SANE consistently learns and recalls the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Gray shaded rectangles show when the agent trains on each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Miner: We additionally demonstrate SANE on Miner, a task where the agent must dig through dirt in two dimensions, collecting diamonds and going to a specified end-goal without getting crushed by rocks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 7 Published at 1st Conference on Lifelong Learning Agents, 2022 Continual Evaluation results are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' SANE overall outperforms the baselines on the first three envi- ronments;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' we see CLEAR and EWC learning and forgetting, static SANE showing more recall than CLEAR but less than SANE, and largely little learning from P&C with the exception of Env 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' However, on Env 2 one of SANE’s seeds fails to learn the task, and on Env 3 all seeds did.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Perhaps CLEAR’s larger buffer effectively provides more exploration, as there is more randomness amongst the batches selected to be trained upon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Table 1 again demonstrates numerically these qualitative results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We see that of the methods that learned the tasks, SANE not only did the best, it also exhibited some backwards transfer (negative forgetting).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Static SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='CLEAR 8x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='CLEAR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 8x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='P&C 8x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Miner: Env 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE Oracle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Static SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='CLEAR 8x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='CLEAR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 8x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='P&C 8x ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Miner: Env 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Figure 4: Results for Continual Evaluation on the Miner task sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We again observe that SANE improves on the baselines at recall across the tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Gray shaded rectangles show when the agent trains on each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Fruitbot: The final Procgen sequence we use is based on Fruitbot, where the environment continuously scrolls and the agent must move left and right to collect fruit, avoid vegetables, and make it through gaps in the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Continual Evaluation results, shown in Figure 5, are less clear-cut than the previous two experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' SANE clearly exceeds baselines on recall on Envs 1 and 3, but remains comparable to the CLEAR 8x baseline on Envs 2 and 4, and struggles on Env 0, only exceeding the baseline in the final cycle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Furthermore for the most part SANE receives a lower maximum score than the CLEAR baselines, with the exception of Env 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Table 1 shows that despite the mixed qualitative results, SANE again exceeds baselines quantitatively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 6 ANALYSIS OF SANE We generate two additional figures to help us analyze our SANE ensembles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The first is a module ID plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' On creation we assign every module a unique identifier: an integer that increases per module created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' This ID is constant through the lifetime of the module, including when other modules get merged into it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We can plot what module is active by plotting its module ID.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' This allows us to see when there are periods of rapid creation (steep regions of the graph), when older modules are re-used, and when modules are being stably activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The second plot is a lineage plot, showing a graph that represents the history of the ensemble, with each node in the graph representing a module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' A blue line indicates that one module spawned another, and a red line indicates that a module was merged into another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Light blue nodes represent modules that are current available to be activated at the current time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' An example lineage plot2 is shown in Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 SINGLE RUN: CLIMBER We start by analyzing a single (non-hand-picked) run of SANE in Climber, to demonstrate the dynamics of learning in a simple environment where behaviors are readily separable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In Figure 6, we aligned a graph of the ID of the currently active module with the reward received at that time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We can see the desired behavior in this case: several new modules are created (a sharp increase in module ID is observed) as performance successively fails to meet expectation, until a suitable module is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Additionally, we can observe that before a task is trained upon, it is likely to use the best module for the current task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Env 3 uses Env 0’s active module (module 0) for the first 3M steps, then Env 1’s module (15) for most of the next 3M, then Env 2’s module (18) for the next 3M, until during its own training period drift is detected and a custom module is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' It is also worth remarking on the decline in Env 3’s performance, which occurs particularly while the task is being trained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' It occurs while a consistent module is being activated, so is not related to the ensembling behaviors of module 2An interactive lineage plot can also be viewed at https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='com/AGI-Labs/continual_rl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 8 Published at 1st Conference on Lifelong Learning Agents,' metadata={'source': 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+page_content=' SANE performs particularly well on Envs 1 and 3, comparable to CLEAR 8x on Envs 2 and 4, and struggles some with Env 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Gray shaded rectangles show when the agent trains on each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M 15M 20M 25M 30M 35M ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Module ID ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Figure 6: Module ID and expected return plots aligned by timestep,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' to show module activation during a single run of Climber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Gray shaded rectangles show when the agent trains on each task.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' creation or merging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Observing the behavior indicates that the agent is jumping into a bat rather than avoiding it, so it would seem it is overfitting to the jumping behavior, possibly as a result of the decreased replay buffer size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 FRUITBOT ANALYSIS Fruitbot performed least well of our experiments, so we dive in further to understand the dynamics at play.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Module Count Ablation: We first discuss the difference in expected return when we vary the number of modules for SANE, as visualized in Figure 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Overall, we observe that the fewer the modules, the higher the maximum scores received.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The one exception is Env 3, where 8 and 16 modules both receive comparable scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' However, in general the fewer the modules the more forgetting is observed as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' This is particularly noticeable on Envs 0, 1, and 4, with more ambiguity on Envs 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' As we observed in the analysis of Climber, when a new task is switched to, we don’t create just a single module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Rather, the critic steadily learns to adapt to the new task;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' each time it passes the vLCB threshold, a new module is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Once the maximum number of modules has been reached, this triggers a merge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Since a merge combines the replay buffers of the two modules, when two “compatible” modules are merged, the resulting policy is more robust than that of the individual modules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' However when two modules representing conflicting behaviors are merged, we see a reduction in performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Taken together, this means that as merging is occurring, more modules in the ensemble will generally be more stable over time, but might be slower to learn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We discuss a concrete example of this in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 SINGLE RUN Here we analyze a single run of Fruitbot, which allows us to see in more detail the dynamics of SANE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We use an ensemble with 8 modules to simplify analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We focus on three important points, labeled A, B, and C in Figure 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In all three cases the module the Environment is using switches to an older module.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' At A (21M timesteps), Env 1 switches from using Module 193 to Module 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' By analyzing the Lineage plot (not shown here due to its large size) we see that 193 merged into Module 150, which then merged into Module 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Thus the continued high performance on this task can be explained by a successful sequence of merges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' At B (27M timesteps), Env 2 switches from using Module 252 to Module 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In this case, Module 252, which is a direct descendent of Module 9, merged into Module 197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Module 197 is at this point still a module available in the ensemble,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' meaning Env 2 began to activate a high-performing previous module (Module 9) instead of the result of the 9 Published at 1st Conference on Lifelong Learning Agents,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 2022 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='40M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (32) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Fruitbot: Env 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (32) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Fruitbot: Env 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (32) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Fruitbot: Env 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (32) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Fruitbot: Env 3 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (16) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (32) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Fruitbot: Env 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Figure 7: 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Fruitbot: Env 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE (8) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Fruitbot: Env 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Figure 8: The Module ID and Expected Return plots for a single run of Fruitbot,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' aligned by timestep to see how modules are getting used and created while Fruitbot is training.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' merge, implying that the critic value of 252 decayed as a result of its merger into Module 197.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' However, performance was rescued by return to a previous module and performance remains high.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' At C (30M timesteps), Env 4 switches from using Module 261 to Module 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Module 10, as we saw in case A, is a module that is well-suited for Env 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In this case, Module 261 merged into Module 262, a descendent of Module 9, which as we saw in case B is well-suited for Env 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Essentially, our 8 module ensemble lacks the capacity to adequately represent all of the behaviors necessary for this sequence of tasks, and start combining policies destructively, resulting in forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' This is mitigated by introducing more modules into our ensemble, as shown in Figure 7 7 CONCLUSION Inspired by the fact that catastrophic forgetting is caused by updating all neurons in a network for all tasks, we propose the creation of self-activating modules to break up a network into modular components that only get updated when they are used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Our experimental results suggest that a dynamic ensemble, which creates modules as necessary and merges them to conserve resources, performs better than a static ensemble where all modules are created up- front.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' By combining these two novel features, we present SANE (Self-Activating Neural Ensembles) for continual reinforcement learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We demonstrate SANE on sequences of Procgen levels that prove particularly challenging for the current state-of-the-art (CLEAR), showing that our method reliably improves the mitigation of catastrophic forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Furthermore, we present a thorough analysis of SANE, showing how modules are created, used, and merged on individual runs of Climber and Fruitbot, to provide a more comprehensive view into the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Limitations and future directions In this paper, we present a straightforward and simple instantiation of SANE, which has some limitations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' First, using the initial observation of the episode to choose which module to activate limits the current method to tasks that are distinguishable immediately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Second, the complete separation of modules precludes transfer, wherein improvement on one task benefits performance on another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Future work to address these issues may include choosing the active module every n steps instead of once at the beginning of an episode, or making a hierarchical version of SANE where similar tasks activate similar paths through the tree while distinct tasks activate non-overlapping paths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 10 Published at 1st Conference on Lifelong Learning Agents, 2022 ACKNOWLEDGEMENTS This work was supported by ONR MURI, ONR Young Investigator Program, and DARPA MCS.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Lifelong learning with dynamically expandable networks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In International Conference on Learning Representations, 2018.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 15 Published at 1st Conference on Lifelong Learning Agents, 2022 Friedemann Zenke, Ben Poole, and Surya Ganguli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Continual learning through synaptic intelligence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In International Conference on Machine Learning, pp.' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Jesse Zhang, Haonan Yu, and Wei Xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Hierarchical reinforcement learning by discovering intrinsic options.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' arXiv preprint arXiv:2101.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='06521, 2021.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Guanyu Zhou, Kihyuk Sohn, and Honglak Lee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Online incremental feature learning with denoising autoencoders.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In Artificial intelligence and statistics, pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 1453–1461.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' PMLR, 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 16 Published at 1st Conference on Lifelong Learning Agents, 2022 A APPENDIX A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 HYPERPARAMETERS Here we give the hyperparameters for our methods;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' see the provided code for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' For convenience, we put all parameters that vary between methods above the line, and those that are consistent below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Hyperparameter Shared Num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' actors 16 Learner threads 1 Unroll length 32 Grad clip 40 Reward clip [−1, 1] Normalize rewards No Entropy cost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='01 Discount factor 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='99 LSTM No Network arch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Nature CNN Learning rate 4e−4 Optimizer RMSProp α = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='99 ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='01 µ = 0 (Climber & Miner) (Fruitbot) Hyperparameter SANE SANE CLEAR P&C EWC Batch size 2 2 2 18 2 Baseline cost 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 EWC λ 30 300 EWC, min task steps 2e5 Fisher samples 100 100 Normalize Fisher Yes No Online EWC γ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='99 KL cost 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 Policy cloning cost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 Value cloning cost 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='005 Replay ratio 8 8 8 Replay buffer size 50k 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5k 400k (per module) (per module) Max num.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' modules 8 32 αu,inf 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 αu,create 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 αl,create 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 Critic cadence T 1000 10000 Critic target update rate τV 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='9 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='9 Uncertainty cost µ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 Table 2: Hyperparameters for all methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The network architecture is the “Nature CNN” model (Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Note that there are two different α parameters for SANE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' One describes the parameter used for inference (αu,inf) and the other describes the parameters used during module creation (αx,create).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Batch size Since two of the methods augment the batch with more data (SANE and CLEAR), there is the question of how to size the batches of the other baselines (EWC, P&C) fairly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' There are two ways to view it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The first is to equalize the total amount of data the optimizer sees per gradient step (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' since the augmented size is 18, collect a batch size of 18 for EWC and P&C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The second is to equalize the amount of new data the optimizer sees (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 2 new trajectories are collected so use a batch size of 2 for EWC and P&C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We compare the differences in Figure 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We can see that a smaller batch (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' training more often) results in learning the tasks more quickly, but the impact on recall is more ambiguous: on Env 0 the smaller batch size recalls less well, but it’s comparable on all other environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 17 Published at 1st Conference on Lifelong Learning Agents, 2022 Worth noting also is that the smaller batch size takes considerably longer to train.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Given these facts we have chosen to use the larger batch size for our EWC and P&C baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 30 (B=2) ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 30 (B=2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 30 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 1 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 30 (B=2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 30 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 30 (B=2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 30 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Figure 9: Comparison of running EWC with a batch size of 2 versus the default (18).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Increasing network size The base architecture used for all methods is the Nature CNN model (Mnih et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2015) with ∼6e5 parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' To make the 8x version, we multiplied the number of filters at every convolutional layer by 6 (for a total of ∼5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='7e6 parameters).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We opted for making the network wider instead of deeper because: (i) conceptually this is more similar to the structure of the SANE ensemble (ii) it does not introduce the possibility of decreased performance due to gradient vanishing or exploding (Srivastava et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 ARCHITECTURE OVERVIEW Here we describe how SANE utilizes the highly parallel IMPALA method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Every SANE module is one instance of an IMPALA agent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' all modules operate independently, with no shared parameters or data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Each IMPALA agent is composed of a set of actors that are constantly collecting data and populating a shared buffer with the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' When a separate learner thread detects that the minimum batch size of trajectories has been collected, it computes the losses and runs a gradient step.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Not needing to pause the actors to update the model provides significant runtime improvements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' SANE augments this basic structure in one primary way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' In order to switch modules, the currently running module is paused (all actors stop collecting data, and all learner threads stop updating the model) every syield seconds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' At this point an activation score is computed for every module and the highest-scoring module is activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Activation means that all actors are restarted, and model updating resumes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' This alternating of module activation and data collection continues until all tasks are complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 HYPERPARAMETER TUNING First, we present the λ tuning graph for EWC in Figure 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We ran using λ values in the range [1, 1e5] with a batch size of 18 (B=18) on the Climber task sequence with 3 seeds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We also ran two experiments with (B=2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We can see that while the lower λ values (1 and 10) learn the tasks better, they are also more inclined to forget, as can be seen particularly on Envs 0 and the first cycle of Env 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Based on these results we chose λ = 300 and a batch size of 2 for the experiments presented in the paper as the best balance of learning and remembering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 300 (B=2) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 1 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 10 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 30 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 100 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 1k (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 10k (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 100 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 1k (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 10k (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 30 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 100 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 1k (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 10k (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 10 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 30 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 100 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 1k (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='EWC 10k (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Figure 10: Comparison of EWC λ variations on the Climber task sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The number represents λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Second, we present the λ tuning graph for P&C in Figure 11, representing values in the range [3, 3e3], over 3 seeds for a batch size of 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We thus chose λ = 30 for our experiments 18 Published at 1st Conference on Lifelong Learning Agents,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 2022 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='P&C 3 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='P&C 30 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='P&C 300 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='P&C 3k (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='P&C 30 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='P&C 300 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='P&C 3k (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 ' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='P&C 3 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='P&C 30 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='P&C 300 (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='P&C 3k (B=18) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Figure 11: Comparison of P&C λ variations on the Climber task sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The number represents λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 PARAMETER ABLATION ON FRUITBOT As shown in Table 2 we use different parameters for Climber/Miner vs Fruitbot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' For simplicity we refer to the former set of parameters as SANE v1 and the latter as SANE v2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Conceptually, the difference between SANE v1 and SANE v2 is that the former trains the critic more aggressively (higher coefficients and fewer frames between target network updates), but also has a much more conservative vLCB, meaning more drift must be observed before a node is created.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We observe that the former performed well on Climber and Miner, but on Fruitbot we obtained higher performance using the smoother critic training of SANE v2, as shown in Figure 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' However, it is worth noting that the ensemble learns and remembers in both cases, even when the parameters vary widely (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' αl,create differs by a factor of 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' However, optimal parameter selection remains an area of future work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE v1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Fruitbot: Env 0 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE v1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Fruitbot: Env 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='40M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE v1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Fruitbot: Env 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='40M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE v1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Fruitbot: Env 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='40M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='−4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE v1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE v2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Fruitbot: Env 4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Figure 12: Comparison of SANE variations on the Fruitbot task sequence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='3 LINEAGE PLOT, FRUITBOT 0 500k 1M 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5M 2M 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5M 3M 0 5 10 15 20 SANE (8) Fruitbot: Env 0 Step Module ID Figure 13: An example Lineage plot, showing how nodes are created and merged while training on Env 0 of Fruitbot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' While we are not able to provide the full Lineage plots used in the analysis above (a graph with hundreds of nodes displays poorly in pdf form), we show an example of what one looks like for the first task of Fruitbot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Node 0 is used initially, but soon spawns a cascading sequence of Nodes that settles for some time on Node 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' There is some churn as nodes are created but merged back into 9, until another burst of creation settles finally on Node 18, though we see in Figure 8 that this does not last either.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 19 X 13 11 14 15 10 16 17 18Published at 1st Conference on Lifelong Learning Agents, 2022 If the policies and critics improved monotonically, we would not see such a chaotic plot, but instead one more like what we see for Climber, in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Essentially, since creation only happens when we detect that a node is worse than its anchor, this pattern of creation represents much noisier learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ALGORITHM PSEUDOCODE FOR SANE Algorithm 1 SANE Require: timestep t, total task timesteps T, state at episode start s0, max allowed module count N, modules M = {M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' , Mk} where Mi contains actor πi, critic Vi, static anchor critic ai, and replay buffer Ri.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 1: while t < T do 2: Mmax = arg maxi(vUCB,i(s0)) ▷ select active module 3: Collect tnew timesteps of data using Mmax 4: t := t + tnew 5: 6: if vUCB,Mmax(s) < va(s) then ▷ negative drift detected, so add module 7: 8: Mnew := clone(Mmax) 9: anew := clone(Vmax) 10: M := {M1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', Mk, Mnew} 11: else 12: if vLCB,m(s) > va(s) then ▷ positive drift detected, so update module 13: amax := clone(Vmax) 14: end if 15: end if 16: if |M| > N then ▷ merge modules 17: gi := avg(batch(Ri)[′observation′]) ▷ compute an avg observation for each module 18: Mi, Mj = arg mini,j ||gi, gj||2 19: Mkeep, Mremove = which of Mi or Mj has been used more and fewer times, respectively 20: Rkeep = {Ri, Rj} 21: M = M − Mremove 22: 23: end if 24: end while A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 FINAL AVERAGE PERFORMANCE TABLES Task SANE Static SANE CLEAR 8x CLEAR EWC 8x PnC 8x SANE Oracle Env 0 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='72 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='29 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='54 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='85 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='14 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='08 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='88 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='06 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='14 Env 1 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='04 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='49 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='32 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='21 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='37 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='05 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='22 Env 2 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='82 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='92 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='95 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='77 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='05 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='57 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='43 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='33 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='91 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='95 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='08 Env 3 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='99 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='28 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='22 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='34 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='02 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='26 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='40 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='42 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='34 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='05 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='59 Table 3: Climber final average performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The Oracle is not considered for the purposes of highlighting the average performance, as it is an idealized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Task SANE Static SANE CLEAR 8x CLEAR EWC 8x PnC 8x SANE Oracle Env 0 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='35 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='61 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='18 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='89 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='35 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='71 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='26 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='03 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='55 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='01 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='41 Env 1 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='43 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='03 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='25 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='07 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='24 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='71 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='58 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='68 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='46 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='66 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='56 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='06 Env 2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='64 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='09 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='46 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='11 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='42 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='90 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='39 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='65 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='62 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='11 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='54 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='49 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='22 Env 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='97 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='05 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='06 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='05 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='06 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='31 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='74 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='87 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='96 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='02 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='93 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='15 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='06 Table 4: Miner final average performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The Oracle is not considered for the purposes of highlighting the average performance, as it is an idealized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 20 Published at 1st Conference on Lifelong Learning Agents, 2022 Task SANE Static SANE CLEAR 8x CLEAR EWC 8x PnC 8x SANE Oracle Env 0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='18 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='46 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='70 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='37 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='77 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='50 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='05 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='77 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='09 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='74 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='84 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='68 Env 1 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='71 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='43 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='45 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='40 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='45 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='26 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='44 −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='91 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='56 −2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='08 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='78 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='85 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='47 Env 2 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='13 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='21 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='34 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='01 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='50 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='52 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='87 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='26 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='87 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='73 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='89 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='68 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='39 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='71 Env 3 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='38 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='84 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='80 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='43 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='52 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='66 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='73 ± 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='36 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='74 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='51 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='16 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='65 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='25 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='88 Env 4 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='36 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='72 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='48 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='95 23.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='86 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='47 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='84 ± 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='06 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='07 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='94 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='37 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='37 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='66 ± 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='75 Table 5: Fruitbot final average performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The Oracle is not considered for the purposes of highlighting the average performance, as it is an idealized model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 2 MODULE EXPERIMENTS Here we present training SANE using only 2 modules (each with 2e5 replay buffer entries) on Climber.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The intent of this experiment is to look at how having fewer modules than tasks impacts performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We can see that while peak performance is generally higher, recall is poor overall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Perhaps unsurprisingly, the curve resembles that of CLEAR: reaching similar maximum values, and observing the same forgetting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE 2 Modules ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE Oracle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='CLEAR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE 2 Modules ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE Oracle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='CLEAR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE 2 Modules ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE Oracle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='CLEAR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='20M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='30M ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE 2 Modules ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='SANE Oracle ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='CLEAR ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Climber: Env 3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Step ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Expected Return ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='Figure 14: Comparison for using SANE with 2 modules to other SANE variants and CLEAR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='7 EXTENDED BACKGROUND In the standard, single-task reinforcement learning scenario, we consider the task T as a discrete-time Markov Decision Process (MDP) consisting of a tuple ⟨S, A, T, r, γ, ρ0, ⟩, with state space S, action space A, state transition probability function T, reward function r, discount factor γ, and probability distribution ρ0 on the initial states S0 ⊂ S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' The goal is to learn a policy π(a|s) which maximizes the expected return, where the return Rt of a state s at timestep t is the discounted sum of rewards over an infinite-horizon trajectory from state s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' For continual RL (Kirkpatrick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Schwarz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Rolnick et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=', 2019), we extend this setting by considering a sequence of N tasks, SN := (T1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' TN), presented to the agent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' This induces a non-stationary learning process as any component of the MDP may change on task switch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' A capable continual RL agent should continue to learn new skills (maintain plasticity), recall prior learned behavior (mitigate catastrophic forgetting), and transfer old abilities to new domains (demonstrate forward transfer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We assume the agent trains on task Ti at timesteps in the interval [Ai, Bi), for ki = Bi − Ai timesteps.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Here Ai and Bi are the task boundaries denoting the start and end, respectively, of task Ti.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Additionally, we may cycle through the tasks M times, which yields full task sequence SNM that has length N · M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='8 ANALYSIS OF ESTIMATED RETURNS In Figure 15 we present SANE’s estimated vtrace return in comparison to the observed vtrace return for the training periods for each task for a single run.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' There is a bit of overestimation in Env 0, more significant overestimation in Env 1, but almost none in Envs 2 and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Env 1 is also where we see the most uncertainty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Note that the first time a transition to a new task occurs, the estimated return is near zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' We believe these estimates to be close enough that using our estimated vtrace returns is preferred over using a history of recent returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' An example of when this is the case is if every episode is different;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' then any window size to average over poses problems: the best module will not be activated and creation will not occur properly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Furthermore, we hope 21 Published at 1st Conference on Lifelong Learning Agents, 2022 to extend SANE such that activation occurs many times during an episode as well, so modules can capture even more fine-grained behaviors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' This fine-grained behavior becomes impossible if we are reliant on final returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Figure 15: Comparison of the true and predicted vtrace returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' Figure 16: Graph of the uncertainty of SANE’s prediction of our vtrace returns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content=' 22 Climber: Env 0 Climber: Eny 1 Climber: Env 2 Climber: Env 3 2 5 2 2 True Return Expected Return Expected Return Return 4 Predicted Return 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 1 00 00 % % 10M 20M 30M 10M 20M 30M 10M 20M 30M 10M 20M 30M Step Step Step StepClimber: Env 0 Climber: Env 1 Climber: Env 2 Climber: Env 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='3 2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='3 Uncertainty Expected Return Expected Return Return I Return 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='2 Expected xpected 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='15 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/SNAyT4oBgHgl3EQfVPeU/content/2301.00141v1.pdf'} +page_content='05 00 % % 00 10M 20M 30M 10M 20M 30M 10M 20M 30M 10M 20M 30M Step Step Step Step' metadata={'source': 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In this paper, we establish continuous bilinear decompositions that arise in the +study of products between elements in martingale Hardy spaces Hp p0 ă p ⩽ 1q and func- +tions in their dual spaces. Our decompositions are based on martingale paraproducts. As a +consequence of our work, we also obtain analogous results for dyadic martingales on spaces of +homogeneous type equipped with a doubling measure. +1. Introduction +The pointwise product of a function in the classical Hardy space H1pRnq and a function of +bounded mean oscillation on Rn need not be in L1pRnq; see e.g. §6.2 in Chapter IV in [24]. +However, using Fefferman’s duality theorem [10] and the fact that the pointwise product of a +BMO-function and a C8 +0 -function is in BMOpRnq, Bonami, Iwaniec, Jones and Zinsmeister +defined in [5] the product f ˆ g of a function f P H1pRnq and a function g P BMOpRnq as a +distribution given by +(1.1) +xf ˆ g, φy :“ xg ¨ φ, fy, +φ P C8 +0 pRnq, +where in the right-hand side of (1.1) the duality between f P H1pRnq and g ¨ φ P BMOpRnq +is employed. Moreover, it is shown in [5] that for any fixed f P H1pRnq there exist two linear +continuous operators Sf from BMOpRnq to L1pRnq and Tf from BMOpRnq to a weighted +Hardy–Orlicz space such that +f ˆ g “ Sfpgq ` Tfpgq +for all g P BMOpRnq; see [5, Theorem 1.6]. +In [4], using wavelet analysis, Bonami, Grellier and Ky showed that there exist two bilinear +continuous operators S from H1pRnq ˆ BMOpRnq to L1pRnq and T from H1pRnq ˆ BMOpRnq +to HlogpRnq such that +f ˆ g “ Spf, gq ` T pf, gq +for all f P H1pRnq and for all g P BMOpRnq; see [4, Theorem 1.1]. The Musielak Hardy–Orlicz +space HlogpRnq is defined as the class consisting of all distributions h on Rn whose grand maximal +function Mh satisfies +ˆ +Rn +|Mhpxq| +logpe ` |x|q ` logpe ` |Mhpxq|qdx ă 8 +and is smaller than the weighted Hardy–Orlicz space appearing in [5]. In fact, as explained in +[5], in view of the results of Nakai and Yabuta [22] on pointwise multipliers of BMOpRnq and +duality, the Musielak Hardy–Orlicz space HlogpRnq is optimal. +2020 Mathematics Subject Classification. Primary: 47A07, 60G42, 60G46. Secondary: 42B30, 46E30, 46F10. +Key words: Paraproducts; Martingales; Hardy–Orlicz spaces; Musielak–Orlicz spaces; Doubling spaces. +O. Bakas is partially supported by the projects CEX2021-001142-S, RYC2018-025477-I, PID2021-122156NB- +I00/AEI/10.13039/501100011033 funded by Agencia Estatal de Investigaci´on and acronym “HAMIP”, Juan de +la Cierva Incorporaci´on IJC2020-043082-I and grant BERC 2022-2025 of the Basque Government. +Y. Zhai acknowledges partial support from ERC project FAnFArE no. 637510 and the region Pays de la Loire. +1 + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +2 +In addition, continuous bilinear decomposition theorems for products of elements in HppRnq, +for 0 ă p ă 1, and their dual spaces were established in [3]. +Using the theory of wavelets on spaces of homogeneous type, which was developed by Auscher +and Hyt¨onen in [1], the aforementioned results have been extended to spaces of homogeneous +type by Liu, Yang and Yuan [15] and Xing, Yang and Liang [29]. More precisely, in [15] and +[29], continuous bilinear decompositions for products between elements in atomic Hardy spaces +Hp +atpΩq (in the sense of Coifman and Weiss [7]) and their dual spaces were established in the case +where p P p +n +n`1, 1s. Here n is defined as the dimension of the homogeneous space Ω. +Recently, in [2], a dyadic variant of the aforementioned results of Bonami, Grellier, and Ky +was established; see [2, Theorem 24], which in turn was used to deduce a periodic version of [4, +Theorem 1.1]; see [2, Theorem 28]. +Motivated by [2], the first part of this article is concerned with the study of multiplication +between Hardy spaces and their dual spaces for martingales on a probability space Ω. More +specifically, we study multiplications between functions in the martingale Hardy space H1pΩq +and its dual space BMOpΩq as stated in our first result, Theorem 1.1. We also investigate the +case 0 ă p ă 1, namely multiplication between elements in HppΩq and their dual spaces, the +so-called martingale Lipschitz spaces Λ1pαpq with αp :“ +1 +p ´ 1, see Theorem 1.2. +Since the +dual space pHppΩqq˚ could be t0u for some irregular martingales, we shall only consider regular +martingales where every σ´algebra Fk in the corresponding filtration is generated by countably +many atoms. +We would like to mention that Yong Jiao, Guangheng Xie, Dachun Yang, and Dejian Zhou +have independently obtained Theorem 1.1, and derived from it interesting applications on the +boundedness of operators involving commutators in [14]. +Theorem 1.1. Let pΩ, F, Pq be a probability space equipped with the filtration tFkukě1. +There exist continuous bilinear operators Π1 : H1pΩq ˆ BMOpΩq Ñ L1pΩq, Π2 : H1pΩq ˆ +BMOpΩq Ñ H1pΩq and Π3 : H1pΩq ˆ BMOpΩq Ñ HΦpΩq such that +f ¨ g “ Π1pf, gq ` Π2pf, gq ` Π3pf, gq +for all f P H1pΩq and g P BMOpΩq, where f ¨ g is in the sense of the pointwise multiplication. +In Theorem 1.1, HΦpΩq is a martingale Hardy–Orlicz space defined in terms of the growth +function Φptq; see Definition 2.15 and (2.3) below. We shall refer to the terms Π2pf, gq and +Π3pf, gq as the martingale paraproducts. +Theorem 1.1 can be regarded as an extension of [2, Theorem 24] to the general case of mar- +tingales. +For 0 ă p ă 1, if f P HppΩq, g P Λ1pαpq and f0 “ g0 “ 0, then their product can be +regarded as a continuous linear functional on L8pΩq X Λ1pαpq. To be more precise, for any +h P L8pΩq X Λ1pαpq, define +xf ˆ g, hy :“ xh ¨ g, fy, +where in the right-hand side the duality between HppΩq and Λ1pαpq is invoked. Note that h ¨ g +belongs to Λ1pαpq since h is a pointwise multiplier on Λ1pαpq (see [21]). +Our following theorem establishes a continuous bilinear decomposition for products between +elements in HppΩq and functions in the dual space Λ1pαpq when 0 ă p ă 1. +Theorem 1.2. Let pΩ, F, Pq be a probability space equipped with the filtration tFkukě1, where +Fk is generated by countably many atoms for any k ě 1. +If HppΩq p0 ă p ă 1q are martingale Hardy spaces, then there exist continuous bilinear +operators Π1 : HppΩq ˆ Λ1pαpq Ñ L1pΩq, Π2 : HppΩq ˆ Λ1pαpq Ñ H1pΩq and Π3 : HppΩq ˆ +Λ1pαpq Ñ HppΩq such that +f ˆ g “ Π1pf, gq ` Π2pf, gq ` Π3pf, gq + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +3 +for all f P HppΩq and g P Λ1pαpq. +In the second part of this paper, we study analogues of Theorems 1.1 and 1.2 for the case of +dyadic martingales on spaces of homogeneous type. Such martingales were first constructed in +[13]. We investigate the corresponding martingale Hardy spaces and extend Mei’s results in [17] +to this general setting. Compared with the probability setting, the case of spaces of homogeneous +type is more difficult to deal with since backward martingales arise, and the underlying measures +on homogeneous spaces may be infinite. +The present paper is organized as follows. In section 2, we set down notation and give some +background on martingale Hardy–Orlicz spaces. In section 3, we prove Theorem 1.1. In section +4, we present a characterization of martingale Lipschitz spaces Λ1pαpq, which is of independent +interest (see Theorem 4.4 and Remark 4.5 below), and then we show Theorem 1.2. The remaining +sections are concerned with spaces of homogeneous type. For the convenience of the reader, in +section 5, we recall some definitions and facts regarding Hardy spaces and Lipschitz spaces on +spaces of homogeneous type in the sense of Coifman and Weiss [7]. In section 6, we give some +new proofs of results in [7] based on martingale methods and the existence of dyadic martingales +on homogeneous spaces. In section 7, we establish analogues of Theorems 1.1 and 1.2 for dyadic +martingales on spaces of homogeneous type; see Theorem 7.7 below. In the last section, we apply +Theorem 7.7 to obtain a decomposition of products of functions in Hardy spaces and their dual +spaces on spaces of homogeneous type. +2. Notation and Background +In this section, we provide some notation and background that will be used in this paper. +2.1. Notation. In several parts of this paper, we consider sums and intersections of quasi- +normed vector spaces. For the convenience of the reader we recall these notions below. +Definition 2.1. Let pX1, } ¨ }X1q, pX2, } ¨ }X2q be two quasi-normed vector spaces and let X be +a topological vector space X such that X1, X2 Ă X continuously. +(1) pX1 X X2, } ¨ }X1Xx2q is the intersection of X1 and X2, where +}x}X1XX2 :“ maxt}x}X1, }x}X2u +for all x P X1 X X2; +(2) pX1 ` X2, } ¨ }X1`X2q is the sum of X1 and X2, where +}x}X1`X2 :“ inft}x1}X1 ` }x2}X2 : x “ x1 ` x2, x1 P X1, x2 P X2u +for all x P X1 ` X2. +For convenience, the sum X1 ` X2 ` ¨ ¨ ¨ ` Xn and the intersection X1 X X2 ` ¨ ¨ ¨ X Xn will +also be denoted by +nř +k“1 +Xk and Şn +k“1 Xk, respectively. +Note that pX1 XX2, }¨}X1XX2q and pX1 `X2, }¨}X1`X2q are both quasi-normed vector spaces. +Moreover, if pX1, } ¨ }X1q and pX2, } ¨ }X2q are Banach spaces, then pX1 X X2, } ¨ }X1XX2q and +pX1 ` X2, } ¨ }X1`X2q are both Banach spaces. +In this article we shall use the following standard notation: A ≲ B (resp. A ≲p B) means +that A ď CB (resp. A ď CpB) for some absolute positive constant C (resp. a positive constant +Cp depending only on a parameter p). If A ≲ B and B ≲ A (resp. A ≲p B and B ≲p A), we +write A « B (resp. A «p B). +Throughout the paper, the terms “homogeneous spaces” and “spaces of homogeneous type” +will be used interchangeably. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +4 +2.2. Musielak–Orlicz-type spaces. We shall first recall some definitions and properties of +Orlicz-type spaces and Musielak–Orlicz-type spaces. In what follows, pΩ, F, µq denotes a σ-finite +measure space. +A function Φ : r0, 8q Ñ r0, 8q is called an Orlicz function if it is strictly positive on p0, 8q, +non-decreasing, unbounded and Φp0q “ 0. A measurable function Ψ : Ω ˆ r0, 8q Ñ r0, 8q is +called a Musielak–Orlicz function if for all x P Ω, Ψpx, ¨q is an Orlicz function. +The Musielak–Orlicz-type space LΨpΩq is the set consisting of all measurable functions f on +Ω such that +ˆ +Ω +Ψpx, λ´1|fpxq|qdµ ă 8 +for some λ ą 0. We equip LΨpΩq with the Luxembourg quasi-norm +}f}LΨpΩq :“ inf +" +λ ą 0 : +ˆ +Ω +Ψpx, λ´1|fpxq|qdµ ⩽ 1 +* +, +f P LΨpΩq. +Let p P R. A Musielak–Orlicz function is said to be of uniformly lower type (respectively, +upper type) p if there exists a positive constant C such that +Ψpx, stq ⩽ CspΨpx, tq +for all x P Ω, t ⩾ 0 and s P p0, 1q (respectively, s P r1, 8q). In particular, if Ψ is of uniformly +lower type p with 0 ă p ă 1 and of uniformly upper type 1 then +(2.1) +Ψpx, ctq «c Ψpx, tq +for all c ą 0. +In the sequel, Ψpx, tq is always assumed to be of uniformly lower type p with 0 ă p ă 1 and +of uniformly upper type 1, and to be continuous in the t variable. For more information on +Musielak–Orlicz spaces, we refer the reader to [4] and [30]. +2.3. Martingales. Let pΩ, F, Pq be a fixed probability space. Given a filtration which consists +of a sequence of σ-algebras +F1 Ă ¨ ¨ ¨ Ă Fk Ă ¨ ¨ ¨ Ă F +such that σpY8 +k“1Fkq “ F, for a random variable f P L1pΩ, F, Pq and k P N`, we set +fk “ E pf | Fkq , +dkf “ fk ´ fk´1, +where we adopt the convention that f0 “ 0. We shall also denote fk by Ekpfq. The sequence +tfkukě0 is called the martingale of f, and tdkfukě1 is called the martingale difference of f. If f +and tfkukě0 are as above, we shall also write f “ tfkukě0. To simplify notation, we write LppΩq +instead of LppΩ, F, Pq, 0 ă p ă 8. +Definition 2.2. If f, tfkukě0 and tdkfukě1 are as above, we define: +(1) the maximal function +f ˚ :“ sup +k⩾0 +|fk|; +(2) the square function +Spfq :“ +˜ 8 +ÿ +k“1 +|dkf|2 +¸ 1 +2 +; +(3) the conditional square function +spfq :“ +˜ 8 +ÿ +k“1 +Ek´1|dkf|2 +¸ 1 +2 +. +There are several types of martingale Hardy spaces, which are defined in terms of maximal +functions, square functions and conditional square functions. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +5 +Definition 2.3. For 1 ď p ă 8, the martingale Hardy spaces hppΩq, HppΩq, Hp +mpΩq are defined +as follows +hppΩq :“ tf P L1pΩq : }f}hp :“ }spfq}p ă 8u, +HppΩq :“ tf P L1pΩq : }f}Hp :“ }Spfq}p ă 8u, +Hp +mpΩq :“ tf P L1pΩq : }f}Hp +m :“ }f ˚}p ă 8u, +respectively. +For 0 ă p ă 1, hppΩq is defined as the completion of the space tf P L1pΩq : }f}hp :“ }spfq}p ă +8u with respect to the norm } ¨ }hp. Similarly, HppΩq is defined as the completion of the space +tf P L1pΩq : }f}Hp :“ }Spfq}p ă 8u with respect to the norm } ¨ }Hp, and Hp +mpΩq is defined as +the completion of the space tf P L1pΩq : }f}Hp +m :“ }f ˚}p ă 8u with respect to the norm } ¨ }Hp +m. +In general, the above three martingale Hardy spaces are different. However, for 1 ď p ă 8, +HppΩq “ Hp +mpΩq (see [6], [9], [26]). +Definition 2.4. (Regular filtration) A filtration is regular if there exists a constant C ą 0 such +that for all k ě 2, Fk P Fk, there exists a Gk P Fk´1 satisfying +Fk Ă Gk, +PpGkq ď C ¨ PpFkq. +In addition, a martingale f “ tfkukě0 with respect to such a regular filtration is called a regular +martingale. +Remark 2.5. Suppose that for a positive random variable f P L1pΩq the corresponding martingale +tfkuk⩾0 is regular. Then for any k ě 2 +fk ⩽ A ¨ fk´1, +where A ą 0 is a constant that depends only on the constant C of Definition 2.4. +See [16] for more information about regular filtrations and martingales. +Remark 2.6. For regular martingale filtrations, HppΩq “ hppΩq “ Hp +mpΩq when 0 ă p ă 8. See +[27], [26] and [16] for more information. +An important aspect of martingale Hardy spaces is that they admit atomic decompositions. +The definition of atoms in the martingale setting is given below. +Definition 2.7. A random variable a : Ω Ñ C is called a martingale simple pp, qq-atom (0 ă +p ď 1, 1 ď q ⩽ 8) if there exist k P N and A P Fk such that +(1) Ekpaq “ 0; +(2) supppaq Ă A; +(3) }a}q ⩽ PpAq +1 +q ´ 1 +p , +where 1 +q :“ 0 when q “ 8 as convention. +Definition 2.8. We define the martingale atomic Hardy spaces Hp,q +at pΩq for 0 ă p ă 1 ⩽ q ⩽ 8 +or p “ 1, 1 ă q ď 8 as follows +Hp,q +at pΩq :“ +# +f “ +8 +ÿ +j“0 +λjaj where aj is a simple pp, qq-atom and +8 +ÿ +j“0 +|λj|p ă 8 ++ +, +where for f P Hp,q +at pΩq +}f}Hp,q +at pΩq :“ inf +# +` 8 +ÿ +j“0 +|λj|p˘ 1 +p : f “ +8 +ÿ +j“0 +λjaj, where aj is a simple pp, qq-atom ++ +. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +6 +It is well-known that hppΩq “ Hp,2 +at pΩq when 0 ă p ď 1 (see [27]). +In particular, if the +martingale filtration is regular, then hppΩq “ Hp,q +at pΩq when 0 ă p ď 1 and 1 ă q ď 8. The +following result is the atomic decomposition of H1pΩq, which follows from the noncommutative +result in [23]. In particular, it reveals the relationship between H1pΩq and h1pΩq and shows that +H1pΩq ‰ h1pΩq for general martingales. +Theorem 2.9. We have H1pΩq “ h1pΩq ` h1 +dpΩq, where h1 +d denotes the diagonal Hardy space of +martingale differences +h1 +dpΩq :“ +# +h P L1pΩq : }h}h1 +dpΩq :“ +8 +ÿ +k“1 +}dkh}1 ă 8 ++ +. +We shall now introduce the martingale BMO and bmo spaces, which are the duals of H1pΩq +and h1pΩq, respectively (see Theorem 2.13 below). +Definition 2.10. Assume f, g P L2pΩq. We say that f is a martingale BMO function if +}f}BMOpΩq :“ sup +n⩾1 +}En|f ´ fn´1|2} +1 +28 ă 8. +We say that g is a martingale bmo function if +}g}bmopΩq :“ sup +n⩾0 +}En|g ´ gn|2} +1 +28 ă 8. +Denote by BMOpΩq and bmopΩq the spaces consisting of all martingale BMO and bmo +functions, respectively. +For regular martingales, BMOpΩq “ bmopΩq. The following result is the so-called martingale +John–Nirenberg inequality and can be found in [12]. +Theorem 2.11. There exists a sufficiently small constant κ ą 0 such that for any f P BMOpΩq +with }f}BMOpΩq ď κ, we have +E +´ +e|f|¯ +⩽ 1. +Remark 2.12. From the martingale John–Nirenberg inequality, we have for any 1 ď p ă 8, +}f}BMOpΩq «p sup +n⩾1 +}En|f ´ fn´1|p} +1 +p +8. +However, the above John–Nirenberg inequality fails for bmopΩq in the general martingale setting. +For the following duality theorem, see [12], [16], [26]. +Theorem 2.13. pH1pΩqq˚ “ BMOpΩq and ph1pΩqq˚ “ bmopΩq. +The following proposition, which can be found in [8] and [12], is a consequence of Theorems +2.9 and 2.13 and it gives a description of the relationship between BMOpΩq and bmopΩq. In +particular, it implies that BMOpΩq ⫋ bmopΩq for general martingales. +Proposition 2.14. Assume f is a martingale BMO function. Then +(2.2) +}f}BMOpΩq « }f}bmopΩq ` sup +kě1 +}dkf}8. +We end this section with the definition of martingale Musielak–Orlicz Hardy spaces and the +generalized H¨older inequality. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +7 +Definition 2.15. The martingale Musielak–Orlicz Hardy space HΨpΩq (where Ψ is described in +Section 2.2) is the space consisting of all martingales f “ tfkukě0 such that the square function +Spfq P LΨpΩq. Moreover, we define the quasi-norm +}f}HΨpΩq :“ }Spfq}LΨpΩq. +If Ψ is replaced by an Orlicz function Φ, the corresponding Hardy–Orlicz space HΦpΩq is defined +in an analogous way. +To obtain the generalized H¨older inequality, we shall introduce a particular Orlicz space LΦpΩq, +where +(2.3) +Φptq :“ +t +logpe ` tq, +t ě 0. +Note that Φ is an Orlicz function of uniformly lower type p p0 ă p ă 1q and upper type 1, which +guarantees that the vector space LΦpΩq is a quasi-normed space. Note that L1pΩq Ă LΦpΩq. +Remark 2.16. It follows from [19] that if f “ tfkukě0 is a regular martingale, then the martin- +gale Hardy–Orlicz space HΦpΩq can also be characterized by martingale maximal functions and +conditional square functions. For any f P HΦpΩq one has +}f}HΦpΩq “ }Spfq}LΦpΩq « }f ˚}LΦpΩq « }spfq}LΦpΩq. +The following lemma is a variant of [5, Proposition 2.1] in the martingale setting. +Lemma 2.17. Assume pΩ, F, Pq is a probability space, f P L1pΩq and g P BMOpΩq. Then +f ¨ g P LΦpΩq and +(2.4) +}f ¨ g}LΦpΩq ≲ }f}1}g}BMOpΩq. +Proof. The proof is similar to the proof of the corresponding Euclidean result and we shall only +outline it here for the convenience of the reader. By [5, Lemma 2.1], one has +(2.5) +st +M ` logpe ` stq ď et´M ` s. +for all M ě 0, s ě 0, t ě 0. +When }f}1 “ 0 or }g}BMOpΩq “ 0, p2.4q trivially holds. +Assume g P BMOpΩq with +}g}BMOpΩq ą 0 and f P L1pΩq with }f}1 ą 0. Let κ be the constant in Theorem 2.11, M “ 0, +t “ +κ|gpxq| +}g}BMOpΩq and s “ |fpxq| +}f}1 . Then by Theorem 2.11 and (2.5), we have +ˆ +Ω +Φ +ˆ +|fpxq ¨ gpxq| +κ´1}f}1}g}BMOpΩq +˙ +dP ď +ˆ +Ω +e +κ|gpxq| +}g}BMOpΩq dP ` +›››› +f +}f}1 +›››› +1 +ď 2. +(2.6) +Hence, from (2.1) we conclude +}f ¨ g}LΦpΩq ≲ κ´1}f}1}g}BMOpΩq, +which completes the proof of the lemma. +□ +We shall refer to (2.4) as the generalized H¨older inequality. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +8 +3. Bilinear decompositions on H1pΩq ˆ BMOpΩq +In this section we prove Theorem 1.1. +Let pΩ, F, Pq be a fixed probability space and let +f P H1pΩq, g P BMOpΩq. If we assume that f and g have finite martingale expansions, then we +may write their pointwise product f ¨ g as follows +(3.1) +f ¨ g “ Π1pf, gq ` Π2pf, gq ` Π3pf, gq, +where +Π1pf, gq :“ +8 +ÿ +k“1 +dkfdkg, +Π2pf, gq :“ +8 +ÿ +k“1 +fk´1dkg +and +Π3pf, gq :“ +8 +ÿ +k“1 +gk´1dkf. +We shall estimate Π1pf, gq, Π2pf, gq, Π3pf, gq separately. To do so, we shall make use of the +atomic decomposition of H1pΩq. It follows from our arguments below that the operators Π1, Π2 +and Π3 are well-defined (in a pointwise sense) on the product space H1pΩq ˆ BMOpΩq. Hence, +the proof of Theorem will follow from the boundedness properties of Π1, Π2 and Π3, (3.1) and +a limiting argument. +In §3.4, we present a direct way to deal with Π3pf, gq, which avoids the use of the atomic +decomposition. +Proof of Theorem 1.1. By Theorem 2.9, there always exist two functions f h and f d such that +f “ f h ` f d, where f h P h1pΩq and f d P h1 +dpΩq. +For any such decomposition of f, since +f h P h1pΩq, there exist tλjuj⩾1 Ă R and simple p1, 2q-atoms +␣ +aj( +j⩾1 such that +(3.2) +f h “ +8 +ÿ +j“1 +λjaj, +}f h}h1pΩq « +8 +ÿ +j“1 +|λj|, +where we assume supppajq Ă Anj and Anj P Fnj with PpAnjq ą 0 for j ě 1. Then +(3.3) +Πipf, gq “ +8 +ÿ +j“1 +λjΠipaj, gq ` Πipf d, gq, +i “ 1, 2, 3. +3.1. Estimates for Π1pf h, gq and Π1pf d, gq. We are going to show that Π1 is a bounded bilinear +operator from H1pΩq ˆ BMOpΩq to L1pΩq. In fact, the boundedness of Π1 follows naturally +from the duality between H1pΩq and BMOpΩq, i.e. Theorem 2.13 (see [12]). For the reader’s +convenience, we give a proof based on atomic decompositions. +We first focus on Π1pf h, gq, which can further be decomposed into atoms as described in (3.2). +It thus suffices to consider +Π1paj, gq “ +8 +ÿ +k“1 +dkajdkg, +which can further be localized as dkaj “ 1Anj dkaj when k ě nj ` 1 since Anj P Fnj, namely +Π1paj, gq “ +8 +ÿ +k“nj`1 +1Anj dkajdkg. +Now, by applying the Cauchy-Schwarz inequality, we derive the estimate + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +9 +}Π1paj, gq}1 “ E +¨ +˝ +ˇˇˇˇˇˇ +8 +ÿ +k“nj`1 +1Anj dkajdkg +ˇˇˇˇˇˇ +˛ +‚ +⩽ +» +–E +¨ +˝ +8 +ÿ +k“nj`1 +|dkaj|2 +˛ +‚ +fi +fl +1 +2 » +–E +¨ +˝ +8 +ÿ +k“nj`1 +1Anj |dkg|2 +˛ +‚ +fi +fl +1 +2 +⩽ }aj}2 +» +–EEnj +¨ +˝ +8 +ÿ +k“nj`1 +1Anj |dkg|2 +˛ +‚ +fi +fl +1 +2 +⩽ PpAnjq´ 1 +2 +» +–E +¨ +˝1Anj Enj +¨ +˝ +8 +ÿ +k“nj`1 +|dkg|2 +˛ +‚ +˛ +‚ +fi +fl +1 +2 +⩽ PpAnjq´ 1 +2 }g}bmopΩqPpAnjq +1 +2 +where the fourth inequalitiy follows from the definition of the atom. Hence, we deduce from the +definition of the bmo´norm that +(3.4) +}Π1paj, gq}1 ď }g}bmopΩq. +By using (3.4) and (3.3), we have by Theorem 2.14 +}Π1pf, gq}1 ⩽ +8 +ÿ +j“1 +|λj|}g}bmopΩq ` +››››› +8 +ÿ +k“1 +dkf ddkg +››››› +1 +≲ }f h}h1}g}bmopΩq ` +ˆ +sup +kě1 +}dkg}8 +˙ ˜ 8 +ÿ +k“1 +}dkf d}1 +¸ +≲ +´ +}f h}h1pΩq ` }f d}hd +1pΩq +¯ +}g}BMOpΩq. +Since the decomposition of f “ f h ` f d is arbitrary, by Theorem 2.9 we conclude +(3.5) +}Π1pf, gq}1 ≲ }f}H1pΩq}g}BMOpΩq. +3.2. Estimates for Π2pf h, gq and Π2pf d, gq. We are going to show that Π2 is a bounded +bilinear operator from H1pΩq ˆ BMOpΩq to H1pΩq. Arguing as in section 3.1, we perform the +localization on each term +Π2paj, gq “ +8 +ÿ +k“1 +aj +k´1dkg “ +8 +ÿ +k“nj`2 +1Anj aj +k´1dkg. +It is easy to verify that +dkpΠ2paj, gqq “ aj +k´1dkg, +k ě nj ` 2 and dkpΠ2paj, gqq “ 0, +1 ⩽ k ⩽ nj ` 1. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +10 +We consider the corresponding square function +S +` +Π2paj, gq +˘ +“ +¨ +˝ +8 +ÿ +k“nj`2 +´ +|aj +k´1|21Anj |dkg|2¯ +˛ +‚ +1 +2 +⩽ |pajq˚| +¨ +˝ +8 +ÿ +k“nj`2 +1Anj +` +|dkg|2˘ +˛ +‚ +1 +2 +. +Then by invoking the Cauchy-Schwarz inequality, we have that +}Π2paj, gq}H1pΩq “ E +“ +S +` +Π2paj, gq +˘‰ +⩽ }pajq˚}2 +» +–E +¨ +˝ +8 +ÿ +k“nj`2 +1Anj p|dkg|2q +˛ +‚ +fi +fl +1 +2 +⩽ }aj}2 +» +–E +¨ +˝1Anj Enj +¨ +˝ +8 +ÿ +k“nj`2 +|dkg|2 +˛ +‚ +˛ +‚ +fi +fl +1 +2 +⩽ PpAnjq´ 1 +2 }g}BMOpΩqPpAnjq +1 +2 +and hence, +(3.6) +}Π2paj, gq}H1pΩq ⩽ }g}BMOpΩq. +Similarly, by Theorem 2.14 +››Π2pf d, gq +›› +H1pΩq “ E +« +S +˜ +Π2p +8 +ÿ +m“1 +dmf d, gq +¸ff +⩽ +8 +ÿ +m“1 +E +“ +S +` +Π2pdmf d, gq +˘‰ +“ +8 +ÿ +m“1 +E +» +–Em +˜ +8 +ÿ +k“m`1 +|dmf d|2|dkg|2 +¸ 1 +2 fi +fl +“ +8 +ÿ +m“1 +E +» +–|dmf d|Em +˜ +8 +ÿ +k“m`1 +|dkg|2 +¸ 1 +2 fi +fl +⩽ +8 +ÿ +m“1 +¨ +˝}dmf d}1 +›››››Em +˜ +8 +ÿ +k“m`1 +|dkg|2 +¸››››› +1 +2 +8 +˛ +‚ +and hence, +(3.7) +››Π2pf d, gq +›› +H1pΩq ⩽ }f d}h1 +dpΩq}g}BMOpΩq. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +11 +By using (3.7), (3.2) and (3.3), we have by Theorem 2.9 +}Π2pf, gq}H1pΩq ď +››Π2pf h, gq +›› +H1pΩq ` +››Π2pf d, gq +›› +H1pΩq +⩽ +8 +ÿ +j“1 +|λj|}g}BMOpΩq ` }f d}hd +1pΩq}g}BMOpΩq +≲ +´ +}f h}h1pΩq ` }f d}h1 +dpΩq +¯ +}g}BMOpΩq. +Since the decomposition of f “ f h ` f d is arbitrary, by Theorem 2.9 we conclude +(3.8) +}Π2pf, gq}H1pΩq ≲ }f}H1pΩq}g}BMOpΩq. +3.3. Estimates for Π3pf h, gq and Π3pf d, gq. We are going to show that Π3 is a bounded bilinear +operator from H1pΩqˆBMOpΩq to HΦpΩq. To this end, we first deal with Π3pf h, gq. Note that +SpΠ3pf h, gqq “ S +˜ 8 +ÿ +k“1 +8 +ÿ +j“1 +λjgk´1dkaj +¸ +ď +8 +ÿ +j“1 +λjS +˜ 8 +ÿ +k“1 +gk´1dkaj +¸ +“ +8 +ÿ +j“1 +λj +¨ +˝ +8 +ÿ +k“nj`1 +|gk´1|2|dkaj|2 +˛ +‚ +1 +2 +ď +8 +ÿ +j“1 +λj +¨ +˝ +8 +ÿ +k“nj`1 +|gk´1 ´ gnj|2|dkaj|2 +˛ +‚ +1 +2 +` +8 +ÿ +j“1 +λj|gnj|Spajq +“: I1 ` I2. +It thus suffices to handle I1 and I2. For I1, we have +EpI1q ⩽ +8 +ÿ +j“1 +|λj|E +¨ +˝ +8 +ÿ +k“nj`1 +1Anj |gk´1 ´ gnj|2|dkaj|2 +˛ +‚ +1 +2 +⩽ +8 +ÿ +j“1 +|λj| +» +—–E +¨ +˝ +8 +ÿ +k“nj`1 +1Anj |gk´1 ´ g|2|dkaj|2 +˛ +‚ +1 +2 +` E +¨ +˝ +8 +ÿ +k“nj`1 +1Anj |g ´ gnj|2|dkaj|2 +˛ +‚ +1 +2 fi +ffifl +⩽ +8 +ÿ +j“1 +|λj| +$ +’ +& +’ +% +PpAnjq +1 +2 +» +–E +¨ +˝ +8 +ÿ +k“nj`1 +|gk´1 ´ g|2|dkaj|2 +˛ +‚ +fi +fl +1 +2 +` E +´ +1Anj |g ´ gnj|Spajq +¯ +, +/ +. +/ +- +⩽ +8 +ÿ +j“1 +|λj| +$ +’ +& +’ +% +PpAnjq +1 +2 +» +–E +¨ +˝ +8 +ÿ +k“nj`1 +|dkaj|2Ekp|gk´1 ´ g|2q +˛ +‚ +fi +fl +1 +2 +` }aj}2PpAnjq +1 +2 }g}BMOpΩq +, +/ +. +/ +- +⩽ 2 +8 +ÿ +j“1 +|λj|PpAnjq +1 +2 }g}BMOpΩq}aj}2 +and so, +(3.9) +EpI1q ≲ }f h}h1pΩq}g}BMOpΩq. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +12 +Next, we obtain an estimate for I2. To this end, notice that +I2 ď +˜ 8 +ÿ +j“1 +1Anj |λj|Spajq +¸ +¨ |g| ` +8 +ÿ +j“1 +|λj|1Anj |gnj ´ g|Spajq +“: I3 ` I4. +Since aj is a simple p1, 2q-atom, we have }1Anj Spajq}1 ď 1 and +››››› +8 +ÿ +j“1 +1Anj |λj|Spajq +››››› +1 +ď +8 +ÿ +j“1 +|λj| ≲ }f h}h1pΩq. +By Lemma 2.17, we have +(3.10) +}I3}LΦpΩq ≲ +››››› +8 +ÿ +j“1 +1Anj |λj|Spajq +››››› +1 +}g}BMOpΩq ≲ }f h}h1pΩq}g}BMOpΩq. +The following estimate is implicit in the proof of (3.9): +(3.11) +EpI4q ⩽ +8 +ÿ +j“1 +|λj|PpAnjq +1 +2 }g}BMOpΩq}aj}2 ≲ }f h}h1pΩq}g}BMOpΩq. +By combining (3.10) and (3.11), we deduce that +(3.12) +}I2}LΦpΩq ≲ }f h}h1pΩq}g}BMOpΩq. +In conclusion, by (3.9) and (3.12) we get +(3.13) +}Π3pf h, gq}HΦpΩq ≲ }f h}h1pΩq}g}BMOpΩq. +It remains to deal with Π3pf d, gq. We have +SpΠ3pf d, gqq “ +˜ 8 +ÿ +k“1 +|gk´1|2|dkf d|2 +¸ 1 +2 +⩽ +8 +ÿ +k“1 +|gk´1||dkf d| +ď +8 +ÿ +k“1 +|gk´1 ´ g||dkf d| ` |g| +˜ 8 +ÿ +k“1 +|dkf d| +¸ +. +By Lemma 2.17, +(3.14) +›››››g +˜ 8 +ÿ +k“1 +|dkf d| +¸››››› +LΦpΩq +≲ +˜ 8 +ÿ +k“1 +}dkf d}1 +¸ +}g}BMOpΩq “ }f d}h1 +dpΩq}g}BMOpΩq. +For the remaining term, we have +E +˜ 8 +ÿ +k“1 +|gk´1 ´ g||dkf d| +¸ +“ E +˜ 8 +ÿ +k“1 +|dkf d|Ek|gk´1 ´ g| +¸ +⩽ }g}BMOpΩq +˜ 8 +ÿ +k“1 +}dkf d}1 +¸ +and so +(3.15) +E +˜ 8 +ÿ +k“1 +|gk´1 ´ g||dkf d| +¸ +⩽ }f d}h1 +dpΩq}g}BMOpΩq. +Hence, by (3.14) and (3.15), we get +(3.16) +}Π3pf d, gq}HΦpΩq ≲ }f d}h1 +dpΩq}g}BMOpΩq. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +13 +By (3.13) and (3.16), we obtain +}Π3pf, gq}HΦpΩq ≲ +´ +}f h}h1pΩq ` }f d}h1 +dpΩq +¯ +}g}BMOpΩq. +Thus we conclude +(3.17) +}Π3pf, gq}HΦpΩq ≲ }f}H1pΩq}g}BMOpΩq. +This completes the proof of Theorem 1.1 +□ +3.4. Another method for handling Π3pf, gq. In this section we present a different method +for dealing with Π3pf, gq, which is much neater and simpler than the one presented above, and +it relies on the following theorem which has been shown in [12]. +Theorem 3.1. If g P BMOpΩq and g0 “ 0, then pg˚q0 ≲ }g}BMOpΩq and g˚ P BMOpΩq. +Moreover, }g˚}BMOpΩq ≲ }g}BMOpΩq. +We begin with a pointwise estimate for SpΠ3pf, gqq. Towards this aim, note that dkpΠ3pf, gqq “ +gk´1dkf, which implies that +SpΠ3pf, gqq “ +˜ 8 +ÿ +k“1 +|gk´1|2|dkf|2 +¸ 1 +2 +⩽ |g˚|Spfq ⩽ J1 ` J2, +where +J1 :“ |g˚ ´ pg˚q0|Spfq +and +J2 :“ Spfq}g}BMOpΩq. +Clearly, +(3.18) +}J2}1 ≲ }f}H1pΩq}g}BMOpΩq. +By Theorem 3.1, we get g˚ P BMOpΩq, and hence by Lemma 2.17 +(3.19) +}J1}LΦpΩq ≲ }g˚}BMOpΩq}Spfq}1 ≲ }f}H1pΩq}g}BMOpΩq. +As SpΠ3pf, gqq ď J1 ` J2, by combining (3.18) with (3.19), and by the fact L1pΩq Ă LΦpΩq, we +conclude +}Π3pf, gq}HΦpΩq “ }SpΠ3pf, gqq}LΦpΩq ≲ }f}H1pΩq}g}BMOpΩq, +as desired. +We would like to end this section with the comparison between our proof and the one provided +in [14]. +Though both arguments heavily rely on the atomic decomposition of H1pΩq, they +further use weak atom decomposition for the diagonal Hardy space while our proof proceeds +more directly. Moreover, the treatment of the most technical term Π3 is significantly simplified +in this section thanks to Theorem 3.1. +4. Bilinear decompositions on HppΩq ˆ Λ1pαpq for 0 ă p ă 1 +In this section, we give a proof of Theorem 1.2. Arguing as in the proof of Theorem 1.1, it +suffices to establish appropriate estimates for the bilinear operators Π1, Π2 and Π3. +Let pΩ, F, Pq be a fixed probability space. If we consider the filtration F0 “ tH, Ωu and +Fk “ F for all k ⩾ 1, then HppΩq “ LppΩq for 0 ă p ă 8. It is well-known that pLppΩqq˚ ‰ t0u +if and only if the probability space pΩ, F, Pq contains at least one atom with non-zero measure +when 0 ă p ă 1. This means that pHppΩqq˚ “ t0u may occur. Therefore, we are only concerned +with regular martingales where every Fk is generated by countably many atoms. +To prove Theorem 1.2, we start with the following lemma, which holds for general martingales +that are not necessarily regular. This shall be familiar to the experts in the area, but we will +enclose the proof here for the sake of completeness. +Lemma 4.1. For any 0 ă p ă 1, we have L1pΩq Ă Hp +mpΩq. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +14 +Proof. By Doob’s maximal inequality, for any f P L1pΩq and for any λ ą 0 we have +(4.1) +Ppf ˚ ą λq ⩽ 1 +λ +ˆ +tf ˚ąλu +|f|dP. +Without loss of generality, we may assume }f}1 ⩽ 1. Then +}f ˚}p +p “ +ˆ +Ω +|f ˚|pdP “ p +ˆ 8 +0 +Pp|f ˚| ą λqλp´1dλ +“ p +ˆ 1 +0 +Ppf ˚ ą λqλp´1dλ ` p +ˆ 8 +1 +Ppf ˚ ą λqλp´1dλ +⩽ p +ˆ 1 +0 +λp´1dλ ` p +ˆ 8 +1 +1 +λ +˜ˆ +tf ˚ąλu +|f|dP +¸ +λp´1dλ +“ 1 ` p +ˆ +tf ˚ą1u +|f| +˜ˆ f ˚ +1 +λp´2dλ +¸ +dP +“ 1 ` +p +1 ´ p +ˆ +tf ˚ą1u +|f| +` +1 ´ |f ˚|p´1˘ +dP +⩽ 1 ` +p +1 ´ p +ˆ +tf ˚ą1u +|f|dP ⩽ +1 +1 ´ p. +This implies that for any f P L1pΩq +}f}Hp +mpΩq ⩽ +` +1 +1 ´ p +˘ 1 +p }f}1, +which yields the desired result. +□ +For regular martingales, we have L1pΩq Ă Hp +mpΩq “ HppΩq “ hppΩq. In what follows, the +martingales are always assumed to be regular and every Fk is generated by countable atoms. +Corollary 4.2. For 0 ă p ă 1 and 1 ď q ď 8, HppΩq “ Hp,q +at pΩq. +Proof. By considering the aforementioned atomic decomposition of HppΩq and Definition 2.8, +we have HppΩq “ Hp,8 +at pΩq. It is easy to see Hp,8 +at pΩq Ă Hp,q +at pΩq Ă Hp,1 +at pΩq. It thus suffices to +show that Hp,1 +at pΩq Ă HppΩq. By Lemma 4.1, if a is a simple pp, 1q-atom, then +}a}HppΩq ≲p }a}1, +which implies that a P HppΩq. Hence, Hp,1 +at pΩq Ă HppΩq and so, HppΩq “ Hp,q +at pΩq. +□ +4.1. Characterization of martingale Lipschitz spaces. In this subsection, we give a char- +acterization of martingale Lipschitz spaces that appears to be new and useful in our argument. +We shall first recall the definition of martingale Lipschitz spaces. For 0 ă p ă 1 define +(4.2) +Λqpαpq :“ +# +f P L2pΩq : }f}Λqpαpq “ sup +ně0 +sup +APFn +PpAq´ 1 +q ´αp +ˆˆ +A +|f ´ fn|qdP +˙ 1 +q +ă 8 ++ +, +where q “ 1 or q “ 2, αp :“ 1 +p ´ 1 ą 0. +In [27], Weisz showed that pHppΩqq˚ “ Λ1pαpq and Λ1pαpq “ Λ2pαpq. +Corollary 4.3. For any g P Λ1pαpq, we have }g ´ g0}8 ≲p }g}Λ1pαpq. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +15 +Proof. By duality and Lemma 4.1, for any f P L2pΩq, +|E +` +fpg ´ g0q +˘ +| “ +ˇˇE pgpf ´ f0qq +ˇˇ ≲p }f}Hp}g}Λ1pαpq ≲p }f}1}g}Λ1pαpq. +The above estimate together with the fact +` +L1` +Ωqq˚ “ L8pΩq yields +}g ´ g0}8 ≲p }g}Λ1pαpq, +which finishes the proof. +□ +By virtue of Corollary 4.3, we have the following property of martingale Lipschitz spaces. +Theorem 4.4. If g P Λ1pαpq, we have }1A ¨ |g ´ gn|}8 ≲p PpAqαp}g}Λ1pαpq, for any n P N and +any A P Fn. +Proof. Note that when PpAq “ 0, the desired result holds trivially. Fix n P N and A P Fn with +PpAq ‰ 0. For k ě 0, let FA +k :“ tB P Fk`n : B Ď Au. Note that the union FA of all FA +k is +exactly tB P F|B Ă Au. Hence, if we define +PApBq :“ PpBq +PpAq +pB P FAq +then pA, FA, PAq is a probability space. Note that for any g P L1pA, FA, PAq one has +Epg|FA +k q “ 1A ¨ Epg|Fk`nq. +Denote Ep¨|FA +k q by EA +k . It is easy to verify tEA +k pgqukě0 is also a regular martingale on pA, FA, PAq. +If g P Λ1pαpq, then for B P FA +k with PpBq ‰ 0, +PApBq´1´αp +ˆˆ +B +|g ´ EA +k pgq|dPA +˙ +“ PpAqαp +ˆ +PpBq´1´αp +ˆˆ +B +|g ´ gk`n|dP +˙˙ +ď PpAqαp}g}Λ1pαpq +which implies that by Corollary 4.3, +}1A ¨ |g ´ gn|}8 “ }1A ¨ |g ´ EA +0 pgq|}8 ≲p PpAqαp}g}Λ1pαpq. +This completes the proof of the theorem. +□ +Remark 4.5. By Theorem 4.4 and (4.2), we conclude that for g P Λ1pαpq we have the character- +ization +(4.3) +}g}Λ1pαpq «p sup +ně0 +sup +APFn +PpAq´αp}1A ¨ |g ´ gn|}8. +Note that the results in [18] can be deduced from (4.3). +4.2. Proof of Theorem 1.2. As in the proof of Theorem 1.1, we divide the proof into three +parts. Without loss of generality, we may assume that f0 “ g0 “ 0. +4.2.1. Estimates for Π1pf, gq and Π3pf, gq. The boundedness of Π1 from HppΩq ˆ Λ1pαpq to +L1pΩq follows directly from the duality between HppΩq and Λ1pαpq, we omit the details. +We shall also prove that Π3 is a bounded bilinear operator from HppΩq ˆ Λ1pαpq to HppΩq. +Note that +(4.4) +SpΠ3pf, gqq2 “ +8 +ÿ +k“1 +|gk´1|2|dkf|2 ⩽ pg˚q2Spfq2. +Hence we conclude from Corollary 4.3 and the L8 boundedness of the maximal function that +(4.5) +}Π3pf, gq}p +HppΩq ≲ }g˚}p +8EpSpfqpq ď }g}p +8}f}p +HppΩq ≲p }f}p +HppΩq}g}p +Λ1pαpq, +as desired. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +16 +4.2.2. Estimates for Π2pf, gq. We will show that Π2 is a bounded bilinear operator from HppΩqˆ +Λ1pαpq to H1pΩq. Note that HppΩq “ hppΩq, and hppΩq admits an atomic decomposition. +Then there exist tλjuj⩾1 Ă R and simple pp, 8q-atoms +␣ +aj( +j⩾1 such that +(4.6) +f “ +8 +ÿ +j“1 +λjaj, +}f}HppΩq «p +˜ 8 +ÿ +j“1 +|λj|p +¸ 1 +p +, +where we assume supppajq Ă Anj and Anj P Fnj with PpAnjq ą 0 for j ě 1. By arguing as in +the corresponding case in the proof of Theorem 1.1, +(4.7) +SpΠ2paj, gqq “ +¨ +˝ +8 +ÿ +k“nj`1 +1Anj |aj +k´1|2|dkg|2 +˛ +‚ +1 +2 +⩽ |pajq˚| +¨ +˝ +8 +ÿ +k“nj`1 +1Anj |dkg|2 +˛ +‚ +1 +2 +. +Hence, +E +“ +SpΠ2paj, gqq +‰ +⩽ }pajq˚}8 +» +—–E +¨ +˝1Anj +8 +ÿ +k“nj`1 +|dkg|2 +˛ +‚ +1 +2 fi +ffifl +⩽ }aj}8 +¨ +˝PpAnjq +¨ +˝E +8 +ÿ +k“nj`1 +|dkg|2 +˛ +‚ +˛ +‚ +1 +2 +⩽ PpAnjq´ 1 +p +´ +PpAnjq}g}2 +Λ2pαpqPpAnjq1`2αp¯ 1 +2 +“ }g}Λ2pαpqPpAnjq´ 1 +p PpAnjq1`αp +⩽ }g}Λ2pαpq ≲p }g}Λ1pαpq, +where the last inequality follows from the condition that αp “ 1 +p ´ 1. As a consequence of the +above estimates, we have that +(4.8) +}Π2pf, gq}p +H1pΩq ⩽ +8 +ÿ +j“1 +|λj|p “ +ESpΠ2paj, gqq +‰p ≲p }f}p +HppΩq}g}p +Λ1pαpq. +This completes the proof of the theorem. +5. Homogeneous spaces +In this section, we introduce some fundamental concepts and important theorems for homo- +geneous spaces, which can be found in [7]. We begin with the definition of homogeneous spaces. +Recall that d is a quasi-metric on Ω if +(1) dpx, xq ě 0, @x P Ω; +(2) dpx, yq “ dpy, xq, @x, y P Ω; +(3) there exists a constant A0 ě 1 such that +(5.1) +dpx, yq ⩽ A0pdpx, zq ` dpz, yqq, +@x, y, z P Ω. +Denote by Bpx, rq :“ ty P Ω : dpy, xq ă ru the open ball centered at x with radius r. In this +paper, all quasi-metric spaces are assumed to have the doubling property: there exists a positive +integer A1 P N such that for every x P Ω and for every r ą 0, the ball Bpx, rq can be covered by +at most A1 balls Bpxi, r +2q for some xi P Ω. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +17 +Definition 5.1. A σ-finite measure space pΩ, F, µq equipped with a quasi-metric d is called a +homogeneous space if µ is a Borel measure of homogeneous type: +(5.2) +0 ă µ pBpx, 2rqq ⩽ 2Cµµ pBpx, rqq ă 8, +@x P Ω, r ą 0, +where the constant Cµ is independent of x and r. +In [7], Coifman and Weiss defined Hardy spaces on homogeneous spaces by regarding their +elements as linear functionals acting on some appropriate quasi-normed spaces. In order to state +the definition of Coifman and Weiss, we need to introduce the notions of atoms, BMO and +Lipschitz spaces on homogeneous spaces. +Definition 5.2. If 0 ă p ⩽ 1 ⩽ q ď 8 and p ă q, we say that a function a is a pp, qq-atom if +(1) supppaq Ă B where B is a ball; +(2) }a}q ⩽ pµpBqq +1 +q ´ 1 +p ; +(3) +´ +Ω adµ “ +´ +B adµ “ 0. +Definition 5.3. A locally integrable function f is called a BMO function if +}f}BMO :“ sup +B +1 +µpBq +ˆ +B +|f ´ fB|dµ ă 8, +where fB :“ +1 +µpBq +´ +B fdµ, and the supremum runs over all balls B. Denote by BMOpµq the +BMO space consisting of all BMO functions. +Definition 5.4. For α ą 0, a locally integrable function l is called a Lipschitz function if +(5.3) +|lpxq ´ lpyq| ⩽ Cα pµpBqqα for any x, y P Ω and any ball B containing x, y. +Moreover, +(5.4) +}l}Lα :“ inftCα : |lpxq ´ lpyq| ⩽ Cα pµpBqqα , @x, y P Bu, +where the infimum runs over all balls B. Denote by Lαpµq the space consisting of all Lipschitz +functions. +It is well-known that each BMO function can be regarded as a continuous linear functional +on the vector space generated by finite linear combinations of p1, qq-atoms for 1 ă q ď 8 (cf. +[7]). Hence we can define the atomic Hardy space H1,q +at pµq p1 ă q ď 8q as follows: +(5.5) +H1,q +at pµq :“ +# +f P pBMOpµqq˚ : f “ +8 +ÿ +j“0 +λjaj, where aj is a p1, qq-atom and +8 +ÿ +j“0 +|λj| ă 8 ++ +endowed with the norm +}f}H1,q +at pµq :“ inf +# 8 +ÿ +j“0 +|λj| : f “ +8 +ÿ +j“0 +λjaj, where aj is a p1, qq-atom ++ +. +Similarly, each Lipschitz function l P Lαppµq can be also regarded as a continuous linear +functional of the vector space generated by finite linear combinations of pp, qq-atoms where 0 ă +p ă 1 ď q ď 8 and αp “ 1 +p ´ 1 (cf. [7]). We define the atomic Hardy spaces Hp,q +at pµq as follows: +(5.6) +Hp,q +at pµq +:“ +# +f P +` +Lαppµq +˘˚ : f “ +8 +ÿ +j“0 +λjaj, where aj is a pp, qq-atom and +8 +ÿ +j“0 +|λj|p ă 8 ++ + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +18 +endowed with the quasi-norm +}f}Hp,q +at pµq :“ inf +$ +& +% +˜ 8 +ÿ +j“0 +|λj|p +¸ 1 +p +: f “ +8 +ÿ +j“0 +λjaj, where aj is a pp, qq-atom +, +. +- . +Although the Hardy spaces vary with p and q according to the above definitions, the following +theorem, which can be found in [7], shows that the Hardy spaces actually depend only on p. +Consequently, this enables us to define the Hardy spaces Hp +atpµq for 0 ă p ⩽ 1 to be any one of +the spaces Hp,q +at pµq for 0 ă p ă q ⩽ 8, 1 ⩽ q ď 8. +Theorem 5.5. Hp,q +at pµq “ Hp,8 +at pµq whenever 0 ă p ď 1 ⩽ q ⩽ 8 and p ă q. +We end this section with the following duality theorem obtained in [7]. +Theorem 5.6. +` +H1 +atpµq +˘˚ “ BMOpµq, and pHp +atpµqq˚ “ Lαppµq for 0 ă p ă 1. +The proofs of Theorem 5.5 and Theorem 5.6 that appeared in [7] are very technical. In the +following sections, by employing martingale methods, we give much simpler proofs of these facts. +Our approach is based on the fact that Hp +atpµq for 0 ă p ď 1 is the finite sum of several dyadic +martingale Hardy spaces. +6. Dyadic systems on homogeneous spaces +In this section, we start with introducing dyadic systems on homogeneous spaces, which first +appeared in the work of Hyt¨onen and Kairema [13]. With the help of these dyadic structures, +we then show that Hp +atpµq is exactly the finite sum of martingale Hardy spaces associated with +some adjacent dyadic martingales, which extends Mei’s result [17] to homogeneous spaces. +The following theorem concerning the existence of dyadic structures is due to Hyt¨onen and +Kairema [13]. +Theorem 6.1. Let Ω denote a homogeneous space. Suppose that the constants 0 ă c0 ⩽ C0 ă 8 +and δ P p0, 1q satisfy +12A3 +0C0δ ⩽ c0, +where A0 is specified in the definition of quasi-metric, see (5.1). +Given a set of reference points tzk +αuα, α P Ak (an index set), for every k P Z, with the +properties that +dpzk +α, zk +βq ⩾ c0δk, pα ‰ βq +min +α dpx, zk +αq ă C0δk, for all x P Ω, +one can construct families of sets ˜Qk +α Ď Qk +α Ď ¯Qk +α, called open, half-open and closed dyadic cubes +respectively, such that: +˜Qk +α and ¯Qk +αare the interior and closure of Qk +α; +(6.1) +if k ⩽ l, then either Ql +β Ď Qk +α or Ql +β X Qk +α “ H; +(6.2) +X “ Ť +α +Qk +α (disjoint union) for all k P Z; +(6.3) +Bpzk +α, c1δkq Ď Qk +α Ď Bpzk +α, C1δkq “: BpQk +αq where c1 “ p3A2 +0q´1c0 and C1 “ 2A0C0; +(6.4) +if k ⩽ l and Ql +β Ď Qk +α then BpQl +βq Ď BpQk +αq. +(6.5) +The open and closed cubes ˜Qk +α and ¯Qk +α depend only on the points zl +β for l ⩾ k. The half-open +cubes Qk +α depend on zl +β for l ⩾ minpk, k0q, where k0 P Z is a preassigned number entering the +construction. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +19 +It is obvious that the construction of the above dyadic systems is not unique, and it depends +on the set of the reference points tzk +αuα. We denote this dyadic system by D “ tQk +αuk,α. Let +Fk “ σptQk +αuαq be the σ-algebra generated by tQk +αuα. Then it is clear that +¨ ¨ ¨ Ă Fk´1 Ă Fk Ă ¨ ¨ ¨ , +which implies that tFkukPZ is a filtration generated by atoms. Let F “ σpYkPZFkq. Note that +each Qk +α is an atom of Fk. +Remark 6.2. The standard dyadic grid on the real line is a dyadic system given by +Fk “ tr2´km, 2´kpm ` 1qq : m P Zu +for all k P Z. +Similarly, an example of a dyadic system on Rn is given by the family of standard dyadic cubes +in Rn. +Recall that, for f P L1 +locpΩ, F, µq, the martingale maximal function, the square function and +the conditional square function of f associated with pFkqkPZ are given by +f ˚ :“ max +kPZ |fk|, +Spfq :“ +˜ÿ +kPZ +|dkf|2 +¸ 1 +2 +and +spfq :“ +˜ÿ +kPZ +Ek´1|dkf|2 +¸ 1 +2 +, +respectively. +Let 0 ă p ď 1. The martingale Hardy space Hp +m,Dpµq is defined as the completion of the space +consisting of all f P L1 +locpΩq such that f ˚ P LppΩq with respect to the quasi-norm }f}Hp +m,Dpµq :“ +}f ˚}p. +We define Hp +Dpµq and hp +Dpµq by the square functions and the conditional square functions +respectively, with the additional assumption that +(6.6) +lim +nÑ´8 +ˆ +Ω +sup +kďn +|fk|pdµ “ 0. +From (6.6), we have +lim +nÑ´8 sup +kďn +|fk| “ 0. +Analogously, define the martingale atomic Hardy spaces Hp,q +at,Dpµq p0 ă p ă 1 ď q ď 8 or p “ +1, 1 ă q ď 8q like Definition 2.8. +In order to show Theorem 5.6, we introduce the dual spaces of these atomic martingale Hardy +spaces. For 0 ă p ă 1, q “ 1 or 2 and αp “ 1 +p ´ 1, define +BMODpµq :“ +# +f P L1 +locpΩ, µq : }f}BMODpµq :“ sup +QPD +1 +µpQq +ˆ +Q +|f ´ fQ|dµ ă 8 ++ +, +ΛD +q pαpq :“ +# +f P L1 +locpΩ, µq : }f}ΛD +q pαpq :“ sup +QPD +µpQq´ 1 +q ´αp +ˆˆ +Q +|f ´ fQ|qdµ +˙ 1 +q +ă 8 ++ +. +The spaces ΛD +q pαpq are called the martingale Lipschitz spaces with respect to D. Note that +ΛD +1 pαpq “ ΛD +2 pαpq. +Arguing as in [27], one can show that +` +H1 +at,Dpµq +˘˚ “ BMODpµq, +and for 0 ă p ă 1, +´ +Hp +at,Dpµq +¯˚ +“ ΛD +q pαpq. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +20 +Remark 6.3. Since every simple pp, qq-atom is locally supported, by Corollary 4.2, we conclude +that for 0 ă p ă 1 ď q ď 8 or p “ 1, 1 ă q ď 8 +Hp,q +at,Dpµq “ Hp,8 +at,Dpµq. +Thus we are only concerned with Hp +at,Dpµq :“ Hp,8 +at,Dpµq. +Proposition 6.4. For 0 ă p ⩽ 1, the martingale Hardy spaces defined above are mutually +equivalent. Namely, Hp +Dpµq “ Hp +m,Dpµq “ hp +Dpµq “ Hp +at,Dpµq. +Proof. Let p P p0, 1s be fixed. First, we show Hp +Dpµq “ Hp +m,Dpµq. Suppose that f P Hp +m,Dpµq. +Then for any n ą 0, by a well-known inequality of Burkholder–Davis–Gundy, +ˆ +Ω +˜ +|f´n|2 ` +nÿ +k“´n`1 +|dkf|2 +¸ p +2 +dµ ≲ +ˆ +Ω +sup +´nďkďn +|fk|pdµ ≲ +ˆ +Ω +pf ˚qpdµ +which yields by letting n Ñ 8 and by Fatou’s lemma +}Spfq}p ≲ }f ˚}p. +Thus Hp +m,Dpµq Ă Hp +Dpµq. +Conversely, if f P Hp +Dpµq, then for n ą 0, +ˆ +Ω +sup +´nďkďn +|fk|pdµ ≲ +ˆ +Ω +˜ +|f´n|2 ` +n +ÿ +k“´n`1 +|dkf|2 +¸ p +2 +dµ, +and hence +(6.7) +ˆ +Ω +sup +´nďkďn +|fk|pdµ ≲ +ˆ +Ω +sup +kď´n +|fk|pdµ ` +ˆ +Ω +|Spfq|pdµ ă 8. +Then by letting n Ñ 8 and applying Fatou’s lemma, we obtain }f ˚}p ă 8 and +}f ˚}p ≲ }Spfq}p. +Therefore, Hp +Dpµq Ă Hp +m,Dpµq and Hp +m,Dpµq “ Hp +Dpµq. +One shows Hp +m,Dpµq “ hp +Dpµq in a completely analogous way. To show hp +Dpµq “ Hp +at,Dpµq, one +can argue by mimicking the corresponding proof in [27] and [26]. We omit the details. +□ +The following theorem can be found in [13] and ensures that there exist enough dyadic cubes +to cover all balls on homogeneous spaces. +Theorem 6.5. Given a set of reference points tzk +αu, k P Z, α P Ak, suppose that there exists +constant δ P p0, 1q that satisfies 96A6 +0δ ⩽ 1. Then there exists a finite collection of families +Dt, t “ 1, 2, ¨ ¨ ¨ , K “ KpA0, A1, δq ă 8, where each Dt is a collection of dyadic cubes, associated +to dyadic points tzk +αu, k P Z, α P Ak, with the properties (6.1)-(6.5) in Theorem 6.1. +In addition, the following property is satisfied: +(6.8) +for every Bpx, rq Ď Ω, there exist t and Q P Dt with Bpx, rq Ď Q and diampQq ⩽ Cr. +The constant C ă 8 in (6.8) only depends on the quasi-metric constant A0 and the parameter +δ. +By virtue of Proposition 6.4 and Theorem 6.5, we have the following theorem, which extends +Mei’s result in [17]. +Theorem 6.6. For 0 ă p ⩽ 1, we have +(6.9) +Hp +atpµq “ +K +ÿ +t“1 +Hp +at,Dtpµq “ +K +ÿ +t“1 +Hp +Dtpµq “ +K +ÿ +t“1 +Hp +m,Dtpµq “ +K +ÿ +t“1 +hp +Dtpµq. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +21 +Proof. Let p P p0, 1s be fixed. In view of Proposition 6.4, it suffices to show Hp +atpµq “ +Kř +t“1 +Hp +at,Dtpµq. +We prove it via comparing the atoms. Let a be a pp, 8q-atom in Hp +atpµq. Then there exists a +ball B such that +supppaq Ă B, }a}8 ⩽ pµpBqq´ 1 +p , +ˆ +B +apxqdµ “ 0. +By Theorem 6.5, there exist t and a cube Q P Dt such that B Ă Q, and µpQq ≲ µpBq. Then +supppaq Ă B Ă Q, }a}8 ⩽ pµpBqq´ 1 +p ≲ pµpQqq´ 1 +p , +ˆ +Q +adµ “ 0, +which implies that a is a constant multiple of a simple pp, 8q-atom in Hp +at,Dtpµq. Thus +(6.10) +Hp +atpµq Ă +K +ÿ +t“1 +Hp +at,Dtpµq. +For any t “ 1, 2, ¨ ¨ ¨ , K and for any given simple pp, 8q-atom b in Hp +at,Dtpµq, there exists +Q P Dt such that +supppbq Ă Q, }b}8 ⩽ pµpQqq´ 1 +p , +ˆ +Q +bdµ “ 0. +By Theorem 6.1, there exists a ball B such that Q Ă B and µpQq ≳ µpBq. Hence +supppbq Ă Q Ă B, }b}8 ⩽ pµpQqq´ 1 +p ≲ pµpBqq´ 1 +p , +ˆ +B +bdµ “ 0, +which implies that a multiple of b is also a pp, 8q-atom in Hp +atpµq, thus +(6.11) +K +ÿ +t“1 +Hp +at,Dtpµq Ă Hp +atpµq. +To complete the proof of the theorem, combine (6.10) and (6.11). +□ +Remark 6.7. Theorem 5.5 follows immediately from Corollary 4.2, Proposition 6.4 and Theorem +6.6, which simplifies the original proof by Coifman and Weiss in [7]. +By duality and Theorem 6.6, we recover the following result of [13], which is an extension of +a result due to Mei [17]: +(6.12) +BMOpµq “ +K +č +t“1 +BMODtpµq. +We will now establish an analogous result for Lαppµq p0 ă p ă 1q. +Theorem 6.8. For 0 ă p ă 1, +Lαppµq “ +K +č +t“1 +ΛDt +2 pαpq. +Proof. By Theorem 6.1, for any Q P Dt (and t “ 1, 2, ¨ ¨ ¨ , K), there exists a ball B such that +Q Ă B and µpBq ≲ µpQq. If f P Lαppµq, then for any x, y P Q, we have +|fpxq ´ fpyq| ď }f}LαppµqµpBqαp ≲ }f}LαppµqµpQqαp. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +22 +We thus have +}f}ΛDt +2 +pαpq ď sup +QPDt pµpQqq´ 1 +2 ´αp +˜ +µpQq´2 +ˆ +Q +ˆˆ +Q +|fpxq ´ fpyq|dµpyq +˙2 +dµpxq +¸ 1 +2 +⩽ sup +QPDt pµpQqq´ 1 +2 ´αp +ˆˆ +Q +}f}2 +Lαppµqµ pQq2αp dµ +˙ 1 +2 +≲ }f}Lαppµq, +which yields +(6.13) +Lαppµq Ă +K +č +t“1 +ΛDt +2 pαpq. +Conversely, let f P +KŞ +t“1 +ΛDt +2 pαpq. For Q P Dt, by Theorem 4.4, +|fpxq ´ fQ| ≲ µpQqαp}f}ΛDt +2 +pαpq +@x P Q, +which implies that for any x, y P Q, +(6.14) +|fpxq ´ fpyq| ≲ µpQqαp}f}ΛDt +2 +pαpq. +For any ball B Ă Ω, by Theorem 6.5, there exist t and Q P Dt such that B Ă Q and +µpQq ≲ µpBq. Then for any x, y P B, by (6.14) +|fpxq ´ fpyq| ≲ µpBqαp}f}ΛDt +2 +pαpq. +Thus +}f}Lαp ≲ +K +ÿ +t“1 +}f}ΛDt +2 +pαpq, +which implies +(6.15) +K +č +t“1 +ΛDt +2 pαpq Ă Lαppµq. +The theorem follows from (6.13) and (6.15). +□ +Remark 6.9. Theorem 6.6 and Theorem 6.8 give a simple proof of Theorem 5.6 originally estab- +lished by Coifman and Weiss [7]: +pHp +atpµqq˚ “ +˜ K +ÿ +t“1 +Hp +at,Dtpµq +¸˚ +“ +K +č +t“1 +pHp +at,Dtpµqq˚ “ +K +č +t“1 +ΛDt +2 pαpq “ Lαppµq. +7. Bilinear decompositions for dyadic martingales on homogeneous spaces +In this section, we focus on bilinear decompositions arising in the study of products between +elements in spaces of dyadic martingales on homogeneous spaces introduced in the previous +section. In the setting of homogeneous spaces, due to their quasi-metrics and measures, the +dyadic martingales behave worse than martingales in probability spaces and the underlying +analysis is more intricate. +In §7.1 we prove appropriate generalized H¨older-type inequalities on homogeneous spaces (see +Lemmas 7.2 and 7.4 below). We then introduce a class of pointwise multipliers of ΛD +1,`pαpq and +BMODpµq; see Theorem 7.5 below. Using Theorem 7.5, we define products between dyadic + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +23 +martingale Hardy spaces on homogeneous spaces and their duals and then, in §7.2 we establish +analogues of the results of Sections 3 and 4 in the setting of homogeneous spaces. +7.1. A generalized H¨older-type inequality. Let 0 ă p ď 1 and D be a dyadic system, +constructed as in Theorem 6.1. The martingale Musielak–Orlicz Hardy spaces HΨp +D pµq consist +of all measurable functions f on pΩ, F, µq such that spfq P LΨppΩq where O P Ω is a fixed point, +and +Ψ1px, tq :“ +t +log pe ` dpx, Oqq ` logpe ` tq, +Ψppx, tq :“ +t +1 ` ttr1 ` µpBpO, dpx, Oqqqsu1´p +p0 ă p ă 1q. +Note that LΨppΩq is a quasi-normed space. +Let M :“ pCµ ` 1q log pe ` dpx, Oqq. By (2.5) we obtain +(7.1) +Ψ1px, stq ≲ pe ` dpx, Oqq´pCµ`1qet ` s ≲ wpxqet ` s, +for all x P Ω, s, t ą 0, +where w : Ω Ñ R` is a weight function with +(7.2) +wpxq ≲ min +! +1, dpx, Oq´pCµ`1q) +. +Let Q0 P F0 be the dyadic cube such that O P Q0. For g P BMODpµq, define +}g}BMOD +`pµq :“ sup +αPA0 +|gQ0α ´ gQ0| +log pe ` dpz0α, Oqq ` |gQ0| ` }g}BMODpµq, +where Q0 +α P F0 is a dyadic cube with its center z0 +α and A0 is the index set in Theorem 6.1. +Denote by BMOD +`pµq the space consisting of all g P BMODpµq such that }g}BMOD +`pµq ă 8. It +is not difficult to verify that } ¨ }BMOD +` pµq is a norm on BMOD +`pµq. +Remark 7.1. If we consider the dyadic martingales on Rn, by taking appropriate cubes Q0 +one shows that if g P BMODpµq, then g P BMOD +`pµq. Note that if g P BMOpµq, then g P +BMOD +`pµq. Moreover, +}g}BMOD +`pµq ≲ }g}BMOpµq ` |gQ0|. +We now introduce the following generalized H¨older inequality for L1pΩ, F, µq and BMOD +`pµq. +Lemma 7.2. If f P L1pΩ, F, µq and g P BMOD +`pµq, then f ¨ g P LΨ1pΩq. Moreover, +(7.3) +}fg}LΨ1pΩq ≲ }f}1}g}BMOD +`pµq. +Proof. Without loss of generality, assume }f}1 ď 1, }g}BMOD +` pµq ď 1 and gQ0 “ 0. It suffices to +show that +ˆ +Ω +Ψ1px, |fpxqgpxq|qdµ ≲ 1. +Let Sk :“ BpO, C0δkqzBpO, C0δk`1q for k ă 0 and S0 :“ BpO, C0q, where δ P p0, 1q is the +constant in Theorem 6.1. Then for each k ď 0, there exists a finite index subset Bk Ă A0 such +that BpO, C0δkq Ă Ť +αPBk +Q0 +α (where Q0 +α P F0) and +ÿ +αPBk +µ +` +Q0 +α +˘ +“ µ +˜ ď +αPBk +Q0 +α +¸ +ď µ +` +BpO, 2A0C0δkq +˘ +≲ δCµk. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +24 +Take s “ ν´1|fpxq|, t “ ν|gpxq| in (7.1), one has +ˆ +Ω +Ψ1px, |fpxqgpxq|qdµ “ +0ÿ +k“´8 +ÿ +αPBk +ˆ +SkXQ0α +Ψ1px, |fpxqgpxq|qdµ +≲ +0ÿ +k“´8 +ÿ +αPBk +ˆ +SkXQ0 +α +wpxqeν|gpxq|dµ ` ν´1}f}1. +Therefore, +(7.4) +ˆ +Ω +Ψ1px, |fpxqgpxq|qdµ ≲ T1 ` ν´1}f}1, +where +T1 :“ +0ÿ +k“´8 +ÿ +αPBk +ˆ +SkXQ0 +α +wpxqeν +ˇˇˇgpxq´gQ0α +ˇˇˇeν +ˇˇˇgQ0α +ˇˇˇdµ. +Let ν :“ mintκ,1u +2 +ą 0 (where κ is defined in Theorem 2.11), by (7.2) and Theorem 4.1, one +has +T1 ≲ +0ÿ +k“´8 +ÿ +αPBk +µpQ0 +αq +` +e ` dpz0 +α, Oq +˘ 1 +2 +δpk`1qpCµ`1q +≲ +0ÿ +k“´8 +ÿ +αPBk +µpQ0 +αqδ +k +2 +δpk`1qpCµ`1q ≲ +0ÿ +k“´8 +δCµkδ +k +2 +δCµk`k +≲ +0ÿ +k“´8 +δ´ 1 +2 k, +and hence +(7.5) +T1 ≲ 1. +Combine (7.4), (7.5) and the fact that ν´1}f}1 ≲ 1, and the proof is complete. +□ +We consider the case 0 ă p ă 1. Define +}g}ΛD +1,`pαpq :“ sup +αPA0 +|gQ0α ´ gQ0| +1 ` µ tB pO, dpz0α, Oqquαp ` |gQ0| ` }g}ΛD +1 pαpq, +Denote by ΛD +1,`pαpq the space consisting of all g P ΛD +1 pαpq such that }g}ΛD +1,`pαpq ă 8. It is easy +to verify that } ¨ }ΛD +1,`pαpq is a norm on ΛD +1,`pαpq. +Remark 7.3. If we consider the dyadic martingales on Rn, by taking appropriate cubes Q0 one +can show that if g P ΛD +1 pαpq, then g P ΛD +1,`pαpq. Note that if g P Lαppµq, then g P ΛD +1,`pαpq. +Moreover, +}g}ΛD +1,`pαpq ≲ }g}Lαppµq ` |gQ0|. +Next we present a generalized H¨older inequality for LppΩ, F, µq and ΛD +1,`pαpq for 0 ă p ă 1. +Lemma 7.4. If f P LppΩ, F, µq and g P ΛD +1,`pαpq for 0 ă p ă 1, then f ¨ g P LΨppµq. Moreover, +(7.6) +}fg}LΨppΩq ≲ }f}p}g}ΛD +1,`pαpq. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +25 +Proof. Without loss of generality, assume }f}p ď 1, }g}ΛD +1,`pαpq ď 1 and gQ0 “ 0. It suffices to +show that +ˆ +Ω +Ψppx, |fpxqgpxq|qdµ ≲ 1. +Take the same family of sets tSkukď0 as above. From Theorem 4.4, we know that for x P Q0 +α, +|gpxq| “ |gpxq ´ gQ0| ⩽ +ˇˇgpxq ´ gQ0α +ˇˇ ` +ˇˇgQ0α ´ gQ0 +ˇˇ +ď +` +µpQ0 +αq +˘αp ` µ +␣ +B +` +O, dpz0 +α, Oq +˘(αp ` 1 +≲ µ +` +BpO, 2A0C0δkq +˘αp ` 1. +Therefore +ˆ +Ω +Ψppx, |fpxqgpxq|qdµ “ +0ÿ +k“´8 +ÿ +αPBk +ˆ +SkXQ0α +|gpxq||fpxq| +1 ` t|gpxq||fpxq| r1 ` µ pBpO, dpx, Oqqsu1´p dµ +≲ +0ÿ +k“´8 +ÿ +αPBk +ˆ +SkXQ0α +|gpxq|p|fpxq|p +1 ` µ pBpO, dpx, Oqq1´p dµ +≲ +0ÿ +k“´8 +ÿ +αPBk +ˆ +SkXQ0α +µ +` +BpO, 2A0C0δkq +˘αpp ` 1 +t1 ` µ pBpO, C0δk`1qu1´p |fpxq|pdµ +≲ 1, +which finishes the proof. +□ +We are now about to present the analogues of the results in Sections 3 and 4 concerning +bilinear decompositions for dyadic martingales on homogeneous spaces. To this end, we need +to define the product between martingale Hardy spaces and their dual spaces first. As in the +probability setting, we regard the product in the sense of distribution as follows: for 0 ă p ă 1, +xf ˆ g, hy :“ xh ¨ g, fy, +f P Hp +at,Dpµq, g P ΛD +1,`pαpq, +where h is a test function such that h ¨ g is in ΛD +1,`pαpq. For p “ 1, we may define the product +between H1 +at,Dpµq and BMODpµq analogously. To this end, we need to introduce some pointwise +multipliers of ΛD +1,`pαpq and BMODpµq. +Denote the space of test functions by Hpαpq p0 ă p ď 1q, and a measurable function h is a +test function if it satisfies the following properties: +(7.7) +|hpxq| ≲ +1 +p1 ` µpBpO, dpx, Oqqqαpq logpe ` dpx, Oqq, +@x P Ω, +and +(7.8) +|hpyq ´ hpzq| ≲ +µpBqαp +p1 ` µrBpO, 1 ` r ` dpcB, Oqqsαpq logpe ` r ` dpcB, Oqq +whenever y, z are both contained in a ball B with center cB and radius r ď dpcB,Oq +2A0 +` 1. +It is obvious that Hpαpq Ă L8pΩq. The following theorem shows that if h P Hpαpq, then h is +a pointwise multiplier of ΛD +1,`pαpq. +Theorem 7.5. For 0 ă p ă 1 and any dyadic system D, Hpαpq is a space of pointwise multipliers +of ΛD +1,`pαpq. For p “ 1, Hp0q is a space of pointwise multipliers of BMOD +`pµq. More precisely, +for any g P ΛD +1,`pαpq and h P Hpαpq, we have +}g ¨ h}ΛD +1,`pαpq ≲ }g}ΛD +1,`pαpq +` +}h}L8pΩq ` 1 +˘ +, + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +26 +and for any g P BMOD +`pµq and h P Hp0q, we have +}g ¨ h}BMOD +` pµq ≲ }g}BMOD +`pµq +` +}h}L8pΩq ` 1 +˘ +. +Proof. First, we consider the case 0 ă p ă 1. +Assume that g P ΛD +1,`pαpq and h P Hpαpq. +According to [20], it suffices to show that +(7.9) +sup +Q +|gQ| +µpQqαp`1 +ˆˆ +Q +|hpxq ´ hQ|dx +˙ +ă 8, +where Q runs over all dyadic cubes in D. +If Q Ă Q0 +β for some β P A0, there exists a collection of cubes Q “ Q0 Ă Q1 Ă ¨ ¨ ¨ Ă QN “ Q0 +β +such that there exists a universal constant 0 ă δ +1 ă 1 with µpQk´1q ď δ +1µpQkq. Hence +|gQ ´ gQ0 +β| ď +N +ÿ +k“1 +|gQk ´ gQk´1| ≲ +N +ÿ +k“1 +µpQkqαp}g}ΛD +1,`pαpq +≲ }g}ΛD +1,`pαpq +N +ÿ +k“1 +ˆ µpQkq +µpQk´1q +tαp´1dt +≲ µpQ0 +βqαp}g}ΛD +1,`pαpq. +Similarly, if Q0 +β Ă Q, we have +|gQ ´ gQ0 +β| ≲ µpQqαp}g}ΛD +1,`pαpq. +By Theorem 6.1, there exists a ball B, with center cB and radius r, such that Q Ă B and +µpBq ≲ µpQq. +If Q0 +β Ă Q and r ą +dpO,cBq +2A0 +` 1, for any x P BpO, rq, we have dpcB, xq ď A0pdpcB, Oq ` +dpO, xqq ă p2A2 +0 ` A0qr. Then µpQq ≳ µpBq ≳ C´p2A2 +0`A0q +µ +µ pBpO, rqq ≳ 1. Similarly, we also +have dpz0 +β, Oq ă p2A2 +0 ` A0qr and µ +! +B +´ +O, dpz0 +β, Oq +¯) +≲ µpBq ≲ µpQq. Thus +|gQ| +µpQqαp`1 +ˆˆ +Q +|hpxq ´ hQ|dx +˙ +≲ +|gQ ´ gQ0 +β| ` |gQ0 +β ´ gQ0| ` |gQ0| +µpQqαp +¨ }h}L8pΩq +≲ +µpQqαp ` µ +! +B +´ +O, dpz0 +β, Oq +¯)αp +` 1 +µpQqαp +¨ }g}ΛD +1,`pαpq}h}L8pΩq +≲ }g}ΛD +1,`pαpq}h}L8pΩq. +If Q0 +β Ă Q and r ď dpO,cBq +2A0 +` 1, for any x P B, we have dpx, Oq ď A0pdpO, cBq ` rq, then +µpQq ≲ µ pBpO, A0pdpO, cBq ` rqq. Thus +|gQ| +µpQqαp`1 +ˆˆ +Q +|hpxq ´ hQ|dx +˙ +≲ +|gQ ´ gQ0 +β| ` |gQ0 +β ´ gQ0| ` |gQ0| +µpQqαp`1 +µpBqαp`1 +p1 ` µrBpO, 1 ` r ` dpO, cBqqsqαp +≲ +´ +µpQqαp ` µ +! +B +´ +O, dpz0 +β, Oq +¯)αp +` 1 +¯ +}g}ΛD +1,`pαpq +p1 ` µrBpO, 1 ` r ` dpO, cBqqsqαp +≲ }g}ΛD +1,`pαpq. +If Q Ă Q0 +β, from Theorem 6.1, we can choose C0 sufficiently small such that C1 “ 2A0C0 ď 1, +then r ď C1 ď dpcB,Oq +2A0 +` 1. For any x P Q0 +β, we have dpO, xq ď A0pdpO, z0 +βq ` C1q. Then +µpQ0 +βq ≲ µ +␣ +B +` +O, A0pdpO, z0 +βq ` C1q +˘( +. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +27 +By a calculation similar to the one presented above, we get the desired result. +Combining the above estimates, we finish our proof for 0 ă p ă 1. The case for p “ 1 is +similar. +□ +Remark 7.6. Note that in Theorem 7.5, the dyadic system D is arbitrary. Then from Theorem +6.8 and (6.12), we conclude that Hpαpq is a space of pointwise multipliers of Lαppµq and Hp0q is +a space of pointwise multipliers of BMOpµq. +7.2. Bilinear decompositions. Assume f P H1 +Dpµq, g P BMOD +`pµq or f P Hp +Dpµq, g P ΛD +1,`pαpq, +0 ă p ă 1. +Denote by Hp +D,finpµq p0 ă p ď 1q the linear space consisting of all functions which can be +written as a finite sum of simple pp, 8q-atoms. Thus if f P Hp +D,finpµq, f is locally supported, +f P L1pΩq X L8pΩq and +´ +Ω fdµ “ 0. Note that Hp +D,finpµq is dense in Hp +Dpµq with respect to the +norm } ¨ }Hp +Dpµq. +In the following, we shall only consider the case where f P Hp +D,finpµq. Then f ¨ g P L1pΩq, and +we can write +(7.10) +f ¨ g “ Π1pf, gq ` Π2pf, gq ` Π3pf, gq, +where +Π1pf, gq :“ +8 +ÿ +k“1 +dkfdkg, +Π2pf, gq :“ +8 +ÿ +k“1 +fk´1dkg +and +Π3pf, gq :“ +8 +ÿ +k“1 +gk´1dkf. +Theorem 7.7. We have the following: +(1) Π1 is a bilinear bounded operator from H1 +Dpµq ˆ BMOD +`pµq to L1pΩq, and +when 0 ă p ă 1, Π1 is a bilinear bounded operator from Hp +Dpµq ˆ ΛD +1,`pαpq to L1pΩq. +(2) Π2 is a bilinear bounded operator from H1 +Dpµq ˆ BMOD +`pµq to H1 +Dpµq, and +when 0 ă p ă 1, Π2 is a bilinear bounded operator from Hp +Dpµq ˆ ΛD +1,`pαpq to H1 +Dpµq. +(3) Π3 is a bilinear bounded operator from H1 +Dpµq ˆ BMOD +`pµq to HΨ1 +D pµq, and +when 0 ă p ă 1, Π3 is a bilinear bounded operator from Hp +Dpµq ˆ ΛD +1,`pαpq to HΨp +D pµq. +Proof. For Π1 and Π2, we can argue as in the corresponding part of the proof of Theorem 1.1. +As for Π3, we can also argue as in the corresponding part of the proof Theorem 1.1, where in the +homogeneous setting one needs to apply Lemma 7.2 and Lemma 7.4. We omit the details. +□ +Remark 7.8. For Π1 and Π2, the condition H1 +Dpµq ˆ BMOD +`pµq and Hp +Dpµq ˆ ΛD +1,`pαpq can be +in fact replaced by H1 +Dpµq ˆ BMODpµq and Hp +Dpµq ˆ ΛD +1 pαpq, respectively. +8. Applications to Homogeneous spaces +In the first part of this section we show that HΨp +D pµq admits an atomic decomposition for +0 ă p ă 1, which allows us to integrate several adjacent dyadic systems on homogeneous spaces. +For a given dyadic system D on Ω, we define the dyadic HΨp +at,D-atom as follows. +Definition 8.1. A measurable function a is said to be an HΨp +at,D-atom if +(i) supppaq Ă Q where Q P D is a cube; +(ii) ´ +Ω adµ “ 0; +(iii) }a}8 ⩽ }1Q}´1 +LΨppΩq. +The atomic dyadic martingale Musielak–Orlicz Hardy spaces HΨp +at,Dpµq p0 ă p ă 1q are defined +in a way analogous to (5.5) and (5.6). We first introduce the space BMOD +Ψppµq, which is a +subspace of continuous linear functionals on finite sums of atoms. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +28 +Definition 8.2. A locally integrable function g is said to be a dyadic BMOD +Ψppµq function +associated with a dyadic system D if +}g}BMOD +Ψppµq :“ sup +kPZ +sup +QPFk +1 +}1Q}LΨppΩq +ˆ +Q +|gpxq ´ gkpxq|dx ă 8. +Then we define the atomic Musielak–Orlicz martingale Hardy spaces HΨp +at,Dpµq as follows: +HΨp +at,Dpµq :“ +# +f P +´ +BMOD +Ψppµq +¯˚ +: f “ +8 +ÿ +i“0 +λiai, where ai is an HΨp +at,Dpµq-atom supported on a cube Qi. ++ +, +where +8 +ÿ +i“0 +ˆ +Qi +Ψppx, |λi|}ai}8qdµ ă 8. +Moreover, +}f}H +Ψp +at,Dpµq :“ inf +# +ρ ą 0 : +8 +ÿ +i“0 +ˆ +Qi +Ψppx, ρ´1|λi|}ai}8qdµ ⩽ 1 ++ +Arguing as in [28], one can show that for 0 ă p ă 1 +(8.1) +HΨp +D pµq “ HΨp +at,Dpµq. +We shall now introduce the atomic Musielak–Orlicz Hardy spaces HΨp +at pµq p0 ă p ď 1q on the +homogeneous space Ω. First, we present the definition of atoms for HΨp +at pµq. +Definition 8.3. A measurable function apxq is said to be an HΨp +at pµq-atom if +(i) supppaq Ă B where B Ă Ω is a ball; +(ii) +´ +Ω adµ “ 0; +(iii) }a}8 ⩽ }1B}´1 +LΨppΩq. +Definition 8.4. A locally integrable function g is said to be a BMOΨppµq function if +}g}BMOΨppµq :“ sup +B +1 +}1B}LΨppΩq +ˆ +B +|gpxq ´ gB|dx ă 8, +where B runs over all balls in Ω. +Definition 8.5. The atomic Musielak–Orlicz Hardy spaces HΨp +at pµq p0 ă p ď 1q are defined as +follows: +HΨp +at pµq :“ +# +f P +` +BMOΨppµq +˘˚ : f “ +8 +ÿ +i“0 +λiai, where ai is an HΨp +at pµq-atom supported on a ball Bi ++ +, +where +8 +ÿ +i“0 +ˆ +Bi +Ψppx, |λi|}ai}8qdµ ă 8. +Moreover, +}f}H +Ψp +at pµq :“ inf +# +ρ ą 0 : +8 +ÿ +i“0 +ˆ +Bi +Ψppx, ρ´1|λi|}ai}8qdµ ⩽ 1 ++ +. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +29 +Let Dt p1 ď t ď Kq be the adjacent systems of Theorem 6.5. By arguing as in the proof of +Theorem 6.6, we have the following: +Lemma 8.6. For 0 ă p ă 1, HΨp +at pµq “ HΨp +at,D1pµq ` HΨp +at,D2pµq ` ¨ ¨ ¨ ` HΨp +at,DKpµq. +Proof. It suffices to show that any dyadic HΨp +Dt -atom a is a constant multiple of an HΨppµq-atom, +and any HΨppµq-atom b is a constant multiple of a dyadic HΨp +Dt -atom. +If B :“ Bpx0, rq, then denote the ball Bpx0, Drq by DB for D ě 1. Denote d :“ dpx0, Oq. +In what follows, CpD, p, A0, Cµq denotes a constant that depends on D, p, A0, Cµ and may differ +from line to line. We first show that if +ˆ +B +1 +1 ` r1 ` µpBpO, dpx, Oqqqs1´p dµpxq “ 1, +then +(8.2) +ˆ +DB +1 +1 ` r1 ` µpBpO, dpx, Oqqqs1´p dµpxq ⩽ CpD, p, A0, Cµq. +Notice that +1 “ +ˆ +B +1 +1 ` r1 ` µpBpO, dpx, Oqqqs1´p dµpxq +⩾ +µpBq +sup +xPB +t1 ` r1 ` µpBpO, dpx, Oqqqs1´pu +⩾ +µpBq +1 ` r1 ` µpBpO, A0pd ` rqqqs1´p , +which implies +µpBq ⩽ 1 ` r1 ` µpBpO, A0pd ` rqqqs1´p. +If d ⩽ 2A0Dr, we have +µpBq ⩽ 1 ` r1 ` µpBpO, A0p2A0D ` 1qrqqs1´p +⩽ 1 ` t1 ` µrBpx0, A0pA0 ` 1qp2A0D ` 1qrqsu1´p +⩽ 1 ` +! +1 ` rA0pA0 ` 1qp2A0D ` 1qsCµ µpBq +)1´p +, +and thus µpBq ⩽ CpD, p, A0, Cµq. +Then +(8.3) +ˆ +DB +1 +1 ` r1 ` µpBpO, dpx, Oqqqs1´p dµpxq ⩽ µpDBq ⩽ DCµµpBq ⩽ CpD, p, A0, Cµq. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +30 +If d ą 2A0Dr, then +ˆ +DB +1 +1 ` r1 ` µpBpO, dpx, Oqqqs1´p dµpxq +⩽ +µpDBq +inf +xPDB t1 ` r1 ` µpBpO, dpx, Oqqqs1´pu +⩽ +DCµµpBq +1 ` r1 ` µpBpO, d{A0 ´ Drqqs1´p , +⩽DCµ t1 ` µrB pO, rA0 ` 1{p2Dqsdqsu1´p ` DCµ +1 ` t1 ` µrBpO, d{p2A0qqsu1´p +⩽ +DCµ +! +1 ` rp2A0 ` 1{DqA0sCµ µrBpO, d{p2A0qqs +)1´p +1 ` t1 ` µrBpO, d{p2A0qqsu1´p +` DCµ. +Hence +(8.4) +ˆ +DB +1 +1 ` r1 ` µpBpO, dpx, Oqqqs1´p dµpxq ⩽ CpD, p, A0, Cµq +Combining (8.3) with (8.4), we get (8.2). +Assume a is an HΨppµq-atom supported on B. By Theorem 6.5, there exist t and a cube +Q P Dt such that B Ă Q and diampQq ⩽ Cr, hence B Ă Q Ă CB. +Note that supppaq Ă Q, +´ +Q apxqdµpxq “ 0 and +}1Q}LΨppµq ⩽ }1CB}LΨppµq ⩽ CpC, p, A0, Cµq}1B}LΨppµq, +which follows from (8.2). Thus +}a}8 ⩽ }1B}´1 +LΨppµq ≲ }1Q}´1 +LΨppµq, +which implies a is a multiple of dyadic HΨp +Dt -atom supported on Q. +For any t “ 1, 2 ¨ ¨ ¨ , K, assume b is a dyadic HΨp +Dt -atom supported on Qk +β. By Theorem 6.1, +there exists two balls such that Bpzk +β, c1δkq Ă Qk +β Ă Bpzk +β, C1δkq. +Thus supppbq Ă Bpzk +β, C1δkq, +´ +Bpzk +β,C1δkq bpxqdµpxq “ 0 and +}1Bpzk +β,C1δkq}LΨppµq ⩽ C +ˆC1 +c1 +, p, A0, Cµ +˙ +}1Bpzk +β,c1δkq}LΨppµq ≲ }1Qk +β}LΨppµq, +which follows from (8.2). Therefore, +}b}8 ⩽ }1Qk +β}´1 +LΨppµq ≲ }1Bpzk +β,C1δkq}´1 +LΨppµq, +which implies b is a multiple of dyadic HΨp-atom supported on Bpzk +β, C1δkq. +□ +Remark 8.7. In [11], Fu, Ma and Yang defined another kind of Musielak–Orlicz Hardy spaces +by grand maximal function and they also proved that these Musielak–Orlicz Hardy spaces are +equivalent to HΨp +at pµq with respect to the corresponding norms when p P p +Cµ +Cµ`1, 1s. +Let B1 :“ BpO, 1q. Define +}g}BMO`pµq :“ |gB1| ` }g}BMOpµq, +for g P BMOpµq, +and +}g}L`,αpµq :“ |gB1| ` }g}Lαppµq, +for g P Lαppµq. +Thus } ¨ }BMO`pµq and } ¨ }L`,αppµq are quasi-norms on BMOpµq and Lαppµq, respectively. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +31 +Theorem 8.8. Let 0 ă p ă 1 and f P Hp +atpµq. There exist three linear continuous operators +Πf +1 : Lαppµq Ñ L1pΩq, Πf +2 : Lαppµq Ñ H1 +atpµq and Πf +3 : Lαppµq Ñ HΨp +at pµq such that +f ¨ g “ Πf +1pgq ` Πf +2pgq ` Πf +3pgq +for all g P Lαppµq, +where Lαppµq is endowed with the quasi-norm } ¨ }L`,αppµq. +Proof. Let f P Hp +atpµq. By Theorem 6.6 there exist f t P Hp +Dtpµq pt “ 1, 2, ¨ ¨ ¨ , Kq such that +f “ f 1 ` f 2 ` ¨ ¨ ¨ ` f K, and +K +ÿ +t“1 +}f t}Hp +Dtpµq « }f}Hp +atpµq. +Define Πf +i pgq :“ +Kř +t“1 +Πipf t, gq for i “ 1, 2, 3 and g P Lαppµq (Πi defined as in Theorem 7.7). Then +f ¨ g “ Πf +1pgq ` Πf +2pgq ` Πf +3pgq. +By Theorem 7.7, Theorem 6.6 and Lemma 8.6, we have +}Πf +1pgq}1 ≲ +K +ÿ +t“1 +}Π1pf t, gq}1 ≲ +K +ÿ +t“1 +}f t}Hp +Dt pµq}g}ΛDt +1,`pαpq ≲ }f}Hp +atpµq}g}L`,αppµq, +}Πf +2pgq}H1 +atpµq ≲ +K +ÿ +t“1 +}Π2pf t, gq}H1 +Dtpµq ≲ +K +ÿ +t“1 +}f t}Hp +Dtpµq}g}ΛDt +1,`pαpq ≲ }f}Hp +atpµq}g}L`,αppµq, +}Πf +3pgq}H +Ψp +at pµq ≲ +K +ÿ +t“1 +}Π3pf t, gq}H +Ψp +Dt pµq ≲ +K +ÿ +t“1 +}f t}Hp +Dtpµq}g}ΛDt +1,`pαpq ≲ }f}Hp +atpµq}g}L`,αppµq. +which finishes the proof. +□ +Remark 8.9. If the homogeneous space pΩ, µq satisfies the reverse doubling condition, then +Lemma 8.6 holds for p “ 1. Then we conclude the following. +Let f P H1 +atpµq. +There exist three linear continuous operators Πf +1 : BMOpµq Ñ L1pΩq, +Πf +2 : BMOpµq Ñ H1 +atpµq and Πf +3 : BMOpµq Ñ HΨ1 +at pµq such that +f ¨ g “ Πf +1pgq ` Πf +2pgq ` Πf +3pgq +for all g P BMOpµq, +where BMOpµq is endowed with the norm } ¨ }BMO`pµq. +Acknowledgments. We thank Professor Quanhua Xu for helpful discussions and suggestions. +O.B. and Y.Z. would like to express their gratitude to Professor Xu for his kind invitation and +hospitality during their visit to Besan¸con in March 2022. +We would also like to thank Yong Jiao, Guangheng Xie, Dachun Yang, Dejian Zhou for +personal communications on their work with us. +References +[1] P. Auscher and T. Hyt¨onen. Orthonormal bases of regular wavelets in spaces of homogeneous type. Appl. +Comput. Harmon. Anal. 34 (2013), no. 2, 266–296; Addendum to Orthonormal bases of regular wavelets in +spaces of homogeneous type. Appl. Comput. Harmon. 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Bilinear decomposition for product of martingale Hardy space H1pΩq +and martingale BMOpΩq. Manuscript. +[15] L. Liu, D. Yang and W. Yuan. Bilinear decompositions for products of Hardy and Lipschitz spaces on spaces +of homogeneous type. Dissertationes Math. 533 (2018), 93 pp. +[16] R. Long. Martingale spaces and inequalities. Peking University Press, Beijing; Friedr. Vieweg & Sohn, +Braunschweig, 1993. iv+346 pp. +[17] T. Mei. BMO is the intersection of two translates of dyadic BMO. C. R. Math. Acad. Sci. Paris, 336 (2003), +no. 12, 1003–1006. +[18] N. G. Meyers. Mean oscillation over cubes and H¨older continuity. Proc. Amer. Math. Soc., 15 (1964), +717–721. +[19] T. Miyamoto, E. Nakai and G. Sadasue. Martingale Orlicz-Hardy spaces. Math. Nachr. 285 (2012), no. 5-6, +670–686. +[20] E. Nakai. Pointwise multipliers on weighted BMO spaces. Studia Math. 125 (1997), no. 1, 35–56. +[21] E. Nakai and G. Sadasue. Pointwise multipliers on martingale Campanato spaces. Studia Math. 220 (2014), +no. 1, 87–100. +[22] E. Nakai and K. Yabuta. Pointwise multipliers for functions of bounded mean oscillation. J. Math. Soc. +Japan 37 (1985), no. 2, 207–218. +[23] M. Perrin. A noncommutative Davis’ decomposition for martingales. J. Lond. Math. Soc. (2) 80 (2009), no. +3, 627–648. +[24] E. M. Stein. Harmonic analysis: real-variable methods, orthogonality, and oscillatory integrals. With the +assistance of Timothy S. Murphy. Princeton Mathematical Series, 43. Monographs in Harmonic Analysis, +III. Princeton University Press, Princeton, NJ, 1993. xiv+695 pp. +[25] B. E. Viviani. An atomic decomposition of the predual of BMOpρq. Rev. Mat. Iberoamericana 3 (1987), +no. 3-4, 401–425 +[26] F. Weisz. Martingale Hardy spaces and their applications in Fourier analysis. Lecture Notes in Mathematics, +1568. Springer-Verlag, Berlin, 1994. viii+218 pp. +[27] F. Weisz. Martingale Hardy spaces for 0 ă p ď 1. Probab. Theory Related Fields 84 (1990), no. 3, 361–376. +[28] G. Xie, Y. Jiao and D. Yang. Martingale Musielak-Orlicz Hardy spaces. Sci. China Math. 62 (2019), no. 8, +1567–1584. +[29] F. Xing, D. Yang and Y. Liang. Products of Functions in BMOpXq and H1 +atpXq via wavelets over spaces +of homogeneous type. J. Fourier Anal. Appl. 23 (2017), no. 4, 919–990. +[30] D. Yang, Y. Liang and L. D. Ky. Real-variable theory of Musielak-Orlicz Hardy spaces. Lecture Notes in +Mathematics, 2182. Springer, Cham, 2017. xiii+466 pp. + +MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS +33 +(O. Bakas) BCAM - Basque Center for Applied Mathematics, 48009 Bilbao, Spain +Email address: obakas@bcamath.org +(Z. Xu) Laboratoire de Math´ematiques, Universit´e de Bourgogne Franche-Comt´e, 25030 Besanc¸on +Cedex, France +Email address: xu.zhendong@univ-fcomte.fr +(Y. Zhai) School of Mathematical and Statistical Sciences, Clemson University, 29634 South Car- +olina, USA +Email address: zhai@clemson.edu +(H. Zhang) Laboratoire de Math´ematiques, Universit´e de Bourgogne Franche-Comt´e, 25030 Besanc¸on +Cedex, France +Email address: hao.zhang@univ-fcomte.fr + diff --git a/ZdFAT4oBgHgl3EQf4B4e/content/tmp_files/load_file.txt b/ZdFAT4oBgHgl3EQf4B4e/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b97f3b496f2d395ad351002ac7f1349e0f0e77cd --- /dev/null +++ b/ZdFAT4oBgHgl3EQf4B4e/content/tmp_files/load_file.txt @@ -0,0 +1,1247 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf,len=1246 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='08723v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='FA] 20 Jan 2023 MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUAL SPACES ODYSSEAS BAKAS, ZHENDONG XU, YUJIA ZHAI, AND HAO ZHANG Abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In this paper, we establish continuous bilinear decompositions that arise in the study of products between elements in martingale Hardy spaces Hp p0 ă p ⩽ 1q and func- tions in their dual spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Our decompositions are based on martingale paraproducts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' As a consequence of our work, we also obtain analogous results for dyadic martingales on spaces of homogeneous type equipped with a doubling measure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Introduction The pointwise product of a function in the classical Hardy space H1pRnq and a function of bounded mean oscillation on Rn need not be in L1pRnq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' §6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2 in Chapter IV in [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' However, using Fefferman’s duality theorem [10] and the fact that the pointwise product of a BMO-function and a C8 0 -function is in BMOpRnq, Bonami, Iwaniec, Jones and Zinsmeister defined in [5] the product f ˆ g of a function f P H1pRnq and a function g P BMOpRnq as a distribution given by (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1) xf ˆ g, φy :“ xg ¨ φ, fy, φ P C8 0 pRnq, where in the right-hand side of (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1) the duality between f P H1pRnq and g ¨ φ P BMOpRnq is employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Moreover, it is shown in [5] that for any fixed f P H1pRnq there exist two linear continuous operators Sf from BMOpRnq to L1pRnq and Tf from BMOpRnq to a weighted Hardy–Orlicz space such that f ˆ g “ Sfpgq ` Tfpgq for all g P BMOpRnq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' see [5, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In [4], using wavelet analysis, Bonami, Grellier and Ky showed that there exist two bilinear continuous operators S from H1pRnq ˆ BMOpRnq to L1pRnq and T from H1pRnq ˆ BMOpRnq to HlogpRnq such that f ˆ g “ Spf, gq ` T pf, gq for all f P H1pRnq and for all g P BMOpRnq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' see [4, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The Musielak Hardy–Orlicz space HlogpRnq is defined as the class consisting of all distributions h on Rn whose grand maximal function Mh satisfies ˆ Rn |Mhpxq| logpe ` |x|q ` logpe ` |Mhpxq|qdx ă 8 and is smaller than the weighted Hardy–Orlicz space appearing in [5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In fact, as explained in [5], in view of the results of Nakai and Yabuta [22] on pointwise multipliers of BMOpRnq and duality, the Musielak Hardy–Orlicz space HlogpRnq is optimal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 2020 Mathematics Subject Classification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Primary: 47A07, 60G42, 60G46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Secondary: 42B30, 46E30, 46F10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Key words: Paraproducts;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Martingales;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Hardy–Orlicz spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Musielak–Orlicz spaces;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Doubling spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Bakas is partially supported by the projects CEX2021-001142-S, RYC2018-025477-I, PID2021-122156NB- I00/AEI/10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='13039/501100011033 funded by Agencia Estatal de Investigaci´on and acronym “HAMIP”, Juan de la Cierva Incorporaci´on IJC2020-043082-I and grant BERC 2022-2025 of the Basque Government.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Zhai acknowledges partial support from ERC project FAnFArE no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 637510 and the region Pays de la Loire.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 1 MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 2 In addition, continuous bilinear decomposition theorems for products of elements in HppRnq, for 0 ă p ă 1, and their dual spaces were established in [3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Using the theory of wavelets on spaces of homogeneous type, which was developed by Auscher and Hyt¨onen in [1], the aforementioned results have been extended to spaces of homogeneous type by Liu, Yang and Yuan [15] and Xing, Yang and Liang [29].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' More precisely, in [15] and [29], continuous bilinear decompositions for products between elements in atomic Hardy spaces Hp atpΩq (in the sense of Coifman and Weiss [7]) and their dual spaces were established in the case where p P p n n`1, 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Here n is defined as the dimension of the homogeneous space Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Recently, in [2], a dyadic variant of the aforementioned results of Bonami, Grellier, and Ky was established;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' see [2, Theorem 24], which in turn was used to deduce a periodic version of [4, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' see [2, Theorem 28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Motivated by [2], the first part of this article is concerned with the study of multiplication between Hardy spaces and their dual spaces for martingales on a probability space Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' More specifically, we study multiplications between functions in the martingale Hardy space H1pΩq and its dual space BMOpΩq as stated in our first result, Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We also investigate the case 0 ă p ă 1, namely multiplication between elements in HppΩq and their dual spaces, the so-called martingale Lipschitz spaces Λ1pαpq with αp :“ 1 p ´ 1, see Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Since the dual space pHppΩqq˚ could be t0u for some irregular martingales, we shall only consider regular martingales where every σ´algebra Fk in the corresponding filtration is generated by countably many atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We would like to mention that Yong Jiao, Guangheng Xie, Dachun Yang, and Dejian Zhou have independently obtained Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, and derived from it interesting applications on the boundedness of operators involving commutators in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let pΩ, F, Pq be a probability space equipped with the filtration tFkukě1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' There exist continuous bilinear operators Π1 : H1pΩq ˆ BMOpΩq Ñ L1pΩq, Π2 : H1pΩq ˆ BMOpΩq Ñ H1pΩq and Π3 : H1pΩq ˆ BMOpΩq Ñ HΦpΩq such that f ¨ g “ Π1pf, gq ` Π2pf, gq ` Π3pf, gq for all f P H1pΩq and g P BMOpΩq, where f ¨ g is in the sense of the pointwise multiplication.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, HΦpΩq is a martingale Hardy–Orlicz space defined in terms of the growth function Φptq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' see Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='15 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3) below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We shall refer to the terms Π2pf, gq and Π3pf, gq as the martingale paraproducts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1 can be regarded as an extension of [2, Theorem 24] to the general case of mar- tingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For 0 ă p ă 1, if f P HppΩq, g P Λ1pαpq and f0 “ g0 “ 0, then their product can be regarded as a continuous linear functional on L8pΩq X Λ1pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' To be more precise, for any h P L8pΩq X Λ1pαpq, define xf ˆ g, hy :“ xh ¨ g, fy, where in the right-hand side the duality between HppΩq and Λ1pαpq is invoked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that h ¨ g belongs to Λ1pαpq since h is a pointwise multiplier on Λ1pαpq (see [21]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Our following theorem establishes a continuous bilinear decomposition for products between elements in HppΩq and functions in the dual space Λ1pαpq when 0 ă p ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let pΩ, F, Pq be a probability space equipped with the filtration tFkukě1, where Fk is generated by countably many atoms for any k ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If HppΩq p0 ă p ă 1q are martingale Hardy spaces, then there exist continuous bilinear operators Π1 : HppΩq ˆ Λ1pαpq Ñ L1pΩq, Π2 : HppΩq ˆ Λ1pαpq Ñ H1pΩq and Π3 : HppΩq ˆ Λ1pαpq Ñ HppΩq such that f ˆ g “ Π1pf, gq ` Π2pf, gq ` Π3pf, gq MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 3 for all f P HppΩq and g P Λ1pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In the second part of this paper, we study analogues of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2 for the case of dyadic martingales on spaces of homogeneous type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Such martingales were first constructed in [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We investigate the corresponding martingale Hardy spaces and extend Mei’s results in [17] to this general setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Compared with the probability setting, the case of spaces of homogeneous type is more difficult to deal with since backward martingales arise, and the underlying measures on homogeneous spaces may be infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The present paper is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In section 2, we set down notation and give some background on martingale Hardy–Orlicz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In section 3, we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In section 4, we present a characterization of martingale Lipschitz spaces Λ1pαpq, which is of independent interest (see Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4 and Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5 below), and then we show Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The remaining sections are concerned with spaces of homogeneous type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For the convenience of the reader, in section 5, we recall some definitions and facts regarding Hardy spaces and Lipschitz spaces on spaces of homogeneous type in the sense of Coifman and Weiss [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In section 6, we give some new proofs of results in [7] based on martingale methods and the existence of dyadic martingales on homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In section 7, we establish analogues of Theorems 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2 for dyadic martingales on spaces of homogeneous type;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' see Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='7 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In the last section, we apply Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='7 to obtain a decomposition of products of functions in Hardy spaces and their dual spaces on spaces of homogeneous type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Notation and Background In this section, we provide some notation and background that will be used in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Notation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In several parts of this paper, we consider sums and intersections of quasi- normed vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For the convenience of the reader we recall these notions below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let pX1, } ¨ }X1q, pX2, } ¨ }X2q be two quasi-normed vector spaces and let X be a topological vector space X such that X1, X2 Ă X continuously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (1) pX1 X X2, } ¨ }X1Xx2q is the intersection of X1 and X2, where }x}X1XX2 :“ maxt}x}X1, }x}X2u for all x P X1 X X2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (2) pX1 ` X2, } ¨ }X1`X2q is the sum of X1 and X2, where }x}X1`X2 :“ inft}x1}X1 ` }x2}X2 : x “ x1 ` x2, x1 P X1, x2 P X2u for all x P X1 ` X2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For convenience, the sum X1 ` X2 ` ¨ ¨ ¨ ` Xn and the intersection X1 X X2 ` ¨ ¨ ¨ X Xn will also be denoted by nř k“1 Xk and Şn k“1 Xk, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that pX1 XX2, }¨}X1XX2q and pX1 `X2, }¨}X1`X2q are both quasi-normed vector spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Moreover, if pX1, } ¨ }X1q and pX2, } ¨ }X2q are Banach spaces, then pX1 X X2, } ¨ }X1XX2q and pX1 ` X2, } ¨ }X1`X2q are both Banach spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In this article we shall use the following standard notation: A ≲ B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' A ≲p B) means that A ď CB (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' A ď CpB) for some absolute positive constant C (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' a positive constant Cp depending only on a parameter p).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If A ≲ B and B ≲ A (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' A ≲p B and B ≲p A), we write A « B (resp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' A «p B).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Throughout the paper, the terms “homogeneous spaces” and “spaces of homogeneous type” will be used interchangeably.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Musielak–Orlicz-type spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We shall first recall some definitions and properties of Orlicz-type spaces and Musielak–Orlicz-type spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In what follows, pΩ, F, µq denotes a σ-finite measure space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' A function Φ : r0, 8q Ñ r0, 8q is called an Orlicz function if it is strictly positive on p0, 8q, non-decreasing, unbounded and Φp0q “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' A measurable function Ψ : Ω ˆ r0, 8q Ñ r0, 8q is called a Musielak–Orlicz function if for all x P Ω, Ψpx, ¨q is an Orlicz function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The Musielak–Orlicz-type space LΨpΩq is the set consisting of all measurable functions f on Ω such that ˆ Ω Ψpx, λ´1|fpxq|qdµ ă 8 for some λ ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We equip LΨpΩq with the Luxembourg quasi-norm }f}LΨpΩq :“ inf " λ ą 0 : ˆ Ω Ψpx, λ´1|fpxq|qdµ ⩽ 1 , f P LΨpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let p P R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' A Musielak–Orlicz function is said to be of uniformly lower type (respectively, upper type) p if there exists a positive constant C such that Ψpx, stq ⩽ CspΨpx, tq for all x P Ω, t ⩾ 0 and s P p0, 1q (respectively, s P r1, 8q).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In particular, if Ψ is of uniformly lower type p with 0 ă p ă 1 and of uniformly upper type 1 then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1) Ψpx, ctq «c Ψpx, tq for all c ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In the sequel, Ψpx, tq is always assumed to be of uniformly lower type p with 0 ă p ă 1 and of uniformly upper type 1, and to be continuous in the t variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For more information on Musielak–Orlicz spaces, we refer the reader to [4] and [30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let pΩ, F, Pq be a fixed probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Given a filtration which consists of a sequence of σ-algebras F1 Ă ¨ ¨ ¨ Ă Fk Ă ¨ ¨ ¨ Ă F such that σpY8 k“1Fkq “ F, for a random variable f P L1pΩ, F, Pq and k P N`, we set fk “ E pf | Fkq , dkf “ fk ´ fk´1, where we adopt the convention that f0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We shall also denote fk by Ekpfq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The sequence tfkukě0 is called the martingale of f, and tdkfukě1 is called the martingale difference of f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If f and tfkukě0 are as above, we shall also write f “ tfkukě0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' To simplify notation, we write LppΩq instead of LppΩ, F, Pq, 0 ă p ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If f, tfkukě0 and tdkfukě1 are as above, we define: (1) the maximal function f ˚ :“ sup k⩾0 |fk|;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (2) the square function Spfq :“ ˜ 8 ÿ k“1 |dkf|2 ¸ 1 2 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (3) the conditional square function spfq :“ ˜ 8 ÿ k“1 Ek´1|dkf|2 ¸ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' There are several types of martingale Hardy spaces, which are defined in terms of maximal functions, square functions and conditional square functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 5 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For 1 ď p ă 8, the martingale Hardy spaces hppΩq, HppΩq, Hp mpΩq are defined as follows hppΩq :“ tf P L1pΩq : }f}hp :“ }spfq}p ă 8u, HppΩq :“ tf P L1pΩq : }f}Hp :“ }Spfq}p ă 8u, Hp mpΩq :“ tf P L1pΩq : }f}Hp m :“ }f ˚}p ă 8u, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For 0 ă p ă 1, hppΩq is defined as the completion of the space tf P L1pΩq : }f}hp :“ }spfq}p ă 8u with respect to the norm } ¨ }hp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Similarly, HppΩq is defined as the completion of the space tf P L1pΩq : }f}Hp :“ }Spfq}p ă 8u with respect to the norm } ¨ }Hp, and Hp mpΩq is defined as the completion of the space tf P L1pΩq : }f}Hp m :“ }f ˚}p ă 8u with respect to the norm } ¨ }Hp m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In general, the above three martingale Hardy spaces are different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' However, for 1 ď p ă 8, HppΩq “ Hp mpΩq (see [6], [9], [26]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (Regular filtration) A filtration is regular if there exists a constant C ą 0 such that for all k ě 2, Fk P Fk, there exists a Gk P Fk´1 satisfying Fk Ă Gk, PpGkq ď C ¨ PpFkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In addition, a martingale f “ tfkukě0 with respect to such a regular filtration is called a regular martingale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Suppose that for a positive random variable f P L1pΩq the corresponding martingale tfkuk⩾0 is regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then for any k ě 2 fk ⩽ A ¨ fk´1, where A ą 0 is a constant that depends only on the constant C of Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' See [16] for more information about regular filtrations and martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For regular martingale filtrations, HppΩq “ hppΩq “ Hp mpΩq when 0 ă p ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' See [27], [26] and [16] for more information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' An important aspect of martingale Hardy spaces is that they admit atomic decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The definition of atoms in the martingale setting is given below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' A random variable a : Ω Ñ C is called a martingale simple pp, qq-atom (0 ă p ď 1, 1 ď q ⩽ 8) if there exist k P N and A P Fk such that (1) Ekpaq “ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (2) supppaq Ă A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (3) }a}q ⩽ PpAq 1 q ´ 1 p , where 1 q :“ 0 when q “ 8 as convention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We define the martingale atomic Hardy spaces Hp,q at pΩq for 0 ă p ă 1 ⩽ q ⩽ 8 or p “ 1, 1 ă q ď 8 as follows Hp,q at pΩq :“ # f “ 8 ÿ j“0 λjaj where aj is a simple pp, qq-atom and 8 ÿ j“0 |λj|p ă 8 + , where for f P Hp,q at pΩq }f}Hp,q at pΩq :“ inf # ` 8 ÿ j“0 |λj|p˘ 1 p : f “ 8 ÿ j“0 λjaj, where aj is a simple pp, qq-atom + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 6 It is well-known that hppΩq “ Hp,2 at pΩq when 0 ă p ď 1 (see [27]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In particular, if the martingale filtration is regular, then hppΩq “ Hp,q at pΩq when 0 ă p ď 1 and 1 ă q ď 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The following result is the atomic decomposition of H1pΩq, which follows from the noncommutative result in [23].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In particular, it reveals the relationship between H1pΩq and h1pΩq and shows that H1pΩq ‰ h1pΩq for general martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We have H1pΩq “ h1pΩq ` h1 dpΩq, where h1 d denotes the diagonal Hardy space of martingale differences h1 dpΩq :“ # h P L1pΩq : }h}h1 dpΩq :“ 8 ÿ k“1 }dkh}1 ă 8 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We shall now introduce the martingale BMO and bmo spaces, which are the duals of H1pΩq and h1pΩq, respectively (see Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='13 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Assume f, g P L2pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We say that f is a martingale BMO function if }f}BMOpΩq :“ sup n⩾1 }En|f ´ fn´1|2} 1 28 ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We say that g is a martingale bmo function if }g}bmopΩq :“ sup n⩾0 }En|g ´ gn|2} 1 28 ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Denote by BMOpΩq and bmopΩq the spaces consisting of all martingale BMO and bmo functions, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For regular martingales, BMOpΩq “ bmopΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The following result is the so-called martingale John–Nirenberg inequality and can be found in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' There exists a sufficiently small constant κ ą 0 such that for any f P BMOpΩq with }f}BMOpΩq ď κ, we have E ´ e|f|¯ ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' From the martingale John–Nirenberg inequality, we have for any 1 ď p ă 8, }f}BMOpΩq «p sup n⩾1 }En|f ´ fn´1|p} 1 p 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' However, the above John–Nirenberg inequality fails for bmopΩq in the general martingale setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For the following duality theorem, see [12], [16], [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' pH1pΩqq˚ “ BMOpΩq and ph1pΩqq˚ “ bmopΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The following proposition, which can be found in [8] and [12], is a consequence of Theorems 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='9 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='13 and it gives a description of the relationship between BMOpΩq and bmopΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In particular, it implies that BMOpΩq ⫋ bmopΩq for general martingales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Assume f is a martingale BMO function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2) }f}BMOpΩq « }f}bmopΩq ` sup kě1 }dkf}8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We end this section with the definition of martingale Musielak–Orlicz Hardy spaces and the generalized H¨older inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 7 Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The martingale Musielak–Orlicz Hardy space HΨpΩq (where Ψ is described in Section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2) is the space consisting of all martingales f “ tfkukě0 such that the square function Spfq P LΨpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Moreover, we define the quasi-norm }f}HΨpΩq :“ }Spfq}LΨpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If Ψ is replaced by an Orlicz function Φ, the corresponding Hardy–Orlicz space HΦpΩq is defined in an analogous way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' To obtain the generalized H¨older inequality, we shall introduce a particular Orlicz space LΦpΩq, where (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3) Φptq :“ t logpe ` tq, t ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that Φ is an Orlicz function of uniformly lower type p p0 ă p ă 1q and upper type 1, which guarantees that the vector space LΦpΩq is a quasi-normed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that L1pΩq Ă LΦpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It follows from [19] that if f “ tfkukě0 is a regular martingale, then the martin- gale Hardy–Orlicz space HΦpΩq can also be characterized by martingale maximal functions and conditional square functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For any f P HΦpΩq one has }f}HΦpΩq “ }Spfq}LΦpΩq « }f ˚}LΦpΩq « }spfq}LΦpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The following lemma is a variant of [5, Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1] in the martingale setting.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Assume pΩ, F, Pq is a probability space, f P L1pΩq and g P BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then f ¨ g P LΦpΩq and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4) }f ¨ g}LΦpΩq ≲ }f}1}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The proof is similar to the proof of the corresponding Euclidean result and we shall only outline it here for the convenience of the reader.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By [5, Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1], one has (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5) st M ` logpe ` stq ď et´M ` s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' for all M ě 0, s ě 0, t ě 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' When }f}1 “ 0 or }g}BMOpΩq “ 0, p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4q trivially holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Assume g P BMOpΩq with }g}BMOpΩq ą 0 and f P L1pΩq with }f}1 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let κ be the constant in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='11, M “ 0, t “ κ|gpxq| }g}BMOpΩq and s “ |fpxq| }f}1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='11 and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5), we have ˆ Ω Φ ˆ |fpxq ¨ gpxq| κ´1}f}1}g}BMOpΩq ˙ dP ď ˆ Ω e κ|gpxq| }g}BMOpΩq dP ` ›››› f }f}1 ›››› 1 ď 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6) Hence, from (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1) we conclude }f ¨ g}LΦpΩq ≲ κ´1}f}1}g}BMOpΩq, which completes the proof of the lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' □ We shall refer to (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4) as the generalized H¨older inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 8 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Bilinear decompositions on H1pΩq ˆ BMOpΩq In this section we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let pΩ, F, Pq be a fixed probability space and let f P H1pΩq, g P BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If we assume that f and g have finite martingale expansions, then we may write their pointwise product f ¨ g as follows (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1) f ¨ g “ Π1pf, gq ` Π2pf, gq ` Π3pf, gq, where Π1pf, gq :“ 8 ÿ k“1 dkfdkg, Π2pf, gq :“ 8 ÿ k“1 fk´1dkg and Π3pf, gq :“ 8 ÿ k“1 gk´1dkf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We shall estimate Π1pf, gq, Π2pf, gq, Π3pf, gq separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' To do so, we shall make use of the atomic decomposition of H1pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It follows from our arguments below that the operators Π1, Π2 and Π3 are well-defined (in a pointwise sense) on the product space H1pΩq ˆ BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Hence, the proof of Theorem will follow from the boundedness properties of Π1, Π2 and Π3, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1) and a limiting argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4, we present a direct way to deal with Π3pf, gq, which avoids the use of the atomic decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='9, there always exist two functions f h and f d such that f “ f h ` f d, where f h P h1pΩq and f d P h1 dpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For any such decomposition of f, since f h P h1pΩq, there exist tλjuj⩾1 Ă R and simple p1, 2q-atoms ␣ aj( j⩾1 such that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2) f h “ 8 ÿ j“1 λjaj, }f h}h1pΩq « 8 ÿ j“1 |λj|, where we assume supppajq Ă Anj and Anj P Fnj with PpAnjq ą 0 for j ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3) Πipf, gq “ 8 ÿ j“1 λjΠipaj, gq ` Πipf d, gq, i “ 1, 2, 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Estimates for Π1pf h, gq and Π1pf d, gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We are going to show that Π1 is a bounded bilinear operator from H1pΩq ˆ BMOpΩq to L1pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In fact, the boundedness of Π1 follows naturally from the duality between H1pΩq and BMOpΩq, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='13 (see [12]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For the reader’s convenience, we give a proof based on atomic decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We first focus on Π1pf h, gq, which can further be decomposed into atoms as described in (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It thus suffices to consider Π1paj, gq “ 8 ÿ k“1 dkajdkg, which can further be localized as dkaj “ 1Anj dkaj when k ě nj ` 1 since Anj P Fnj, namely Π1paj, gq “ 8 ÿ k“nj`1 1Anj dkajdkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Now,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' by applying the Cauchy-Schwarz inequality,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' we derive the estimate MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 9 }Π1paj,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' gq}1 “ E ¨ ˝ ˇˇˇˇˇˇ 8 ÿ k“nj`1 1Anj dkajdkg ˇˇˇˇˇˇ ˛ ‚ ⩽ » –E ¨ ˝ 8 ÿ k“nj`1 |dkaj|2 ˛ ‚ fi fl 1 2 » –E ¨ ˝ 8 ÿ k“nj`1 1Anj |dkg|2 ˛ ‚ fi fl 1 2 ⩽ }aj}2 » –EEnj ¨ ˝ 8 ÿ k“nj`1 1Anj |dkg|2 ˛ ‚ fi fl 1 2 ⩽ PpAnjq´ 1 2 » –E ¨ ˝1Anj Enj ¨ ˝ 8 ÿ k“nj`1 |dkg|2 ˛ ‚ ˛ ‚ fi fl 1 2 ⩽ PpAnjq´ 1 2 }g}bmopΩqPpAnjq 1 2 where the fourth inequalitiy follows from the definition of the atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Hence, we deduce from the definition of the bmo´norm that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4) }Π1paj, gq}1 ď }g}bmopΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3), we have by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='14 }Π1pf, gq}1 ⩽ 8 ÿ j“1 |λj|}g}bmopΩq ` ››››› 8 ÿ k“1 dkf ddkg ››››› 1 ≲ }f h}h1}g}bmopΩq ` ˆ sup kě1 }dkg}8 ˙ ˜ 8 ÿ k“1 }dkf d}1 ¸ ≲ ´ }f h}h1pΩq ` }f d}hd 1pΩq ¯ }g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Since the decomposition of f “ f h ` f d is arbitrary, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='9 we conclude (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5) }Π1pf, gq}1 ≲ }f}H1pΩq}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Estimates for Π2pf h, gq and Π2pf d, gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We are going to show that Π2 is a bounded bilinear operator from H1pΩq ˆ BMOpΩq to H1pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Arguing as in section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, we perform the localization on each term Π2paj, gq “ 8 ÿ k“1 aj k´1dkg “ 8 ÿ k“nj`2 1Anj aj k´1dkg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It is easy to verify that dkpΠ2paj, gqq “ aj k´1dkg, k ě nj ` 2 and dkpΠ2paj, gqq “ 0, 1 ⩽ k ⩽ nj ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 10 We consider the corresponding square function S ` Π2paj, gq ˘ “ ¨ ˝ 8 ÿ k“nj`2 ´ |aj k´1|21Anj |dkg|2¯ ˛ ‚ 1 2 ⩽ |pajq˚| ¨ ˝ 8 ÿ k“nj`2 1Anj ` |dkg|2˘ ˛ ‚ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then by invoking the Cauchy-Schwarz inequality, we have that }Π2paj, gq}H1pΩq “ E “ S ` Π2paj, gq ˘‰ ⩽ }pajq˚}2 » –E ¨ ˝ 8 ÿ k“nj`2 1Anj p|dkg|2q ˛ ‚ fi fl 1 2 ⩽ }aj}2 » –E ¨ ˝1Anj Enj ¨ ˝ 8 ÿ k“nj`2 |dkg|2 ˛ ‚ ˛ ‚ fi fl 1 2 ⩽ PpAnjq´ 1 2 }g}BMOpΩqPpAnjq 1 2 and hence, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6) }Π2paj, gq}H1pΩq ⩽ }g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Similarly, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='14 ››Π2pf d, gq ›› H1pΩq “ E « S ˜ Π2p 8 ÿ m“1 dmf d, gq ¸ff ⩽ 8 ÿ m“1 E “ S ` Π2pdmf d, gq ˘‰ “ 8 ÿ m“1 E » –Em ˜ 8 ÿ k“m`1 |dmf d|2|dkg|2 ¸ 1 2 fi fl “ 8 ÿ m“1 E » –|dmf d|Em ˜ 8 ÿ k“m`1 |dkg|2 ¸ 1 2 fi fl ⩽ 8 ÿ m“1 ¨ ˝}dmf d}1 ›››››Em ˜ 8 ÿ k“m`1 |dkg|2 ¸››››› 1 2 8 ˛ ‚ and hence, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='7) ››Π2pf d, gq ›› H1pΩq ⩽ }f d}h1 dpΩq}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 11 By using (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='7), (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3), we have by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='9 }Π2pf, gq}H1pΩq ď ››Π2pf h, gq ›› H1pΩq ` ››Π2pf d, gq ›› H1pΩq ⩽ 8 ÿ j“1 |λj|}g}BMOpΩq ` }f d}hd 1pΩq}g}BMOpΩq ≲ ´ }f h}h1pΩq ` }f d}h1 dpΩq ¯ }g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Since the decomposition of f “ f h ` f d is arbitrary, by Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='9 we conclude (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='8) }Π2pf, gq}H1pΩq ≲ }f}H1pΩq}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Estimates for Π3pf h, gq and Π3pf d, gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We are going to show that Π3 is a bounded bilinear operator from H1pΩqˆBMOpΩq to HΦpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' To this end, we first deal with Π3pf h, gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that SpΠ3pf h, gqq “ S ˜ 8 ÿ k“1 8 ÿ j“1 λjgk´1dkaj ¸ ď 8 ÿ j“1 λjS ˜ 8 ÿ k“1 gk´1dkaj ¸ “ 8 ÿ j“1 λj ¨ ˝ 8 ÿ k“nj`1 |gk´1|2|dkaj|2 ˛ ‚ 1 2 ď 8 ÿ j“1 λj ¨ ˝ 8 ÿ k“nj`1 |gk´1 ´ gnj|2|dkaj|2 ˛ ‚ 1 2 ` 8 ÿ j“1 λj|gnj|Spajq “: I1 ` I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It thus suffices to handle I1 and I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For I1, we have EpI1q ⩽ 8 ÿ j“1 |λj|E ¨ ˝ 8 ÿ k“nj`1 1Anj |gk´1 ´ gnj|2|dkaj|2 ˛ ‚ 1 2 ⩽ 8 ÿ j“1 |λj| » —–E ¨ ˝ 8 ÿ k“nj`1 1Anj |gk´1 ´ g|2|dkaj|2 ˛ ‚ 1 2 ` E ¨ ˝ 8 ÿ k“nj`1 1Anj |g ´ gnj|2|dkaj|2 ˛ ‚ 1 2 fi ffifl ⩽ 8 ÿ j“1 |λj| $ ’ & ’ % PpAnjq 1 2 » –E ¨ ˝ 8 ÿ k“nj`1 |gk´1 ´ g|2|dkaj|2 ˛ ‚ fi fl 1 2 ` E ´ 1Anj |g ´ gnj|Spajq ¯ , / .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' / ⩽ 8 ÿ j“1 |λj| $ ’ & ’ % PpAnjq 1 2 » –E ¨ ˝ 8 ÿ k“nj`1 |dkaj|2Ekp|gk´1 ´ g|2q ˛ ‚ fi fl 1 2 ` }aj}2PpAnjq 1 2 }g}BMOpΩq , / .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' / ⩽ 2 8 ÿ j“1 |λj|PpAnjq 1 2 }g}BMOpΩq}aj}2 and so, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='9) EpI1q ≲ }f h}h1pΩq}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 12 Next, we obtain an estimate for I2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' To this end, notice that I2 ď ˜ 8 ÿ j“1 1Anj |λj|Spajq ¸ ¨ |g| ` 8 ÿ j“1 |λj|1Anj |gnj ´ g|Spajq “: I3 ` I4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Since aj is a simple p1, 2q-atom, we have }1Anj Spajq}1 ď 1 and ››››› 8 ÿ j“1 1Anj |λj|Spajq ››››› 1 ď 8 ÿ j“1 |λj| ≲ }f h}h1pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='17, we have (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='10) }I3}LΦpΩq ≲ ››››› 8 ÿ j“1 1Anj |λj|Spajq ››››› 1 }g}BMOpΩq ≲ }f h}h1pΩq}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The following estimate is implicit in the proof of (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='9): (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='11) EpI4q ⩽ 8 ÿ j“1 |λj|PpAnjq 1 2 }g}BMOpΩq}aj}2 ≲ }f h}h1pΩq}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='10) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='11), we deduce that (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='12) }I2}LΦpΩq ≲ }f h}h1pΩq}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In conclusion, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='9) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='12) we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='13) }Π3pf h, gq}HΦpΩq ≲ }f h}h1pΩq}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It remains to deal with Π3pf d, gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We have SpΠ3pf d, gqq “ ˜ 8 ÿ k“1 |gk´1|2|dkf d|2 ¸ 1 2 ⩽ 8 ÿ k“1 |gk´1||dkf d| ď 8 ÿ k“1 |gk´1 ´ g||dkf d| ` |g| ˜ 8 ÿ k“1 |dkf d| ¸ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='17, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='14) ›››››g ˜ 8 ÿ k“1 |dkf d| ¸››››› LΦpΩq ≲ ˜ 8 ÿ k“1 }dkf d}1 ¸ }g}BMOpΩq “ }f d}h1 dpΩq}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For the remaining term, we have E ˜ 8 ÿ k“1 |gk´1 ´ g||dkf d| ¸ “ E ˜ 8 ÿ k“1 |dkf d|Ek|gk´1 ´ g| ¸ ⩽ }g}BMOpΩq ˜ 8 ÿ k“1 }dkf d}1 ¸ and so (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='15) E ˜ 8 ÿ k“1 |gk´1 ´ g||dkf d| ¸ ⩽ }f d}h1 dpΩq}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Hence, by (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='14) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='15), we get (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='16) }Π3pf d, gq}HΦpΩq ≲ }f d}h1 dpΩq}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 13 By (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='13) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='16), we obtain }Π3pf, gq}HΦpΩq ≲ ´ }f h}h1pΩq ` }f d}h1 dpΩq ¯ }g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Thus we conclude (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='17) }Π3pf, gq}HΦpΩq ≲ }f}H1pΩq}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' This completes the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1 □ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Another method for handling Π3pf, gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In this section we present a different method for dealing with Π3pf, gq, which is much neater and simpler than the one presented above, and it relies on the following theorem which has been shown in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If g P BMOpΩq and g0 “ 0, then pg˚q0 ≲ }g}BMOpΩq and g˚ P BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Moreover, }g˚}BMOpΩq ≲ }g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We begin with a pointwise estimate for SpΠ3pf, gqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Towards this aim, note that dkpΠ3pf, gqq “ gk´1dkf, which implies that SpΠ3pf, gqq “ ˜ 8 ÿ k“1 |gk´1|2|dkf|2 ¸ 1 2 ⩽ |g˚|Spfq ⩽ J1 ` J2, where J1 :“ |g˚ ´ pg˚q0|Spfq and J2 :“ Spfq}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Clearly, (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='18) }J2}1 ≲ }f}H1pΩq}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, we get g˚ P BMOpΩq, and hence by Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='17 (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='19) }J1}LΦpΩq ≲ }g˚}BMOpΩq}Spfq}1 ≲ }f}H1pΩq}g}BMOpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' As SpΠ3pf, gqq ď J1 ` J2, by combining (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='18) with (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='19), and by the fact L1pΩq Ă LΦpΩq, we conclude }Π3pf, gq}HΦpΩq “ }SpΠ3pf, gqq}LΦpΩq ≲ }f}H1pΩq}g}BMOpΩq, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We would like to end this section with the comparison between our proof and the one provided in [14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Though both arguments heavily rely on the atomic decomposition of H1pΩq, they further use weak atom decomposition for the diagonal Hardy space while our proof proceeds more directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Moreover, the treatment of the most technical term Π3 is significantly simplified in this section thanks to Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Bilinear decompositions on HppΩq ˆ Λ1pαpq for 0 ă p ă 1 In this section, we give a proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Arguing as in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, it suffices to establish appropriate estimates for the bilinear operators Π1, Π2 and Π3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let pΩ, F, Pq be a fixed probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If we consider the filtration F0 “ tH, Ωu and Fk “ F for all k ⩾ 1, then HppΩq “ LppΩq for 0 ă p ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It is well-known that pLppΩqq˚ ‰ t0u if and only if the probability space pΩ, F, Pq contains at least one atom with non-zero measure when 0 ă p ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' This means that pHppΩqq˚ “ t0u may occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Therefore, we are only concerned with regular martingales where every Fk is generated by countably many atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' To prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2, we start with the following lemma, which holds for general martingales that are not necessarily regular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' This shall be familiar to the experts in the area, but we will enclose the proof here for the sake of completeness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For any 0 ă p ă 1, we have L1pΩq Ă Hp mpΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 14 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By Doob’s maximal inequality, for any f P L1pΩq and for any λ ą 0 we have (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1) Ppf ˚ ą λq ⩽ 1 λ ˆ tf ˚ąλu |f|dP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Without loss of generality, we may assume }f}1 ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then }f ˚}p p “ ˆ Ω |f ˚|pdP “ p ˆ 8 0 Pp|f ˚| ą λqλp´1dλ “ p ˆ 1 0 Ppf ˚ ą λqλp´1dλ ` p ˆ 8 1 Ppf ˚ ą λqλp´1dλ ⩽ p ˆ 1 0 λp´1dλ ` p ˆ 8 1 1 λ ˜ˆ tf ˚ąλu |f|dP ¸ λp´1dλ “ 1 ` p ˆ tf ˚ą1u |f| ˜ˆ f ˚ 1 λp´2dλ ¸ dP “ 1 ` p 1 ´ p ˆ tf ˚ą1u |f| ` 1 ´ |f ˚|p´1˘ dP ⩽ 1 ` p 1 ´ p ˆ tf ˚ą1u |f|dP ⩽ 1 1 ´ p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' This implies that for any f P L1pΩq }f}Hp mpΩq ⩽ ` 1 1 ´ p ˘ 1 p }f}1, which yields the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' □ For regular martingales, we have L1pΩq Ă Hp mpΩq “ HppΩq “ hppΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In what follows, the martingales are always assumed to be regular and every Fk is generated by countable atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For 0 ă p ă 1 and 1 ď q ď 8, HppΩq “ Hp,q at pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By considering the aforementioned atomic decomposition of HppΩq and Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='8, we have HppΩq “ Hp,8 at pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It is easy to see Hp,8 at pΩq Ă Hp,q at pΩq Ă Hp,1 at pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It thus suffices to show that Hp,1 at pΩq Ă HppΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, if a is a simple pp, 1q-atom, then }a}HppΩq ≲p }a}1, which implies that a P HppΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Hence, Hp,1 at pΩq Ă HppΩq and so, HppΩq “ Hp,q at pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' □ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Characterization of martingale Lipschitz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In this subsection, we give a char- acterization of martingale Lipschitz spaces that appears to be new and useful in our argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We shall first recall the definition of martingale Lipschitz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For 0 ă p ă 1 define (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2) Λqpαpq :“ # f P L2pΩq : }f}Λqpαpq “ sup ně0 sup APFn PpAq´ 1 q ´αp ˆˆ A |f ´ fn|qdP ˙ 1 q ă 8 + , where q “ 1 or q “ 2, αp :“ 1 p ´ 1 ą 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In [27], Weisz showed that pHppΩqq˚ “ Λ1pαpq and Λ1pαpq “ Λ2pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For any g P Λ1pαpq, we have }g ´ g0}8 ≲p }g}Λ1pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 15 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By duality and Lemma 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, for any f P L2pΩq, |E ` fpg ´ g0q ˘ | “ ˇˇE pgpf ´ f0qq ˇˇ ≲p }f}Hp}g}Λ1pαpq ≲p }f}1}g}Λ1pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The above estimate together with the fact ` L1` Ωqq˚ “ L8pΩq yields }g ´ g0}8 ≲p }g}Λ1pαpq, which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' □ By virtue of Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3, we have the following property of martingale Lipschitz spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If g P Λ1pαpq, we have }1A ¨ |g ´ gn|}8 ≲p PpAqαp}g}Λ1pαpq, for any n P N and any A P Fn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that when PpAq “ 0, the desired result holds trivially.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Fix n P N and A P Fn with PpAq ‰ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For k ě 0, let FA k :“ tB P Fk`n : B Ď Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that the union FA of all FA k is exactly tB P F|B Ă Au.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Hence, if we define PApBq :“ PpBq PpAq pB P FAq then pA, FA, PAq is a probability space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that for any g P L1pA, FA, PAq one has Epg|FA k q “ 1A ¨ Epg|Fk`nq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Denote Ep¨|FA k q by EA k .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It is easy to verify tEA k pgqukě0 is also a regular martingale on pA, FA, PAq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If g P Λ1pαpq, then for B P FA k with PpBq ‰ 0, PApBq´1´αp ˆˆ B |g ´ EA k pgq|dPA ˙ “ PpAqαp ˆ PpBq´1´αp ˆˆ B |g ´ gk`n|dP ˙˙ ď PpAqαp}g}Λ1pαpq which implies that by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3, }1A ¨ |g ´ gn|}8 “ }1A ¨ |g ´ EA 0 pgq|}8 ≲p PpAqαp}g}Λ1pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' □ Remark 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4 and (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2), we conclude that for g P Λ1pαpq we have the character- ization (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3) }g}Λ1pαpq «p sup ně0 sup APFn PpAq´αp}1A ¨ |g ´ gn|}8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that the results in [18] can be deduced from (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' As in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, we divide the proof into three parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Without loss of generality, we may assume that f0 “ g0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Estimates for Π1pf, gq and Π3pf, gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The boundedness of Π1 from HppΩq ˆ Λ1pαpq to L1pΩq follows directly from the duality between HppΩq and Λ1pαpq, we omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We shall also prove that Π3 is a bounded bilinear operator from HppΩq ˆ Λ1pαpq to HppΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4) SpΠ3pf, gqq2 “ 8 ÿ k“1 |gk´1|2|dkf|2 ⩽ pg˚q2Spfq2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Hence we conclude from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3 and the L8 boundedness of the maximal function that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5) }Π3pf, gq}p HppΩq ≲ }g˚}p 8EpSpfqpq ď }g}p 8}f}p HppΩq ≲p }f}p HppΩq}g}p Λ1pαpq, as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 16 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Estimates for Π2pf, gq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We will show that Π2 is a bounded bilinear operator from HppΩqˆ Λ1pαpq to H1pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that HppΩq “ hppΩq, and hppΩq admits an atomic decomposition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then there exist tλjuj⩾1 Ă R and simple pp, 8q-atoms ␣ aj( j⩾1 such that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6) f “ 8 ÿ j“1 λjaj, }f}HppΩq «p ˜ 8 ÿ j“1 |λj|p ¸ 1 p , where we assume supppajq Ă Anj and Anj P Fnj with PpAnjq ą 0 for j ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By arguing as in the corresponding case in the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='7) SpΠ2paj, gqq “ ¨ ˝ 8 ÿ k“nj`1 1Anj |aj k´1|2|dkg|2 ˛ ‚ 1 2 ⩽ |pajq˚| ¨ ˝ 8 ÿ k“nj`1 1Anj |dkg|2 ˛ ‚ 1 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Hence, E “ SpΠ2paj, gqq ‰ ⩽ }pajq˚}8 » —–E ¨ ˝1Anj 8 ÿ k“nj`1 |dkg|2 ˛ ‚ 1 2 fi ffifl ⩽ }aj}8 ¨ ˝PpAnjq ¨ ˝E 8 ÿ k“nj`1 |dkg|2 ˛ ‚ ˛ ‚ 1 2 ⩽ PpAnjq´ 1 p ´ PpAnjq}g}2 Λ2pαpqPpAnjq1`2αp¯ 1 2 “ }g}Λ2pαpqPpAnjq´ 1 p PpAnjq1`αp ⩽ }g}Λ2pαpq ≲p }g}Λ1pαpq, where the last inequality follows from the condition that αp “ 1 p ´ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' As a consequence of the above estimates, we have that (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='8) }Π2pf, gq}p H1pΩq ⩽ 8 ÿ j“1 |λj|p “ ESpΠ2paj, gqq ‰p ≲p }f}p HppΩq}g}p Λ1pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' This completes the proof of the theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Homogeneous spaces In this section, we introduce some fundamental concepts and important theorems for homo- geneous spaces, which can be found in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We begin with the definition of homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Recall that d is a quasi-metric on Ω if (1) dpx, xq ě 0, @x P Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (2) dpx, yq “ dpy, xq, @x, y P Ω;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (3) there exists a constant A0 ě 1 such that (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1) dpx, yq ⩽ A0pdpx, zq ` dpz, yqq, @x, y, z P Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Denote by Bpx, rq :“ ty P Ω : dpy, xq ă ru the open ball centered at x with radius r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In this paper, all quasi-metric spaces are assumed to have the doubling property: there exists a positive integer A1 P N such that for every x P Ω and for every r ą 0, the ball Bpx, rq can be covered by at most A1 balls Bpxi, r 2q for some xi P Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 17 Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' A σ-finite measure space pΩ, F, µq equipped with a quasi-metric d is called a homogeneous space if µ is a Borel measure of homogeneous type: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2) 0 ă µ pBpx, 2rqq ⩽ 2Cµµ pBpx, rqq ă 8, @x P Ω, r ą 0, where the constant Cµ is independent of x and r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In [7], Coifman and Weiss defined Hardy spaces on homogeneous spaces by regarding their elements as linear functionals acting on some appropriate quasi-normed spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In order to state the definition of Coifman and Weiss, we need to introduce the notions of atoms, BMO and Lipschitz spaces on homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If 0 ă p ⩽ 1 ⩽ q ď 8 and p ă q, we say that a function a is a pp, qq-atom if (1) supppaq Ă B where B is a ball;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (2) }a}q ⩽ pµpBqq 1 q ´ 1 p ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (3) ´ Ω adµ “ ´ B adµ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' A locally integrable function f is called a BMO function if }f}BMO :“ sup B 1 µpBq ˆ B |f ´ fB|dµ ă 8, where fB :“ 1 µpBq ´ B fdµ, and the supremum runs over all balls B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Denote by BMOpµq the BMO space consisting of all BMO functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Definition 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For α ą 0, a locally integrable function l is called a Lipschitz function if (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3) |lpxq ´ lpyq| ⩽ Cα pµpBqqα for any x, y P Ω and any ball B containing x, y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Moreover, (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4) }l}Lα :“ inftCα : |lpxq ´ lpyq| ⩽ Cα pµpBqqα , @x, y P Bu, where the infimum runs over all balls B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Denote by Lαpµq the space consisting of all Lipschitz functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It is well-known that each BMO function can be regarded as a continuous linear functional on the vector space generated by finite linear combinations of p1, qq-atoms for 1 ă q ď 8 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Hence we can define the atomic Hardy space H1,q at pµq p1 ă q ď 8q as follows: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5) H1,q at pµq :“ # f P pBMOpµqq˚ : f “ 8 ÿ j“0 λjaj, where aj is a p1, qq-atom and 8 ÿ j“0 |λj| ă 8 + endowed with the norm }f}H1,q at pµq :“ inf # 8 ÿ j“0 |λj| : f “ 8 ÿ j“0 λjaj, where aj is a p1, qq-atom + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Similarly, each Lipschitz function l P Lαppµq can be also regarded as a continuous linear functional of the vector space generated by finite linear combinations of pp, qq-atoms where 0 ă p ă 1 ď q ď 8 and αp “ 1 p ´ 1 (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' [7]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We define the atomic Hardy spaces Hp,q at pµq as follows: (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6) Hp,q at pµq :“ # f P ` Lαppµq ˘˚ : f “ 8 ÿ j“0 λjaj, where aj is a pp, qq-atom and 8 ÿ j“0 |λj|p ă 8 + MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 18 endowed with the quasi-norm }f}Hp,q at pµq :“ inf $ & % ˜ 8 ÿ j“0 |λj|p ¸ 1 p : f “ 8 ÿ j“0 λjaj, where aj is a pp, qq-atom , .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Although the Hardy spaces vary with p and q according to the above definitions, the following theorem, which can be found in [7], shows that the Hardy spaces actually depend only on p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Consequently, this enables us to define the Hardy spaces Hp atpµq for 0 ă p ⩽ 1 to be any one of the spaces Hp,q at pµq for 0 ă p ă q ⩽ 8, 1 ⩽ q ď 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Hp,q at pµq “ Hp,8 at pµq whenever 0 ă p ď 1 ⩽ q ⩽ 8 and p ă q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We end this section with the following duality theorem obtained in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' ` H1 atpµq ˘˚ “ BMOpµq, and pHp atpµqq˚ “ Lαppµq for 0 ă p ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The proofs of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5 and Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6 that appeared in [7] are very technical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In the following sections, by employing martingale methods, we give much simpler proofs of these facts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Our approach is based on the fact that Hp atpµq for 0 ă p ď 1 is the finite sum of several dyadic martingale Hardy spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Dyadic systems on homogeneous spaces In this section, we start with introducing dyadic systems on homogeneous spaces, which first appeared in the work of Hyt¨onen and Kairema [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' With the help of these dyadic structures, we then show that Hp atpµq is exactly the finite sum of martingale Hardy spaces associated with some adjacent dyadic martingales, which extends Mei’s result [17] to homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The following theorem concerning the existence of dyadic structures is due to Hyt¨onen and Kairema [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let Ω denote a homogeneous space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Suppose that the constants 0 ă c0 ⩽ C0 ă 8 and δ P p0, 1q satisfy 12A3 0C0δ ⩽ c0, where A0 is specified in the definition of quasi-metric, see (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Given a set of reference points tzk αuα, α P Ak (an index set), for every k P Z, with the properties that dpzk α, zk βq ⩾ c0δk, pα ‰ βq min α dpx, zk αq ă C0δk, for all x P Ω, one can construct families of sets ˜Qk α Ď Qk α Ď ¯Qk α, called open, half-open and closed dyadic cubes respectively, such that: ˜Qk α and ¯Qk αare the interior and closure of Qk α;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1) if k ⩽ l, then either Ql β Ď Qk α or Ql β X Qk α “ H;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2) X “ Ť α Qk α (disjoint union) for all k P Z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3) Bpzk α, c1δkq Ď Qk α Ď Bpzk α, C1δkq “: BpQk αq where c1 “ p3A2 0q´1c0 and C1 “ 2A0C0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4) if k ⩽ l and Ql β Ď Qk α then BpQl βq Ď BpQk αq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5) The open and closed cubes ˜Qk α and ¯Qk α depend only on the points zl β for l ⩾ k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The half-open cubes Qk α depend on zl β for l ⩾ minpk, k0q, where k0 P Z is a preassigned number entering the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 19 It is obvious that the construction of the above dyadic systems is not unique, and it depends on the set of the reference points tzk αuα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We denote this dyadic system by D “ tQk αuk,α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let Fk “ σptQk αuαq be the σ-algebra generated by tQk αuα.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then it is clear that ¨ ¨ ¨ Ă Fk´1 Ă Fk Ă ¨ ¨ ¨ , which implies that tFkukPZ is a filtration generated by atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let F “ σpYkPZFkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that each Qk α is an atom of Fk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The standard dyadic grid on the real line is a dyadic system given by Fk “ tr2´km, 2´kpm ` 1qq : m P Zu for all k P Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Similarly, an example of a dyadic system on Rn is given by the family of standard dyadic cubes in Rn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Recall that, for f P L1 locpΩ, F, µq, the martingale maximal function, the square function and the conditional square function of f associated with pFkqkPZ are given by f ˚ :“ max kPZ |fk|, Spfq :“ ˜ÿ kPZ |dkf|2 ¸ 1 2 and spfq :“ ˜ÿ kPZ Ek´1|dkf|2 ¸ 1 2 , respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let 0 ă p ď 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The martingale Hardy space Hp m,Dpµq is defined as the completion of the space consisting of all f P L1 locpΩq such that f ˚ P LppΩq with respect to the quasi-norm }f}Hp m,Dpµq :“ }f ˚}p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We define Hp Dpµq and hp Dpµq by the square functions and the conditional square functions respectively, with the additional assumption that (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6) lim nÑ´8 ˆ Ω sup kďn |fk|pdµ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' From (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6), we have lim nÑ´8 sup kďn |fk| “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Analogously, define the martingale atomic Hardy spaces Hp,q at,Dpµq p0 ă p ă 1 ď q ď 8 or p “ 1, 1 ă q ď 8q like Definition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In order to show Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6, we introduce the dual spaces of these atomic martingale Hardy spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For 0 ă p ă 1, q “ 1 or 2 and αp “ 1 p ´ 1, define BMODpµq :“ # f P L1 locpΩ, µq : }f}BMODpµq :“ sup QPD 1 µpQq ˆ Q |f ´ fQ|dµ ă 8 + , ΛD q pαpq :“ # f P L1 locpΩ, µq : }f}ΛD q pαpq :“ sup QPD µpQq´ 1 q ´αp ˆˆ Q |f ´ fQ|qdµ ˙ 1 q ă 8 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The spaces ΛD q pαpq are called the martingale Lipschitz spaces with respect to D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that ΛD 1 pαpq “ ΛD 2 pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Arguing as in [27], one can show that ` H1 at,Dpµq ˘˚ “ BMODpµq, and for 0 ă p ă 1, ´ Hp at,Dpµq ¯˚ “ ΛD q pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 20 Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Since every simple pp, qq-atom is locally supported, by Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2, we conclude that for 0 ă p ă 1 ď q ď 8 or p “ 1, 1 ă q ď 8 Hp,q at,Dpµq “ Hp,8 at,Dpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Thus we are only concerned with Hp at,Dpµq :“ Hp,8 at,Dpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For 0 ă p ⩽ 1, the martingale Hardy spaces defined above are mutually equivalent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Namely, Hp Dpµq “ Hp m,Dpµq “ hp Dpµq “ Hp at,Dpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let p P p0, 1s be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' First, we show Hp Dpµq “ Hp m,Dpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Suppose that f P Hp m,Dpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then for any n ą 0, by a well-known inequality of Burkholder–Davis–Gundy, ˆ Ω ˜ |f´n|2 ` nÿ k“´n`1 |dkf|2 ¸ p 2 dµ ≲ ˆ Ω sup ´nďkďn |fk|pdµ ≲ ˆ Ω pf ˚qpdµ which yields by letting n Ñ 8 and by Fatou’s lemma }Spfq}p ≲ }f ˚}p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Thus Hp m,Dpµq Ă Hp Dpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Conversely, if f P Hp Dpµq, then for n ą 0, ˆ Ω sup ´nďkďn |fk|pdµ ≲ ˆ Ω ˜ |f´n|2 ` n ÿ k“´n`1 |dkf|2 ¸ p 2 dµ, and hence (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='7) ˆ Ω sup ´nďkďn |fk|pdµ ≲ ˆ Ω sup kď´n |fk|pdµ ` ˆ Ω |Spfq|pdµ ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then by letting n Ñ 8 and applying Fatou’s lemma, we obtain }f ˚}p ă 8 and }f ˚}p ≲ }Spfq}p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Therefore, Hp Dpµq Ă Hp m,Dpµq and Hp m,Dpµq “ Hp Dpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' One shows Hp m,Dpµq “ hp Dpµq in a completely analogous way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' To show hp Dpµq “ Hp at,Dpµq, one can argue by mimicking the corresponding proof in [27] and [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' □ The following theorem can be found in [13] and ensures that there exist enough dyadic cubes to cover all balls on homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Given a set of reference points tzk αu, k P Z, α P Ak, suppose that there exists constant δ P p0, 1q that satisfies 96A6 0δ ⩽ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then there exists a finite collection of families Dt, t “ 1, 2, ¨ ¨ ¨ , K “ KpA0, A1, δq ă 8, where each Dt is a collection of dyadic cubes, associated to dyadic points tzk αu, k P Z, α P Ak, with the properties (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1)-(6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5) in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In addition, the following property is satisfied: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='8) for every Bpx, rq Ď Ω, there exist t and Q P Dt with Bpx, rq Ď Q and diampQq ⩽ Cr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The constant C ă 8 in (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='8) only depends on the quasi-metric constant A0 and the parameter δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By virtue of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5, we have the following theorem, which extends Mei’s result in [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For 0 ă p ⩽ 1, we have (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='9) Hp atpµq “ K ÿ t“1 Hp at,Dtpµq “ K ÿ t“1 Hp Dtpµq “ K ÿ t“1 Hp m,Dtpµq “ K ÿ t“1 hp Dtpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 21 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let p P p0, 1s be fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In view of Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4, it suffices to show Hp atpµq “ Kř t“1 Hp at,Dtpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We prove it via comparing the atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let a be a pp, 8q-atom in Hp atpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then there exists a ball B such that supppaq Ă B, }a}8 ⩽ pµpBqq´ 1 p , ˆ B apxqdµ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5, there exist t and a cube Q P Dt such that B Ă Q, and µpQq ≲ µpBq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then supppaq Ă B Ă Q, }a}8 ⩽ pµpBqq´ 1 p ≲ pµpQqq´ 1 p , ˆ Q adµ “ 0, which implies that a is a constant multiple of a simple pp, 8q-atom in Hp at,Dtpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Thus (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='10) Hp atpµq Ă K ÿ t“1 Hp at,Dtpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For any t “ 1, 2, ¨ ¨ ¨ , K and for any given simple pp, 8q-atom b in Hp at,Dtpµq, there exists Q P Dt such that supppbq Ă Q, }b}8 ⩽ pµpQqq´ 1 p , ˆ Q bdµ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, there exists a ball B such that Q Ă B and µpQq ≳ µpBq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Hence supppbq Ă Q Ă B, }b}8 ⩽ pµpQqq´ 1 p ≲ pµpBqq´ 1 p , ˆ B bdµ “ 0, which implies that a multiple of b is also a pp, 8q-atom in Hp atpµq, thus (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='11) K ÿ t“1 Hp at,Dtpµq Ă Hp atpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' To complete the proof of the theorem, combine (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='10) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' □ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5 follows immediately from Corollary 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2, Proposition 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6, which simplifies the original proof by Coifman and Weiss in [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By duality and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6, we recover the following result of [13], which is an extension of a result due to Mei [17]: (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='12) BMOpµq “ K č t“1 BMODtpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We will now establish an analogous result for Lαppµq p0 ă p ă 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For 0 ă p ă 1, Lαppµq “ K č t“1 ΛDt 2 pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, for any Q P Dt (and t “ 1, 2, ¨ ¨ ¨ , K), there exists a ball B such that Q Ă B and µpBq ≲ µpQq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If f P Lαppµq, then for any x, y P Q, we have |fpxq ´ fpyq| ď }f}LαppµqµpBqαp ≲ }f}LαppµqµpQqαp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 22 We thus have }f}ΛDt 2 pαpq ď sup QPDt pµpQqq´ 1 2 ´αp ˜ µpQq´2 ˆ Q ˆˆ Q |fpxq ´ fpyq|dµpyq ˙2 dµpxq ¸ 1 2 ⩽ sup QPDt pµpQqq´ 1 2 ´αp ˆˆ Q }f}2 Lαppµqµ pQq2αp dµ ˙ 1 2 ≲ }f}Lαppµq, which yields (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='13) Lαppµq Ă K č t“1 ΛDt 2 pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Conversely, let f P KŞ t“1 ΛDt 2 pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For Q P Dt, by Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4, |fpxq ´ fQ| ≲ µpQqαp}f}ΛDt 2 pαpq @x P Q, which implies that for any x, y P Q, (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='14) |fpxq ´ fpyq| ≲ µpQqαp}f}ΛDt 2 pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For any ball B Ă Ω, by Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5, there exist t and Q P Dt such that B Ă Q and µpQq ≲ µpBq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then for any x, y P B, by (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='14) |fpxq ´ fpyq| ≲ µpBqαp}f}ΛDt 2 pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Thus }f}Lαp ≲ K ÿ t“1 }f}ΛDt 2 pαpq, which implies (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='15) K č t“1 ΛDt 2 pαpq Ă Lαppµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The theorem follows from (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='13) and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' □ Remark 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6 and Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='8 give a simple proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6 originally estab- lished by Coifman and Weiss [7]: pHp atpµqq˚ “ ˜ K ÿ t“1 Hp at,Dtpµq ¸˚ “ K č t“1 pHp at,Dtpµqq˚ “ K č t“1 ΛDt 2 pαpq “ Lαppµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Bilinear decompositions for dyadic martingales on homogeneous spaces In this section, we focus on bilinear decompositions arising in the study of products between elements in spaces of dyadic martingales on homogeneous spaces introduced in the previous section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In the setting of homogeneous spaces, due to their quasi-metrics and measures, the dyadic martingales behave worse than martingales in probability spaces and the underlying analysis is more intricate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1 we prove appropriate generalized H¨older-type inequalities on homogeneous spaces (see Lemmas 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4 below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We then introduce a class of pointwise multipliers of ΛD 1,`pαpq and BMODpµq;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' see Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Using Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5, we define products between dyadic MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 23 martingale Hardy spaces on homogeneous spaces and their duals and then, in §7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2 we establish analogues of the results of Sections 3 and 4 in the setting of homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' A generalized H¨older-type inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let 0 ă p ď 1 and D be a dyadic system, constructed as in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The martingale Musielak–Orlicz Hardy spaces HΨp D pµq consist of all measurable functions f on pΩ, F, µq such that spfq P LΨppΩq where O P Ω is a fixed point, and Ψ1px, tq :“ t log pe ` dpx, Oqq ` logpe ` tq, Ψppx, tq :“ t 1 ` ttr1 ` µpBpO, dpx, Oqqqsu1´p p0 ă p ă 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that LΨppΩq is a quasi-normed space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let M :“ pCµ ` 1q log pe ` dpx, Oqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5) we obtain (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1) Ψ1px, stq ≲ pe ` dpx, Oqq´pCµ`1qet ` s ≲ wpxqet ` s, for all x P Ω, s, t ą 0, where w : Ω Ñ R` is a weight function with (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2) wpxq ≲ min !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 1, dpx, Oq´pCµ`1q) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let Q0 P F0 be the dyadic cube such that O P Q0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For g P BMODpµq, define }g}BMOD `pµq :“ sup αPA0 |gQ0α ´ gQ0| log pe ` dpz0α, Oqq ` |gQ0| ` }g}BMODpµq, where Q0 α P F0 is a dyadic cube with its center z0 α and A0 is the index set in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Denote by BMOD `pµq the space consisting of all g P BMODpµq such that }g}BMOD `pµq ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It is not difficult to verify that } ¨ }BMOD ` pµq is a norm on BMOD `pµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If we consider the dyadic martingales on Rn, by taking appropriate cubes Q0 one shows that if g P BMODpµq, then g P BMOD `pµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that if g P BMOpµq, then g P BMOD `pµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Moreover, }g}BMOD `pµq ≲ }g}BMOpµq ` |gQ0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We now introduce the following generalized H¨older inequality for L1pΩ, F, µq and BMOD `pµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If f P L1pΩ, F, µq and g P BMOD `pµq, then f ¨ g P LΨ1pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Moreover, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3) }fg}LΨ1pΩq ≲ }f}1}g}BMOD `pµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Without loss of generality, assume }f}1 ď 1, }g}BMOD ` pµq ď 1 and gQ0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It suffices to show that ˆ Ω Ψ1px, |fpxqgpxq|qdµ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let Sk :“ BpO, C0δkqzBpO, C0δk`1q for k ă 0 and S0 :“ BpO, C0q, where δ P p0, 1q is the constant in Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then for each k ď 0, there exists a finite index subset Bk Ă A0 such that BpO, C0δkq Ă Ť αPBk Q0 α (where Q0 α P F0) and ÿ αPBk µ ` Q0 α ˘ “ µ ˜ ď αPBk Q0 α ¸ ď µ ` BpO, 2A0C0δkq ˘ ≲ δCµk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 24 Take s “ ν´1|fpxq|, t “ ν|gpxq| in (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1), one has ˆ Ω Ψ1px, |fpxqgpxq|qdµ “ 0ÿ k“´8 ÿ αPBk ˆ SkXQ0α Ψ1px, |fpxqgpxq|qdµ ≲ 0ÿ k“´8 ÿ αPBk ˆ SkXQ0 α wpxqeν|gpxq|dµ ` ν´1}f}1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Therefore, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4) ˆ Ω Ψ1px, |fpxqgpxq|qdµ ≲ T1 ` ν´1}f}1, where T1 :“ 0ÿ k“´8 ÿ αPBk ˆ SkXQ0 α wpxqeν ˇˇˇgpxq´gQ0α ˇˇˇeν ˇˇˇgQ0α ˇˇˇdµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let ν :“ mintκ,1u 2 ą 0 (where κ is defined in Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='11), by (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2) and Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, one has T1 ≲ 0ÿ k“´8 ÿ αPBk µpQ0 αq ` e ` dpz0 α, Oq ˘ 1 2 δpk`1qpCµ`1q ≲ 0ÿ k“´8 ÿ αPBk µpQ0 αqδ k 2 δpk`1qpCµ`1q ≲ 0ÿ k“´8 δCµkδ k 2 δCµk`k ≲ 0ÿ k“´8 δ´ 1 2 k, and hence (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5) T1 ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Combine (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4), (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5) and the fact that ν´1}f}1 ≲ 1, and the proof is complete.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' □ We consider the case 0 ă p ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Define }g}ΛD 1,`pαpq :“ sup αPA0 |gQ0α ´ gQ0| 1 ` µ tB pO, dpz0α, Oqquαp ` |gQ0| ` }g}ΛD 1 pαpq, Denote by ΛD 1,`pαpq the space consisting of all g P ΛD 1 pαpq such that }g}ΛD 1,`pαpq ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It is easy to verify that } ¨ }ΛD 1,`pαpq is a norm on ΛD 1,`pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If we consider the dyadic martingales on Rn, by taking appropriate cubes Q0 one can show that if g P ΛD 1 pαpq, then g P ΛD 1,`pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that if g P Lαppµq, then g P ΛD 1,`pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Moreover, }g}ΛD 1,`pαpq ≲ }g}Lαppµq ` |gQ0|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Next we present a generalized H¨older inequality for LppΩ, F, µq and ΛD 1,`pαpq for 0 ă p ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If f P LppΩ, F, µq and g P ΛD 1,`pαpq for 0 ă p ă 1, then f ¨ g P LΨppµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Moreover, (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6) }fg}LΨppΩq ≲ }f}p}g}ΛD 1,`pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 25 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Without loss of generality, assume }f}p ď 1, }g}ΛD 1,`pαpq ď 1 and gQ0 “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It suffices to show that ˆ Ω Ψppx, |fpxqgpxq|qdµ ≲ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Take the same family of sets tSkukď0 as above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' From Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4, we know that for x P Q0 α, |gpxq| “ |gpxq ´ gQ0| ⩽ ˇˇgpxq ´ gQ0α ˇˇ ` ˇˇgQ0α ´ gQ0 ˇˇ ď ` µpQ0 αq ˘αp ` µ ␣ B ` O, dpz0 α, Oq ˘(αp ` 1 ≲ µ ` BpO, 2A0C0δkq ˘αp ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Therefore ˆ Ω Ψppx, |fpxqgpxq|qdµ “ 0ÿ k“´8 ÿ αPBk ˆ SkXQ0α |gpxq||fpxq| 1 ` t|gpxq||fpxq| r1 ` µ pBpO, dpx, Oqqsu1´p dµ ≲ 0ÿ k“´8 ÿ αPBk ˆ SkXQ0α |gpxq|p|fpxq|p 1 ` µ pBpO, dpx, Oqq1´p dµ ≲ 0ÿ k“´8 ÿ αPBk ˆ SkXQ0α µ ` BpO, 2A0C0δkq ˘αpp ` 1 t1 ` µ pBpO, C0δk`1qu1´p |fpxq|pdµ ≲ 1, which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' □ We are now about to present the analogues of the results in Sections 3 and 4 concerning bilinear decompositions for dyadic martingales on homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' To this end, we need to define the product between martingale Hardy spaces and their dual spaces first.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' As in the probability setting, we regard the product in the sense of distribution as follows: for 0 ă p ă 1, xf ˆ g, hy :“ xh ¨ g, fy, f P Hp at,Dpµq, g P ΛD 1,`pαpq, where h is a test function such that h ¨ g is in ΛD 1,`pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For p “ 1, we may define the product between H1 at,Dpµq and BMODpµq analogously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' To this end, we need to introduce some pointwise multipliers of ΛD 1,`pαpq and BMODpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Denote the space of test functions by Hpαpq p0 ă p ď 1q, and a measurable function h is a test function if it satisfies the following properties: (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='7) |hpxq| ≲ 1 p1 ` µpBpO, dpx, Oqqqαpq logpe ` dpx, Oqq, @x P Ω, and (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='8) |hpyq ´ hpzq| ≲ µpBqαp p1 ` µrBpO, 1 ` r ` dpcB, Oqqsαpq logpe ` r ` dpcB, Oqq whenever y, z are both contained in a ball B with center cB and radius r ď dpcB,Oq 2A0 ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It is obvious that Hpαpq Ă L8pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The following theorem shows that if h P Hpαpq, then h is a pointwise multiplier of ΛD 1,`pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For 0 ă p ă 1 and any dyadic system D, Hpαpq is a space of pointwise multipliers of ΛD 1,`pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For p “ 1, Hp0q is a space of pointwise multipliers of BMOD `pµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' More precisely, for any g P ΛD 1,`pαpq and h P Hpαpq, we have }g ¨ h}ΛD 1,`pαpq ≲ }g}ΛD 1,`pαpq ` }h}L8pΩq ` 1 ˘ , MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 26 and for any g P BMOD `pµq and h P Hp0q, we have }g ¨ h}BMOD ` pµq ≲ }g}BMOD `pµq ` }h}L8pΩq ` 1 ˘ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' First, we consider the case 0 ă p ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Assume that g P ΛD 1,`pαpq and h P Hpαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' According to [20], it suffices to show that (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='9) sup Q |gQ| µpQqαp`1 ˆˆ Q |hpxq ´ hQ|dx ˙ ă 8, where Q runs over all dyadic cubes in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If Q Ă Q0 β for some β P A0, there exists a collection of cubes Q “ Q0 Ă Q1 Ă ¨ ¨ ¨ Ă QN “ Q0 β such that there exists a universal constant 0 ă δ 1 ă 1 with µpQk´1q ď δ 1µpQkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Hence |gQ ´ gQ0 β| ď N ÿ k“1 |gQk ´ gQk´1| ≲ N ÿ k“1 µpQkqαp}g}ΛD 1,`pαpq ≲ }g}ΛD 1,`pαpq N ÿ k“1 ˆ µpQkq µpQk´1q tαp´1dt ≲ µpQ0 βqαp}g}ΛD 1,`pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Similarly, if Q0 β Ă Q, we have |gQ ´ gQ0 β| ≲ µpQqαp}g}ΛD 1,`pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, there exists a ball B, with center cB and radius r, such that Q Ă B and µpBq ≲ µpQq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If Q0 β Ă Q and r ą dpO,cBq 2A0 ` 1, for any x P BpO, rq, we have dpcB, xq ď A0pdpcB, Oq ` dpO, xqq ă p2A2 0 ` A0qr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then µpQq ≳ µpBq ≳ C´p2A2 0`A0q µ µ pBpO, rqq ≳ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Similarly, we also have dpz0 β, Oq ă p2A2 0 ` A0qr and µ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' B ´ O, dpz0 β, Oq ¯) ≲ µpBq ≲ µpQq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Thus |gQ| µpQqαp`1 ˆˆ Q |hpxq ´ hQ|dx ˙ ≲ |gQ ´ gQ0 β| ` |gQ0 β ´ gQ0| ` |gQ0| µpQqαp ¨ }h}L8pΩq ≲ µpQqαp ` µ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' B ´ O, dpz0 β, Oq ¯)αp ` 1 µpQqαp ¨ }g}ΛD 1,`pαpq}h}L8pΩq ≲ }g}ΛD 1,`pαpq}h}L8pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If Q0 β Ă Q and r ď dpO,cBq 2A0 ` 1, for any x P B, we have dpx, Oq ď A0pdpO, cBq ` rq, then µpQq ≲ µ pBpO, A0pdpO, cBq ` rqq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Thus |gQ| µpQqαp`1 ˆˆ Q |hpxq ´ hQ|dx ˙ ≲ |gQ ´ gQ0 β| ` |gQ0 β ´ gQ0| ` |gQ0| µpQqαp`1 µpBqαp`1 p1 ` µrBpO, 1 ` r ` dpO, cBqqsqαp ≲ ´ µpQqαp ` µ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' B ´ O, dpz0 β, Oq ¯)αp ` 1 ¯ }g}ΛD 1,`pαpq p1 ` µrBpO, 1 ` r ` dpO, cBqqsqαp ≲ }g}ΛD 1,`pαpq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If Q Ă Q0 β, from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, we can choose C0 sufficiently small such that C1 “ 2A0C0 ď 1, then r ď C1 ď dpcB,Oq 2A0 ` 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For any x P Q0 β, we have dpO, xq ď A0pdpO, z0 βq ` C1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then µpQ0 βq ≲ µ ␣ B ` O, A0pdpO, z0 βq ` C1q ˘( .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 27 By a calculation similar to the one presented above, we get the desired result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Combining the above estimates, we finish our proof for 0 ă p ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The case for p “ 1 is similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' □ Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5, the dyadic system D is arbitrary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then from Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='8 and (6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='12), we conclude that Hpαpq is a space of pointwise multipliers of Lαppµq and Hp0q is a space of pointwise multipliers of BMOpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Bilinear decompositions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Assume f P H1 Dpµq, g P BMOD `pµq or f P Hp Dpµq, g P ΛD 1,`pαpq, 0 ă p ă 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Denote by Hp D,finpµq p0 ă p ď 1q the linear space consisting of all functions which can be written as a finite sum of simple pp, 8q-atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Thus if f P Hp D,finpµq, f is locally supported, f P L1pΩq X L8pΩq and ´ Ω fdµ “ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that Hp D,finpµq is dense in Hp Dpµq with respect to the norm } ¨ }Hp Dpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In the following, we shall only consider the case where f P Hp D,finpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then f ¨ g P L1pΩq, and we can write (7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='10) f ¨ g “ Π1pf, gq ` Π2pf, gq ` Π3pf, gq, where Π1pf, gq :“ 8 ÿ k“1 dkfdkg, Π2pf, gq :“ 8 ÿ k“1 fk´1dkg and Π3pf, gq :“ 8 ÿ k“1 gk´1dkf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We have the following: (1) Π1 is a bilinear bounded operator from H1 Dpµq ˆ BMOD `pµq to L1pΩq, and when 0 ă p ă 1, Π1 is a bilinear bounded operator from Hp Dpµq ˆ ΛD 1,`pαpq to L1pΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (2) Π2 is a bilinear bounded operator from H1 Dpµq ˆ BMOD `pµq to H1 Dpµq, and when 0 ă p ă 1, Π2 is a bilinear bounded operator from Hp Dpµq ˆ ΛD 1,`pαpq to H1 Dpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (3) Π3 is a bilinear bounded operator from H1 Dpµq ˆ BMOD `pµq to HΨ1 D pµq, and when 0 ă p ă 1, Π3 is a bilinear bounded operator from Hp Dpµq ˆ ΛD 1,`pαpq to HΨp D pµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For Π1 and Π2, we can argue as in the corresponding part of the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' As for Π3, we can also argue as in the corresponding part of the proof Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, where in the homogeneous setting one needs to apply Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2 and Lemma 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We omit the details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' □ Remark 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For Π1 and Π2, the condition H1 Dpµq ˆ BMOD `pµq and Hp Dpµq ˆ ΛD 1,`pαpq can be in fact replaced by H1 Dpµq ˆ BMODpµq and Hp Dpµq ˆ ΛD 1 pαpq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Applications to Homogeneous spaces In the first part of this section we show that HΨp D pµq admits an atomic decomposition for 0 ă p ă 1, which allows us to integrate several adjacent dyadic systems on homogeneous spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For a given dyadic system D on Ω, we define the dyadic HΨp at,D-atom as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' A measurable function a is said to be an HΨp at,D-atom if (i) supppaq Ă Q where Q P D is a cube;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (ii) ´ Ω adµ “ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (iii) }a}8 ⩽ }1Q}´1 LΨppΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The atomic dyadic martingale Musielak–Orlicz Hardy spaces HΨp at,Dpµq p0 ă p ă 1q are defined in a way analogous to (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We first introduce the space BMOD Ψppµq, which is a subspace of continuous linear functionals on finite sums of atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 28 Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' A locally integrable function g is said to be a dyadic BMOD Ψppµq function associated with a dyadic system D if }g}BMOD Ψppµq :“ sup kPZ sup QPFk 1 }1Q}LΨppΩq ˆ Q |gpxq ´ gkpxq|dx ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then we define the atomic Musielak–Orlicz martingale Hardy spaces HΨp at,Dpµq as follows: HΨp at,Dpµq :“ # f P ´ BMOD Ψppµq ¯˚ : f “ 8 ÿ i“0 λiai, where ai is an HΨp at,Dpµq-atom supported on a cube Qi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' + , where 8 ÿ i“0 ˆ Qi Ψppx, |λi|}ai}8qdµ ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Moreover, }f}H Ψp at,Dpµq :“ inf # ρ ą 0 : 8 ÿ i“0 ˆ Qi Ψppx, ρ´1|λi|}ai}8qdµ ⩽ 1 + Arguing as in [28], one can show that for 0 ă p ă 1 (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1) HΨp D pµq “ HΨp at,Dpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We shall now introduce the atomic Musielak–Orlicz Hardy spaces HΨp at pµq p0 ă p ď 1q on the homogeneous space Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' First, we present the definition of atoms for HΨp at pµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' A measurable function apxq is said to be an HΨp at pµq-atom if (i) supppaq Ă B where B Ă Ω is a ball;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (ii) ´ Ω adµ “ 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (iii) }a}8 ⩽ }1B}´1 LΨppΩq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' A locally integrable function g is said to be a BMOΨppµq function if }g}BMOΨppµq :“ sup B 1 }1B}LΨppΩq ˆ B |gpxq ´ gB|dx ă 8, where B runs over all balls in Ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Definition 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' The atomic Musielak–Orlicz Hardy spaces HΨp at pµq p0 ă p ď 1q are defined as follows: HΨp at pµq :“ # f P ` BMOΨppµq ˘˚ : f “ 8 ÿ i“0 λiai, where ai is an HΨp at pµq-atom supported on a ball Bi + , where 8 ÿ i“0 ˆ Bi Ψppx, |λi|}ai}8qdµ ă 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Moreover, }f}H Ψp at pµq :“ inf # ρ ą 0 : 8 ÿ i“0 ˆ Bi Ψppx, ρ´1|λi|}ai}8qdµ ⩽ 1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 29 Let Dt p1 ď t ď Kq be the adjacent systems of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By arguing as in the proof of Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6, we have the following: Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For 0 ă p ă 1, HΨp at pµq “ HΨp at,D1pµq ` HΨp at,D2pµq ` ¨ ¨ ¨ ` HΨp at,DKpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' It suffices to show that any dyadic HΨp Dt -atom a is a constant multiple of an HΨppµq-atom, and any HΨppµq-atom b is a constant multiple of a dyadic HΨp Dt -atom.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If B :“ Bpx0, rq, then denote the ball Bpx0, Drq by DB for D ě 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Denote d :“ dpx0, Oq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In what follows, CpD, p, A0, Cµq denotes a constant that depends on D, p, A0, Cµ and may differ from line to line.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We first show that if ˆ B 1 1 ` r1 ` µpBpO, dpx, Oqqqs1´p dµpxq “ 1, then (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2) ˆ DB 1 1 ` r1 ` µpBpO, dpx, Oqqqs1´p dµpxq ⩽ CpD, p, A0, Cµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Notice that 1 “ ˆ B 1 1 ` r1 ` µpBpO, dpx, Oqqqs1´p dµpxq ⩾ µpBq sup xPB t1 ` r1 ` µpBpO, dpx, Oqqqs1´pu ⩾ µpBq 1 ` r1 ` µpBpO, A0pd ` rqqqs1´p , which implies µpBq ⩽ 1 ` r1 ` µpBpO, A0pd ` rqqqs1´p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If d ⩽ 2A0Dr, we have µpBq ⩽ 1 ` r1 ` µpBpO, A0p2A0D ` 1qrqqs1´p ⩽ 1 ` t1 ` µrBpx0, A0pA0 ` 1qp2A0D ` 1qrqsu1´p ⩽ 1 ` !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 1 ` rA0pA0 ` 1qp2A0D ` 1qsCµ µpBq )1´p , and thus µpBq ⩽ CpD, p, A0, Cµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3) ˆ DB 1 1 ` r1 ` µpBpO, dpx, Oqqqs1´p dµpxq ⩽ µpDBq ⩽ DCµµpBq ⩽ CpD, p, A0, Cµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 30 If d ą 2A0Dr, then ˆ DB 1 1 ` r1 ` µpBpO, dpx, Oqqqs1´p dµpxq ⩽ µpDBq inf xPDB t1 ` r1 ` µpBpO, dpx, Oqqqs1´pu ⩽ DCµµpBq 1 ` r1 ` µpBpO, d{A0 ´ Drqqs1´p , ⩽DCµ t1 ` µrB pO, rA0 ` 1{p2Dqsdqsu1´p ` DCµ 1 ` t1 ` µrBpO, d{p2A0qqsu1´p ⩽ DCµ !' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 1 ` rp2A0 ` 1{DqA0sCµ µrBpO, d{p2A0qqs )1´p 1 ` t1 ` µrBpO, d{p2A0qqsu1´p ` DCµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Hence (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4) ˆ DB 1 1 ` r1 ` µpBpO, dpx, Oqqqs1´p dµpxq ⩽ CpD, p, A0, Cµq Combining (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='3) with (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='4), we get (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Assume a is an HΨppµq-atom supported on B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='5, there exist t and a cube Q P Dt such that B Ă Q and diampQq ⩽ Cr, hence B Ă Q Ă CB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Note that supppaq Ă Q, ´ Q apxqdµpxq “ 0 and }1Q}LΨppµq ⩽ }1CB}LΨppµq ⩽ CpC, p, A0, Cµq}1B}LΨppµq, which follows from (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Thus }a}8 ⩽ }1B}´1 LΨppµq ≲ }1Q}´1 LΨppµq, which implies a is a multiple of dyadic HΨp Dt -atom supported on Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' For any t “ 1, 2 ¨ ¨ ¨ , K, assume b is a dyadic HΨp Dt -atom supported on Qk β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='1, there exists two balls such that Bpzk β, c1δkq Ă Qk β Ă Bpzk β, C1δkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Thus supppbq Ă Bpzk β, C1δkq, ´ Bpzk β,C1δkq bpxqdµpxq “ 0 and }1Bpzk β,C1δkq}LΨppµq ⩽ C ˆC1 c1 , p, A0, Cµ ˙ }1Bpzk β,c1δkq}LΨppµq ≲ }1Qk β}LΨppµq, which follows from (8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Therefore, }b}8 ⩽ }1Qk β}´1 LΨppµq ≲ }1Bpzk β,C1δkq}´1 LΨppµq, which implies b is a multiple of dyadic HΨp-atom supported on Bpzk β, C1δkq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' □ Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' In [11], Fu, Ma and Yang defined another kind of Musielak–Orlicz Hardy spaces by grand maximal function and they also proved that these Musielak–Orlicz Hardy spaces are equivalent to HΨp at pµq with respect to the corresponding norms when p P p Cµ Cµ`1, 1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let B1 :“ BpO, 1q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Define }g}BMO`pµq :“ |gB1| ` }g}BMOpµq, for g P BMOpµq, and }g}L`,αpµq :“ |gB1| ` }g}Lαppµq, for g P Lαppµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Thus } ¨ }BMO`pµq and } ¨ }L`,αppµq are quasi-norms on BMOpµq and Lαppµq, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 31 Theorem 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let 0 ă p ă 1 and f P Hp atpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' There exist three linear continuous operators Πf 1 : Lαppµq Ñ L1pΩq, Πf 2 : Lαppµq Ñ H1 atpµq and Πf 3 : Lαppµq Ñ HΨp at pµq such that f ¨ g “ Πf 1pgq ` Πf 2pgq ` Πf 3pgq for all g P Lαppµq, where Lαppµq is endowed with the quasi-norm } ¨ }L`,αppµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let f P Hp atpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6 there exist f t P Hp Dtpµq pt “ 1, 2, ¨ ¨ ¨ , Kq such that f “ f 1 ` f 2 ` ¨ ¨ ¨ ` f K, and K ÿ t“1 }f t}Hp Dtpµq « }f}Hp atpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Define Πf i pgq :“ Kř t“1 Πipf t, gq for i “ 1, 2, 3 and g P Lαppµq (Πi defined as in Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='7).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then f ¨ g “ Πf 1pgq ` Πf 2pgq ` Πf 3pgq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' By Theorem 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='7, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6 and Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6, we have }Πf 1pgq}1 ≲ K ÿ t“1 }Π1pf t, gq}1 ≲ K ÿ t“1 }f t}Hp Dt pµq}g}ΛDt 1,`pαpq ≲ }f}Hp atpµq}g}L`,αppµq, }Πf 2pgq}H1 atpµq ≲ K ÿ t“1 }Π2pf t, gq}H1 Dtpµq ≲ K ÿ t“1 }f t}Hp Dtpµq}g}ΛDt 1,`pαpq ≲ }f}Hp atpµq}g}L`,αppµq, }Πf 3pgq}H Ψp at pµq ≲ K ÿ t“1 }Π3pf t, gq}H Ψp Dt pµq ≲ K ÿ t“1 }f t}Hp Dtpµq}g}ΛDt 1,`pαpq ≲ }f}Hp atpµq}g}L`,αppµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' which finishes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' □ Remark 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' If the homogeneous space pΩ, µq satisfies the reverse doubling condition, then Lemma 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='6 holds for p “ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Then we conclude the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Let f P H1 atpµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' There exist three linear continuous operators Πf 1 : BMOpµq Ñ L1pΩq, Πf 2 : BMOpµq Ñ H1 atpµq and Πf 3 : BMOpµq Ñ HΨ1 at pµq such that f ¨ g “ Πf 1pgq ` Πf 2pgq ` Πf 3pgq for all g P BMOpµq, where BMOpµq is endowed with the norm } ¨ }BMO`pµq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Acknowledgments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We thank Professor Quanhua Xu for helpful discussions and suggestions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' and Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' would like to express their gratitude to Professor Xu for his kind invitation and hospitality during their visit to Besan¸con in March 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' We would also like to thank Yong Jiao, Guangheng Xie, Dachun Yang, Dejian Zhou for personal communications on their work with us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' References [1] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Auscher and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Hyt¨onen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Orthonormal bases of regular wavelets in spaces of homogeneous type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Harmon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 34 (2013), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 2, 266–296;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Addendum to Orthonormal bases of regular wavelets in spaces of homogeneous type.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Comput.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Harmon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Anal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 39 (2015), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 3, 568–569.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' [2] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Bakas, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Pott, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Rodr´ıguez-L´opez and A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Sola.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Notes on Hlog: structural properties, dyadic variants, and bilinear H1-BMO mappings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Ark.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Mat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 60 (2022), no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 2, 231–275.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' [3] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Bonami, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Cao, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Ky, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Liu, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Yang and W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Yuan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Multiplication between Hardy spaces and their dual spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Pures Appl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' (9) 131 (2019), 130–170.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 32 [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Bonami, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Grellier and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Paraproducts and products of functions in BMOpRnq and H1pRnq through wavelets.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' [17] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Mei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' BMO is the intersection of two translates of dyadic BMO.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 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+page_content=' Mean oscillation over cubes and H¨older continuity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Amer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=', 15 (1964), 717–721.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' [19] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Miyamoto, E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Nakai and G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Sadasue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Martingale Orlicz-Hardy spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Nachr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' 285 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Liang and L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Ky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Real-variable theory of Musielak-Orlicz Hardy spaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Lecture Notes in Mathematics, 2182.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Springer, Cham, 2017.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' xiii+466 pp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' MULTIPLICATION BETWEEN ELEMENTS IN MARTINGALE HARDY SPACES AND THEIR DUALS 33 (O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Bakas) BCAM - Basque Center for Applied Mathematics, 48009 Bilbao, Spain Email address: obakas@bcamath.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='org (Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Xu) Laboratoire de Math´ematiques, Universit´e de Bourgogne Franche-Comt´e, 25030 Besanc¸on Cedex, France Email address: xu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='zhendong@univ-fcomte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='fr (Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Zhai) School of Mathematical and Statistical Sciences, Clemson University, 29634 South Car- olina, USA Email address: zhai@clemson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='edu (H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content=' Zhang) Laboratoire de Math´ematiques, Universit´e de Bourgogne Franche-Comt´e, 25030 Besanc¸on Cedex, France Email address: hao.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='zhang@univ-fcomte.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} +page_content='fr' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFAT4oBgHgl3EQf4B4e/content/2301.08723v1.pdf'} diff --git a/ZdFLT4oBgHgl3EQfWC9a/vector_store/index.faiss b/ZdFLT4oBgHgl3EQfWC9a/vector_store/index.faiss new file mode 100644 index 0000000000000000000000000000000000000000..8592b1d39388f0693b1118e85314be5e1c2dd584 --- /dev/null +++ b/ZdFLT4oBgHgl3EQfWC9a/vector_store/index.faiss @@ -0,0 +1,3 @@ +version https://git-lfs.github.com/spec/v1 +oid sha256:5275ec6cbe1f76146b27dfbcdabf6ffb93b98ec5f2f2adf70e293a08f206fd0e +size 3014701 diff --git a/ZdFRT4oBgHgl3EQfPzc_/content/tmp_files/2301.13518v1.pdf.txt b/ZdFRT4oBgHgl3EQfPzc_/content/tmp_files/2301.13518v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..1dcd0d865a9b952452b4404b85c62456e039e447 --- /dev/null +++ b/ZdFRT4oBgHgl3EQfPzc_/content/tmp_files/2301.13518v1.pdf.txt @@ -0,0 +1,951 @@ +arXiv:2301.13518v1 [math.NT] 31 Jan 2023 +Relations between values of arithmetic Gevrey series, +and applications to values of the Gamma function +S. Fischler and T. Rivoal +February 1, 2023 +Abstract +We investigate the relations between the rings E, G and D of values taken at alge- +braic points by arithmetic Gevrey series of order either −1 (E-functions), 0 (analytic +continuations of G-functions) or 1 (renormalization of divergent series solutions at +∞ of E-operators) respectively. We prove in particular that any element of G can +be written as multivariate polynomial with algebraic coefficients in elements of E +and D, and is the limit at infinity of some E-function along some direction. This +prompts to defining and studying the notion of mixed functions, which generalizes +simultaneously E-functions and arithmetic Gevrey series of order 1. Using natural +conjectures for arithmetic Gevrey series of order 1 and mixed functions (which are +analogues of a theorem of Andr´e and Beukers for E-functions) and the conjecture +D ∩ E = Q (but not necessarily all these conjectures at the same time), we deduce a +number of interesting Diophantine results such as an analogue for mixed functions of +Beukers’ linear independence theorem for values of E-functions, the transcendance +of the values of the Gamma function and its derivatives at all non-integral algebraic +numbers, the transcendance of Gompertz constant as well as the fact that Euler’s +constant is not in E. +1 +Introduction +A power series �∞ +n=0 +an +n! xn ∈ Q[[x]] is said to be an E-function when it is solution of a linear +differential equation over Q(x) (holonomic), and |σ(an)| (for any σ ∈ Gal(Q/Q)) and the +least common denominator of a0, a1, . . . , an grow at most exponentially in n. They were +defined and studied by Siegel in 1929, who also defined the class of G-functions: a power +series �∞ +n=0 anxn ∈ Q[[x]] is said to be a G-function when �∞ +n=0 +an +n! xn is an E-function. In +this case, �∞ +n=0 n!anzn ∈ Q[[z]] is called an Э-function, following the terminology intro- +duced by Andr´e in [1]. E-functions are entire, while G-functions have a positive radius of +convergence, which is finite except for polynomials. Here and below, we see Q as embedded +into C. Following Andr´e again, E-functions, G-functions and Э-fonctions are exactly arith- +metic Gevrey series of order s = −1, 0, 1 respectively. Actually Andr´e defines arithmetic +1 + +Gevrey series of any order s ∈ Q, but the set of values at algebraic points is the same for +a given s ̸= 0 as for s/|s| using [1, Corollaire 1.3.2]. +Э-functions are divergent series, unless they are polynomials. Given an Э-function f +and any θ ∈ R, except finitely many values mod 2π (namely anti-Stokes directions of +f), one can perform Ramis’ 1-summation of f(1/z) in the direction θ, which coincides +in this setting with Borel-Laplace summation (see [14] or [9]). This provides a function +denoted by fθ(1/z), holomorphic on the open subset of C consisting in all z ̸= 0 such that +θ− π +2 −ε < arg z < θ+ π +2 +ε for some ε > 0, of which f(1/z) is the asymptotic expansion in +this sector (called a large sector bisected by θ). Of course f(1/z) can be extended further +by analytic continuation, but this asymptotic expansion may no longer be valid. When an +Э-function is denoted by fj, we shall denote by fj,θ or fj;θ its 1-summation and we always +assume (implicitly or explicitly) that θ is not an anti-Stokes direction. +In [8], [9] and [10, §4.3], we have studied the sets G, E and D defined respectively as +the sets of all the values taken by all (analytic continuations of) G-functions at algebraic +points, of all the values taken by all E-functions at algebraic points and of all values fθ(1) +where f is an Э-function (θ = 0 if it is not an anti-Stokes direction, and θ > 0 is very small +otherwise.) These three sets are countable sub-rings of C that all contain Q; conjecturally, +they are related to the set of periods and exponential periods, see §3. (The ring D is +denoted by Э in [10].) +We shall prove the following result in §3. +Theorem 1. Every element of G can be written as a multivariate polynomial (with coef- +ficients in Q) in elements of E and D. +Moreover, G coincides with the set of all convergent integrals +� ∞ +0 F(x)dx where F is +an E-function, or equivalently with the set of all finite limits of E-functions at ∞ along +some direction. +Above, a convergent integral +� ∞ +0 F(x)dx means a finite limit of the E-function +� z +0 F(x)dx +as z → ∞ along some direction; this explains the equivalence of both statements. +We refer to Eq. (3.2) in §3 for an expression of log(2) as a polynomial in elements in E +and D; the number π could be similarly expressed by considering z and iz instead of z and +2z there. Examples of the last statement are the identities (see [12] for the second one): +� +∞ +0 +sin(x) +x +dx = π +2 +and +� +∞ +0 +J0(ix)e−3xdx = +√ +6 +96π3Γ +� 1 +24 +� +Γ +� 5 +24 +� +Γ +� 7 +24 +� +Γ +�11 +24 +� +. +It is notoriously difficult to prove/disprove that a given element of G is transcendental; +it is known that a Siegel-Shidlovskii type theorem for G-functions can not hold mutatis +mutandis. Theorem 1 suggests that an alternative approach to the study of the Diophantine +properties of elements of G can be through a better understanding of joint study of the +elements of E and D, modulo certain conjectures to begin with. Our applications will +not be immediately directed to the elements of G but rather to the understanding of the +(absence of) relations between the elements of E and D. +2 + +It seems natural (see [9, p. 37]) to conjecture that E ∩ G = Q, and also that G ∩ D = +Q, though both properties seem currently out of reach. In this paper, we suggest (see §2) +a possible approach towards the following analogous conjecture. +Conjecture 1. We have E ∩ D = Q. +In §2 we shall make a functional conjecture, namely Conjecture 3, that implies Conjec- +ture 1. We also prove that Conjecture 1 has very important consequences, as the following +result shows. +Theorem 2. Assume that Conjecture 1 holds. Then Γ(s)(a) is a transcendental number +for any rational number a > 0 and any integer s ≥ 0, except of course if s = 0 and a ∈ N. +One of the aims of this paper is to show that combining Э- and E-functions may lead +to very important results in transcendental number theory. Let us recall now briefly the +main known results on E-functions. +Point (i) in the following result is due to Andr´e [2] for E-functions with rational Taylor +coefficients, and to Beukers [6] in the general case. Andr´e used this property to obtain a +new proof of the Siegel-Shidlovskii Theorem, and Beukers to prove an optimal refinement +of this theorem (namely, (ii) below). +Theorem A. (i) [Andr´e, Beukers] If an E-function F(z) is such that F(1) = 0, then F (z) +z−1 +is an E-function. +(ii) [Beukers] Let F(z) := t(f1(z), . . . , fn(z)) be a vector of E-functions solution of a +differential system F ′(z) = A(z)F(z) for some matrix A(z) ∈ Mn(Q(z)). +Let ξ ∈ Q +∗ which is not a pole of a coefficient of A. Let P ∈ Q[X1, . . . , Xn] be a +homogeneous polynomial such that +P(f1(ξ), . . . , fn(ξ)) = 0. +Then there exists Q ∈ Q[Z, X1, . . . , Xn], homogeneous in the Xi, such that +Q(z, f1(z), . . . , fn(z)) = 0 identically and P(X1, . . . , Xn) = Q(ξ, X1, . . . , Xn). +In particular, we have +trdegQ(f1(ξ), . . . , fn(ξ)) = trdegQ(z)(f1(z), . . . , fn(z)). +The Siegel-Shidlovskii Theorem itself is the final statement about equality of transcen- +dence degrees. +In this paper we state conjectural analogues of these results, involving Э-functions. +The principal difficulty is that these functions are divergent power series, and the exact +analogue of Theorem A is meaningless. +Andr´e discussed the situation in [2] and even +though he did not formulate exactly the following conjecture, it seems plausible to us. +From it, we will show how to deduce an analogue of the Siegel-Shidlovskii theorem for +Э-functions. Ferguson [7, p. 171, Conjecture 1] essentially stated this conjecture when f(z) +has rational coefficients and when θ = 0. +3 + +Conjecture 2. Let f(z) be an Э-function and θ ∈ (−π/2, π/2) be such that fθ(1) = 0. +Then f(z) +z−1 is an Э-function. +In other words, the conclusion of this conjecture asserts that +z +1−zf(1/z) is an Э-function +in 1/z; this is equivalent to f(1/z) +z−1 being an Э-function in 1/z (since we have f(1/z) +z−1 = O(1/z) +unconditionally as |z| → ∞). +Following Beukers’ proof [6] yields the following result (see [3, §4.6] for a related con- +jecture). +Theorem 3. Assume that Conjecture 2 holds. +Let f(z) := t(f1(z), . . . , fn(z)) be a vector of Э-functions solution of a differential system +f′(z) = A(z)f(z) for some matrix A(z) ∈ Mn(Q(z)). +Let ξ ∈ Q +∗ and θ ∈ (arg(ξ) − +π/2, arg(ξ) + π/2) ; assume that ξ is not a pole of a coefficient of A, and that θ is anti- +Stokes for none of the fj. +Let P ∈ Q[X1, . . . , Xn] be a homogeneous polynomial such that +P(f1,θ(1/ξ), . . . , fn,θ(1/ξ)) = 0. +Then there exists Q ∈ Q[Z, X1, . . . , Xn], homogeneous in the Xi, such that +Q(z, f1(z), . . . , fn(z)) = 0 identically and P(X1, . . . , Xn) = Q(1/ξ, X1, . . . , Xn). +In particular, we have +trdegQ(f1,θ(1/ξ), . . ., fn,θ(1/ξ)) = trdegQ(z)(f1(z), . . . , fn(z)). +As an application of Theorem 3, we shall prove the following corollary. Note that under +his weaker version of Conjecture 2, Ferguson [7, p. 171, Theorem 2] proved that Gompertz’s +constant is an irrational number. +Corollary 1. Assume that Conjecture 2 holds. Then for any α ∈ Q, α > 0, and any +s ∈ Q \ Z≥0, the number +� ∞ +0 (t + α)se−tdt is a transcendental number. +In particular, Gompertz’s constant δ := +� ∞ +0 e−t/(t + 1)dt is a transcendental number. +In this text we suggest an approach towards Conjecture 1, based on the new notion +of mixed functions which enables one to consider E- and Э-functions at the same time. +In particular we shall state a conjecture about such functions, namely Conjecture 3 in §2, +which implies both Conjecture 1 and Conjecture 2. The following result is a motivation +for this approach. +Proposition 1. Assume that both Conjectures 1 and 2 hold. Then neither Euler’s constant +γ := −Γ′(1) nor Γ(a) (with a ∈ Q+ \ N) are in E. +It is likely that none of these numbers is in G, but (as far as we know) there is no +“functional” conjecture like Conjecture 3 that implies this. It is also likely that none is in +D as well, but we don’t know if this can be deduced from Conjecture 3. +4 + +The structure of this paper is as follows. In §2 we define and study mixed functions, a +combination of E- and Э-functions. Then in §3 we express any value of a G-function as a +polynomial in values of E- and Э-functions, thereby proving Theorem 1. We study deriva- +tives of the Γ function at rational points in §4, and prove Theorem 2 and Proposition 1. +At last, §5 is devoted to adapting Beukers’ method to our setting: this approach yields +Theorem 3 and Corollary 1. +2 +Mixed functions +2.1 +Definition and properties +In view of Theorem 1, it is natural to study polynomials in E- and Э-functions. We can +prove a Diophantine result that combines both Theorems A(ii) and 3 but under a very com- +plicated polynomial generalization of Conjecture 2. We opt here for a different approach +to mixing E- and Э-functions for which very interesting Diophantine consequences can be +deduced from a very easy to state conjecture which is more in the spirit of Conjecture 2. +We refer to §2.3 for proofs of all properties stated in this section (including Lemma 1 and +Proposition 2), except Theorem 4. +Definition 1. We call mixed (arithmetic Gevrey) function any formal power series +� +n∈Z +anzn +such that � +n≥0 anzn is an E-function in z, and � +n≥1 a−nz−n is an Э-function in 1/z. +In other words, a mixed function is defined as a formal sum Ψ(z) = F(z)+f(1/z) where +F is an E-function and f is an Э-function. In particular, such a function is zero if, and +only if, both F and f are constants such that F + f = 0; obviously, F and f are uniquely +determined by Ψ upon assuming (for instance) that f(0) = 0. The set of mixed functions +is a Q-vector space stable under multiplication by zn for any n ∈ Z. Unless f(z) is a +polynomial, such a function Ψ(z) = F(z) + f(1/z) is purely formal: there is no z ∈ C such +that f(1/z) is a convergent series. However, choosing a direction θ which is not anti-Stokes +for f allows one to evaluate Ψθ(z) = F(z) + fθ(1/z) at any z in a large sector bisected by +θ. Here and below, such a direction will be said not anti-Stokes for Ψ and whenever we +write fθ or Ψθ we shall assume implicitly that θ is not anti-Stokes. +Definition 1 is a formal definition, but one may identify a mixed function with the +holomorphic function it defines on a given large sector by means of the following lemma. +Lemma 1. Let Ψ be a mixed function, and θ ∈ R be a non-anti-Stokes direction for Ψ. +Then Ψθ is identically zero (as a holomorphic function on a large sector bisected by θ) if, +and only if, Ψ is equal to zero (as a formal power series in z and 1/z). +5 + +Any mixed function Ψ(z) = F(z) + f(1/z) is solution of an E-operator. Indeed, this +follows from applying [1, Theorem 6.1] twice: there exist an E-operator L such that +L(f(1/z)) = 0, and an E-operator M such that M(L(F(z))) = 0 (because L(F(z)) is an +E-function). Hence ML(F(z) + f(1/z)) = 0 and by [1, p. 720, §4.1], ML is an E-operator. +We formulate the following conjecture, which implies both Conjecture 1 and Conjec- +ture 2. +Conjecture 3. Let Ψ(z) be an mixed function, and θ ∈ (−π/2, π/2) be such that Ψθ(1) = 0. +Then Ψ(z) +z−1 is an mixed function. +The conclusion of this conjecture is that Ψ(z) = (z − 1)Ψ1(z) for some mixed function +Ψ1. This conclusion can be made more precise as follows; see §2.3 for the proof. +Proposition 2. Let Ψ(z) = F(z) + f(1/z) be an mixed function, and θ ∈ (−π/2, π/2) be +such that Ψθ(1) = 0. Assume that Conjecture 3 holds for Ψ and θ. +Then both F(1) and fθ(1) are algebraic, and f(1/z)−fθ(1) +z−1 +is an Э-function. +Of course, in the conclusion of this proposition, one may assert also that F (z)−F (1) +z−1 +is an +E-function using Theorem A(i). +Conjecture 3 already has far reaching Diophantine consequences: Conjecture 2 and +Theorem 2 stated in the introduction, and also the following result that encompasses +Theorem 3 in the linear case. +Theorem 4. Assume that Conjecture 3 holds. +Let Ψ(z) := t(Ψ1(z), . . . , Ψn(z)) be a vector of mixed functions solution of a differential +system Ψ′(z) = A(z)Ψ(z) for some matrix A(z) ∈ Mn(Q(z)). +Let ξ ∈ Q +∗ and θ ∈ +(arg(ξ) − π/2, arg(ξ) + π/2) ; assume that ξ is not a pole of a coefficient of A, and that θ +is anti-Stokes for none of the Ψj. +Let λ1, . . . , λn ∈ Q be such that +n +� +i=1 +λiΨi,θ(ξ) = 0. +Then there exist L1, . . . , Ln ∈ Q[z] such that +n +� +i=1 +Li(z)Ψi(z) = 0 identically and Li(ξ) = λi for any i. +In particular, we have +rkQ(Ψ1,θ(ξ), . . . , Ψn,θ(ξ)) = rkQ(z)(Ψ1(z), . . . , Ψn(z)). +The proof of Theorem 4 follows exactly the linear part of the proof of Theorem 3 (see +§5.1), which is based on [6, §3]. The only difference is that Э-functions have to be replaced +6 + +with mixed functions, and Conjecture 2 with Conjecture 3. In particular Proposition 4 +stated in §5.1 remains valid with these modifications. +However a product of mixed functions is not, in general, a mixed function. Therefore +the end of [6, §3] does not adapt to mixed functions, and there is no hope to obtain in this +way a result on the transcendence degree of a field generated by values of mixed functions. +As an application of Theorem 4, we can consider the mixed functions 1, eβz and f(1/z) := +�∞ +n=0(−1)nn!z−n, where β is a fixed non-zero algebraic number. These three functions are +linearly independent over C(z) and form a solution of a differential system with only 0 for +singularity (because (f(1/z))′ = (1 + 1/z)f(1/z) − 1), hence for any α ∈ Q, α > 0 and any +̺ ∈ Q +∗, the numbers 1, e̺, f0(1/α) := +� ∞ +0 e−t/(1 + αt)dt are Q-linearly independent (for a +fixed α, take β = ̺/α). +2.2 +Values of mixed functions +We denote by MG the set of values Ψθ(1), where Ψ is a mixed function and θ = 0 if it is +not anti-Stokes, θ > 0 is sufficiently small otherwise. This set is obviously equal to E + D. +Proposition 3. For every integer s ≥ 0 and every a ∈ Q+, a ̸= 0, we have Γ(s)(a) ∈ +e−1MG. +This results follows immediately from Eq. (4.4) below (see §4.2), written in the form +Γ(s)(a) = e−1� +(−1)ses!Ea,s+1(−1) + fa,s+1;0(1) +� +, +because ezEa,s+1(−z) is an E-function and fa,s+1;0(1) is the 1-summation in the direction +0 of an Э-function. +It would be interesting to know if Γ(s)(a) belongs to MG. We did not succeed in proving +it does, and we believe it does not. Indeed, for instance if we want to prove that γ ∈ MG, +a natural strategy would be to construct an E-function F(z) with asymptotic expansion +of the form γ + log(z) + f(1/z) in a large sector, and then to evaluate at z = 1. However +this strategy cannot work since there is no such E-function (see the footnote in the proof +of Lemma 1 in §2.3). +2.3 +Proofs concerning mixed functions +To begin with, let us take Proposition 2 for granted and prove that Conjecture 3 implies +both Conjecture 1 and Conjecture 2. Concerning Conjecture 2 it is clear. To prove that it +implies Conjecture 1, let ξ ∈ D, i.e. ξ = fθ(1) is the 1-summation of an Э-function f(z) in +the direction θ = 0 if it is not anti-Stokes, and θ > 0 close to 0 otherwise. Assume that ξ +is also in E: we have ξ = F(1) for some E-function F(z). Therefore, Ψ(z) = F(z) − f(1/z) +is a mixed function such that Ψθ(1) = 0. By Conjecture 3 and Proposition 2, we have +ξ = fθ(1) ∈ Q. This concludes the proof that Conjecture 3 implies Conjecture 1. +7 + +Let us prove Proposition 2 now. Assuming that Conjecture 3 holds for Ψ and θ, there +exists a mixed function Ψ1(z) = F1(z) + f1(1/z) such that Ψ(z) = (z − 1)Ψ1(z). We have +F(z) − (z − 1)F1(z) + f(1/z) − (z − 1)f1(1/z) = 0 +(2.1) +as a formal power series in z and 1/z. Now notice that z − 1 = z(1 − 1 +z), and that we +may assume f and f1 to have zero constant terms. +Denote by α the constant term of +f(1/z) − z(1 − 1 +z)f1(1/z). Then we have +F(z) − (z − 1)F1(z) + α + f2(1/z) = 0 +for some Э-function f2 without constant term, so that f2 = 0, F(z) = (z − 1)F1(z) − α and +F(1) = −α ∈ Q. This implies fθ(1) = α, and f(1/z)−fθ(1) +z−1 += f1(1/z) is an Э-function since +f2 = 0. This concludes the proof of Proposition 2. +At last, let us prove Lemma 1. We write Ψ(z) = F(z) + f(1/z) and assume that Ψθ +is identically zero. Modifying θ slightly if necessary, we may assume that the asymptotic +expansion −f(1/z) of F(z) in a large sector bisected by θ is given explicitly by [9, Theorem 5] +applied to F(z) − F(0); recall that such an asymptotic expansion is unique (see [9]). As +in [9] we let g(z) = �∞ +n=1 anz−n−1 where the coefficients an are given by F(z) − F(0) = +�∞ +n=1 +an +n! zn. For any σ ∈ C\{0} there is no contribution in eσz in the asymptotic expansion +of F(z), so that g(z) is holomorphic at σ. At σ = 0, the local expansion of g is of the +form g(z) = h1(z) + h2(z) log(z) with G-functions h1 and h2, and the coefficients of h2 +are related to those of f; however we shall not use this special form (1). Now recall that +g(z) = G(1/z)/z where G is a G-function; then G is entire and has moderate growth at +infinity (because ∞ is a regular singularity of G), so it is a polynomial due to Liouville’s +theorem. This means that F(z) is a polynomial in z. Recall that asymptotic expansions +in large sectors are unique. Therefore both F and f are constant functions, and F + f = 0. +This concludes the proof of Lemma 1. +3 +Proof of Theorem 1: values of G-functions +In this section we prove Theorem 1. Let us begin with an example, starting with the +relation proved in [15, Proposition 1] for z ∈ C \ (−∞, 0]: +γ + log(z) = zE1,2(−z) − e−zf1,2;0(1/z) +(3.1) +where E1,2 is an E-function, and f1,2 is an Э-function, both defined below in §4.2. +Apply Eq. (3.1) at both z and 2z, and then substract one equation from the other. +This provides a relation of the form +log(2) = F(z) + e−zf1;0(1/z) + e−2zf2;0(1/z) +(3.2) +1Actually we are proving that the asymptotic expansion of a non-polynomial E-function is never a C- +linear combination of functions zα logk(z)f(1/z) with α ∈ Q, k ∈ N and Э-functions f: some exponentials +have to appear. +8 + +valid in a large sector bisected by 0, with an E-function F and Э-functions f1 and f2. +Choosing arbitrarily a positive real algebraic value of z yields an explicit expression of +log(2) ∈ G as a multivariate polynomial in elements of E and D. But this example shows +also that a polynomial in E- and Э-functions may be constant eventhough there does not +seem to be any obvious reason. In particular, the functions 1, F(z), e−zf1;0(1/z), and +e−2zf2;0(1/z) are linearly dependent over C. However we see no reason why they would +be linearly dependent over Q. This could be a major drawback to combine in E- and +Э-functions, since functions that are linearly dependent over C but not over Q can not +belong to any Picard-Vessiot extension over Q. +Let us come now to the proof of Theorem 1. We first prove the second part, which runs +as follows (it is reproduced from the unpublished note [16]). +From the stability of the class of E-functions by +d +dz and +� z +0 , we deduce that the set of +convergent integrals +� ∞ +0 F(x)dx of E-functions and the set of finite limits of E-functions +along some direction as z → ∞ are the same. Theorem 2(iii) in [9] implies that if an +E-function has a finite limit as z → ∞ along some direction, then this limit must be in G. +Conversely, let β ∈ G. By Theorem 1 in [8], there exists a G-function G(z) = �∞ +n=0 anzn +of radius of convergence ≥ 2 (say) such that G(1) = β. Let F(z) = �∞ +n=0 +an +n! zn be the +associated E-function. Then for any z such that Re(z) > 1 +2, we have +1 +zG +�1 +z +� += +� +∞ +0 +e−xzF(x)dx. +Hence, β = +� +∞ +0 +e−xF(x)dx where e−zF(z) is an E-function. +We shall now prove the first part of Theorem 1. In fact, we shall prove a slightly more +general result, namely Theorem 5 below. We first recall a few notations. Denote by S the +G-module generated by all derivatives Γ(s)(a) (with s ∈ N and a ∈ Q\ Z≤0), and by V the +S-module generated by E. Recall that G, S and V are rings. Conjecturally, G = P[1/π] +and V = Pe[1/π] where P and Pe are the ring of periods and the ring of exponential +periods over Q respectively (see [8, §2.2] and [10, §4.3]). We have proved in [10, Theorem +3] that V is the S-module generated by the numbers eρχ, with ρ ∈ Q and χ ∈ D. +Theorem 5. The ring V is the ring generated by E and D. In particular, all values of +G-functions belong to the ring generated by E and D. +In other words, the elements of V are exactly the sums of products ab with a ∈ E and +b ∈ D. +Proof of Theorem 5. We already know that V is a ring, and that it contains E and D. To +prove the other inclusion, denote by U the ring generated by E and D. Using Proposition 3 +proved in §2.2 and the functional equation of Γ, we have Γ(s)(a) ∈ U for any s ∈ N and +any a ∈ Q \ Z≤0. Therefore for proving that V ⊂ U, it is enough to prove that G ⊂ U. +Let ξ ∈ G. Using [11, Theorem 3] there exists an E-function F(z) such that for any for +any θ ∈ [−π, π) outside a finite set, ξ is a coefficient of the asymptotic expansion of F(z) +9 + +in a large sector bisected by θ. As the proof of [11, Theorem 3] shows, we can assume that +ξ is the coefficient of ez in this expansion. +Denote by L an E-operator of which F is a solution, and by µ its order. Andr´e has +proved [1] that there exists a basis (H1(z), . . . , Hµ(z)) of formal solutions of L at infinity +such that for any j, e−ρjzHj(z) ∈ NGA{1/z}Q +1 for some algebraic number ρj. We recall +that elements of NGA{1/z}Q +1 are arithmetic Nilsson-Gevrey series of order 1 with algebraic +coefficients, i.e. Q-linear combinations of functions zk(log z)ℓf(1/z) with k ∈ Q, ℓ ∈ N +and Э-functions f. Expanding in this basis the asymptotic expansion of F(z) in a large +sector bisected by θ (denoted by �F), there exist complex numbers κ1, . . . , κd such that +�F(z) = κ1H1(z) + . . . + κµHµ(z). Then we have ξ = κ1c1 + . . . + κµcµ, where cj is the +coefficient of ez in Hj(z) ∈ eρjzNGA{1/z}Q +1 . We have cj = 0 if ρj ̸= 1, and otherwise cj is +the constant coefficient of e−zHj(z): in both cases cj is an algebraic number. Therefore to +conclude the proof that ξ ∈ U, it is enough to prove that κ1, . . . , κµ ∈ U. +For simplicity let us prove that κ1 ∈ U. Given solutions F1, . . . , Fµ of L, we denote +by W(F1, . . . , Fµ) the corresponding wronskian matrix. Then for any z in a large sector +bisected by θ we have +κ1 = det W(F(z), H2,θ(z), . . . , Hµ,θ(z)) +det W(H1,θ(z), . . . , Hµ,θ(z)) +where Hj,θ(z) is the 1-sommation of Hj(z) in this sector. The determinant in the denomi- +nator belongs to eazNGA{1/z}Q +1 with a = ρ1+. . .+ρµ ∈ Q. As the proof of [10, Theorem 6] +shows, there exist b, c ∈ Q, with c ̸= 0, such that +det W(H1,θ(z), . . . , Hµ,θ(z)) = czbeaz. +We take z = 1, and choose θ = 0 if it is not anti-Stokes for L (and θ > 0 sufficiently small +otherwise). Then we have +κ1 = c−1e−a� +det W(F(z), H2,θ(z), . . . , Hµ,θ(z)) +� +|z=1 ∈ U. +This concludes the proof. +Remark 1. The second part of Theorem 1 suggests the following comments. +It would +be interesting to have a better understanding (in terms of E, G and D) of the set of +convergent integrals +� ∞ +0 R(x)F(x)dx where R is a rational function in Q(x) and F is an +E-function, which are thus in G when R = 1 (see [16] for related considerations). Indeed, +classical examples of such integrals are +� +∞ +0 +cos(x) +1+x2 dx = π/(2e) ∈ πE, Euler’s constant +� +∞ +0 +1−(1+x)e−x +x(1+x) +dx = γ ∈ E + e−1D (using Eq. (3.1) and [20, p. 248, Example 2]) and +Gompertz constant δ := +� +∞ +0 +e−x +1+xdx ∈ D. A large variety of behaviors can thus be expected +here. +10 + +For instance, using various explicit formulas in [13, Chapters 6.5–6.7], it can be proved +that +� +∞ +0 +R(x)J0(x)dx ∈ G + E + γE + log(Q +⋆)E +for any R(x) ∈ Q(x) without poles on [0, +∞), where J0(x) = �∞ +n=0(ix/2)2n/n!2 is a Bessel +function. +A second class of examples is when R(x)F(x) is an even function without poles on +[0, +∞) and such that lim|x|→∞,Im(x)≥0 x2R(x)F(x) = 0. Then by the residue theorem, +� +∞ +0 +R(x)F(x)dx = iπ +� +ρ, Im(ρ)>0 +Resx=ρ +� +R(x)F(x) +� +∈ πE +where the summation is over the poles of R in the upper half plane. +4 +Derivatives of the Γ function at rational points +In this section we prove Theorem 2 and Proposition 1 stated in the introduction, dealing +with Γ(s)(a). +To begin with, we define E-functions Ea,s(z) in §4.1 and prove a linear +independence result concerning these functions. +Then we prove in §4.2 a formula for +Γ(s)(a), namely Eq. (4.4), involving Ea,s+1(−1) and the 1-summation of an Э-function. +This enables us to prove Theorem 2 in §4.3 and Proposition 1 in §4.4. +4.1 +Linear independence of a family of E-functions +To study derivatives of the Γ function at rational points, we need the following lemma. +For s ≥ 1 and a ∈ Q \ Z≤0, we consider the E-function Ea,s(z) := �∞ +n=0 +zn +n!(n+a)s. +Lemma 2. (i) For any a ∈ Q \ Z and any s ≥ 1, the functions +1, ez, ezEa,1(−z), ezEa,2(−z), . . . , ezEa,s(−z) +are linearly independent over C(z). +(ii) For any a ∈ N∗ and any s ≥ 2, the functions +1, ez, ezEa,2(−z), . . . , ezEa,s(−z) +are linearly independent over C(z). +Remark 2. Part (i) of the lemma is false if a ∈ N∗ because 1, ez, ezEa,1(−z) are Q(z)-linearly +dependent in this case (see the proof of Part (ii) below). +11 + +Proof. (i) For simplicity, we set ψs(z) := ezEa,s(−z). We proceed by induction on s ≥ 1. +Let us first prove the case s = 1. The derivative of ψ1(z) is (1 + (z − a)ψ1(z))/z. Let +us assume the existence of a relation ψ1(z) = u(z)ez + v(z) with u, v ∈ C(z) (a putative +relation U(z) + V (z)ez + W(z)ψ1(z) = 0 forces W ̸= 0 because ez /∈ C(z)). Then after +differentiation of both sides, we end up with +1 + (z − a)ψ1(z) +z += +� +u(z) + u′(z) +� +ez + v′(z). +Hence, +1 + (z − a) +� +u(z)ez + v(z) +� +z += +� +u(z) + u′(z) +� +ez + v′(z). +Since ez /∈ C(z), the function v(z) is a rational solution of the differential equation zv′(z) = +(z − a)v(z) + 1: v(z) cannot be identically 0, and it cannot be a polynomial (the degrees +do not match on both sides). It must then have a pole at some point ω, of order d ≥ 1 +say. We must have ω = 0 because otherwise the order of the pole at ω of zv′(z) is d + 1 +while the order of the pole of (z − a)v(z) + 1 is at most d. Writing v(z) = � +n≥−d vnzn +with v−d ̸= 0 and comparing the term in z−d of zv′(z) and (z − a)v(z) + 1, we obtain that +d = a. This forces a to be an integer ≥ 1, which is excluded. Hence, 1, ez, ezEa,1(−z) are +C(z)-linearly independent. +Let us now assume that the case s − 1 ≥ 1 holds. Let us assume the existence of a +relation over C(z) +ψs(z) = v(z) + u0(z)ez + +s−1 +� +j=1 +uj(z)ψj(z). +(4.1) +(A putative relation V (z) + U0(z)ez + �s +j=1 Uj(z)ψj(z) = 0 forces Us ̸= 0 by the induction +hypothesis). Differentiating (4.1) and because ψ′ +j(z) = (1− a +z)ψj(z)+ 1 +zψj−1(z) for all j ≥ 1 +(where we have let ψ0(z) = 1), we have +A(z)ψs(z) + 1 +zψs−1(z) = v′(z) + +� +u0(z) + u′ +0(z) +� +ez + +s−1 +� +j=1 +u′ +j(z)ψj(z) ++ +s−1 +� +j=1 +uj(z) +� +A(z)ψj(z) + 1 +zψj−1(z) +� +, +(4.2) +where A(z) := 1 −a/z. Substituting the right-hand side of (4.1) for ψs(z) on the left-hand +side of (4.2), we then deduce that +v′(z) − A(z)v(z) + +� +u′ +0(z) + (1 − A(z))u0(z) +� +ez ++ 1 +z(z − a)u1(z)ψ1(z) + +s−1 +� +j=1 +u′ +j(z)ψj(z) + 1 +z +s−1 +� +j=1 +uj(z)ψj−1(z) − 1 +zψs−1(z) = 0. +12 + +This is a non-trivial C(z)-linear relation between 1, ez, ψ1(z), ψ2(z), . . . , ψs−1(z) because +the coefficient of ψs−1(z) is u′ +s−1(z) −1/z and it is not identically 0 because u′ +s−1(z) cannot +have a pole of order 1. But by the induction hypothesis, we know that such a relation is +impossible. +(ii) The proof can be done by induction on s ≥ 2 similarily. In the case s = 2, assume +the existence of a relation ψ2(z) = u(z)ez +v(z) with u(z), v(z) ∈ C(z). By differentiation, +we obtain +� +1 − a +z +� +ψ2(z) = −1 +zψ1(z) + +� +u(z) + u′(z) +� +ez + v′(z). +By induction on a ≥ 1, we have ψ1(z) = (a−1)!ez/za +w(z) for some w(z) ∈ Q(z). Hence, +we have +� +1 − a +z +� +u(z) = − +�(a − 1)! +za+1 ++ 1 +� +u(z) + u′(z) +which is not possible. Let us now assume that the case s − 1 ≥ 2 holds, as well as the +existence of a relation over C(z) +ψs(z) = v(z) + u0(z)ez + +s−1 +� +j=2 +uj(z)ψj(z). +(4.3) +We proceed exactly as above by differentiation of both sides of (4.3). Using the relation +ψ′ +j(z) = (1− a +z)ψj(z)+ 1 +zψj−1(z) for all j ≥ 2 and the fact that ψ1(z) = (a−1)!ez/za+w(z), +we obtain a relation �v(z) + �u0(z)ez + �s−1 +j=2 �uj(z)ψj(z) = 0 where �us−1(z) = u′ +s−1(z) − +1/z cannot be identically 0. The induction hypothesis rules out the existence of such a +relation. +4.2 +A formula for Γ(s)(a) +Let z > 0 and a ∈ Q+, a ̸= 0. We have +Γ(s)(a) = +� ∞ +0 +ta−1 log(t)se−tdt = +� z +0 +ta−1 log(t)se−tdt + +� ∞ +z +ta−1 log(t)se−tdt. +On the one hand, +� z +0 +ta−1 log(t)se−tdt = +∞ +� +n=0 +(−1)n +n! +� z +0 +ta+n−1 log(t)sdt += +∞ +� +n=0 +(−1)n +n! +s +� +k=0 +(−1)k +s! +(s − k)! +zn+a log(z)s−k +(n + a)k+1 += +s +� +k=0 +(−1)ks! +(s − k)!za log(z)s−kEa,k+1(−z); +13 + +recall that Ea,j(z) = �∞ +n=0 +zn +n!(n+a)j . On the other hand, +� ∞ +z +ta−1 log(t)se−tdt = e−z +� ∞ +0 +(t + z)a−1 log(t + z)se−tdt += za−1e−z +s +� +k=0 +�s +k +� +log(z)s−k +� ∞ +0 +(1 + t/z)a−1 log(1 + t/z)ke−tdt. +Now z > 0 so that +fa,k+1;0(z) := +� ∞ +0 +(1 + tz)a−1 log(1 + tz)ke−tdt = 1 +z +� ∞ +0 +(1 + x)a−1 log(1 + x)ke−x/zdx +is the 1-summation at the origin in the direction 0 of the Э-function +∞ +� +n=0 +n!ua,k,nzn, +where the sequence (ua,k,n)n≥0 ∈ QN is defined by the expansion of the G-function: +(1 + x)a−1 log(1 + x)k = +∞ +� +n=0 +ua,k,nxn. +Note that if k = 0 and a ∈ N∗, then ua,k,n = 0 for any n ≥ a, and fa,k+1;0(1/z) is a +polynomial in 1/z. Therefore, we have for any z > 0: +Γ(s)(a) = +s +� +k=0 +(−1)ks! +(s − k)!za log(z)s−kEa,k+1(−z) + za−1e−z +s +� +k=0 +�s +k +� +log(z)s−kfa,k+1;0(1/z). +In particular, for z = 1, this relation reads +Γ(s)(a) = (−1)ss!Ea,s+1(−1) + e−1fa,s+1;0(1). +(4.4) +Since γ = −Γ′(1) we obtain as a special case the formula +γ = E1,2(−1) − e−1f1,2;0(1), +(4.5) +which is also a special case of Eq. (3.1) proved in [15]. +4.3 +Proof of Theorem 2 +Let us assume that Γ(s)(a) ∈ Q for some a ∈ Q+ \ N and s ≥ 0. +Then ezΓ(s)(a) + +(−1)s+1s!ezEa,s+1(−z) is an E-function. The relation eΓ(s)(a) + (−1)s+1s!eEa,s+1(−1) = +fa,s+1;0(1) proved at the end of §4.2 shows that α := eΓ(s)(a)+(−1)s+1s!eEa,s+1(−1) ∈ E∩D. +14 + +Hence α is in Q by Conjecture 1 and we have a non-trivial Q-linear relation between 1, e +and eEa,s+1(−1): we claim that this is not possible. Indeed, consider the vector +Y (z) := t(1, ez, ezEa,1(−z), . . . , ezEa,s+1(−z)). +It is solution of a differential system Y ′(z) = M(z)Y (z) where 0 is the only pole of +M(z) ∈ Ms+3(Q(z)) (see the computations in the proof of Lemma 2 above). Since the +components of Y (z) are Q(z)-linearly independent by Lemma 2(i), we deduce from Beuk- +ers’ [6, Corollary 1.4] that +1, e, eEa,1(−1), . . . , eEa,s+1(−1) +are Q-linearly independent, and in particular that 1, e and eEa,s+1(−1) are Q-linearly +independent. This concludes the proof if a ∈ Q+ \ N. +Let us assume now that Γ(s)(a) ∈ Q for some a ∈ N∗ and s ≥ 1. Then ezΓ(s)(a) + +(−1)s+1s!ezEa,s+1(−z) is an E-function. +The relation Γ(s)(a) + (−1)s+1s!Ea,s+1(−1) = +e−1fa,s+1;0(1) shows that α := eΓ(s)(a)+(−1)s+1s!eEa,s+1(−1) ∈ E∩D. Hence α is in Q by +Conjecture 1 and we have a non-trivial Q-linear relation between 1, e and eEa,s+1(−1): we +claim that this is not possible. Indeed, consider the vector Y (z) := t(1, ez, ezEa,2(−z), . . . , +ezEa,s+1(−z)): it is solution of a differential system Y ′(z) = M(z)Y (z) where 0 is the only +pole of M(z) ∈ Ms+2(Q(z)). Since the components of Y (z) are Q(z)-linearly independent +by Lemma 2(ii), we deduce again from Beukers’ theorem that +1, e, eEa,2(−1), . . . , eEa,s+1(−1) +are Q-linearly independent, and in particular that 1, e and eEa,s+1(−1) are Q-linearly +independent. This concludes the proof of Theorem 2. +4.4 +Proof of Proposition 1 +Recall that Eq. (4.5) proved in §4.2 reads eE1,2(−1)−eγ = f1,2;0(1). Assuming that γ ∈ E, +the left-hand side is in E while the right-hand side is in D. Hence both sides are in Q by +Conjecture 1. Note that, by integration by parts, +f1,2;0(1) = +� ∞ +0 +log(1 + t)e−tdt = +� ∞ +0 +e−t +1 + tdt +is Gompertz’s constant. +Hence, by Corollary 1 (which holds under Conjecture 2), the +number f1,2;0(1) is not in Q. Consequently, γ /∈ E. +Similarly, Eq. (4.4) with a ∈ Q \ Z and s = 0 reads eΓ(a) − eEa,1(−1) = fa,1;0(1). +Assuming that Γ(a) ∈ E, the left-hand side is in E while the right-hand side is in D. Hence +both sides are in Q by Conjecture 1. But by Corollary 1 (which holds under Conjecture 2), +the number fa,1;0(1) = +� ∞ +0 (1 + t)a−1e−tdt is not in Q. Hence, Γ(a) /∈ E. +15 + +5 +Application of Beukers’ method and consequence +In this section we prove Theorem 3 and Corollary 1 stated in the introduction. +5.1 +Proof of Theorem 3 +The proof of Theorem 3 is based on the arguments given in [6], except that E-functions +have to be replaced with Э-functions, and 1-summation in non-anti-Stokes directions is +used for evaluations. Conjecture 2 is used as a substitute for Theorem A(i). +The main step is the following result, the proof of which is analogous to the end of the +proof of [6, Corollary 2.2]. +Proposition 4. Assume that Conjecture 2 holds. +Let f be an Э-function, ξ ∈ Q +∗ and θ ∈ (arg(ξ) − π/2, arg(ξ) + π/2). Assume that θ is +not anti-Stokes for f, and that fθ(1/ξ) = 0. Denote by Ly = 0 a differential equation, of +minimal order, satisfied by f(1/z). +Then all solutions of Ly = 0 are holomorphic and vanish at ξ; the differential operator +L has an apparent singularity at ξ. +To deduce Theorem 3 from Proposition 4, it is enough to follow [6, §3]. +5.2 +Proof of Corollary 1 +Let s ∈ Q \ Z≥0. The Э-function f(z) := �∞ +n=0 s(s − 1) . . . (s − n + 1)zn is solution of the +inhomogeneous differential equation z2f′(z)+(1−sz)f(z)−1 = 0, which can be immediately +transformed into a differential system satisfied by the vector of Э-functions t(1, f(z)). The +coefficients of the matrix have only 0 as pole. Moreover, f(z) is a transcendental function +because s /∈ Z≥0. Hence, by Theorem 3, f0(1/α) /∈ Q when α ∈ Q, α > 0, because 0 is not +an anti-Stokes direction of f(z). It remains to observe that this 1-sommation is +� ∞ +0 +(1 + tz)se−tdt. +References +[1] Y. Andr´e, S´eries Gevrey de type arithm´etique I. Th´eor`emes de puret´e et de dualit´e, +Annals of Math. 151 (2000), 705–740. +[2] Y. Andr´e, S´eries Gevrey de type arithm´etique II. Transcendance sans transcendance, +Annals of Math. 151 (2000), 741–756. +[3] Y. Andr´e, Arithmetic Gevrey series and transcendence. A survey, J. Th´eor. Nombres +Bordeaux 15 (2003), 1–10. +16 + +[4] D. Bertrand, F. Beukers, ´Equations diff´erentielles lin´eaires et majorations de multi- +plicit´es, Annales scientifiques ENS 18.1 (1985), 181–192. +[5] D. Bertrand, V. Chirskii, J. Yebbou, Effective estimates for global relations on Euler- +type series, Annales de la Facult´e des sciences de Toulouse : Math´ematiques, S´er. 6, +13.2 (2004), 241–260. +[6] F. Beukers, A refined version of the Siegel-Shidlovskii theorem, Annals of Math. 163 +(2006), 369–379. +[7] T. Ferguson, Algebraic properties of Э-functions, J. Number Theory 229 (2021), 168– +178. +[8] S. Fischler, T. Rivoal, On the values of G-functions, Commentarii Math. Helv. 89.2 +(2014), 313–341. +[9] S. Fischler, T. Rivoal, Arithmetic theory of E-operators, Journal de l’´Ecole polytech- +nique – Math´ematiques 3 (2016), 31–65. +[10] S. Fischler, T. Rivoal, Microsolutions of differential operators and values of arithmetic +Gevrey series, American J. of Math. 140.2 (2018), 317–348. +[11] S. Fischler, T. Rivoal, On Siegel’s problem for E-functions, preprint arxiv 1910.06817 +[math.NT], Rendiconti Sem. Math. Univ. Padova, to appear. +[12] M. L. Glasser, I. J. Zucker, Extended Watson integrals for the cubic lattices, Proc. +Nat. Acad. Sci. U.S.A. 74.5 (1977), 1800—1801. +[13] I. S. Gradshteyn, I. M. Ryzhik, Table of Integrals, Series, and Products, trans- +lated from the Russian, edited by A. Jeffrey and D. Zwillinger, Amsterdam: Else- +vier/Academic Press, 7th edition (english), 1176 pp., 2007. +[14] J. P. Ramis, S´eries divergentes et th´eories asymptotiques, Panoramas et Synth`eses, +no. 21, Soc. Math. France, Paris, 1993. +[15] T. Rivoal, +On the arithmetic nature of the values of the Gamma function, Euler’s +constant et Gompertz’s constant, Michigan Math. Journal 61 (2012), 239–254. +[16] T. Rivoal, Is Euler’s constant a value of an arithmetic special function?, unpublished +note (2017), 10 pages, https://hal.archives-ouvertes.fr/hal-01619235 +[17] A. B. Shidlovskii, On a criterion for algebraic independence of the values of a class of +integral functions, Izvestia Akad. Nauk SSSR 23 (1959), 35–66. +[18] C. L. Siegel, ¨Uber einige Anwendungen diophantischer Approximationen, vol. 1 S. +Abhandlungen Akad., Berlin, 1929. +17 + +[19] C. L. Siegel, Transcendental Numbers, Annals of Mathematical Studies 16, Princeton +University Press, 1949. +[20] E. T. Whittaker, G. N. Watson, A course of Modern Analysis, 4th edition, Cambridge +Mathematical Library, 1996. +S. Fischler, Universit´e Paris-Saclay, CNRS, Laboratoire de math´ematiques d’Orsay, 91405 +Orsay, France. +T. Rivoal, Universit´e Grenoble Alpes, CNRS, Institut Fourier, CS 40700, 38058 Grenoble +cedex 9, France. +Keywords: +E-functions, Э-functions, G-functions, Gamma function, Siegel-Shidlovskii +Theorem. +MSC 2020: 11J91 (Primary), 33B15 (Secondary) +18 + diff --git a/ZdFRT4oBgHgl3EQfPzc_/content/tmp_files/load_file.txt b/ZdFRT4oBgHgl3EQfPzc_/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5bcd749029f46fb9f95ab2e0813870780ba40b65 --- /dev/null +++ b/ZdFRT4oBgHgl3EQfPzc_/content/tmp_files/load_file.txt @@ -0,0 +1,705 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf,len=704 +page_content='arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='13518v1 [math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='NT] 31 Jan 2023 Relations between values of arithmetic Gevrey series, and applications to values of the Gamma function S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Fischler and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Rivoal February 1, 2023 Abstract We investigate the relations between the rings E, G and D of values taken at alge- braic points by arithmetic Gevrey series of order either −1 (E-functions), 0 (analytic continuations of G-functions) or 1 (renormalization of divergent series solutions at ∞ of E-operators) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We prove in particular that any element of G can be written as multivariate polynomial with algebraic coefficients in elements of E and D, and is the limit at infinity of some E-function along some direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This prompts to defining and studying the notion of mixed functions, which generalizes simultaneously E-functions and arithmetic Gevrey series of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Using natural conjectures for arithmetic Gevrey series of order 1 and mixed functions (which are analogues of a theorem of Andr´e and Beukers for E-functions) and the conjecture D ∩ E = Q (but not necessarily all these conjectures at the same time),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' we deduce a number of interesting Diophantine results such as an analogue for mixed functions of Beukers’ linear independence theorem for values of E-functions,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' the transcendance of the values of the Gamma function and its derivatives at all non-integral algebraic numbers,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' the transcendance of Gompertz constant as well as the fact that Euler’s constant is not in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 1 Introduction A power series �∞ n=0 an n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' xn ∈ Q[[x]] is said to be an E-function when it is solution of a linear differential equation over Q(x) (holonomic), and |σ(an)| (for any σ ∈ Gal(Q/Q)) and the least common denominator of a0, a1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , an grow at most exponentially in n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' They were defined and studied by Siegel in 1929, who also defined the class of G-functions: a power series �∞ n=0 anxn ∈ Q[[x]] is said to be a G-function when �∞ n=0 an n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' xn is an E-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In this case, �∞ n=0 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='anzn ∈ Q[[z]] is called an Э-function, following the terminology intro- duced by Andr´e in [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' E-functions are entire, while G-functions have a positive radius of convergence, which is finite except for polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Here and below, we see Q as embedded into C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Following Andr´e again, E-functions, G-functions and Э-fonctions are exactly arith- metic Gevrey series of order s = −1, 0, 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Actually Andr´e defines arithmetic 1 Gevrey series of any order s ∈ Q, but the set of values at algebraic points is the same for a given s ̸= 0 as for s/|s| using [1, Corollaire 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Э-functions are divergent series, unless they are polynomials.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Given an Э-function f and any θ ∈ R, except finitely many values mod 2π (namely anti-Stokes directions of f), one can perform Ramis’ 1-summation of f(1/z) in the direction θ, which coincides in this setting with Borel-Laplace summation (see [14] or [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This provides a function denoted by fθ(1/z), holomorphic on the open subset of C consisting in all z ̸= 0 such that θ− π 2 −ε < arg z < θ+ π 2 +ε for some ε > 0, of which f(1/z) is the asymptotic expansion in this sector (called a large sector bisected by θ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Of course f(1/z) can be extended further by analytic continuation, but this asymptotic expansion may no longer be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' When an Э-function is denoted by fj, we shall denote by fj,θ or fj;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='θ its 1-summation and we always assume (implicitly or explicitly) that θ is not an anti-Stokes direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In [8], [9] and [10, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='3], we have studied the sets G, E and D defined respectively as the sets of all the values taken by all (analytic continuations of) G-functions at algebraic points, of all the values taken by all E-functions at algebraic points and of all values fθ(1) where f is an Э-function (θ = 0 if it is not an anti-Stokes direction, and θ > 0 is very small otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=') These three sets are countable sub-rings of C that all contain Q;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' conjecturally, they are related to the set of periods and exponential periods, see §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (The ring D is denoted by Э in [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=') We shall prove the following result in §3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Every element of G can be written as a multivariate polynomial (with coef- ficients in Q) in elements of E and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Moreover, G coincides with the set of all convergent integrals � ∞ 0 F(x)dx where F is an E-function, or equivalently with the set of all finite limits of E-functions at ∞ along some direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Above, a convergent integral � ∞ 0 F(x)dx means a finite limit of the E-function � z 0 F(x)dx as z → ∞ along some direction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' this explains the equivalence of both statements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We refer to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2) in §3 for an expression of log(2) as a polynomial in elements in E and D;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' the number π could be similarly expressed by considering z and iz instead of z and 2z there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Examples of the last statement are the identities (see [12] for the second one): � +∞ 0 sin(x) x dx = π 2 and � +∞ 0 J0(ix)e−3xdx = √ 6 96π3Γ � 1 24 � Γ � 5 24 � Γ � 7 24 � Γ �11 24 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' It is notoriously difficult to prove/disprove that a given element of G is transcendental;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' it is known that a Siegel-Shidlovskii type theorem for G-functions can not hold mutatis mutandis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Theorem 1 suggests that an alternative approach to the study of the Diophantine properties of elements of G can be through a better understanding of joint study of the elements of E and D, modulo certain conjectures to begin with.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Our applications will not be immediately directed to the elements of G but rather to the understanding of the (absence of) relations between the elements of E and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 2 It seems natural (see [9, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 37]) to conjecture that E ∩ G = Q, and also that G ∩ D = Q, though both properties seem currently out of reach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In this paper, we suggest (see §2) a possible approach towards the following analogous conjecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We have E ∩ D = Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In §2 we shall make a functional conjecture, namely Conjecture 3, that implies Conjec- ture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We also prove that Conjecture 1 has very important consequences, as the following result shows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Assume that Conjecture 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then Γ(s)(a) is a transcendental number for any rational number a > 0 and any integer s ≥ 0, except of course if s = 0 and a ∈ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' One of the aims of this paper is to show that combining Э- and E-functions may lead to very important results in transcendental number theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let us recall now briefly the main known results on E-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Point (i) in the following result is due to Andr´e [2] for E-functions with rational Taylor coefficients, and to Beukers [6] in the general case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Andr´e used this property to obtain a new proof of the Siegel-Shidlovskii Theorem, and Beukers to prove an optimal refinement of this theorem (namely, (ii) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Theorem A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (i) [Andr´e, Beukers] If an E-function F(z) is such that F(1) = 0, then F (z) z−1 is an E-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (ii) [Beukers] Let F(z) := t(f1(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , fn(z)) be a vector of E-functions solution of a differential system F ′(z) = A(z)F(z) for some matrix A(z) ∈ Mn(Q(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let ξ ∈ Q ∗ which is not a pole of a coefficient of A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let P ∈ Q[X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Xn] be a homogeneous polynomial such that P(f1(ξ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , fn(ξ)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then there exists Q ∈ Q[Z, X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Xn], homogeneous in the Xi, such that Q(z, f1(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , fn(z)) = 0 identically and P(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Xn) = Q(ξ, X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In particular, we have trdegQ(f1(ξ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , fn(ξ)) = trdegQ(z)(f1(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , fn(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The Siegel-Shidlovskii Theorem itself is the final statement about equality of transcen- dence degrees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In this paper we state conjectural analogues of these results, involving Э-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The principal difficulty is that these functions are divergent power series, and the exact analogue of Theorem A is meaningless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Andr´e discussed the situation in [2] and even though he did not formulate exactly the following conjecture, it seems plausible to us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' From it, we will show how to deduce an analogue of the Siegel-Shidlovskii theorem for Э-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Ferguson [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 171, Conjecture 1] essentially stated this conjecture when f(z) has rational coefficients and when θ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 3 Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let f(z) be an Э-function and θ ∈ (−π/2, π/2) be such that fθ(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then f(z) z−1 is an Э-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In other words, the conclusion of this conjecture asserts that z 1−zf(1/z) is an Э-function in 1/z;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' this is equivalent to f(1/z) z−1 being an Э-function in 1/z (since we have f(1/z) z−1 = O(1/z) unconditionally as |z| → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Following Beukers’ proof [6] yields the following result (see [3, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='6] for a related con- jecture).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Theorem 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Assume that Conjecture 2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let f(z) := t(f1(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , fn(z)) be a vector of Э-functions solution of a differential system f′(z) = A(z)f(z) for some matrix A(z) ∈ Mn(Q(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let ξ ∈ Q ∗ and θ ∈ (arg(ξ) − π/2, arg(ξ) + π/2) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' assume that ξ is not a pole of a coefficient of A, and that θ is anti- Stokes for none of the fj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let P ∈ Q[X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Xn] be a homogeneous polynomial such that P(f1,θ(1/ξ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , fn,θ(1/ξ)) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then there exists Q ∈ Q[Z, X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Xn], homogeneous in the Xi, such that Q(z, f1(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , fn(z)) = 0 identically and P(X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Xn) = Q(1/ξ, X1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Xn).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In particular, we have trdegQ(f1,θ(1/ξ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=', fn,θ(1/ξ)) = trdegQ(z)(f1(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , fn(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' As an application of Theorem 3, we shall prove the following corollary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Note that under his weaker version of Conjecture 2, Ferguson [7, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 171, Theorem 2] proved that Gompertz’s constant is an irrational number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Assume that Conjecture 2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then for any α ∈ Q, α > 0, and any s ∈ Q \\ Z≥0, the number � ∞ 0 (t + α)se−tdt is a transcendental number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In particular, Gompertz’s constant δ := � ∞ 0 e−t/(t + 1)dt is a transcendental number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In this text we suggest an approach towards Conjecture 1, based on the new notion of mixed functions which enables one to consider E- and Э-functions at the same time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In particular we shall state a conjecture about such functions, namely Conjecture 3 in §2, which implies both Conjecture 1 and Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The following result is a motivation for this approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Assume that both Conjectures 1 and 2 hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then neither Euler’s constant γ := −Γ′(1) nor Γ(a) (with a ∈ Q+ \\ N) are in E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' It is likely that none of these numbers is in G, but (as far as we know) there is no “functional” conjecture like Conjecture 3 that implies this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' It is also likely that none is in D as well, but we don’t know if this can be deduced from Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 4 The structure of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In §2 we define and study mixed functions, a combination of E- and Э-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then in §3 we express any value of a G-function as a polynomial in values of E- and Э-functions, thereby proving Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We study deriva- tives of the Γ function at rational points in §4, and prove Theorem 2 and Proposition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' At last, §5 is devoted to adapting Beukers’ method to our setting: this approach yields Theorem 3 and Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 2 Mixed functions 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1 Definition and properties In view of Theorem 1, it is natural to study polynomials in E- and Э-functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We can prove a Diophantine result that combines both Theorems A(ii) and 3 but under a very com- plicated polynomial generalization of Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We opt here for a different approach to mixing E- and Э-functions for which very interesting Diophantine consequences can be deduced from a very easy to state conjecture which is more in the spirit of Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We refer to §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='3 for proofs of all properties stated in this section (including Lemma 1 and Proposition 2), except Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Definition 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We call mixed (arithmetic Gevrey) function any formal power series � n∈Z anzn such that � n≥0 anzn is an E-function in z, and � n≥1 a−nz−n is an Э-function in 1/z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In other words, a mixed function is defined as a formal sum Ψ(z) = F(z)+f(1/z) where F is an E-function and f is an Э-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In particular, such a function is zero if, and only if, both F and f are constants such that F + f = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' obviously, F and f are uniquely determined by Ψ upon assuming (for instance) that f(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The set of mixed functions is a Q-vector space stable under multiplication by zn for any n ∈ Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Unless f(z) is a polynomial, such a function Ψ(z) = F(z) + f(1/z) is purely formal: there is no z ∈ C such that f(1/z) is a convergent series.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' However, choosing a direction θ which is not anti-Stokes for f allows one to evaluate Ψθ(z) = F(z) + fθ(1/z) at any z in a large sector bisected by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Here and below, such a direction will be said not anti-Stokes for Ψ and whenever we write fθ or Ψθ we shall assume implicitly that θ is not anti-Stokes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Definition 1 is a formal definition, but one may identify a mixed function with the holomorphic function it defines on a given large sector by means of the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let Ψ be a mixed function, and θ ∈ R be a non-anti-Stokes direction for Ψ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then Ψθ is identically zero (as a holomorphic function on a large sector bisected by θ) if, and only if, Ψ is equal to zero (as a formal power series in z and 1/z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 5 Any mixed function Ψ(z) = F(z) + f(1/z) is solution of an E-operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Indeed, this follows from applying [1, Theorem 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1] twice: there exist an E-operator L such that L(f(1/z)) = 0, and an E-operator M such that M(L(F(z))) = 0 (because L(F(z)) is an E-function).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Hence ML(F(z) + f(1/z)) = 0 and by [1, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 720, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1], ML is an E-operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We formulate the following conjecture, which implies both Conjecture 1 and Conjec- ture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let Ψ(z) be an mixed function, and θ ∈ (−π/2, π/2) be such that Ψθ(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then Ψ(z) z−1 is an mixed function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The conclusion of this conjecture is that Ψ(z) = (z − 1)Ψ1(z) for some mixed function Ψ1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This conclusion can be made more precise as follows;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' see §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='3 for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let Ψ(z) = F(z) + f(1/z) be an mixed function, and θ ∈ (−π/2, π/2) be such that Ψθ(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Assume that Conjecture 3 holds for Ψ and θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then both F(1) and fθ(1) are algebraic, and f(1/z)−fθ(1) z−1 is an Э-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Of course, in the conclusion of this proposition, one may assert also that F (z)−F (1) z−1 is an E-function using Theorem A(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Conjecture 3 already has far reaching Diophantine consequences: Conjecture 2 and Theorem 2 stated in the introduction, and also the following result that encompasses Theorem 3 in the linear case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Theorem 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Assume that Conjecture 3 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let Ψ(z) := t(Ψ1(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Ψn(z)) be a vector of mixed functions solution of a differential system Ψ′(z) = A(z)Ψ(z) for some matrix A(z) ∈ Mn(Q(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let ξ ∈ Q ∗ and θ ∈ (arg(ξ) − π/2, arg(ξ) + π/2) ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' assume that ξ is not a pole of a coefficient of A, and that θ is anti-Stokes for none of the Ψj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let λ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , λn ∈ Q be such that n � i=1 λiΨi,θ(ξ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then there exist L1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Ln ∈ Q[z] such that n � i=1 Li(z)Ψi(z) = 0 identically and Li(ξ) = λi for any i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In particular, we have rkQ(Ψ1,θ(ξ), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Ψn,θ(ξ)) = rkQ(z)(Ψ1(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Ψn(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The proof of Theorem 4 follows exactly the linear part of the proof of Theorem 3 (see §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1), which is based on [6, §3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The only difference is that Э-functions have to be replaced 6 with mixed functions, and Conjecture 2 with Conjecture 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In particular Proposition 4 stated in §5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1 remains valid with these modifications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' However a product of mixed functions is not, in general, a mixed function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Therefore the end of [6, §3] does not adapt to mixed functions, and there is no hope to obtain in this way a result on the transcendence degree of a field generated by values of mixed functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' As an application of Theorem 4, we can consider the mixed functions 1, eβz and f(1/z) := �∞ n=0(−1)nn!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='z−n, where β is a fixed non-zero algebraic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' These three functions are linearly independent over C(z) and form a solution of a differential system with only 0 for singularity (because (f(1/z))′ = (1 + 1/z)f(1/z) − 1), hence for any α ∈ Q, α > 0 and any ̺ ∈ Q ∗, the numbers 1, e̺, f0(1/α) := � ∞ 0 e−t/(1 + αt)dt are Q-linearly independent (for a fixed α, take β = ̺/α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2 Values of mixed functions We denote by MG the set of values Ψθ(1), where Ψ is a mixed function and θ = 0 if it is not anti-Stokes, θ > 0 is sufficiently small otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This set is obviously equal to E + D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Proposition 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' For every integer s ≥ 0 and every a ∈ Q+, a ̸= 0, we have Γ(s)(a) ∈ e−1MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This results follows immediately from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='4) below (see §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2), written in the form Γ(s)(a) = e−1� (−1)ses!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='Ea,s+1(−1) + fa,s+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1) � , because ezEa,s+1(−z) is an E-function and fa,s+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1) is the 1-summation in the direction 0 of an Э-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' It would be interesting to know if Γ(s)(a) belongs to MG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We did not succeed in proving it does, and we believe it does not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Indeed, for instance if we want to prove that γ ∈ MG, a natural strategy would be to construct an E-function F(z) with asymptotic expansion of the form γ + log(z) + f(1/z) in a large sector, and then to evaluate at z = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' However this strategy cannot work since there is no such E-function (see the footnote in the proof of Lemma 1 in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='3 Proofs concerning mixed functions To begin with, let us take Proposition 2 for granted and prove that Conjecture 3 implies both Conjecture 1 and Conjecture 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Concerning Conjecture 2 it is clear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' To prove that it implies Conjecture 1, let ξ ∈ D, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' ξ = fθ(1) is the 1-summation of an Э-function f(z) in the direction θ = 0 if it is not anti-Stokes, and θ > 0 close to 0 otherwise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Assume that ξ is also in E: we have ξ = F(1) for some E-function F(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Therefore, Ψ(z) = F(z) − f(1/z) is a mixed function such that Ψθ(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' By Conjecture 3 and Proposition 2, we have ξ = fθ(1) ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This concludes the proof that Conjecture 3 implies Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 7 Let us prove Proposition 2 now.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Assuming that Conjecture 3 holds for Ψ and θ, there exists a mixed function Ψ1(z) = F1(z) + f1(1/z) such that Ψ(z) = (z − 1)Ψ1(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We have F(z) − (z − 1)F1(z) + f(1/z) − (z − 1)f1(1/z) = 0 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1) as a formal power series in z and 1/z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Now notice that z − 1 = z(1 − 1 z), and that we may assume f and f1 to have zero constant terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Denote by α the constant term of f(1/z) − z(1 − 1 z)f1(1/z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then we have F(z) − (z − 1)F1(z) + α + f2(1/z) = 0 for some Э-function f2 without constant term, so that f2 = 0, F(z) = (z − 1)F1(z) − α and F(1) = −α ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This implies fθ(1) = α, and f(1/z)−fθ(1) z−1 = f1(1/z) is an Э-function since f2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This concludes the proof of Proposition 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' At last, let us prove Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We write Ψ(z) = F(z) + f(1/z) and assume that Ψθ is identically zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Modifying θ slightly if necessary, we may assume that the asymptotic expansion −f(1/z) of F(z) in a large sector bisected by θ is given explicitly by [9, Theorem 5] applied to F(z) − F(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' recall that such an asymptotic expansion is unique (see [9]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' As in [9] we let g(z) = �∞ n=1 anz−n−1 where the coefficients an are given by F(z) − F(0) = �∞ n=1 an n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' zn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' For any σ ∈ C\\{0} there is no contribution in eσz in the asymptotic expansion of F(z), so that g(z) is holomorphic at σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' At σ = 0, the local expansion of g is of the form g(z) = h1(z) + h2(z) log(z) with G-functions h1 and h2, and the coefficients of h2 are related to those of f;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' however we shall not use this special form (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Now recall that g(z) = G(1/z)/z where G is a G-function;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' then G is entire and has moderate growth at infinity (because ∞ is a regular singularity of G), so it is a polynomial due to Liouville’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This means that F(z) is a polynomial in z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Recall that asymptotic expansions in large sectors are unique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Therefore both F and f are constant functions, and F + f = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This concludes the proof of Lemma 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 3 Proof of Theorem 1: values of G-functions In this section we prove Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let us begin with an example, starting with the relation proved in [15, Proposition 1] for z ∈ C \\ (−∞, 0]: γ + log(z) = zE1,2(−z) − e−zf1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1/z) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1) where E1,2 is an E-function, and f1,2 is an Э-function, both defined below in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Apply Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1) at both z and 2z, and then substract one equation from the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This provides a relation of the form log(2) = F(z) + e−zf1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1/z) + e−2zf2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1/z) (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2) 1Actually we are proving that the asymptotic expansion of a non-polynomial E-function is never a C- linear combination of functions zα logk(z)f(1/z) with α ∈ Q, k ∈ N and Э-functions f: some exponentials have to appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 8 valid in a large sector bisected by 0, with an E-function F and Э-functions f1 and f2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Choosing arbitrarily a positive real algebraic value of z yields an explicit expression of log(2) ∈ G as a multivariate polynomial in elements of E and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' But this example shows also that a polynomial in E- and Э-functions may be constant eventhough there does not seem to be any obvious reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In particular, the functions 1, F(z), e−zf1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1/z), and e−2zf2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1/z) are linearly dependent over C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' However we see no reason why they would be linearly dependent over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This could be a major drawback to combine in E- and Э-functions, since functions that are linearly dependent over C but not over Q can not belong to any Picard-Vessiot extension over Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let us come now to the proof of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We first prove the second part, which runs as follows (it is reproduced from the unpublished note [16]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' From the stability of the class of E-functions by d dz and � z 0 , we deduce that the set of convergent integrals � ∞ 0 F(x)dx of E-functions and the set of finite limits of E-functions along some direction as z → ∞ are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Theorem 2(iii) in [9] implies that if an E-function has a finite limit as z → ∞ along some direction, then this limit must be in G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Conversely, let β ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' By Theorem 1 in [8], there exists a G-function G(z) = �∞ n=0 anzn of radius of convergence ≥ 2 (say) such that G(1) = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let F(z) = �∞ n=0 an n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' zn be the associated E-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then for any z such that Re(z) > 1 2, we have 1 zG �1 z � = � +∞ 0 e−xzF(x)dx.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Hence, β = � +∞ 0 e−xF(x)dx where e−zF(z) is an E-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We shall now prove the first part of Theorem 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In fact, we shall prove a slightly more general result, namely Theorem 5 below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We first recall a few notations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Denote by S the G-module generated by all derivatives Γ(s)(a) (with s ∈ N and a ∈ Q\\ Z≤0), and by V the S-module generated by E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Recall that G, S and V are rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Conjecturally, G = P[1/π] and V = Pe[1/π] where P and Pe are the ring of periods and the ring of exponential periods over Q respectively (see [8, §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2] and [10, §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='3]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We have proved in [10, Theorem 3] that V is the S-module generated by the numbers eρχ, with ρ ∈ Q and χ ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The ring V is the ring generated by E and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In particular, all values of G-functions belong to the ring generated by E and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In other words, the elements of V are exactly the sums of products ab with a ∈ E and b ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Proof of Theorem 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We already know that V is a ring, and that it contains E and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' To prove the other inclusion, denote by U the ring generated by E and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Using Proposition 3 proved in §2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2 and the functional equation of Γ, we have Γ(s)(a) ∈ U for any s ∈ N and any a ∈ Q \\ Z≤0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Therefore for proving that V ⊂ U, it is enough to prove that G ⊂ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let ξ ∈ G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Using [11, Theorem 3] there exists an E-function F(z) such that for any for any θ ∈ [−π, π) outside a finite set, ξ is a coefficient of the asymptotic expansion of F(z) 9 in a large sector bisected by θ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' As the proof of [11, Theorem 3] shows, we can assume that ξ is the coefficient of ez in this expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Denote by L an E-operator of which F is a solution, and by µ its order.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Andr´e has proved [1] that there exists a basis (H1(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Hµ(z)) of formal solutions of L at infinity such that for any j, e−ρjzHj(z) ∈ NGA{1/z}Q 1 for some algebraic number ρj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We recall that elements of NGA{1/z}Q 1 are arithmetic Nilsson-Gevrey series of order 1 with algebraic coefficients, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Q-linear combinations of functions zk(log z)ℓf(1/z) with k ∈ Q, ℓ ∈ N and Э-functions f.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Expanding in this basis the asymptotic expansion of F(z) in a large sector bisected by θ (denoted by �F), there exist complex numbers κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , κd such that �F(z) = κ1H1(z) + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' + κµHµ(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then we have ξ = κ1c1 + .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' + κµcµ, where cj is the coefficient of ez in Hj(z) ∈ eρjzNGA{1/z}Q 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We have cj = 0 if ρj ̸= 1, and otherwise cj is the constant coefficient of e−zHj(z): in both cases cj is an algebraic number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Therefore to conclude the proof that ξ ∈ U, it is enough to prove that κ1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , κµ ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' For simplicity let us prove that κ1 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Given solutions F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Fµ of L, we denote by W(F1, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Fµ) the corresponding wronskian matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then for any z in a large sector bisected by θ we have κ1 = det W(F(z), H2,θ(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Hµ,θ(z)) det W(H1,θ(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Hµ,θ(z)) where Hj,θ(z) is the 1-sommation of Hj(z) in this sector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The determinant in the denomi- nator belongs to eazNGA{1/z}Q 1 with a = ρ1+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='+ρµ ∈ Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' As the proof of [10, Theorem 6] shows, there exist b, c ∈ Q, with c ̸= 0, such that det W(H1,θ(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Hµ,θ(z)) = czbeaz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We take z = 1, and choose θ = 0 if it is not anti-Stokes for L (and θ > 0 sufficiently small otherwise).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then we have κ1 = c−1e−a� det W(F(z), H2,θ(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , Hµ,θ(z)) � |z=1 ∈ U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This concludes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Remark 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The second part of Theorem 1 suggests the following comments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' It would be interesting to have a better understanding (in terms of E, G and D) of the set of convergent integrals � ∞ 0 R(x)F(x)dx where R is a rational function in Q(x) and F is an E-function, which are thus in G when R = 1 (see [16] for related considerations).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Indeed, classical examples of such integrals are � +∞ 0 cos(x) 1+x2 dx = π/(2e) ∈ πE, Euler’s constant � +∞ 0 1−(1+x)e−x x(1+x) dx = γ ∈ E + e−1D (using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1) and [20, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 248, Example 2]) and Gompertz constant δ := � +∞ 0 e−x 1+xdx ∈ D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' A large variety of behaviors can thus be expected here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 10 For instance, using various explicit formulas in [13, Chapters 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='5–6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='7], it can be proved that � +∞ 0 R(x)J0(x)dx ∈ G + E + γE + log(Q ⋆)E for any R(x) ∈ Q(x) without poles on [0, +∞), where J0(x) = �∞ n=0(ix/2)2n/n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2 is a Bessel function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' A second class of examples is when R(x)F(x) is an even function without poles on [0, +∞) and such that lim|x|→∞,Im(x)≥0 x2R(x)F(x) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then by the residue theorem, � +∞ 0 R(x)F(x)dx = iπ � ρ, Im(ρ)>0 Resx=ρ � R(x)F(x) � ∈ πE where the summation is over the poles of R in the upper half plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 4 Derivatives of the Γ function at rational points In this section we prove Theorem 2 and Proposition 1 stated in the introduction, dealing with Γ(s)(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' To begin with, we define E-functions Ea,s(z) in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1 and prove a linear independence result concerning these functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then we prove in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2 a formula for Γ(s)(a), namely Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='4), involving Ea,s+1(−1) and the 1-summation of an Э-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This enables us to prove Theorem 2 in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='3 and Proposition 1 in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1 Linear independence of a family of E-functions To study derivatives of the Γ function at rational points, we need the following lemma.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' For s ≥ 1 and a ∈ Q \\ Z≤0, we consider the E-function Ea,s(z) := �∞ n=0 zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (n+a)s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Lemma 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (i) For any a ∈ Q \\ Z and any s ≥ 1, the functions 1, ez, ezEa,1(−z), ezEa,2(−z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , ezEa,s(−z) are linearly independent over C(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (ii) For any a ∈ N∗ and any s ≥ 2, the functions 1, ez, ezEa,2(−z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , ezEa,s(−z) are linearly independent over C(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Remark 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Part (i) of the lemma is false if a ∈ N∗ because 1, ez, ezEa,1(−z) are Q(z)-linearly dependent in this case (see the proof of Part (ii) below).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 11 Proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (i) For simplicity, we set ψs(z) := ezEa,s(−z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We proceed by induction on s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let us first prove the case s = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The derivative of ψ1(z) is (1 + (z − a)ψ1(z))/z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let us assume the existence of a relation ψ1(z) = u(z)ez + v(z) with u, v ∈ C(z) (a putative relation U(z) + V (z)ez + W(z)ψ1(z) = 0 forces W ̸= 0 because ez /∈ C(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then after differentiation of both sides, we end up with 1 + (z − a)ψ1(z) z = � u(z) + u′(z) � ez + v′(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Hence, 1 + (z − a) � u(z)ez + v(z) � z = � u(z) + u′(z) � ez + v′(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Since ez /∈ C(z), the function v(z) is a rational solution of the differential equation zv′(z) = (z − a)v(z) + 1: v(z) cannot be identically 0, and it cannot be a polynomial (the degrees do not match on both sides).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' It must then have a pole at some point ω, of order d ≥ 1 say.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We must have ω = 0 because otherwise the order of the pole at ω of zv′(z) is d + 1 while the order of the pole of (z − a)v(z) + 1 is at most d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Writing v(z) = � n≥−d vnzn with v−d ̸= 0 and comparing the term in z−d of zv′(z) and (z − a)v(z) + 1, we obtain that d = a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This forces a to be an integer ≥ 1, which is excluded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Hence, 1, ez, ezEa,1(−z) are C(z)-linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let us now assume that the case s − 1 ≥ 1 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let us assume the existence of a relation over C(z) ψs(z) = v(z) + u0(z)ez + s−1 � j=1 uj(z)ψj(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1) (A putative relation V (z) + U0(z)ez + �s j=1 Uj(z)ψj(z) = 0 forces Us ̸= 0 by the induction hypothesis).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Differentiating (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1) and because ψ′ j(z) = (1− a z)ψj(z)+ 1 zψj−1(z) for all j ≥ 1 (where we have let ψ0(z) = 1), we have A(z)ψs(z) + 1 zψs−1(z) = v′(z) + � u0(z) + u′ 0(z) � ez + s−1 � j=1 u′ j(z)ψj(z) + s−1 � j=1 uj(z) � A(z)ψj(z) + 1 zψj−1(z) � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2) where A(z) := 1 −a/z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Substituting the right-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1) for ψs(z) on the left-hand side of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2), we then deduce that v′(z) − A(z)v(z) + � u′ 0(z) + (1 − A(z))u0(z) � ez + 1 z(z − a)u1(z)ψ1(z) + s−1 � j=1 u′ j(z)ψj(z) + 1 z s−1 � j=1 uj(z)ψj−1(z) − 1 zψs−1(z) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 12 This is a non-trivial C(z)-linear relation between 1, ez, ψ1(z), ψ2(z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , ψs−1(z) because the coefficient of ψs−1(z) is u′ s−1(z) −1/z and it is not identically 0 because u′ s−1(z) cannot have a pole of order 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' But by the induction hypothesis, we know that such a relation is impossible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (ii) The proof can be done by induction on s ≥ 2 similarily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In the case s = 2, assume the existence of a relation ψ2(z) = u(z)ez +v(z) with u(z), v(z) ∈ C(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' By differentiation, we obtain � 1 − a z � ψ2(z) = −1 zψ1(z) + � u(z) + u′(z) � ez + v′(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' By induction on a ≥ 1, we have ψ1(z) = (a−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='ez/za +w(z) for some w(z) ∈ Q(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Hence, we have � 1 − a z � u(z) = − �(a − 1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' za+1 + 1 � u(z) + u′(z) which is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let us now assume that the case s − 1 ≥ 2 holds, as well as the existence of a relation over C(z) ψs(z) = v(z) + u0(z)ez + s−1 � j=2 uj(z)ψj(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='3) We proceed exactly as above by differentiation of both sides of (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Using the relation ψ′ j(z) = (1− a z)ψj(z)+ 1 zψj−1(z) for all j ≥ 2 and the fact that ψ1(z) = (a−1)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='ez/za+w(z), we obtain a relation �v(z) + �u0(z)ez + �s−1 j=2 �uj(z)ψj(z) = 0 where �us−1(z) = u′ s−1(z) − 1/z cannot be identically 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The induction hypothesis rules out the existence of such a relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2 A formula for Γ(s)(a) Let z > 0 and a ∈ Q+, a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' We have Γ(s)(a) = � ∞ 0 ta−1 log(t)se−tdt = � z 0 ta−1 log(t)se−tdt + � ∞ z ta−1 log(t)se−tdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' On the one hand, � z 0 ta−1 log(t)se−tdt = ∞ � n=0 (−1)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' � z 0 ta+n−1 log(t)sdt = ∞ � n=0 (−1)n n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' s � k=0 (−1)k s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (s − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' zn+a log(z)s−k (n + a)k+1 = s � k=0 (−1)ks!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (s − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='za log(z)s−kEa,k+1(−z);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 13 recall that Ea,j(z) = �∞ n=0 zn n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (n+a)j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' On the other hand, � ∞ z ta−1 log(t)se−tdt = e−z � ∞ 0 (t + z)a−1 log(t + z)se−tdt = za−1e−z s � k=0 �s k � log(z)s−k � ∞ 0 (1 + t/z)a−1 log(1 + t/z)ke−tdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Now z > 0 so that fa,k+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(z) := � ∞ 0 (1 + tz)a−1 log(1 + tz)ke−tdt = 1 z � ∞ 0 (1 + x)a−1 log(1 + x)ke−x/zdx is the 1-summation at the origin in the direction 0 of the Э-function ∞ � n=0 n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='ua,k,nzn, where the sequence (ua,k,n)n≥0 ∈ QN is defined by the expansion of the G-function: (1 + x)a−1 log(1 + x)k = ∞ � n=0 ua,k,nxn.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Note that if k = 0 and a ∈ N∗, then ua,k,n = 0 for any n ≥ a, and fa,k+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1/z) is a polynomial in 1/z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Therefore, we have for any z > 0: Γ(s)(a) = s � k=0 (−1)ks!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (s − k)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='za log(z)s−kEa,k+1(−z) + za−1e−z s � k=0 �s k � log(z)s−kfa,k+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1/z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' In particular, for z = 1, this relation reads Γ(s)(a) = (−1)ss!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='Ea,s+1(−1) + e−1fa,s+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='4) Since γ = −Γ′(1) we obtain as a special case the formula γ = E1,2(−1) − e−1f1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1), (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='5) which is also a special case of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1) proved in [15].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='3 Proof of Theorem 2 Let us assume that Γ(s)(a) ∈ Q for some a ∈ Q+ \\ N and s ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then ezΓ(s)(a) + (−1)s+1s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='ezEa,s+1(−z) is an E-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The relation eΓ(s)(a) + (−1)s+1s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='eEa,s+1(−1) = fa,s+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1) proved at the end of §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2 shows that α := eΓ(s)(a)+(−1)s+1s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='eEa,s+1(−1) ∈ E∩D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 14 Hence α is in Q by Conjecture 1 and we have a non-trivial Q-linear relation between 1, e and eEa,s+1(−1): we claim that this is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Indeed, consider the vector Y (z) := t(1, ez, ezEa,1(−z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , ezEa,s+1(−z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' It is solution of a differential system Y ′(z) = M(z)Y (z) where 0 is the only pole of M(z) ∈ Ms+3(Q(z)) (see the computations in the proof of Lemma 2 above).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Since the components of Y (z) are Q(z)-linearly independent by Lemma 2(i), we deduce from Beuk- ers’ [6, Corollary 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='4] that 1, e, eEa,1(−1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , eEa,s+1(−1) are Q-linearly independent, and in particular that 1, e and eEa,s+1(−1) are Q-linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This concludes the proof if a ∈ Q+ \\ N.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let us assume now that Γ(s)(a) ∈ Q for some a ∈ N∗ and s ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then ezΓ(s)(a) + (−1)s+1s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='ezEa,s+1(−z) is an E-function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The relation Γ(s)(a) + (−1)s+1s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='Ea,s+1(−1) = e−1fa,s+1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1) shows that α := eΓ(s)(a)+(−1)s+1s!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='eEa,s+1(−1) ∈ E∩D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Hence α is in Q by Conjecture 1 and we have a non-trivial Q-linear relation between 1, e and eEa,s+1(−1): we claim that this is not possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Indeed, consider the vector Y (z) := t(1, ez, ezEa,2(−z), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , ezEa,s+1(−z)): it is solution of a differential system Y ′(z) = M(z)Y (z) where 0 is the only pole of M(z) ∈ Ms+2(Q(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Since the components of Y (z) are Q(z)-linearly independent by Lemma 2(ii), we deduce again from Beukers’ theorem that 1, e, eEa,2(−1), .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' , eEa,s+1(−1) are Q-linearly independent, and in particular that 1, e and eEa,s+1(−1) are Q-linearly independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' This concludes the proof of Theorem 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='4 Proof of Proposition 1 Recall that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='5) proved in §4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2 reads eE1,2(−1)−eγ = f1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Assuming that γ ∈ E, the left-hand side is in E while the right-hand side is in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Hence both sides are in Q by Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Note that, by integration by parts, f1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1) = � ∞ 0 log(1 + t)e−tdt = � ∞ 0 e−t 1 + tdt is Gompertz’s constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Hence, by Corollary 1 (which holds under Conjecture 2), the number f1,2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1) is not in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Consequently, γ /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Similarly, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='4) with a ∈ Q \\ Z and s = 0 reads eΓ(a) − eEa,1(−1) = fa,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Assuming that Γ(a) ∈ E, the left-hand side is in E while the right-hand side is in D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Hence both sides are in Q by Conjecture 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' But by Corollary 1 (which holds under Conjecture 2), the number fa,1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='0(1) = � ∞ 0 (1 + t)a−1e−tdt is not in Q.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Hence, Γ(a) /∈ E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 15 5 Application of Beukers’ method and consequence In this section we prove Theorem 3 and Corollary 1 stated in the introduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='1 Proof of Theorem 3 The proof of Theorem 3 is based on the arguments given in [6], except that E-functions have to be replaced with Э-functions, and 1-summation in non-anti-Stokes directions is used for evaluations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Conjecture 2 is used as a substitute for Theorem A(i).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The main step is the following result, the proof of which is analogous to the end of the proof of [6, Corollary 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Proposition 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Assume that Conjecture 2 holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Let f be an Э-function, ξ ∈ Q ∗ and θ ∈ (arg(ξ) − π/2, arg(ξ) + π/2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Assume that θ is not anti-Stokes for f, and that fθ(1/ξ) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Denote by Ly = 0 a differential equation, of minimal order, satisfied by f(1/z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Then all solutions of Ly = 0 are holomorphic and vanish at ξ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' the differential operator L has an apparent singularity at ξ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' To deduce Theorem 3 from Proposition 4, it is enough to follow [6, §3].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content='2 Proof of Corollary 1 Let s ∈ Q \\ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The Э-function f(z) := �∞ n=0 s(s − 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' (s − n + 1)zn is solution of the inhomogeneous differential equation z2f′(z)+(1−sz)f(z)−1 = 0, which can be immediately transformed into a differential system satisfied by the vector of Э-functions t(1, f(z)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' The coefficients of the matrix have only 0 as pole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Moreover, f(z) is a transcendental function because s /∈ Z≥0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Hence, by Theorem 3, f0(1/α) /∈ Q when α ∈ Q, α > 0, because 0 is not an anti-Stokes direction of f(z).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' It remains to observe that this 1-sommation is � ∞ 0 (1 + tz)se−tdt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' References [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Andr´e, S´eries Gevrey de type arithm´etique I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Th´eor`emes de puret´e et de dualit´e, Annals of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 151 (2000), 705–740.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' [2] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Andr´e, S´eries Gevrey de type arithm´etique II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Transcendance sans transcendance, Annals of Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' 151 (2000), 741–756.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' Keywords: E-functions, Э-functions, G-functions, Gamma function, Siegel-Shidlovskii Theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} +page_content=' MSC 2020: 11J91 (Primary), 33B15 (Secondary) 18' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ZdFRT4oBgHgl3EQfPzc_/content/2301.13518v1.pdf'} diff --git a/_dAzT4oBgHgl3EQfvf3k/content/tmp_files/2301.01709v1.pdf.txt b/_dAzT4oBgHgl3EQfvf3k/content/tmp_files/2301.01709v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..2593d64c7e78ed1b57c1edb06889f19a589076c3 --- /dev/null +++ b/_dAzT4oBgHgl3EQfvf3k/content/tmp_files/2301.01709v1.pdf.txt @@ -0,0 +1,2699 @@ +Novel topological black holes from thermodynamics and +deforming horizons +Jinbo Yang 1,2,3 +1Department of Astronomy, School of Physics and Materials Science, +Guangzhou University, Guangzhou 510006, P.R.China +2Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan +3HKUST Jockey Club Institute for Advanced Study, +The Hong Kong University of Science and Technology, +Clear Water Bay, Hong Kong, P.R.China +ABSTRACT +Two novel topological black hole exact solutions with unusual shapes of the horizon in +the 4-dimensional Einstein-scalars theory are constructed. The application of the unified +first law dramatically simplifies the construction. The article re-derives such first law from +the Einstein equations for a spacetime with product structure +¯ +M(2) × ˆ +M(D−2). The non- +constant Ricci scalar ˆR of the sub-manifold ˆ +M(D−2) remarkably leads to two versions of the +Misner-Sharp mass and the unified first law. Both versions are applied to generate topo- +logical black holes and to discuss their significance. Unusual shapes of ˆ +M(D−2) in solutions +naturally require topological charges. It gives the first glimpse of non-explored aspects of +the first law of black hole thermodynamics. +Keywords: topological black hole, unusual shape, unified first law, Misner-Sharp mass, +topological charge +yangjinbophy@gmail.com +arXiv:2301.01709v1 [gr-qc] 4 Jan 2023 + +Contents +1 +Introduction +3 +2 +Unified first law including unusual shape +4 +2.1 +Unified first law from Einstein field equation . . . . . . . . . . . . . . . . . . +5 +2.2 +Misner-Sharp mass as conserved charge +. . . . . . . . . . . . . . . . . . . . +8 +3 +Topological black holes from the unified first law +9 +3.1 +Topological black holes sourced by sigma model . . . . . . . . . . . . . . . . +10 +3.2 +Additional sources . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +4 +The first law including topological charge +21 +4.1 +Topological charge as the portal of +ˆ +M(D−2) . . . . . . . . . . . . . . . . . . +22 +4.2 +Unusual shapes and topological charges +. . . . . . . . . . . . . . . . . . . . +24 +5 +Summary and discussion +27 +Appendix A: Calculate Γλ +µν and Rλ +ρµν +29 +Appendix B: Calculate ∗J and ∗K +31 +Appendix C: General Eddington-Finkelstein coordinates +32 +Appendix D: Design the field space metric GIJ +34 +2 + +1 +Introduction +In several decades, black hole physics, especially black hole thermodynamics, has brought us +deep insights into theoretical physics [1–9]. The mainly concerning shape of the black hole +has a spherical topology, supported by the topological theorem for Einstein gravity [10]. +According to the theorem, the horizon of a 4-dimensional asymptotically flat black hole +must be topologically spherical. Further focusing on the perfectly spherical black hole is +an appropriate starting point for detailed research of black hole evolution. A spherically +symmetric spacetime in 4-dimensional Einstein gravity has a quasi-local mass called Misner- +Sharp (MS) mass [11,12]. +mMS = +r +2GN +(1 − gµν∇µr∇νr) , +(1.1) +where GN is the 4-dimensional Newton constant, and r is the areal radius which gives the +area of a sphere A = 4πr2. This mass reduces to the Arnowitt-Deser-Misner(ADM) mass +in the asymptotic flat background and is significant for formulating the unified first law in +Einstein gravity [13]. The MS mass and the unified first law are widely applied in the context +of primordial black hole formation [14–19], detailed study for Hawking evaporation [20–24] +and the first law on the apparent horizon of an expanding universe [25,26]. These concepts +also inspire a novel approach to generate spherical black hole solutions of general relativity +from thermodynamics [27]. +Nevertheless, many black objects beyond the spherical horizon in higher dimensional +supergravity or string theory have been discovered [28–33]. One kind of black object is +the topological black hole. The part of its metric describing the horizon shape is not a +sphere but rather an Einstein manifold [34–37]. +Widely studied topological black holes +have planar or hyperbolic horizons [38–40]. The hyperbolic counterpart of the Schwarzchild +black hole can be viewed as a gravitational description of S-brane [41], and the planar one +is widely applied since it generally describes a specific boundary phase in the context of +AdS/CFT correspondence [42, 43]. The parameters of a planar Reissner-Nordstr¨om(RN)- +AdS black hole satisfy the Gibbs-Duhem relationship. The unified first law and the approach +of deriving solutions from thermodynamics also have been generalized to include planar +and hyperbolic cases in the context of modified gravity [44–47]. Ref. [48, 49] introduces +the concept of topological charge for the topological RN-AdS black hole to preserve the +Gibbs-Duhem-like relation. They suggest that the topological charge is the last charge for +the first law of thermodynamics of a maximally symmetric black hole. +The article explores the possibility of replacing the spherical part with an unusual shape, +3 + +not limited to maximally symmetric space or Einstein manifold. Suppose the part replacing +the sphere is an independent manifold with metric ˆgij(x), where x donate the point in the +independent manifold, and i, j, k are the corresponding indexes. If the manifold is maxi- +mally symmetric, the Rienman tensor from ˆgij(x) is ˆRijkl = k(ˆgikˆgjl − ˆgilˆgjk). An Einstein +manifold satisfies a weaker condition ˆRij(x) = λˆgij(x), where λ is a constant [35]. Its Ricci +scalar ˆR is a constant. This article concerns the situation with non-constant ˆR(x). The +original unified first law only focuses on the maximally symmetric situation. However, the +unusual shape with non-constant ˆR(x) implies two versions of the unified first law, which +is discussed by re-deriving the law from Einstein equations in section 2. Remarkably, ap- +plying unified first laws simplifies constructing concrete examples of topological black holes +with unusual shapes. The method used here improves the approach of generating solutions +from thermodynamics in Ref. [27]. Novel topological black holes with unusual shapes in +4-dimensional Einstein-scalars theory are obtained through this method. Successfully con- +structing solutions with the non-constant ˆR(x) seems to give a more natural motivation for +introducing the topological charge defined in [48,49], but it also challenges the topological +significance of such a charge. Nevertheless, novel solutions open a new window to inves- +tigate how parameters describing the shape of the horizon join the first law of black hole +thermodynamics. +The article is organized as follows. Section 2 revisits the unified first law of general +relativity by dimensional reduction, an explicitly covariant method. Additionally, it is shown +that Misner-Sharp mass is a conserved charge relating to the Kodama vector, and a more +general horizon shape leads to two versions for formulating the unified first law. Section +3 constructs novel topological black hole solutions. The improved approach to generating +solutions from thermodynamics is discussed in detail. Embedding plots are displayed to +visualize the horizon shape. Contributions from other sources are also attached. Metrics +for novel solutions with charge and the cosmological constant are given. Section 4 focuses +on the topological charge to discuss how such a charge should relate to the shape of the +horizon from the viewpoint of the unified first law. Section 5 gives a summary and discusses +other open issues. +2 +Unified first law including unusual shape +This section re-derives the unified first law from the Einstein equations and shows that +Misner-Sharp mass is a conserved charge. Consider a D-dimensional spacetime with the +4 + +following line element +ds2 = gµν(X)dXµdXν = Iab(u)duadub + r2(u)ˆgij(x)dxidxj . +(2.1) +This spacetime has a product structure +¯ +M(2) × ˆ +M(D−2). Coordinates frame {Xµ} of the +whole spacetime is specified as {ua, xi}. It is beneficial to choose the following viewpoint. +Coordinates ua label the point in the 2-dimensional manifold ¯ +M(2), while xi label the point +in the (D − 2)-dimensional manifold +ˆ +M(D−2). Both manifolds have its independent metric +Iab(u) and ˆgij(x). The areal radius r(u) is a scalar function in spacetime, also a function in +¯ +M(2). Its value means enlarging the unit ˆ +M(D−2) in r times. It is worth emphasizing again +that the manifold +ˆ +M(D−2) is not limited to the maximally symmetric space investigated +in Ref [13,44]. It is not presuming that the independent Ricci scalar ˆR(x) of +ˆ +M(D−2) is a +constant. +It is also assumed that the manifold +ˆ +M(D−2) has a finite (D − 2)-dimensional volume +Ω(D−2), or just the finite part of the +ˆ +M(D−2) with volume Ω(D−2) is concerned. +2.1 +Unified first law from Einstein field equation +Next, consider the Hilbert-Einstein action in the higher dimension +S = +� �MD−2 +Pl +2 +R + Lm +� √−g dDx , +(2.2) +where Lm is the Lagrangian for matter fields. It leads to Einstein equations Rµν −Rgµν/2 = +M2−D +Pl +Tµν . +Appendix A shows tricks for obtaining the Levi-Civita connection and the +Riemann tensor of the metric (2.1). +Using Eq (A.14),(A.15) and (A.16), the Einstein +equations become as +¯Rab − (D − 2) +¯∇a ¯∇br +r +− R +2 Iab = M2−D +Pl +Tab(u, x) , +(2.3) +ˆRij − ˆgij +� +r ¯∇2r + (D − 3)Irr� +− R +2 r2ˆgij = M2−D +Pl +Tij(u, x) , +(2.4) +where Irr is ¯∇ar ¯∇ar for short; ¯∇ and ˆ∇ is the Levi-Civita connection of ¯ +M(2) and ˆ +M(D−2) +respectively. Even the ab components Tab of the energy-momentum tensor may depend on +x due to the x-dependent ˆR(x). Nevertheless, the unified first law is directly derived from +Eq.(2.3). Since the Einstein tensor of a 2-dimensional metric always vanishes1, there is +¯Rab = ¯RIab/2. Then, contracting the Eq.(2.3) with ¯∇br gives +¯R − R +2 +¯∇ar − (D − 2) +¯∇aIrr +2r += M2−D +Pl +Tab(u, x) ¯∇br . +(2.5) +1This feature relates to the fact that any 2-dimensional metric is conformally flat, see [50,51] +5 + +On the other hand, the trace of Eq.(2.3) gives +¯R − R − D − 2 +r +¯∇2r = −2M2−D +Pl +W(u, x) , +(2.6) +where the trace of Tab contributes to the so-called work term [13], +W(u, x) = −1 +2IabTab(u, x) . +(2.7) +The work term (2.7) used in this article probably depends on x. It is necessary to reduce +the order of Eq.(2.6). This relys on re-writting ¯∇2r in terms of W(u, x) and K(x) − Irr(u). +Substitute the D-dimensional Ricci scalar (A.16) into Eq. (2.6), there is +(D − 2) +¯∇2r +r += −2M2−D +Pl +W(u, x) + (K(x) − Irr)(D − 2)(D − 3) +r2 +, +(2.8) +where the K is proportional to the ˆR as follow2, +K(x) = +ˆR(x) +(D − 2)(D − 3) . +(2.9) +Then, using the Eq.(2.6) and Eq.(2.8), the Eq.(2.5) implies +D − 2 +2rD−2 ¯∇a(rD−3(K(x) − Irr)) = M2−D +Pl +(Tab∇br + 2W∇ar) . +(2.10) +Noting that the trick ¯∇aK(x) = ∂K(x)/∂ua = 0 is applied to simplify the left-hand side +(LHS). The appearance of rD−2 hints that the (D − 2)-dimensional volume for the sub- +manifold at the fixed ua is relevant. It naturally leads to the following definition +MMS(u, x) = (D − 2)Ω(D−2) +2 M2−D +Pl +rD−3(u)(K(x) − Irr(u)) , +(2.11) +in which the Ω(D−2) represents the (D − 2)-dimensional volume of +ˆ +M(D−2) with unit-size. +In 4-dimensional spherical case, since there are Ω(2) = 4π, M2 +Pl = 8πGN, and K is always +normalized as 1, Eq. (2.11) reduces to the original definition (1.1). The (D − 2)-volume for +a general sub-manifold at fixed ua is A(D−2) = Ω(D−2)rD−2, and the Eq. (2.10) leads to +¯∇aMMS(u, x) = A(D−2)Tab(u, x) ¯∇br + 2W(u, x) ¯∇aV(D−1) , +(2.12) +in which V(D−1) = Ω(D−2)rD−1/(D − 1) is the (D − 1)-dimensional volume for the region +surrounded by the the (D − 2)-dimensional surface with size A(D−2). +The last step to finish deriving needs to introduce the energy supply vector field. Ref. [13] +defines it as +ψa = T ab(u, x) ¯∇br + W(u, x) ¯∇ar . +(2.13) +2Such notation is ill-defined if D = 3. The +ˆ +M(D−2) only has one dimension such that the ˆR vanishes. +The case of D = 3 is ignored in this article since the main concern here is the x-dependent ˆR. +6 + +The ψa does not depend on x, although it relys on Tab(u, x). It is the vector along +¯ +M(2) +defined by contracting ¯∇ar with the trace-less part of T ab. Since components (2.3) of the +Einstein equation force the traceless part of Tab becomes +Tab − 1 +2(TcdIcd)Iab = −MD−2 +Pl +D − 2 +r +� ¯∇a ¯∇b r − 1 +2 +¯∇2r Iab +� +, +(2.14) +the ψa reduces to +ψa = MD−2 +Pl +D − 2 +2 r +� +( ¯∇2r) ¯∇a r − ¯∇a Irr� +, +(2.15) +which is x-independent. Finally, rewrite the Eq. (2.10) as +¯∇aMMS(u, x) = A(D−2)ψa + W(u, x) ¯∇aV(D−1) . +(2.16) +It completes the derivation of the united first law of Einstein gravity from the Eq. (2.3). +The x-dependence happens at MMS(u, x) and W(u, x) in the unified first law (2.16). Once +restrict +ˆ +M(D−2) as the Einstein manifold including the original case in Refs. [13], the x- +dependence of Eq. (2.16) goes out. An alternative approach is integrating out x on the +compact +ˆ +M(D−2), or its finite part if the volume of the whole +ˆ +M(D−2) is ill-defined. To +formulate this, one should define the average MS mass as +mMS(u) = (Ω(D−2))−1 +� +ˆ +M(D−2) +MMS(u, x) +� +ˆg(x) dD−2x += (D − 2)Ω(D−2) +2 M2−D +Pl +rD−3(k − Irr) , +(2.17) +in which k is the average K in the sense of +k = (Ω(D−2))−1 +� +ˆ +M(D−2) +K(x) +� +ˆg(x) dD−2x , +(2.18) +and define the average work term +w(u) = (Ω(D−2))−1 +� +ˆ +M(D−2) +W(u, x) +� +ˆg(x) dD−2x . +(2.19) +The unified first law has the following average version +¯∇amMS = A(D−2)ψa + w ¯∇aV(D−1) . +(2.20) +Therefore, the unusual shape requires two versions of the unified first law, the non-average +one (2.16) and the average one (2.20). They share the same Aψ term. It is the reason that +this article does not introduce the MS mass “density” in Eq. (2.11), though the “density” +seems competitive with the holographic viewpoint when treating ˆ +M(D−2) as the holographic +screen [48,49]. The next subsection applies the language of differential form to clarify that +the non-average MS mass MMS relates to a (D − 2)-form field, while the average MS mass +mMS as a conserved charge should be the integral of such (D − 2)-form field on +ˆ +M(D−2). +7 + +2.2 +Misner-Sharp mass as conserved charge +The so-called Kodama vector field is introduced for the spacetime with the metric (2.1) +in which the +ˆ +M(D−2) part is the maximal symmetric space [12,13,52]. In this article, the +requirement of the maximal symmetry is released, any spacetime with metric (2.1) has a +Kodama vector field. Suppose ¯ϵab is the volume 2-form of +¯ +M(2), then the Kodama vector +field is given by +Kµ∂µ = −¯ϵab ¯∇br +∂ +∂ua , +(2.21) +where ¯ϵab = IacIbd¯ϵcd. The Kodama vector field does not have i components, i.e., Ki = 0. +The vector field also orthogonal to the ∇µr because Kµ∇µr = Ka ¯∇ar = 0. Combining +these facts with the Levi-Civita connection (A.6), non-vanishing components of ∇µKν are +∇aKb = ¯∇aKb . +(2.22) +Therefore, it is straightforward to show that +∇µKµ = −¯ϵab ¯∇a ¯∇br = 0, +(2.23) +i.e., the Kodama vector field is divergence-free. Then define a conserved current as following +Jµ = −T µνKν . +(2.24) +The vector field Jµ is also divergence free without i components due to the Einstein equation. +More concretely, ∇µJµ = 0 is derived by +∇µ(−T µνKν) = −D − 2 +r +¯ϵbc( ¯∇a ¯∇cr)( ¯∇a ¯∇br) = 0 , +(2.25) +while Ji = 0 is due to Rai = 0. +Lowering the index of vector fields Kµ, Jµ leads to one-form fields K = KµdXµ, J = +JµdXµ. Divergence free implies ∗d ∗ K = 0 and ∗d ∗ J = 0. Therefore, the Hodge dual +(D − 1)-forms ∗K and ∗J are closed , such that there are locally exact (D − 2)-forms QK +and QJ which satisfy ∗K = dQK and ∗J = dQJ. Integrating ∗K and ∗J on a slice Σ with +a boundary ∂Σ leads to two conserve charges +QK = +� +Σ +∗K = +� +∂Σ +QK , +QJ = +� +Σ +∗J = +� +∂Σ +QJ , +(2.26) +The Hodge dual (D − 1)-forms ∗K and ∗J are calculated in detail in the Appendix. B. +The result for ∗K shows a straightforward relationship QK = rD−1ˆϵ/(D − 1) in which ˆϵ is +8 + +the volume (D − 2)-form for the unit +ˆ +M(D−2). Suppose the slice Σ only has one connected +boundary ∂Σ located at a fixed u, then the K charge is +QK = +� +∂Σ +rD−1 +D − 1 ˆϵ = Ω(D−2) +D − 1 rD−1 , +(2.27) +Compare the result (B.11) with Eq. (2.12),(2.13), ∗J should be +∗J = rD−2(ψa + W ¯∇ar) dua ∧ ˆϵ = +¯∇aMMS +Ω(D−2) +dua ∧ ˆϵ , +(2.28) +Thus, the locally exact (D − 2)-form should be QJ = (MMS/Ω(D−2))ˆϵ, and the conserve +charge QJ at ∂Σ is +QJ = +� +∂Σ +MMS +Ω(D−2) +ˆϵ = mMS , +(2.29) +exactly the averaged Misner-Sharp mass (2.17). +3 +Topological black holes from the unified first law +This section applies the unified first law to generate solutions. The method is developed from +Ref. [27] and applied for constructing exact solutions with non-constant ˆR(x). Scalar fields +described by the sigma model offer the matter source. Subsection 1 gives the construction +of such solutions. Additional matter sources, like the cosmological constant, Maxwell field, +and in-falling null matters, are discussed in subsection 2. +Only the unified first law itself is not enough to find solutions since it requires one part +of information about Eq. (2.3). The complementary part needs to introduce the geometric +surface gravity [13], defined as +κgeo = 1 +2 ¯∇2r . +(3.1) +It is x-independent. Moreover, the Eq. (2.8) gives the following relationship +κgeo(u) = D − 3 +D − 2 +M2−D +Pl +Ω(D−2) +MMS(u, x) +rD−2(u) +− M2−D +Pl +D − 2 r(u) W(u, x) , +(3.2) +relating the geometric surface gravity with the MS mass and work term. The fact about +x-independence of κgeo(u) even allows replacing the RHS by corresponding average version +mMS(u) and w(u). +For generating solutions, specific coordinates may make the task easier. The approach +of generating solutions from thermodynamics in Ref. [27] prefers the orthogonal coordinates +{t, r} for the metric Iab. The following takes a different choice. Generally, the 2-dimensional +line element d¯s2 for +¯ +M(2) can be formulated as +d¯s2 = Iabduadub = − f(v, r) +σ2(v, r)dv2 + 2dvdr +σ(v, r) , +(3.3) +9 + +in the general Eddington-Finkelstein coordinates, see appendix C. The geometric surface +gravity in terms of functions f and σ is +κgeo = f′ +2 − σ′ +σ +f +2 , +(3.4) +where f′ = ∂f/∂r and σ′ = ∂σ/∂r. Latter application of such a relation will show that it +dramatically simplifies constructing solutions. +3.1 +Topological black holes sourced by sigma model +This subsection considers the sigma model, which contains multiple scalar fields. As the +matter field, its Lagrangian is +Lφ = −1 +2GIJ(φ)∂µφI∂µφJ − V (φ) . +(3.5) +Variation of δφI gives equations of motion (EOM) of φI, the Klein-Gordon (KG) equations +in curved field space +∇2φI + Γ I +φ JK∇λφJ∇λφK = GIJ ∂V +∂φJ , +(3.6) +where ∇2φ is ∇λ∇λφ for short, and Γ I +φ JK represents the field space connection. Variation +of δgµν leads to the σ-model’s energy-momentum tensor Tµν = GIJ∂µφI∂νφJ + gµνLφ . In +the +¯ +M(2) × ˆ +M(D−2) spacetime, it is decomposed as +Tab = ¯Φab − 1 +2 +¯ΦIab − Iab( +ˆΦ +2r2 + V ) , +Tij = ˆΦij − 1 +2 +ˆΦˆgij − r2ˆgij( +¯Φ +2 + V ) , +(3.7) +where ¯Φab = GIJ∂aφI∂bφJ, ˆΦij = GIJ∂iφI∂jφJ, while ¯Φ, ˆΦ are their trace ¯Φ = Iab ¯Φab, +ˆΦ = ˆgij ˆΦij. The term ¯Φab − 1 +2 ¯ΦIab is the trace-less part of Tφ ab while the term contained ˆΦ +and the potential V contribute to the trace part, so the energy supply vector and the work +term are3 +ψa +φ = ¯Φab ¯∇br − 1 +2 +¯Φ ¯∇ar , +Wφ = +ˆΦ +2r2 + V . +(3.8) +Components Tφ ai = 0 are due to the Einstein equations. +Obviously, ∂aφI = 0 or +∂iφJ = 0 automatically satisfies these conditions. This article chose the simpler ∂aφI = 0 +since the energy supply vector vanishes in this situation. Then, the non-averaged unified +first law (2.16) leads to +MMS(r, x) = ˜ +M(x) + +ˆΦ(x) Ω(D−2) +2(D − 3) +rD−3 + V (x) Ω(D−2) +(D − 1) +rD−1 , +(3.9) +3For a scalar, covariant derivative and usual derivative are the same, namely, ¯∇aφ = ∂aφ and ˆ∇iφ = ∂iφ. +10 + +The relationship between MMS and the function Irr implies +Irr = K(x) − +M2−D +Pl +ˆΦ(x) +(D − 2)(D − 3) − 2M2−D +Pl +(D − 2) +˜ +M(x) +Ω(D−2) +1 +rD−3 − +2M2−D +Pl +V (x) +(D − 1)(D − 2)r2 , +(3.10) +Since Irr does not depend on x, the combination K(x)− M2−D +Pl +ˆΦ(x) +(D−2)(D−3), the function ˜ +M(x) and +the value of the potential V (x) should be constants. Donate them c, M, Vmin respectively. +The mark ”min” reminds one that it is reasonable to suppose φI(x) locate at the mini- +mum of the potential. Then, the function Irr can be rewritten as c − 2M2−D +Pl +(D−2) +M +Ω(D−2) +1 +rD−3 − +2M2−D +Pl +Vmin +(D−1)(D−2)r2, while the average Misner-Sharp mass is +mMS(r) = M + (k − c) (D − 2) Ω(D−2) +2M2−D +Pl +rD−3 + Vmin Ω(D−2) +(D − 1) +rD−1 . +(3.11) +The left-hand side (LHS) of Eq.(2.3) gives a consistent result. Since the only x-dependent +term at is ˆRIab/(2r2), which comes from the term RIab/(2r2), the V (φ(x)) should be a +constant, while the x-dependent Ricci scalar ˆR(x) should be +ˆR(x) = M2−D +Pl +ˆΦ(x) + C , +(3.12) +where the constant C can be identified as (D − 2)(D − 3) c. If introduce the averaged ˆΦ as +oφ = +� +ˆ +M(D−2) ˆΦ(x) +� +ˆg(x) dD−2x +Ω(D−2)(D − 2)(D − 3) +, +(3.13) +then Eq.(3.12) becomes c = k − M2−D +Pl +oφ. Both the Ricci scalar of +ˆ +M(D−2) and the ˆΦ from +scalars contribute to the constant term in Irr. According to Eq. (3.2), the geometric surface +gravity is +κgeo = D − 3 +D − 2 +M2−D +Pl +Ω(D−2) +M +rD−2(u) − +2M2−D +Pl +Vmin +(D − 1)(D − 2)r . +(3.14) +It also equals to f′/2 since f(r) = Irr(r). Thus, the function σ in the line element (3.3) +should satisfy σ′ = 0, such that it only depends on the null time v or be a nonzero constant. +Introduce the coordinates transformation, +dts = +dv +σ(v) + dr +f(r) . +(3.15) +Then, the unified first law and the geometric surface gravity fix the Iab(u)duadub part as +d¯s2 = − (c − 2M2−D +Pl +(D − 2) +M +Ω(D−2) +1 +rD−3 − +2M2−D +Pl +Vmin +(D − 1)(D − 2)r2)dt2 +s ++ +dr2 +c − 2M2−D +Pl +(D−2) +M +Ω(D−2) +1 +rD−3 − 2M2−D +Pl +Vmin +(D−1)(D−2)r2 +. +(3.16) +11 + +On the other hand, there are still freedoms for choosing ˆgij. It becomes the question of +how to embed +ˆ +M(D−2) to the vacuum manifold of the sigma model. The relavant EOMs +are Eq.(3.6) and Eq.(2.4). The Eq.(3.6) reduces to +ˆ∇2φI + Γ I +φ JK ˆ∇iφJ ˆ∇iφK = 0 , +(3.17) +due to ∂aφI = 0 and the constant minimum of the potential, while the Eq.(2.4) becomes +ˆRij − 1 +2 +ˆRˆgij + (D − 3)(D − 4) c +2 ˆgij = M2−D +Pl +(ˆΦij − 1 +2 +ˆΦˆgij) . +(3.18) +The ˆΦij can be treated as the induced metric on the image of +ˆ +M(D−2). In principle, one +can design a suitable field space metric GIJ for an arbitrary shape of +ˆ +M(D−2) such that +Eq.(3.17) and Eq.(3.18) are satisfied. Appendix D gives examples for designing the GIJ to +let the global monopoles family become exact solutions, repeating results of Ref. [53]. +The 4 dimension situation trivializes Eq.(3.18) further. Its LHS totally vanishes because +ˆ +M(D−2) becomes a 2-dimensional manifold such that ˆRij = ˆRˆgij/2. Meanwhile, the RHS +is proportional to the trace-less part of the induced metric ˆΦij. It also vanishes since any +2-dimensional Riemann metric is conformally the same. +ˆΦij as a non-degenerate metric +requires at least two scalar fields. For simplicity, the field space metric is chosen as GIJ = +δIJ, I, J = 1, 2, then the kinetic term of the model is +− 1 +2δIJ ∇µφI∇µφJ = −1 +2(∇µφ1∇µφ1 + ∇µφ2∇µφ2) , +(3.19) +Therefore, Eq.(3.17) is just the Laplacian equation ˆ∇2φI = 0 on the +ˆ +M2 manifold. Again, +any two-dimensional Riemann metric is conformally flat, so it seems to become easier as- +suming ˆgijdxidxj = e−f(α,β)(dα2 + dβ2). +A simple calculation shows that φ1 ∝ α and +φ2 ∝ β are zero modes of the Laplacian ˆ∇2, no matter what the function f(α, β) is. Thus, +the ansatz φ1 = p α, φ2 = p β implies ˆΦijdxidxj = p2(dα2 + dβ2), and automatically solves +the KG equations, but still left the problem of finding f(α, β). +The main constraint for the function f(α, β) comes from Eq.(3.12). A direct calculation +shows +ˆR(α, β) = ef�∂2f +∂α2 + ∂2f +∂β2 +� += 2M−2 +Pl p2 ef + 2c . +(3.20) +One way to attach the problem of finding f is to reduce the number of variables. Changing +the coordinates {α, β} in the the polar coordinates {ρ, ϕ} through α = γ cos ϕ, β = γ sin ϕ, +the line element of ˆgij becomes e−f(dγ2 + γ2dϕ2). Assuming f only depends on γ highly +simplifies the case. The alternative method is to focus on the case of c = 0, such that the +simplest solution is f ∝ α2 + β2. Based on these considerations, two novel 4-dimensional +topological black holes with unusual shapes will be given in the following. +12 + +Type I +It seems beneficial to compare with the general metric of the 2-dimensional maximal sym- +metric space dρ2/(1 − kρ2) + ρ2dϕ. The expression inspires the following ansatz +dρ2 +1 − cρ2 − ψ(ρ) + ρ2dϕ , +(3.21) +for +ˆ +M(D−2) metric, and +φ1 = a +� ρ +dξ +ξ +� +1 − c ξ2 − ψ(ξ) +, +φ2 = a ϕ , +(3.22) +for scalar fields. Using c instead of k is to avoid that k does not mean the average of K(x) +anymore. Then, Eq.(3.12) leads to +dψ(ρ) +dρ += 2M−2 +Pl p2 +ρ +. +(3.23) +Therefore, after adjusting parameters by re-scaling coordinates, the solution becomes +ds2 = − (c − M−2 +Pl M +Ω(2) r +− Vminr2 +3M2 +Pl +)dt2 + +dr2 +c − M−2 +Pl M +Ω(2) r − Vminr2 +3M2 +Pl ++ r2( +dρ2 +1 − cρ2 − 2a2 log ρ + ρ2dϕ2) , +(3.24) +and corresponding scalars are +φ1(ρ) = a MPl +� ρ +dξ +ξ +� +1 − c ξ2 − 2a2 log ξ +, +φ2(ϕ) = a MPl ϕ , +(3.25) +The metric is similar to the topological black hole in (A)dS background. The Iabduadub +part is the same. However, the shape of the horizon is different if a ̸= 0. The coordinate +ϕ can be periodical. The simplest choice is to identify ϕ + 2π with ϕ. It also introduces a +period 2πaMPl for the second scalar field φ2 such that the field space of the model is a flat +cylinder. The Ricci scalar of the unit +ˆ +M2 and the trace of the induced metric ˆΦij are +ˆR(ρ) = 2a2 +ρ2 + 2c , +ˆΦ(ρ) = 2a2MPl2 +ρ2 +, +(3.26) +Thus, the shape of +ˆ +M2 is controlled by two parameters a and c. The square root of the +determinant of the metric is √ˆg = ρ/ +� +1 − cρ2 − 2a2 log ρ. Such an expression makes it +difficult to calculate the 2-dimensional volume of the unit +ˆ +M2 exactly. Instead, it seems +beneficial to study the shape by numerical method. As an example, setting c = 1.1, a = 0.2, +the function 1−cρ2−2a2 log ρ is plotted in Fig.1. The root of this function is ρmax ≃ 0.9552. +13 + +Figure 1: Function 1 − cρ2 − 2a2 log ρ, c = 1.1, a = 0.2. +Then the numerical integral of 2π +� ρmax +0 +√ˆgdρ gives Ω2 ≃ 5.5717. Unfortunately, integrals +like +� ρmax +0 +ˆR√ˆg and +� ρmax +0 +ˆΦ√ˆg blow up near ρ = 0. These divergences cause the average +Misner-Sharp mass mMS and the average work term wφ of the whole ˆ +M2 are not well defined +until introducing a small cut-off for ρ. +One can embed the ˆ +M2 into the 3-dimensional flat space. Considering the metric of the +unit +ˆ +M2, +dρ2 +1 − cρ2 − 2a2 log ρ + ρ2dϕ2 = +cρ2 + 2a2 log ρ +1 − cρ2 − 2a2 log ρdρ2 + dρ2 + ρ2dϕ2 , +(3.27) +and setting the same data of a, c, then plot the function (cρ2+2a2 log ρ)/(1−cρ2−2a2 log ρ) +in Fig.2. +Figure 2: Function +cρ2+2a2 log ρ +1−cρ2−2a2 log ρ, c = 1.1, a = 0.2. +14 + +2.5 +2.0 +1.5 +(d)0l +1.0 +2 +0 +0.5 +- +0.0 +-0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +p2.0 +1.5 +1.0 +20. +0.5 +0.0 +0.5 +0.0 +0.2 +0.4 +0.6 +0.8 +pThere exists a turning point implying that the above function changes its sign. The +numerical result is ρturn ≃ 0.2971. Therefore, the metric in the ρturn < ρ < ρmax region can +be embedded into the Euclidean space, while the 0 < ρ < ρturn part can be only embedded +into the Minkowski space. The embedding diagrams are given in Fig.3. The metric (3.24) +Figure 3: Embedding diagrams for the metric (3.24). The left plot shows the Euclidean part, while +the medium plot shows the Minkowski part. The right plot shows the embedding diagram connects +those two parts and completes another symmetric side. Parameters are set as c = 1.1, a = 0.2. +can describe other unusual shapes by adjusting values of a and c. One can expect cases +of c = 0 and a ̸= 0 will warp the plane, and c < 0 and a ̸= 0 will deform the hyperbolic +surface. Such other cases are ignored in this article and left for future investigation. +Type II +The alternative metric with an unusual shape is +ds2 = −(−M−2 +Pl M +Ω(2) r +− Vminr2 +3M2 +Pl +)dt2 + +dr2 +− M−2 +Pl M +Ω(2) r − Vminr2 +3M2 +Pl ++ r2e−(x2+y2)(dx2 + dy2) . +(3.28) +The ˆgijdxidxj is a conformal plane where the conformal factor is a Gaussian function. Thus, +the volume of the unit +ˆ +M2 is Ω2 = π. When M > 0 and Vmin < 0, the metric (3.28) has +the same Iabduadub part with the planar Schwarzchild-AdS black hole, such that it is also +a topological black hole in this case. Two scalar fields supporting this unusual shape are +φ1(x) = +√ +2MPl x , +φ2(y) = +√ +2MPl y . +(3.29) +The Ricci scalar of the unit +ˆ +M2 and the trace of the induced metric ˆΦij are +ˆR(x, y) = 4ex2+y2 , +ˆΦ(x, y) = 4M2 +Plex2+y2 , +(3.30) +Since the √ˆg = e−(x2+y2), the integral of ˆR(x, y) or ˆΦ(x, y) on the whole is divergence. +Therefore the average Misner-Sharp and the average work term still lack definition. One +15 + +can simply propose a cut-off L for the integral region to preserve the validity of the average +unified first law. It is worth noting that the 2-volume Ω2 should be related to the error +function erf(L) = (2/√π) +� L +0 e−x2dx in this case. More concretely, Ω2 = π erf(L)2, k = +16L2/(π erf(L)2), such that +mMS = 16L2M2 +Pl r + M + π erf(L)2Vmin r3 +3 +, +wφ = +8M2 +PlL2 +π erf(L)2 r2 + Vmin , +(3.31) +which have L2 divergence if setting L → ∞. +To draw the embedding diagram, consider coordinates transformation x = ρ cos ϕ, y = +ρ sin ϕ, the line element of unit +ˆ +M2 becomes e−ρ2dρ2 + e−ρ2ρ2dϕ2. Then compare with the +polar coordinates of a plane, identify ˆr = ρe−ρ2/2, rewritten the unit +ˆ +M2 line element as +follows, +e−ρ2dρ2 + e−ρ2ρ2dϕ2 = (2 − ρ2)ρ2e−ρ2dρ2 + (dˆr +dρ)2dρ2 + ˆr2dϕ2 , +(3.32) +where the sign of function (2 − ρ2)ρ2e−ρ2 determines whether the metric can be embedded +into Euclidean space or Minkowski space. The plot of the function is shown in Fig.4. The +Figure 4: Function ρ e−ρ2/2 and (2 − ρ2)ρ2e−ρ2. The latter one is positive in 0 < ρ < +√ +2 and +negative in ρ > +√ +2. +turning point is ρturn = +√ +2. The Euclidean region is 0 < ρ < +√ +2 while the Minkowski +region is ρ > +√ +2. The embedding diagrams are shown in Fig.5. +16 + +0.6 +0.5 +0.4 +22 +0.3 +0.2 +0.1 +0.0 +0 +2 +3 +4 +p0.4 +0.3 +(0. +0.2 +0.1 +0.0 +-0.1 +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +3.5 +pFigure 5: Embedding diagrams for the metric (3.28). Similar to Fig.3, the left two plots show the +Euclidean part and the Minkowski part separately. The right-most plot connects them. As ρ tends +to infinity, the Minkowski part tends to the center peak. +3.2 +Additional sources +Cosmological constant +The cosmological constant, which can be regarded as the vacuum energy density, is the +simplest “matter” source. Its Lagrangian is +LΛ = −MD−2 +Pl +Λ , +(3.33) +which gives the Λgµν term at the LHS of Einstein equation, or classically effective energy- +momentum tensor Tµν = −MD−2 +Pl +Λgµν when putting at the RHS. Such an energy-momentum +tensor contributes a constant energy density in space. It also shifts the value of the potential +V → V + MD−2 +Pl +Λ. Thus, only turning off the spacetime dependence of scalars in Eq.(3.8), +the energy supply vector and the work term of the cosmological constant are directly read +as +ψa +Λ = 0 , +WΛ = MD−2 +Pl +Λ . +(3.34) +Such a constant work term contributes the MD−2 +Pl +Λ ¯∇aV(D−1) term to the unified first law, +and the energy supply vector vanishes. Therefore, the cosmological constant plays the role +of pressure for a thermodynamics system. There are more interesting issues in the context +of cosmology if treating Λ as a pressure [25, 26, 54, 55]. An alternative view is that the +cosmological constant should be put at the LHS, such that it provides a −MD−2 +Pl +Λ V(D−1) +correction to the MS mass, as in Ref. [44]. +17 + +Maxwell field +The Lagrangian of a Maxwell field is +Lem = −1 +4FµνF µν . +(3.35) +It gives the energy-momentum tensor Tµν = FµλF λ +ν +− 1 +4gµνFλρF λρ and the source-free +Maxwell equations are +dF = 0 , +d ∗ F = 0 , +(3.36) +in terms of differential forms. The F is the 2-form field strength, while ∗F is the Hodge +dual of F. The generalized Coulomb field in higher dimension +Fab = +Q +Ω(D−2) rD−2 ¯ϵab , +Fij = 0 , +(3.37) +has already solved the Maxwell equations. Obviously, dF = 0 since the factor only depends +on r and any +¯ +M(2) 1-form p = padua satisfies p ∧ ¯ϵ = 0. The Hodge dual (D − 2)-form is +∗F = Qˆϵ/Ω(D−2), so d ∗ F ∝ dˆϵ = 0. Thus, despite what the Iabduadub part is, Eq.(3.37) +must be the solution of source-free Maxwell equations. Then, integrating the dual (D − 2)- +form ∗F on +ˆ +M(D−2) gives the electric charge +Q = +� +ˆ +M(D−2) +∗F , +(3.38) +which confirms the choice of the 1/Ω(D−2) factor. Then, the energy-momentum tensor for +such Coulumb field F = Q¯ϵ/(Ω(D−2) rD−2) is decomposed as +Tab = − +Q2 +2Ω2 +(D−2) +1 +r2(D−2) Iab , +Tij = +Q2 +2Ω2 +(D−2) +1 +r2(D−3) ˆgij , +(3.39) +where F 2 = FabF ab. +ψa +em = 0 , +Wem = +Q2 +2Ω2 +(D−2) +1 +r2(D−2) . +(3.40) +For the 4-dimensional spherical case, Eq.(3.40) reduces to the result in Ref. [13]. Again, the +energy supply vector vanishes, and the work term only changes when varying r. +If every matter source considered in the theory contributes i) only r-dependent average +work term w(i)(r); and ii) vanishing energy supply vector ψa +(i) = 0, then the Iabduadub +part of the spacetime +¯ +M(2) × ˆ +M(D−2) is already determined as −Irrdt2 + dr2/Irr, and the +function Irr should be +Irr = k − 2M2−D +Pl +(D − 2) +M +Ω(D−2) +1 +rD−3 − +� +i +2M2−D +Pl +(D − 2) rD−3 +� r +w(i)(ξ)ξD−2dξ . +(3.41) +18 + +Similar tricks for constructing solutions in the previous subsection are still valid for the +proof. The average unified first law under conditions (i) and (ii) gives +¯∇amMS = +� +i +w(i)(r) Ω(D−2)rD−2 ¯∇ar , +(3.42) +such that directly integrating r implies that the average MS mass is +mMS(r) = M + +� +i +Ω(D−2) +� r +w(i)(ξ)ξD−2dξ . +(3.43) +It constricts the function Irr taking the form of Eq.(3.41). Concretely, the cosmological +contributes − +2Λ r2 +(D−1)(D−2), while the Maxwell field contributes +M2−D +Pl +(D−2)(D−3) +Q2 +Ω2 +(D−2) r−2(D−3) to +Irr. Eq.(3.42) also implies +1 +2 +dIrr +dr += D − 3 +D − 2 +M2−D +Pl +Ω(D−2) +mMS(r) +rD−2 +− M2−D +Pl +D − 2 r +� +i +w(i)(r) , +(3.44) +The RHS coincides with the geometric surface gravity relation (3.2). Thus, in the GEF +coordinates {v, r}, there is σ′ = 0 again. Finally, the coordinates transformation (3.15) +explicitly shows that the Iabduadub part must be −Irrdt2 +dr2/Irr. It completes the proof. +Taking the case of vacuum, both ψa and w vanishes. Meanwhile, for spherical symmetry, +the ˆgij is detemeined4 and the value of k should be 1. The theorem reduces to Birkhoff’s +theorem. In such a sense, this theorem generalizes Birkhoff’s theorem. +Apply the above conclusion, one obtains charged novel topological black hole solutions +ds2 = − (c − M−2 +Pl M +Ω(2) r ++ +M−2 +Pl Q2 +2Ω2 +(2) r2(D−3) − Λr2 +3 )dt2 ++ +dr2 +c − M−2 +Pl M +Ω(2) r + +M−2 +Pl Q2 +2Ω2 +(2) r2(D−3) − Λr2 +3 ++ r2( +dρ2 +1 − cρ2 − 2a2 log ρ + ρ2dϕ2) , +(3.45) +and +ds2 = − (−M−2 +Pl M +Ω(2) r ++ +M−2 +Pl Q2 +2Ω2 +(2) r2(D−3) − Λr2 +3 )dt2 ++ +dr2 +− M−2 +Pl M +Ω(2) r + +M−2 +Pl Q2 +2Ω2 +(2) r2(D−3) − Λr2 +3 ++ r2e−(x2+y2)(dx2 + dy2) , +(3.46) +where the minimum of the potential is absorbed into the cosmological constant Λ. +4In the complete proof, there is one part to proof that spherical symmetry demanding the spacetime +contains an orbit of SO(3) group, such that the metric has the form of Eq.(2.1), see Ref. [56]. Such a step +is skipped in this article. +19 + +Irreducible mass and usable energy +Suppose rH is the location of the black hole horizon, since there should be Irr(rH) = 0, +the averaged MS mass at the horizon is mMS = (D − 2)Ω(D−2)MD−2 +Pl +krD−3 +H +/2. Modestly, +in the case of k > 0, the MS mass mMS should not decrease from the viewpoint of the area +theorem [56]. It seems reasonable to define (D − 2)Ω(D−2)MD−2 +Pl +krD−3 +H +/2 as irreducible +mass mirr. It is natural to further define the usable energy as musable(r) = mMS(r) − mirr +outside the horizon r > rH. The mass parameter does not appear in musable(r). Instead, +such a new definition is determined by the difference of the work term between location r +and horizon rH. Ref. [57] discussed the possibility of a Schwarzchild black hole as a battery. +One can regard that their charging process turns the rest energy of in-falling material into +the usable energy of the black hole. Suitable discharging will extract such energy. It seems +the concept of usable energy at least works in the case of the asymptotic flat RN black hole. +Non-vanishing energy supply vector +The case of ∂aφI ̸= 0 violates the condition ψa = 0 such that the above generalization +of Birkhoff’s theorem becomes invalid. It explains that a static spherical solution with a +non-constant σ(r) is due to r-dependent scalars [58]. Another situation violating condition +(ii) ψa = 0 is the appearance of a Vaidya mass, i.e., a null time-dependent function for +the mass parameter. More concretely, in the GEF coordinates {v, r}, the Iabduadub part +satisfies +− +� +(k − ωD M(v) +rD−3 +− +� +i +2M2−D +Pl +(D − 2) rD−3 +� r +w(i)(ξ)ξD−2dξ) +� +dv2 ± 2dvdr , +(3.47) +such that the average unified first law gives +¯∇amMS = ˙M ¯∇av + +� +i +w(i)(r) Ω(D−2)rD−2 ¯∇ar , +(3.48) +where ˙M is dM/dv. Thus, the function M(v) implies the non-vanishing energy supply vector +ψa = ˙M ¯∇av/A(D−2) without changing contributions from other matter fields. Eq.(3.44) is +still valid if change the derivative dIrr/dr as ∂Irr/∂r. Therefore, the work term of the source +of Vaidya mass must vanish. It hints at finding the corresponding energy-momentum tensor +T vr = ψa ¯∇av = 0 , +T rr = ψa ¯∇ar = ± +˙M +Ω(D−2) rD−2 , +T vv = 0 . +(3.49) +The result for component T vv is obtained by using the vanishing work term W = 0 again. At +the horizon, the energy supply vector is interpreted as a heat flow [13], so it is plausible that +20 + +the vector ψa does not vanish in dynamical cases, like the approach of obtaining evolving +spacetime by multiplying a time-dependent conformal factor to the static solution [18]. +4 +The first law including topological charge +Ref. [48,49] introduce the concept of topological charge for a topological RN-AdS black hole +to preserve the Gibbs-Duhem-like relation. Using the notation of this article, the explicit +expression of the metric of a topological RN-AdS black hole is +ds2 = −fdt2 + dr2 +f ++ r2ˆgijdxidxj , +(4.1) +where +f = k − 2M2−D +Pl +(D − 2) +M +Ω(D−2) +1 +rD−3 + +M2−D +Pl +(D − 2)(D − 3) +Q2 +Ω2 +(D−2) +1 +r2(D−3) + r2 +l2 , +(4.2) +and l is the AdS radius. They consider formally changing k continually, then define the +topological charge as +ε = Ω(D−2) +� +|k| +D−2 . +(4.3) +Treat r = rc as a holographic screen, label f(rc), df(rc)/dr as fc, f′ +c. The Brown-York +tensor on the screen leads to the energy density ϵ and the pressure p as +ϵ = C − D − 2 +M2−D +Pl +√fc +rc +, +p = D − 3 +M2−D +Pl +√fc +rc ++ +1 +2M2−D +Pl +f′ +c +√fc +− C , +(4.4) +where C is a constant from holographic renormalization [48,49,59]. They define the total +“volume”5 and energy of the whole screen as +V = Ω(D−2)rD−2 +c +, +E = ϵV . +(4.5) +However, the entropy is chosen as horizon entropy +S = +2π +M2−D +Pl +Ω(D−2)rD−2 +H +, +(4.6) +in which rH satisfies f(rH) = 0, determines the location of the Killing horizon. Replacing +M2−D +Pl +as 8π, the entropy reduces to the well-known one quarter of the horizon area. After +suitably defining the temperature T +T = TH +√fc += +1 +2π√fc +�rH +l2 + (D − 3) 2M2−D +Pl +(D − 2) +M +Ω(D−2) +1 +rD−2 +H +− M2−D +Pl +D − 2 +Q2 +Ω2 +(D−2) +1 +r2D−5 +H +� +, +(4.7) +5It is worth noting that the “volume” V is the (D − 2)-dimensional area A(D−2) rather than the (D − 1)- +dimensional volume V(D−1) in this article. +21 + +the chemical potential µ for the electric charge Q +µ = − +1 +(D − 3) Ω(D−2) +√fc +� +Q +rD−3 +c +− +Q +rD−3 +H +� +, +(4.8) +and the potential ζ for the topological charge ε +ζ = sgn(k) +ε +4−D +D−2 Ω +D−4 +D−2 +(D−2) +M2−D +Pl +rD−3 +H +− rD−3 +c +√fc +, +(4.9) +on the screen6, not only the first law of thermodynamics dE + pdV = TdS + µdQ + ζdε is +obtained, the Gibbs-Duhem-like relation +E + pV = TS + µQ + ζε , +(4.10) +still holds for non-planar (k ̸= 0) cases. It is the motivation of Ref. [48,49] introducing the +topological charge. This charge is suggested as the last charge of a black hole with maximal +symmetry. On the other hand, Ref. [60] introduces the chemical potential for the number +of colors, which is applied to study the phase structure of a topological black hole [61]. The +number of colors seems to have similar behavior with the topological charge under scale +transformation of +ˆ +M(D−2), but their connection is still unclear. +It is worth mentioning details about scaling +ˆ +M(D−2). +The power (D − 2)/2 in the +definition (4.3) implies that the topological charge is also a k-normalized volume of MD−2 or +the part of MD−2 in interest. For the maximally symmetric ˆ +M(D−2) with a non-normalized +and nonzero k, re-scale the metric ˆgij → l2 ˆgij to set |k| as 1, then the additional scale l +can always be absorbed by adjusting other parameters in Irr. Therefore, the maximally +symmetric +ˆ +M(D−2) with a non-normalized k does not distinguish it from the k-normalized +situation. +Modestly, if introducing a topological charge for MD−2, it seems better to +consider the MD−2 as an Einstein manifold without maximal symmetry. +4.1 +Topological charge as the portal of +ˆ +M(D−2) +Nevertheless, the parameter k plays the role of a portal connecting the manifold +¯ +M(2) and +shape parameters of +ˆ +M(D−2). In higher dimensions, even if the Ricci scalar of +ˆ +M(D−2) +is a constant, the manifold +ˆ +M(D−2) is not necessary to have maximal symmetry. As an +example, consider the following 4-dimensional Einstein manifold satisfied ˆRij = kˆgij, +dˆs2 = +∆ˆr +ˆr2 − ˆa2 cos2 ˆθ +� +dˆτ − ˆa sin2 ˆθ +1 − k ˆa2 +3 +dˆφ +�2 + (ˆr2 − ˆa2 cos2 ˆθ) +�dˆr2 +∆ˆr ++ dˆθ2 +∆ˆθ +� ++ +∆ˆθ sin2 ˆθ +ˆr2 − ˆa2 cos2 ˆθ +� +ˆadˆτ + ˆr2 − ˆa2 +1 − k ˆa2 +3 +dˆφ +�2 , +(4.11) +6The notation for factors of M and Q is different with Refs. [48,49]. +22 + +where +∆ˆr = (ˆr2 − ˆa2)(1 − kˆr2 +3 ) − 2 ˆmˆr , +∆ˆθ = 1 − kˆa2 +3 cos2 ˆθ . +(4.12) +This is the Euclideanized Kerr-(A)dS gravitational instanton [62]. The 4-dimensional Planck +scale MPl(4) is set as 1. The additional hat mark reminds us that the instanton would be +regarded as the +ˆ +M(D−2) manifold. Then it is not hard to construct the following metric in +6-dimensional Einstein gravity. +ds2 = −(k − ω6 M +r3 +)dt2 + +dr2 +k − ω6 M +r3 ++ r2dˆs2 , +(4.13) +where ω6 can be treated as a suitable constant in 6-dimension when the Eq.(4.11) is the +metric ˆgij of the +ˆ +M(4). This is a vacuum solution of the 6-dimensional Einstein equations. +At least the situation of M > 0 and k > 0 can describe a black hole. The shape of its +horizon is a 4-dimensional Kerr-dS instanton. +It is reasonable to allow continuously varying k for the Einstein manifold. +The pa- +rameters of the Euclidean spacetime satisfy the following Smarr-like relationship due to +∆ˆr = 0.7 +ˆ +M2 = +ˆS(1 − k ˆS +3π )2 +4π +− π ˆJ2 +ˆS ++ k ˆJ2 +3 +, +(4.14) +where +ˆ +M = +ˆm +(1 − k ˆa2 +3 )2 , +ˆJ = +ˆmˆa +(1 − k ˆa2 +3 )2 , +ˆS = π ˆr2 ++ + ˆa2 +1 − k ˆa2 +3 +, +(4.15) +in which the ˆr+ is the larger root of the equation ∆ˆr = 0. Treating ˆ +M as the function of +ˆS, ˆJ, k, the variation of the Smarr-like relation leads to the first law +δ ˆ +M = ˆTδ ˆS + ˆΩδ ˆJ + ˆΘδk , +(4.16) +where +ˆT = 12π4 ˆJ2 + 3π2 ˆS2 − 4πk ˆS3 + k2 ˆS4 +24π3 ˆ +M ˆS2 +, ˆΩ = +ˆJ(k ˆS − 3π) +3 ˆ +M ˆS +, +ˆΘ = 6π3 ˆJ2 − 3π ˆS2 + k ˆS3 +36π3 ˆ +M +. +(4.17) +Thus, there should be the first law for the 6-dimensional black hole, +δM = TδS + r3 +H +ω6 ˆΘ +(δ ˆ +M − ˆTδ ˆS − ˆΩHδ ˆJ) . +(4.18) +where +rH = (ω6M +k +)1/3 , +T = κ +2π , +S = +A +4G6 +. +(4.19) +7The J2 term in Euclidean signature has a different sign from the one in Lorentz signature of Ref. [62]. +23 + +Parameters of the Einstein manifold can join the first law of black hole thermodynamics for +the whole +¯ +M(2) × ˆ +M(D−2). The δk plays the role of a portal. From such a point of view, +the concrete definition (4.3) does not matter. For convenience, this article still states that +continuously varying k introduces the topological charge. +4.2 +Unusual shapes and topological charges +Reminding that the role of k for the average unified first law is similar to the role of K +for the non-average first law, it may be beneficial to consider the non-average law to figure +out natural interpretation of the topological charge, since changing the direction, i.e., the +position in +ˆ +M(D−2) naturally causes continually varying K. Such a consideration requires +clarifying the relationship between the unified first law, and the usual first law of black hole +thermodynamics. +The unified first law states a relation in spacetime, while the usual one deals with the +variation of global parameters. To handle the tension between two viewpoints of the first +law, consider a finite falling energy package that goes through the trapping horizon8, shown +in fig.6, in such a way that the horizon begins as a Killing horizon, and finally settles down as +a new Killing horizon. To ensure the unified first law is valid, it is assumed that the ˆ +M(D−2) +part keeps unchanging. The trapping horizon only changes its size during the process. Select +a vector za tangent to the ¯ +M(2) and the trapping horizon, i.e., (za ¯∇aIrr)H = 0, and donate +f′ as (za ¯∇af)H, then according to Eq.(2.15), contracting za with the A(D−2)ψa generates the +MD−2 +Pl +κgeo A′ +(D−2) term, the equivalent expression of the heat flow term TdS. The tunneling +approach for Hawking radiation in dynamical spacetime confirms the relationship between +κgeo at the horizon and the horizon temperature T = κH/(2π) [20–24]. Remarkably, two +versions of the unified first law Eq.(2.16) and Eq.(2.20) become +M′ +MS − WH V ′ +(D−1) = m′ +MS − wH V ′ +(D−1) = +κgeo A′ +(D−2) +M2−D +Pl +, +(4.20) +which is regarded as the quasi-local first law. The average first law implies the integral +mMS,f − mMS,i − +� +TH +wH dVH = +1 +M2−D +Pl +� +TH +κH dAH , +(4.21) +where mMS,f is the final average MS mass on the trapping horizon, mMS,i is the initial +one, lower subscribes H represent taking value on the horizon. Suppose the change is tiny +8The trapping horizon is defined as a hypersurface foliated by marginal surfaces, so it has vanishing +expansion. The equation Irr = 0 determines the location of a trapping horizon, see appendix C. +24 + +Figure 6: The time coordinate v is the null retarded time, while ¯t is given by v − r. Such a sketch +describes a falling process changing the size of the trapping horizon. The red curve represents the +evolving tapping horizon. Gray arrows portray the in-falling energy package. Both the initial and +final moment is marked by gray dotted lines while Killing horizons are indicated by black lines. +enough such that every global parameter becomes M + δM and Q(i) + δQ(i) at the final +state. Then the (4.21) reduces to +δmMS(rH) − wH δVH = κH δAH +M2−D +Pl +. +(4.22) +The term about δk is omitted since the shape of ˆ +M(D−2) is fixed during the falling process. +It is significant that the term κH δAH appears. To indicate that the Eq.(4.20) is the quasi- +local viewpoint for the first law, although it will be less general, calculate δmMS(rH) of +Eq.(3.43). Keeping every small change of global parameters except k, the result is +δmMS(rH) = δM + Ω(D−2) +� +i +� � rH ∂w(i)(ξ) +∂Q(i) +ξD−2dξ +� +δQ(i) + wH δVH . +(4.23) +Identify the corresponding potential for the charge Q(i) as +Φ(i),H = −Ω(D−2) +� rH ∂w(i)(ξ) +∂Q(i) +ξD−2dξ , +(4.24) +such that the relation +δmMS(rH) − wH δVH = δM − +� +i +Φ(i),HδQ(i) , +(4.25) +25 + +2is obtained. Eq.(4.25) and Eq.(4.22) together give the usual first law formulated by variation +of global parameters. Thus, the result (4.25) shows how to connect the quasi-local viewpoint +(LHS) with the global viewpoint (RHS). +Now, back to the issue of x-dependence of K. It is only the K that contributes the +x-dependence of the non-average MMS by definition (2.11). +Consider a tangent vector +hµ∂/∂Xµ = za∂/∂ua + yi∂/∂xi on the trapping horizon. +Such vector not only has ua +components za but also has xi component yi. Then redefine f′ as hµ∂µf taking value on +the horizon, the non-average Eq.(4.20) is modified as +M′ +MS − (D − 2)Ω(D−2) rD−3 +H +2M2−D +Pl +K′ − WH V ′ +(D−1) = +κgeo A′ +(D−2) +M2−D +Pl +. +(4.26) +Again, considering integrating Eq.(4.26) along h and supposing only a small change of global +parameters, all derivatives f′ are replaced by δf, including K′. Keeping δk in the usual +first law or the average unified first law is only a formal operation. However, from the +non-average point of view, allowing δK is naturally achieved by changing the direction in +ˆ +M(D−2). It opens up the consideration of defining topological charge (4.3) by K. +The K term responds to the x-dependence in MMS, relating to the LHS of the non- +average unified first law (2.16). For the RHS side, as is discussed in section 2, the energy +supply vector does not depend on x, so the only term responding to the x-dependence in +MMS must appear in the work term with the correct power of r. Eq.(3.2) helps again. It +specifies that the relevant work term should be proportional to r−2. Multi-scalars offer +such work term. Concrete examples are novel topological black holes constructed in section +3 and global monopole solutions. The contribution from the r−2 work term cancels the +x-dependence from K(x) in the charged solution (3.45) and (3.46). The type II solution +(3.46) have the same Iabduadub part with the planar black hole, while the type I solution +(3.45) describes diverse situations replacing k by c in Irr. However, it is unclear what the +topological significance is if replace k with c in the definition (4.3). +An alternative approach is to define topological charge in a concrete context. In novel +solutions and global monopole solutions in appendix D, scalar fields φI map the sub-manifold +ˆ +M(D−2) to the vacuum manifold of the sigma model. Thus, another choice for defining the +topological charge is the degree of map for scalars, which is proportional to the integral of +the volume element of the vacuum manifold +deg(φ) ∝ +� +vac +ϵIJ...K dφI ∧ dφJ ∧ · · · ∧ dφK . +(4.27) +Here, it is assumed an appropriate compactification was chosen to ensure the integral is +finite. The definition includes situations discussed in Ref. [53] in principle. Such a definition +26 + +is topological because the degree is homotopy invariant. The x-dependence K rely on the +x-dependence work term with r−2 power. One can treat the non-average unified first law +should contain both “topological charges” from curvature K and degree of map from scalars +simultaneously. Two topological charges satisfy some relation through c if concerning the +ˆ +M(D−2) part. The x-dependence is canceled while the constant c may be retained. In this +sense, it is the c playing the role of the portal. It seems that introducing topological charges +only touches the corner of an iceberg. Exact solutions (3.45) and (3.46) remind us that the +constant term in Irr still hides rich structures. +5 +Summary and discussion +This article re-derives the unified first law from Einstein equations in detail. Distinguish- +ing from the original law, sub-manifold +ˆ +M(D−2) is no limit to the maximally symmetric +space, but rather an arbitrary manifold including dependent-on-position Ricci scalar ˆR(x). +Such unusual shapes require two versions of MS masses and unified first laws. The non- +average law keeps x-dependence, while the average law is from integrating out positions x +on +ˆ +M(D−2). +Two types of novel 4-dimensional topological black hole solutions in Einstein-scalars the- +ory are constructed. The improved approach of generating solutions from thermodynamics +simplifies the construction. Horizon shapes of these black holes are visualized by plotting +embedding diagrams in one higher dimension space. Both cannot be entirely embedded +into a single Euclidean or Minkowski space. The improved approach further gives a simple +method to introduce cosmological constant, electric charge, and Vaidya mass. +The non-constant K(x) gives natural motivation to consider topological charge since its +value changes when moving the position x in +ˆ +M(D−2). Novel solutions show that K(x) is +achieved by x-dependent scalars. Contribution for Irr from K(x) and ˆΦ(x) cancel out, but +they left a constant. The residual constant seems to challenge the original motivation for +introducing the topological charge. A modest viewpoint is viewing the residual constant as +a portal to parameters about the horizon shape for the first law. +The next task is to investigate how to add angular momentum to topological black holes +obtained here. It has been successfully done for planar and hyperbolic solutions in Ref. +[63–66]. We expect a systematic method to generate spinning solutions from the solutions +with the metric (2.1), at least type I and type II solutions. Once corresponding spinning +solutions are obtained, they are worth testing the validity of Kerr/CFT corresponding [67], +27 + +and may even give some new ideas about black hole evaporation and entropy of horizon. +Recent progress in quantum gravity has calculated the entropy of horizon with arbitrary +shape [68]. Solutions here offer concrete examples. It is unclear whether the non-asymptotic +flat or dS features forbid the production of topological black holes with unusual shapes. It +is worth exploring these possibilities like Ref. [34]. +An alternative approach to pushing forward such an investigation is considering modified +gravity. Generalizing the MS mass and the unified first law to modified gravity theories +has been studied in Ref. [44–46], but they are still only concerned about the maximal +symmetry. Especially the MS mass for the Gauss-Bonet gravity has non-linear terms of +about k [44], such that the relation between the average unified first law and the non- +average law may be non-trivial. Despite the difficulties, the investigation of this article +suggests that the generating solutions approach for modified gravity theories [69–71] may +still be worth studying. One can expect the corresponding improved approach benefit of +finding new topological black holes with unusual horizon shapes. +Novel topological black holes and the construction method may imply some unexpected +application in the holographic context. At first, the topological charge, or the color charge +introduced in Ref. [60] enriches phase structures of topological black hole [61]. Such previous +researches are interested in maximal symmetry. Nevertheless, novel topological black holes +imply an inhomogeneous AdS boundary. The entrance of parameters about the shape in +the first law would enlarge the black hole phase structures. It seems studying black hole +phase transition would also inspire some new ideas for cosmology [54, 55]. Secondly, Ref. +[61] emphasizes the sub-system aspect for formulating the thermodynamics when dealing +with the non-compact nature of the plane and the hyperbolic surface. +The sub-system +consideration inspires the possibility that the metric of ˆgij describes the unusual shape with +richer inhomogeneous structures. +The whole +ˆ +M(D−2) may be too inhomogeneous to be +described by a single coordinates frame {xi}. Instead, the entire spacetime may need many +small pieces to write down the line-element (2.1). Different patches are not the same but +relate to each other by scaling r → lr if they overlap. +Applying the AdS/CFT corresponding to the type I and type II topological AdS black +holes, the boundary field theory should live in the non-Euclidean space. The AdS boundary +with unusual shape means that novel solutions are not asymptotic AdS spacetime in a +precise sense. Solutions do not reduce to asymptotically flat spacetime when turning off +the cosmological constant. Thus, the topological theorem in Ref. [10] is not violated. It is +natural to expect this is the holographic description for some boundary solidary material +28 + +in unusual shapes, although the part embedded in Minkowski space seems less plausible. +The construction in section 3 implies inverse engineering. For a given +ˆ +M(D−2), one can +design an appropriate field space metric GIJ for the sigma model to achieve the metric ˆgij +of ˆ +M(D−2). The warp ˆgij should respond to a non-vanishing stress tensor Tij, which may be +similar to the previous study about holographic viscoelastic hydrodynamics [72]. Moreover, +it is remarkable that the ansatz (3.25) and (3.29) for scalars is similar to the ansatz that +appeared in the holographic axion model [73]. Several scalars have VEV proportional to +coordinates. Completing the inverse engineering technique may give a systematic method +to find analytic models, opening up a new field of holographic solids. +Appendix A: Calculate Γλ +µν and Rλ +ρµν +This appendix shows an efficient method for calculating the Levi-Civita connection and +Riemann tensor. Usually, the components of the Levi-Civita connection for a given metric, +are calculated by +Γλ +µν = 1 +2gλσ(∂gµσ +∂xν + ∂gνσ +∂xµ − ∂gµν +∂xσ ) , +(A.1) +The geodesic equations give hints to finding the trick. Considering the geodesic equations +d2xλ +dτ 2 + Γλ +µν +dxµ +dτ +dxν +dτ = d2xλ +dτ 2 + gλσ(dgσν +dτ +dxν +dτ − 1 +2 +∂gµν +∂xσ +dxµ +dτ +dxν +dτ ) = 0 , +(A.2) +in which dgσν/dτ = (∂gσν/∂xµ)(dxµ/dτ). one can extract the following structure +Γλ +µνdxµdxν = gλσ(dgσνdxν − 1 +2 +∂ds2 +∂xσ ) , +(A.3) +which can be viewed as a coordinate-dependent rank-2 symmetric tensor, in which ∂ds2/∂xσ +is the short notation of (∂gµν/∂xσ)dxµdxν. Since the coordinates frame is fixed under a +particular calculation, one can simply treat gσν as several functions of xµ and dgσν represents +their differential. The trick is to calculate the structure Eq. (A.3) rather than to calculate +components of Eq. (A.1) one by one. Now we use this trick to calculate the Levi-Civita +connection of the metric (2.1). The components of its inverse metric are +gab = Iab , +gij = ˆgij +r2 . +(A.4) +Thus, Eq. (A.3) gives +Γa +µνdxµdxν = ¯Γa +bcdubduc − (r ¯∇ar)ˆgijdxidxj , +Γi +µνdxµdxν = 2 +¯∇ar +r duadxi + ˆΓi +jkdxjdxk . +(A.5) +29 + +Here, the property ∂r/∂ua = ¯∇ar is used. Noticing that product terms like duadxi are the +short notation for symmetric tensor product (duadxi ⊗ dxidua)/2, every component can be +correctly read as +Γa +bc = ¯Γa +bc , Γa +ij = −rIab ∂r +∂ub ˆgij , Γi +aj = 1 +r +∂r +∂ua δi +j , Γi +jk = ˆΓi +jk . +(A.6) +It is worth noting that the components Γa +bc and Γi +jk are just the independent Levi-Civita +connection of +¯ +M(2) and +ˆ +M(D−2) respectively. +The author would like to introduce the +covariant differential operator ¯∇a for +¯ +M(2) and ˆ∇i for +ˆ +M(D−2). The areal radius r(u) can +be treated as a scalar field in ¯ +M(2). The notation ¯∇ar also means ∂r/∂ua while ¯∇ar means +Iab(∂r/∂ub). +Once the connection was obtained, the Riemann tensor can be calculated through +Rλ +ρµν = ∂Γλ +νρ/∂xµ + Γλ +µσΓσ +νρ − (µ ↔ ν). It can also be treated as several 2-forms due +to the anti-symmetry of exchanging µ and ν, +1 +2Rλ +ρµνdxµ ∧ dxν = dΓλ +νρ ∧ dxν + (Γλ +µσdxµ) ∧ (Γσ +νρdxν) . +(A.7) +In order to simplify the notation, label 1 +2Rλ +ρµνdxµ ∧ dxν as Ωλ +ρ and Γλ +µσdxµ as Aλ +ρ , then +Ωλ +ρ = dAλ +ρ + Aλ +σ ∧ Aσ +ρ . +(A.8) +These 1-forms Aλ +ρ can be viewed as the connection 1-forms for the coordinates tetrad +(∂µ)ν = δν +µ while Ωλ +ρ are their curvature 2-forms. Concretely, 1-forms Aλ +ρ for the metric +(2.1) are +Aa +b = ¯Γa +cb duc = ¯Aa +b , +Aa +i = − (r ¯∇ar)ˆgijdxj , +Ai +a = +¯∇ar +r dxi , +Ai +j = +¯∇ar +r δi +jdua + ˆΓi +kjdxk = dr +r δi +j + ˆAi +j . +(A.9) +Then one obtains curvature 2-forms by applying Eq .(A.8). Firstly, +Ωa +b = dAa +b + Aa +c ∧ Ac +b + Aa +i ∧ Ai +b = ¯Ωa +b , +(A.10) +since term containing ˆgijdxi ∧ dxj vanishes. Notice that Ωai = IabΩb +i = −Ωia = −r2ˆgijΩj +a, +calculating Ωi +a can avoid dealing with dˆgij here: +Ωi +a = +¯∇a ¯∇br +r +dub ∧ dxi , +(A.11) +therefore, +Ωa +i = −r ¯∇a ¯∇br ˆgij dub ∧ dxj , +(A.12) +30 + +The final 2-form Ωi +j is +Ωi +j = ˆΩi +j − Irrδi +k ˆgjl dxk ∧ dxl = 1 +2 +� ˆRi +jkl − Irr(δi +k ˆgjl − δi +l ˆgjk) +� +dxk ∧ dxl , +(A.13) +where the term Irr is the short notation for Iab ¯∇ar ¯∇br. Reminding dxµ ∧ dxν = dxµ ⊗ +dxν − dxν ⊗ dxµ, one can read the components of Riemann tensor as +Ra +bcd = ¯Ra +bcd , +Ra +ibj = −Ra +ijb = −r( ¯∇a ¯∇br) ˆgij , +Ri +ajb = −Ri +abj = − +¯∇a ¯∇br +r +δi +j , +Ri +jkl = ˆRi +jkl − Irr(δi +kˆgjl − δi +lˆgjk) . +(A.14) +The same result can be found in Ref. [44]. Contracting δi +i = D −2, components of the Ricci +tensor are +Rab = ¯Rab − (D − 2) +¯∇a ¯∇br +r +, Rij = ˆRij − ˆgij +� +r ¯∇2r + (D − 3)Irr� +, +(A.15) +in which the ¯∇2r is ¯∇a ¯∇ar for short. The Ricci scalar is +R = IabRab + ˆgij +r2 Rij = ¯R − 2(D − 2) +¯∇2r +r ++ +ˆR +r2 − (D − 2)(D − 3)Irr +r2 . +(A.16) +Appendix B: Calculate ∗J and ∗K +Then the author will calculate the Hodge dual ∗K and ∗J directly. One can read the volume +D-form ϵ from the metric (2.1) as following +ϵ = rD−2� +−I ˆg du1 ∧ du2 ∧ dx1 ∧ · · · ∧ dxn−2 , +(B.1) +where I, ˆg is the determinate of Iab, ˆgij respectly. Meanwhile, the volume 2-form ¯ϵ for ¯ +M(2) +is +¯ϵ = +√ +−I du1 ∧ du2 , +(B.2) +and the volume (D − 2)-form ˆϵ for +ˆ +M(D−2) is +ˆϵ = +� +ˆg dx1 ∧ · · · ∧ dxD−2 . +(B.3) +This notation makes the following relation obvious, +ϵ = rD−2¯ϵ ∧ ˆϵ . +(B.4) +It is still necessary to use the notation of tensor components to calculate the Hodge dual, +ϵab i1i2...iD−2 = rD−2¯ϵab ˆϵi1i2...iD−2 . +(B.5) +31 + +Then, ∗J is given by +∗ J = rD−2Ja¯ϵabdub ∧ ˆϵ = rD−2T ac(¯ϵab¯ϵcd) ¯∇dr dub ∧ ˆϵ , +(B.6) +Noting that tensor notation of the volume 2-form ¯ϵab can be expressed as +¯ϵab = +√ +−I¯εab , +(B.7) +in which the symbol ¯εab satisfies ¯ε12 = −¯ε21 = 1 and ¯ε11 = ¯ε22 = 0 . Then the ¯ϵab is +¯ϵab = −( +√ +−I)−1¯εab , +(B.8) +where the symbol ¯εab also share the same pattern with ¯εab, i.e., ¯ε12 = −¯ε21 = 1 and +¯ε11 = ¯ε22 = 0, such that +¯ϵab¯ϵcd = −¯εab¯εcd = −δa +cδb +d + δa +dδb +c . +(B.9) +Lowering the ab indexes leads to +¯ϵab¯ϵcd = −IacIbd + IadIbc . +(B.10) +Thus, the (D − 1)-form ∗J becomes +∗J =rD−2(Tab ¯∇ar + 2W ¯∇br) dub ∧ ˆϵ , +(B.11) +As for ∗K, it can be obtained by simply replacing the T ab in ∗J by −Iab. Thus, +∗K = rD−2 ¯∇br dub ∧ ˆϵ = d( rD−1 +D − 1) ∧ ˆϵ . +(B.12) +Appendix C: General Eddington-Finkelstein coordinates +The 2-dimensional sub-spacetime must permits double null coordinates {u, v} such that the +line element becomes +ds2 = −Ω2(u, v)dudv + r2(u, v)ˆgij(x)dxidxj . +(C.1) +In the region where ¯∇ar does not vanish, r itself can be a coordinate, such that one can +change to other coordinates frame like {v, r}. +Since dr = r,u du + r,v dv where r,u, r,v +represents ∂r/∂u, ∂r/∂v, the line element becomes +ds2 = Ω2 r,v +r,u +dv2 − Ω2 +r,u +drdv + r2ˆgij(x)dxidxj . +(C.2) +32 + +The line element (C.2) is still general. +The coordinates frame {v, r} is called general +Eddington-Finkelstein (GEF) coordinates in this article. Define functions σ(v, r) = −2r,u/Ω2 +and f(v, r) = −4r,ur,v/Ω2, metric components under the GEF coordinates are +gvv = Ivv = − f(v, r) +σ2(v, r) , +grr = Irr = 0 , +gvr = grv = Ivr = Irv = +1 +σ(v, r) , +gij = r2ˆgij , +(C.3) +while inverse metric components are +gvv = Ivv = 0 , +grr = Irr = f(v, r) , +gvr = grv = Ivr = Irv = σ(v, r) , +gij = ˆgij +r2 . +(C.4) +In general, the shape of spacetime +¯ +M(2) × ˆ +M(D−2) with a given metric ˆgij for the unit +ˆ +M(D−2) is described by two functions. In double null coordinates, they are Ω(u, v) and +r(u, v), while in GEF coordinates, they are σ(v, r) and f(v, r). +The usage of function +f(v, r) is convenient since it picks up the important function Irr = ∇µr∇µr. Further, the +determinant of Iab in the GEF coordinates is simply I = −σ2. Then +√ +−I equals to σ−1 up +to a sign. Therefore, the Laplacian of r in +¯ +M(2) is +¯∇2r = σ∂(σ−1Ivr) +∂v ++ σ∂(σ−1Irr) +∂r += f′ − σ′ +σ f , +(C.5) +where f′ = ∂f/∂r and σ′ = ∂σ/∂r. The result leads to a simple expression (3.4) for the +geometric surface gravity (3.1) in terms of f and σ. +The next task is to calculate expansions of null vector fields tangent to +¯ +M(2). Specify +fields as +kµ +∂ +∂Xµ = ∂ +∂v + f +2σ +∂ +∂r , +lµ +∂ +∂Xµ = −σ ∂ +∂r , +(C.6) +and require the condition kµlν = −1. Assuming the increasing direction of v is future, +kµ and lν are all future-pointed. +Their expansion can be easily calculated by the trick +θ(k) = (D − 2) ka ¯∇ar/r without dealing with gµν∇µkν +θ(k) = (D − 2) f +2σr , +θ(l) = −(D − 2) σ +r . +(C.7) +Such a method is also used in Ref. [26]. The hypersurface Irr = f = 0 leads to θ(k) = 0, +thus determining a trapping horizon, which is defined as a hypersurface foliated by marginal +surfaces [12, 13]. A marginal surface is a 2-codimensional spatial surface with vanishing +expansion. One can further classify types of trapping horizons according to the behavior of +θ(l) and Llθ(k), see Refs. [12,13]. +33 + +It seems there is a problem that the vector lµ vanishes at location σ = 0. However, +such location also implies a singular line element. To handle this issue, let us back to the +non-singular double null coordinates. Another choice for two null vector fields is +˜kµ +∂ +∂Xµ = +√ +2 +Ω +∂ +∂v , +˜lµ +∂ +∂Xµ = +√ +2 +Ω +∂ +∂u . +(C.8) +Their expansions are +θ(˜k) = +√ +2(D − 2) +Ω r +r,v , +θ(˜l) = +√ +2(D − 2) +Ω r +r,u . +(C.9) +Apply the coordinates transformation and carefully deal with the chain rule +∂ +∂vDN += +∂ +∂vGEF ++ r,v +∂ +∂r , +∂ +∂u = r,u +∂ +∂r , +(C.10) +where vGEF donates the coordinate v in GEF frame and vDN donates the v in double null +frame, one confirms the following relation +˜kµ = +√ +2 +Ω kµ , +˜lµ = Ω +√ +2 lµ , +(C.11) +such that +θ(˜k) = +√ +2 +Ω θ(k) , +θ(˜l) = Ω +√ +2 θ(l) . +(C.12) +Since Ω ̸= 0, σ = 0 is just a coordinate singularity, implying that r,u = 0, which is another +trapping horizon which satisfies θ(l) = 0. However, the relationship +Irr = −2( +r +D − 2)2θ(k)θ(l) , +(C.13) +is covariant. The normalization condition kµlµ forces that the construction θ(k)θ(l) does not +change under a regular rescaling like Eq.(C.11). Switching the role of u and v changes the +concerned null vector field from ˜k to ˜l. Thus, the equation Irr = 0 always picks up the +hypersurface with vanishing expansion (θ(k) = 0 or θ(l) = 0), i.e., the trapping horizon. +Appendix D: Design the field space metric GIJ +This appendix gives examples of designing the field space metric GIJ for a given ˆgij. Ref. +[53] modifies the field space metric to let the global monopole becomes an exact solution +and further generalizes it to planar and hyperbolic situations in higher dimension. Such +research serves as a good example of designing. This appendix aims to repeat results in +Ref. [53] through the approach of this article. Ref. [47] overstated the approach of generating +34 + +solutions from thermodynamics. It was thought that the approach cannot derive the global +monopole solution. It is going to show the validity of the improved approach. +Consider a D dimensional spacetime with metric (2.1) and the ansatz φI = η nI(x). +As what has been discussed in section 3, such ansatz implies the vanishing energy supply +vector and the work term depending on ua only through r. Thus, according to the theorem +in subsection 3.3, the line element Iabduadub must be −f(r)dt2 +dr2/f(r) with the function +f(r) = c − 2M2−D +Pl +(D − 2) +M +Ω(D−2) +1 +rD−3 − +2M2−D +Pl +Vmin +(D − 1)(D − 2)r2 . +(D.1) +The only problem is to seek the suitable field space metric GIJ. +Suppose the ˆgij part is a (D − 2)-dimensional sphere. The simplest choice is GIJ = δIJ +in (D − 1) dimensions where I, J = 1, 2, . . . , D − 1. Demanding δIJnInJ = 1 respects the +symmetry of rotation, no matter in +ˆ +M(D−2) or the field space. It means embedding the +unit sphere ˆgijdxidxj into a higher dimensional Euclidean space δIJdφIdφJ, such that +δIJ ∂inI ∂jnJ = ˆgij , +(D.2) +However, a flat field space metric δIJ does not satisfy the goal of exact solutions [53]. It +inspires the choice of +GIJ = F(X) δIJ , +(D.3) +where X = δIJφIφJ. Then the induced metric due to φI = η nI(x) is +ˆΦij = F(η2) η2 ∂inI∂jnI = F(η2) η2 ˆgij . +(D.4) +Thus, the work term of the model is +Wφ = (D − 2) η2 F(η2) +2r2 ++ Vmin , +(D.5) +which implies c = k − M2−D +Pl +η2F(η2)/(D − 3) according to Eq.(3.12). The final check is the +Klein-Gordon (KG) equations. Since the field space metric GIJ is conformally flat, the field +space connection is +Γ I +φ JK = 1 +F +dF +dX (φJ δI +K + φK δI +J − φIδJK) . +(D.6) +Thus, the KG equations give +ˆ∇2nI(x) = +ˆΦ nI +F(η2)( dF +dX )X=η2 . +(D.7) +Tricks in appendix A lead to ˆ∇2nI = −(D − 2)nI. Consider the following line element +ds2 = f(ρ)dρ2 + ρ2ˆgijdxidxj , +(D.8) +35 + +in which the Iab part is only one dimensional. +Since the ˆgij is supposed as a (D − 2)- +dimensional metric in this article, the metric (D.8) has D − 1 dimensions. +Results in +appendix A implies +Γρ +ρρ = 1 +2f +df +dρ , Γa +ij = − ρ +f ˆgij , Γi +aj = 1 +ρδi +j , Γi +jk = ˆΓi +jk . +(D.9) +and +Rρ +iρj = − ρ +2f2 +df +dρ ˆgij , Ri +ρjρ = +1 +2ρf +df +dρδi +j , Ri +jkl = ˆRi +jkl − 1 +f (δi +kˆgjl − δi +lˆgjk) . +(D.10) +If f = 1 and the ˆgijdxidxj part is a sphere, the metric (D.8) describes a Euclidean space +ds2 = δIJdXIdXJ in spherical coordinates. Thus, flatness requires vanishing curvature, so +there should be ˆRi +jkl = δi +kˆgjl−δi +lˆgjk. The case for a Minkowski space ds2 = ηIJdXIdXJ has +f = −1 and the ˆgijdxidxj part is a hyperbolic surface, such that ˆRi +jkl = −(δi +kˆgjl − δi +lˆgjk). +This is a quick way to ensure the curvature of the maximum symmetric space is ˆRi +jkl = +k(δi +kˆgjl − δi +lˆgjk). As for the Laplacian, its expression under coordinates {ρ, xi} is +∇I∇Iφ = +1 +fρD−2 +∂ +∂ρ(ρD−2 ∂φ +∂ρ) + +ˆ∇2φ +ρ2 , +(D.11) +where ∇I donates the covariant derivative with respect to the metric (D.8) . If choosing +the scalar φ as one of the coordinates XI = ρnI, then ∇J∇JXI = 0. In coordinates {ρ, xi}, +the vanishing Laplacian in a flat space implies +ˆ∇2nI = −D − 2 +f +nI . +(D.12) +For sphere, f = 1, so ˆ∇2nI = −(D − 2)nI. Therefore, to be consistent, there should be +( dF +dX )X=η2 = − 1 +η2 . +(D.13) +For simplicity, one can further require F(η2) = 1. These are conditions for the field space +metric F(X)δIJ. +Such an argument repeats the results of Ref. [53] for the spherical case. For the hyper- +bolic case, ˆgijdxidxj has k = −1. It can only be embedded into one higher dimensional +Minkowski space rather than the Euclidean space. The above argument is valid by simply +replacing δIJ with ηIJ. As for the planar case k = 0, there is no need to introduce higher +dimensional field space. Simply using the ansatz φI = ηxI leads to a similar result. The +unified expression for the metric is +ds2 = − (k − M2−D +Pl +η2 +D − 3 +− 2M2−D +Pl +(D − 2) +M +Ω(D−2) +1 +rD−3 − Vminr2 +3MD−2 +Pl +)dt2 ++ +dr2 +k − M2−D +Pl +η2 +D−3 +− 2M2−D +Pl +(D−2) +M +Ω(D−2) +1 +rD−3 − Vminr2 +3MD−2 +Pl ++ r2ˆgijdxidxj , +(D.14) +36 + +where k = −1, 0, 1. It completes deriving solutions and finding suitable field space metrics, +and obtains the same results with Ref. [53]. +Acknowledgments +The author thanks Hyat Huang and Hongwei Tan for their helpful and inspiring discus- +sion, and the support of Mayumi Aoki and Ryoko Nishikawa regarding personnel affairs at +Kanazawa University. 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Astron. 64 (2021), +no. 7 270001, [2101.01892]. +42 + diff --git a/_dAzT4oBgHgl3EQfvf3k/content/tmp_files/load_file.txt b/_dAzT4oBgHgl3EQfvf3k/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..5b9bf8e8f23354fd5bf4fb8555280af7fc2a59bd --- /dev/null +++ b/_dAzT4oBgHgl3EQfvf3k/content/tmp_files/load_file.txt @@ -0,0 +1,1675 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf,len=1674 +page_content='Novel topological black holes from thermodynamics and deforming horizons Jinbo Yang 1,2,3 1Department of Astronomy, School of Physics and Materials Science, Guangzhou University, Guangzhou 510006, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='China 2Institute for Theoretical Physics, Kanazawa University, Kanazawa 920-1192, Japan 3HKUST Jockey Club Institute for Advanced Study, The Hong Kong University of Science and Technology, Clear Water Bay, Hong Kong, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='China ABSTRACT Two novel topological black hole exact solutions with unusual shapes of the horizon in the 4-dimensional Einstein-scalars theory are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The application of the unified first law dramatically simplifies the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The article re-derives such first law from the Einstein equations for a spacetime with product structure ¯ M(2) × ˆ M(D−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The non- constant Ricci scalar ˆR of the sub-manifold ˆ M(D−2) remarkably leads to two versions of the Misner-Sharp mass and the unified first law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Both versions are applied to generate topo- logical black holes and to discuss their significance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Unusual shapes of ˆ M(D−2) in solutions naturally require topological charges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It gives the first glimpse of non-explored aspects of the first law of black hole thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Keywords: topological black hole, unusual shape, unified first law, Misner-Sharp mass, topological charge yangjinbophy@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='com arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='01709v1 [gr-qc] 4 Jan 2023 Contents 1 Introduction 3 2 Unified first law including unusual shape 4 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1 Unified first law from Einstein field equation .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 8 3 Topological black holes from the unified first law 9 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1 Topological black holes sourced by sigma model .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 17 4 The first law including topological charge 21 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1 Topological charge as the portal of ˆ M(D−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 22 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2 Unusual shapes and topological charges .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 24 5 Summary and discussion 27 Appendix A: Calculate Γλ µν and Rλ ρµν 29 Appendix B: Calculate ∗J and ∗K 31 Appendix C: General Eddington-Finkelstein coordinates 32 Appendix D: Design the field space metric GIJ 34 2 1 Introduction In several decades, black hole physics, especially black hole thermodynamics, has brought us deep insights into theoretical physics [1–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The mainly concerning shape of the black hole has a spherical topology, supported by the topological theorem for Einstein gravity [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' According to the theorem, the horizon of a 4-dimensional asymptotically flat black hole must be topologically spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Further focusing on the perfectly spherical black hole is an appropriate starting point for detailed research of black hole evolution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' A spherically symmetric spacetime in 4-dimensional Einstein gravity has a quasi-local mass called Misner- Sharp (MS) mass [11,12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' mMS = r 2GN (1 − gµν∇µr∇νr) , (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1) where GN is the 4-dimensional Newton constant, and r is the areal radius which gives the area of a sphere A = 4πr2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' This mass reduces to the Arnowitt-Deser-Misner(ADM) mass in the asymptotic flat background and is significant for formulating the unified first law in Einstein gravity [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The MS mass and the unified first law are widely applied in the context of primordial black hole formation [14–19], detailed study for Hawking evaporation [20–24] and the first law on the apparent horizon of an expanding universe [25,26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' These concepts also inspire a novel approach to generate spherical black hole solutions of general relativity from thermodynamics [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Nevertheless, many black objects beyond the spherical horizon in higher dimensional supergravity or string theory have been discovered [28–33].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' One kind of black object is the topological black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The part of its metric describing the horizon shape is not a sphere but rather an Einstein manifold [34–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Widely studied topological black holes have planar or hyperbolic horizons [38–40].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The hyperbolic counterpart of the Schwarzchild black hole can be viewed as a gravitational description of S-brane [41], and the planar one is widely applied since it generally describes a specific boundary phase in the context of AdS/CFT correspondence [42, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The parameters of a planar Reissner-Nordstr¨om(RN)- AdS black hole satisfy the Gibbs-Duhem relationship.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The unified first law and the approach of deriving solutions from thermodynamics also have been generalized to include planar and hyperbolic cases in the context of modified gravity [44–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [48, 49] introduces the concept of topological charge for the topological RN-AdS black hole to preserve the Gibbs-Duhem-like relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' They suggest that the topological charge is the last charge for the first law of thermodynamics of a maximally symmetric black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The article explores the possibility of replacing the spherical part with an unusual shape, 3 not limited to maximally symmetric space or Einstein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Suppose the part replacing the sphere is an independent manifold with metric ˆgij(x), where x donate the point in the independent manifold, and i, j, k are the corresponding indexes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' If the manifold is maxi- mally symmetric, the Rienman tensor from ˆgij(x) is ˆRijkl = k(ˆgikˆgjl − ˆgilˆgjk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' An Einstein manifold satisfies a weaker condition ˆRij(x) = λˆgij(x), where λ is a constant [35].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Its Ricci scalar ˆR is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' This article concerns the situation with non-constant ˆR(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The original unified first law only focuses on the maximally symmetric situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' However, the unusual shape with non-constant ˆR(x) implies two versions of the unified first law, which is discussed by re-deriving the law from Einstein equations in section 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Remarkably, ap- plying unified first laws simplifies constructing concrete examples of topological black holes with unusual shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The method used here improves the approach of generating solutions from thermodynamics in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Novel topological black holes with unusual shapes in 4-dimensional Einstein-scalars theory are obtained through this method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Successfully con- structing solutions with the non-constant ˆR(x) seems to give a more natural motivation for introducing the topological charge defined in [48,49], but it also challenges the topological significance of such a charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Nevertheless, novel solutions open a new window to inves- tigate how parameters describing the shape of the horizon join the first law of black hole thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The article is organized as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Section 2 revisits the unified first law of general relativity by dimensional reduction, an explicitly covariant method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Additionally, it is shown that Misner-Sharp mass is a conserved charge relating to the Kodama vector, and a more general horizon shape leads to two versions for formulating the unified first law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Section 3 constructs novel topological black hole solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The improved approach to generating solutions from thermodynamics is discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Embedding plots are displayed to visualize the horizon shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Contributions from other sources are also attached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Metrics for novel solutions with charge and the cosmological constant are given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Section 4 focuses on the topological charge to discuss how such a charge should relate to the shape of the horizon from the viewpoint of the unified first law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Section 5 gives a summary and discusses other open issues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 2 Unified first law including unusual shape This section re-derives the unified first law from the Einstein equations and shows that Misner-Sharp mass is a conserved charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Consider a D-dimensional spacetime with the 4 following line element ds2 = gµν(X)dXµdXν = Iab(u)duadub + r2(u)ˆgij(x)dxidxj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1) This spacetime has a product structure ¯ M(2) × ˆ M(D−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Coordinates frame {Xµ} of the whole spacetime is specified as {ua, xi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is beneficial to choose the following viewpoint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Coordinates ua label the point in the 2-dimensional manifold ¯ M(2), while xi label the point in the (D − 2)-dimensional manifold ˆ M(D−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Both manifolds have its independent metric Iab(u) and ˆgij(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The areal radius r(u) is a scalar function in spacetime, also a function in ¯ M(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Its value means enlarging the unit ˆ M(D−2) in r times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is worth emphasizing again that the manifold ˆ M(D−2) is not limited to the maximally symmetric space investigated in Ref [13,44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is not presuming that the independent Ricci scalar ˆR(x) of ˆ M(D−2) is a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is also assumed that the manifold ˆ M(D−2) has a finite (D − 2)-dimensional volume Ω(D−2), or just the finite part of the ˆ M(D−2) with volume Ω(D−2) is concerned.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1 Unified first law from Einstein field equation Next, consider the Hilbert-Einstein action in the higher dimension S = � �MD−2 Pl 2 R + Lm � √−g dDx , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2) where Lm is the Lagrangian for matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It leads to Einstein equations Rµν −Rgµν/2 = M2−D Pl Tµν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Appendix A shows tricks for obtaining the Levi-Civita connection and the Riemann tensor of the metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Using Eq (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='14),(A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='15) and (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='16), the Einstein equations become as ¯Rab − (D − 2) ¯∇a ¯∇br r − R 2 Iab = M2−D Pl Tab(u, x) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) ˆRij − ˆgij � r ¯∇2r + (D − 3)Irr� − R 2 r2ˆgij = M2−D Pl Tij(u, x) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='4) where Irr is ¯∇ar ¯∇ar for short;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' ¯∇ and ˆ∇ is the Levi-Civita connection of ¯ M(2) and ˆ M(D−2) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Even the ab components Tab of the energy-momentum tensor may depend on x due to the x-dependent ˆR(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Nevertheless, the unified first law is directly derived from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Since the Einstein tensor of a 2-dimensional metric always vanishes1, there is ¯Rab = ¯RIab/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Then, contracting the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) with ¯∇br gives ¯R − R 2 ¯∇ar − (D − 2) ¯∇aIrr 2r = M2−D Pl Tab(u, x) ¯∇br .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5) 1This feature relates to the fact that any 2-dimensional metric is conformally flat, see [50,51] 5 On the other hand, the trace of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) gives ¯R − R − D − 2 r ¯∇2r = −2M2−D Pl W(u, x) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6) where the trace of Tab contributes to the so-called work term [13], W(u, x) = −1 2IabTab(u, x) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='7) The work term (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='7) used in this article probably depends on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is necessary to reduce the order of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' This relys on re-writting ¯∇2r in terms of W(u, x) and K(x) − Irr(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Substitute the D-dimensional Ricci scalar (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='16) into Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6), there is (D − 2) ¯∇2r r = −2M2−D Pl W(u, x) + (K(x) − Irr)(D − 2)(D − 3) r2 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8) where the K is proportional to the ˆR as follow2, K(x) = ˆR(x) (D − 2)(D − 3) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='9) Then, using the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8), the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5) implies D − 2 2rD−2 ¯∇a(rD−3(K(x) − Irr)) = M2−D Pl (Tab∇br + 2W∇ar) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='10) Noting that the trick ¯∇aK(x) = ∂K(x)/∂ua = 0 is applied to simplify the left-hand side (LHS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The appearance of rD−2 hints that the (D − 2)-dimensional volume for the sub- manifold at the fixed ua is relevant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It naturally leads to the following definition MMS(u, x) = (D − 2)Ω(D−2) 2 M2−D Pl rD−3(u)(K(x) − Irr(u)) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='11) in which the Ω(D−2) represents the (D − 2)-dimensional volume of ˆ M(D−2) with unit-size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' In 4-dimensional spherical case, since there are Ω(2) = 4π, M2 Pl = 8πGN, and K is always normalized as 1, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='11) reduces to the original definition (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The (D − 2)-volume for a general sub-manifold at fixed ua is A(D−2) = Ω(D−2)rD−2, and the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='10) leads to ¯∇aMMS(u, x) = A(D−2)Tab(u, x) ¯∇br + 2W(u, x) ¯∇aV(D−1) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='12) in which V(D−1) = Ω(D−2)rD−1/(D − 1) is the (D − 1)-dimensional volume for the region surrounded by the the (D − 2)-dimensional surface with size A(D−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The last step to finish deriving needs to introduce the energy supply vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [13] defines it as ψa = T ab(u, x) ¯∇br + W(u, x) ¯∇ar .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='13) 2Such notation is ill-defined if D = 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The ˆ M(D−2) only has one dimension such that the ˆR vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The case of D = 3 is ignored in this article since the main concern here is the x-dependent ˆR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 6 The ψa does not depend on x, although it relys on Tab(u, x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is the vector along ¯ M(2) defined by contracting ¯∇ar with the trace-less part of T ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Since components (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) of the Einstein equation force the traceless part of Tab becomes Tab − 1 2(TcdIcd)Iab = −MD−2 Pl D − 2 r � ¯∇a ¯∇b r − 1 2 ¯∇2r Iab � , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='14) the ψa reduces to ψa = MD−2 Pl D − 2 2 r � ( ¯∇2r) ¯∇a r − ¯∇a Irr� , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='15) which is x-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Finally, rewrite the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='10) as ¯∇aMMS(u, x) = A(D−2)ψa + W(u, x) ¯∇aV(D−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='16) It completes the derivation of the united first law of Einstein gravity from the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The x-dependence happens at MMS(u, x) and W(u, x) in the unified first law (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Once restrict ˆ M(D−2) as the Einstein manifold including the original case in Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [13], the x- dependence of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='16) goes out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' An alternative approach is integrating out x on the compact ˆ M(D−2), or its finite part if the volume of the whole ˆ M(D−2) is ill-defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' To formulate this, one should define the average MS mass as mMS(u) = (Ω(D−2))−1 � ˆ M(D−2) MMS(u, x) � ˆg(x) dD−2x = (D − 2)Ω(D−2) 2 M2−D Pl rD−3(k − Irr) , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='17) in which k is the average K in the sense of k = (Ω(D−2))−1 � ˆ M(D−2) K(x) � ˆg(x) dD−2x , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='18) and define the average work term w(u) = (Ω(D−2))−1 � ˆ M(D−2) W(u, x) � ˆg(x) dD−2x .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='19) The unified first law has the following average version ¯∇amMS = A(D−2)ψa + w ¯∇aV(D−1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='20) Therefore, the unusual shape requires two versions of the unified first law, the non-average one (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='16) and the average one (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' They share the same Aψ term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is the reason that this article does not introduce the MS mass “density” in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='11), though the “density” seems competitive with the holographic viewpoint when treating ˆ M(D−2) as the holographic screen [48,49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The next subsection applies the language of differential form to clarify that the non-average MS mass MMS relates to a (D − 2)-form field, while the average MS mass mMS as a conserved charge should be the integral of such (D − 2)-form field on ˆ M(D−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 7 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2 Misner-Sharp mass as conserved charge The so-called Kodama vector field is introduced for the spacetime with the metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1) in which the ˆ M(D−2) part is the maximal symmetric space [12,13,52].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' In this article, the requirement of the maximal symmetry is released, any spacetime with metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1) has a Kodama vector field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Suppose ¯ϵab is the volume 2-form of ¯ M(2), then the Kodama vector field is given by Kµ∂µ = −¯ϵab ¯∇br ∂ ∂ua , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='21) where ¯ϵab = IacIbd¯ϵcd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The Kodama vector field does not have i components, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=', Ki = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The vector field also orthogonal to the ∇µr because Kµ∇µr = Ka ¯∇ar = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Combining these facts with the Levi-Civita connection (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6), non-vanishing components of ∇µKν are ∇aKb = ¯∇aKb .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='22) Therefore, it is straightforward to show that ∇µKµ = −¯ϵab ¯∇a ¯∇br = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='23) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=', the Kodama vector field is divergence-free.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Then define a conserved current as following Jµ = −T µνKν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='24) The vector field Jµ is also divergence free without i components due to the Einstein equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' More concretely, ∇µJµ = 0 is derived by ∇µ(−T µνKν) = −D − 2 r ¯ϵbc( ¯∇a ¯∇cr)( ¯∇a ¯∇br) = 0 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='25) while Ji = 0 is due to Rai = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Lowering the index of vector fields Kµ, Jµ leads to one-form fields K = KµdXµ, J = JµdXµ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Divergence free implies ∗d ∗ K = 0 and ∗d ∗ J = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Therefore, the Hodge dual (D − 1)-forms ∗K and ∗J are closed , such that there are locally exact (D − 2)-forms QK and QJ which satisfy ∗K = dQK and ∗J = dQJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Integrating ∗K and ∗J on a slice Σ with a boundary ∂Σ leads to two conserve charges QK = � Σ ∗K = � ∂Σ QK , QJ = � Σ ∗J = � ∂Σ QJ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='26) The Hodge dual (D − 1)-forms ∗K and ∗J are calculated in detail in the Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The result for ∗K shows a straightforward relationship QK = rD−1ˆϵ/(D − 1) in which ˆϵ is 8 the volume (D − 2)-form for the unit ˆ M(D−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Suppose the slice Σ only has one connected boundary ∂Σ located at a fixed u, then the K charge is QK = � ∂Σ rD−1 D − 1 ˆϵ = Ω(D−2) D − 1 rD−1 , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='27) Compare the result (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='11) with Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='12),(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='13), ∗J should be ∗J = rD−2(ψa + W ¯∇ar) dua ∧ ˆϵ = ¯∇aMMS Ω(D−2) dua ∧ ˆϵ , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='28) Thus, the locally exact (D − 2)-form should be QJ = (MMS/Ω(D−2))ˆϵ, and the conserve charge QJ at ∂Σ is QJ = � ∂Σ MMS Ω(D−2) ˆϵ = mMS , (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='29) exactly the averaged Misner-Sharp mass (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='17).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 3 Topological black holes from the unified first law This section applies the unified first law to generate solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The method is developed from Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [27] and applied for constructing exact solutions with non-constant ˆR(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Scalar fields described by the sigma model offer the matter source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Subsection 1 gives the construction of such solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Additional matter sources, like the cosmological constant, Maxwell field, and in-falling null matters, are discussed in subsection 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Only the unified first law itself is not enough to find solutions since it requires one part of information about Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The complementary part needs to introduce the geometric surface gravity [13], defined as κgeo = 1 2 ¯∇2r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1) It is x-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Moreover, the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8) gives the following relationship κgeo(u) = D − 3 D − 2 M2−D Pl Ω(D−2) MMS(u, x) rD−2(u) − M2−D Pl D − 2 r(u) W(u, x) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2) relating the geometric surface gravity with the MS mass and work term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The fact about x-independence of κgeo(u) even allows replacing the RHS by corresponding average version mMS(u) and w(u).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' For generating solutions, specific coordinates may make the task easier.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The approach of generating solutions from thermodynamics in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [27] prefers the orthogonal coordinates {t, r} for the metric Iab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The following takes a different choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Generally, the 2-dimensional line element d¯s2 for ¯ M(2) can be formulated as d¯s2 = Iabduadub = − f(v, r) σ2(v, r)dv2 + 2dvdr σ(v, r) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) 9 in the general Eddington-Finkelstein coordinates, see appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The geometric surface gravity in terms of functions f and σ is κgeo = f′ 2 − σ′ σ f 2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='4) where f′ = ∂f/∂r and σ′ = ∂σ/∂r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Latter application of such a relation will show that it dramatically simplifies constructing solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1 Topological black holes sourced by sigma model This subsection considers the sigma model, which contains multiple scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' As the matter field, its Lagrangian is Lφ = −1 2GIJ(φ)∂µφI∂µφJ − V (φ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5) Variation of δφI gives equations of motion (EOM) of φI, the Klein-Gordon (KG) equations in curved field space ∇2φI + Γ I φ JK∇λφJ∇λφK = GIJ ∂V ∂φJ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6) where ∇2φ is ∇λ∇λφ for short, and Γ I φ JK represents the field space connection.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Variation of δgµν leads to the σ-model’s energy-momentum tensor Tµν = GIJ∂µφI∂νφJ + gµνLφ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' In the ¯ M(2) × ˆ M(D−2) spacetime, it is decomposed as Tab = ¯Φab − 1 2 ¯ΦIab − Iab( ˆΦ 2r2 + V ) , Tij = ˆΦij − 1 2 ˆΦˆgij − r2ˆgij( ¯Φ 2 + V ) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='7) where ¯Φab = GIJ∂aφI∂bφJ, ˆΦij = GIJ∂iφI∂jφJ, while ¯Φ, ˆΦ are their trace ¯Φ = Iab ¯Φab, ˆΦ = ˆgij ˆΦij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The term ¯Φab − 1 2 ¯ΦIab is the trace-less part of Tφ ab while the term contained ˆΦ and the potential V contribute to the trace part, so the energy supply vector and the work term are3 ψa φ = ¯Φab ¯∇br − 1 2 ¯Φ ¯∇ar , Wφ = ˆΦ 2r2 + V .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8) Components Tφ ai = 0 are due to the Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Obviously, ∂aφI = 0 or ∂iφJ = 0 automatically satisfies these conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' This article chose the simpler ∂aφI = 0 since the energy supply vector vanishes in this situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Then, the non-averaged unified first law (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='16) leads to MMS(r, x) = ˜ M(x) + ˆΦ(x) Ω(D−2) 2(D − 3) rD−3 + V (x) Ω(D−2) (D − 1) rD−1 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='9) 3For a scalar, covariant derivative and usual derivative are the same, namely, ¯∇aφ = ∂aφ and ˆ∇iφ = ∂iφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 10 The relationship between MMS and the function Irr implies Irr = K(x) − M2−D Pl ˆΦ(x) (D − 2)(D − 3) − 2M2−D Pl (D − 2) ˜ M(x) Ω(D−2) 1 rD−3 − 2M2−D Pl V (x) (D − 1)(D − 2)r2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='10) Since Irr does not depend on x, the combination K(x)− M2−D Pl ˆΦ(x) (D−2)(D−3), the function ˜ M(x) and the value of the potential V (x) should be constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Donate them c, M, Vmin respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The mark ”min” reminds one that it is reasonable to suppose φI(x) locate at the mini- mum of the potential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Then, the function Irr can be rewritten as c − 2M2−D Pl (D−2) M Ω(D−2) 1 rD−3 − 2M2−D Pl Vmin (D−1)(D−2)r2, while the average Misner-Sharp mass is mMS(r) = M + (k − c) (D − 2) Ω(D−2) 2M2−D Pl rD−3 + Vmin Ω(D−2) (D − 1) rD−1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='11) The left-hand side (LHS) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) gives a consistent result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Since the only x-dependent term at is ˆRIab/(2r2), which comes from the term RIab/(2r2), the V (φ(x)) should be a constant, while the x-dependent Ricci scalar ˆR(x) should be ˆR(x) = M2−D Pl ˆΦ(x) + C , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='12) where the constant C can be identified as (D − 2)(D − 3) c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' If introduce the averaged ˆΦ as oφ = � ˆ M(D−2) ˆΦ(x) � ˆg(x) dD−2x Ω(D−2)(D − 2)(D − 3) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='13) then Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='12) becomes c = k − M2−D Pl oφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Both the Ricci scalar of ˆ M(D−2) and the ˆΦ from scalars contribute to the constant term in Irr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' According to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2), the geometric surface gravity is κgeo = D − 3 D − 2 M2−D Pl Ω(D−2) M rD−2(u) − 2M2−D Pl Vmin (D − 1)(D − 2)r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='14) It also equals to f′/2 since f(r) = Irr(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Thus, the function σ in the line element (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) should satisfy σ′ = 0, such that it only depends on the null time v or be a nonzero constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Introduce the coordinates transformation, dts = dv σ(v) + dr f(r) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='15) Then, the unified first law and the geometric surface gravity fix the Iab(u)duadub part as d¯s2 = − (c − 2M2−D Pl (D − 2) M Ω(D−2) 1 rD−3 − 2M2−D Pl Vmin (D − 1)(D − 2)r2)dt2 s + dr2 c − 2M2−D Pl (D−2) M Ω(D−2) 1 rD−3 − 2M2−D Pl Vmin (D−1)(D−2)r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='16) 11 On the other hand, there are still freedoms for choosing ˆgij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It becomes the question of how to embed ˆ M(D−2) to the vacuum manifold of the sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The relavant EOMs are Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='(2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6) reduces to ˆ∇2φI + Γ I φ JK ˆ∇iφJ ˆ∇iφK = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='17) due to ∂aφI = 0 and the constant minimum of the potential, while the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='4) becomes ˆRij − 1 2 ˆRˆgij + (D − 3)(D − 4) c 2 ˆgij = M2−D Pl (ˆΦij − 1 2 ˆΦˆgij) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='18) The ˆΦij can be treated as the induced metric on the image of ˆ M(D−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' In principle, one can design a suitable field space metric GIJ for an arbitrary shape of ˆ M(D−2) such that Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='17) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='18) are satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Appendix D gives examples for designing the GIJ to let the global monopoles family become exact solutions, repeating results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The 4 dimension situation trivializes Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='18) further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Its LHS totally vanishes because ˆ M(D−2) becomes a 2-dimensional manifold such that ˆRij = ˆRˆgij/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Meanwhile, the RHS is proportional to the trace-less part of the induced metric ˆΦij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It also vanishes since any 2-dimensional Riemann metric is conformally the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' ˆΦij as a non-degenerate metric requires at least two scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' For simplicity, the field space metric is chosen as GIJ = δIJ, I, J = 1, 2, then the kinetic term of the model is − 1 2δIJ ∇µφI∇µφJ = −1 2(∇µφ1∇µφ1 + ∇µφ2∇µφ2) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='19) Therefore, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='17) is just the Laplacian equation ˆ∇2φI = 0 on the ˆ M2 manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Again, any two-dimensional Riemann metric is conformally flat, so it seems to become easier as- suming ˆgijdxidxj = e−f(α,β)(dα2 + dβ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' A simple calculation shows that φ1 ∝ α and φ2 ∝ β are zero modes of the Laplacian ˆ∇2, no matter what the function f(α, β) is.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Thus, the ansatz φ1 = p α, φ2 = p β implies ˆΦijdxidxj = p2(dα2 + dβ2), and automatically solves the KG equations, but still left the problem of finding f(α, β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The main constraint for the function f(α, β) comes from Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' A direct calculation shows ˆR(α, β) = ef�∂2f ∂α2 + ∂2f ∂β2 � = 2M−2 Pl p2 ef + 2c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='20) One way to attach the problem of finding f is to reduce the number of variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Changing the coordinates {α, β} in the the polar coordinates {ρ, ϕ} through α = γ cos ϕ, β = γ sin ϕ, the line element of ˆgij becomes e−f(dγ2 + γ2dϕ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Assuming f only depends on γ highly simplifies the case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The alternative method is to focus on the case of c = 0, such that the simplest solution is f ∝ α2 + β2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Based on these considerations, two novel 4-dimensional topological black holes with unusual shapes will be given in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 12 Type I It seems beneficial to compare with the general metric of the 2-dimensional maximal sym- metric space dρ2/(1 − kρ2) + ρ2dϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The expression inspires the following ansatz dρ2 1 − cρ2 − ψ(ρ) + ρ2dϕ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='21) for ˆ M(D−2) metric, and φ1 = a � ρ dξ ξ � 1 − c ξ2 − ψ(ξ) , φ2 = a ϕ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='22) for scalar fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Using c instead of k is to avoid that k does not mean the average of K(x) anymore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Then, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='12) leads to dψ(ρ) dρ = 2M−2 Pl p2 ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='23) Therefore, after adjusting parameters by re-scaling coordinates, the solution becomes ds2 = − (c − M−2 Pl M Ω(2) r − Vminr2 3M2 Pl )dt2 + dr2 c − M−2 Pl M Ω(2) r − Vminr2 3M2 Pl + r2( dρ2 1 − cρ2 − 2a2 log ρ + ρ2dϕ2) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='24) and corresponding scalars are φ1(ρ) = a MPl � ρ dξ ξ � 1 − c ξ2 − 2a2 log ξ , φ2(ϕ) = a MPl ϕ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='25) The metric is similar to the topological black hole in (A)dS background.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The Iabduadub part is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' However, the shape of the horizon is different if a ̸= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The coordinate ϕ can be periodical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The simplest choice is to identify ϕ + 2π with ϕ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It also introduces a period 2πaMPl for the second scalar field φ2 such that the field space of the model is a flat cylinder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The Ricci scalar of the unit ˆ M2 and the trace of the induced metric ˆΦij are ˆR(ρ) = 2a2 ρ2 + 2c , ˆΦ(ρ) = 2a2MPl2 ρ2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='26) Thus, the shape of ˆ M2 is controlled by two parameters a and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The square root of the determinant of the metric is √ˆg = ρ/ � 1 − cρ2 − 2a2 log ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Such an expression makes it difficult to calculate the 2-dimensional volume of the unit ˆ M2 exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Instead, it seems beneficial to study the shape by numerical method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' As an example, setting c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2, the function 1−cρ2−2a2 log ρ is plotted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The root of this function is ρmax ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='9552.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 13 Figure 1: Function 1 − cρ2 − 2a2 log ρ, c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Then the numerical integral of 2π � ρmax 0 √ˆgdρ gives Ω2 ≃ 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5717.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Unfortunately, integrals like � ρmax 0 ˆR√ˆg and � ρmax 0 ˆΦ√ˆg blow up near ρ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' These divergences cause the average Misner-Sharp mass mMS and the average work term wφ of the whole ˆ M2 are not well defined until introducing a small cut-off for ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' One can embed the ˆ M2 into the 3-dimensional flat space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Considering the metric of the unit ˆ M2, dρ2 1 − cρ2 − 2a2 log ρ + ρ2dϕ2 = cρ2 + 2a2 log ρ 1 − cρ2 − 2a2 log ρdρ2 + dρ2 + ρ2dϕ2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='27) and setting the same data of a, c, then plot the function (cρ2+2a2 log ρ)/(1−cρ2−2a2 log ρ) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Figure 2: Function cρ2+2a2 log ρ 1−cρ2−2a2 log ρ, c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 14 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5 (d)0l 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='0 2 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2 p2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='0 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8 pThere exists a turning point implying that the above function changes its sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The numerical result is ρturn ≃ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2971.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Therefore, the metric in the ρturn < ρ < ρmax region can be embedded into the Euclidean space, while the 0 < ρ < ρturn part can be only embedded into the Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The embedding diagrams are given in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The metric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='24) Figure 3: Embedding diagrams for the metric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The left plot shows the Euclidean part, while the medium plot shows the Minkowski part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The right plot shows the embedding diagram connects those two parts and completes another symmetric side.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Parameters are set as c = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1, a = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' can describe other unusual shapes by adjusting values of a and c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' One can expect cases of c = 0 and a ̸= 0 will warp the plane, and c < 0 and a ̸= 0 will deform the hyperbolic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Such other cases are ignored in this article and left for future investigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Type II The alternative metric with an unusual shape is ds2 = −(−M−2 Pl M Ω(2) r − Vminr2 3M2 Pl )dt2 + dr2 − M−2 Pl M Ω(2) r − Vminr2 3M2 Pl + r2e−(x2+y2)(dx2 + dy2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='28) The ˆgijdxidxj is a conformal plane where the conformal factor is a Gaussian function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Thus, the volume of the unit ˆ M2 is Ω2 = π.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' When M > 0 and Vmin < 0, the metric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='28) has the same Iabduadub part with the planar Schwarzchild-AdS black hole, such that it is also a topological black hole in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Two scalar fields supporting this unusual shape are φ1(x) = √ 2MPl x , φ2(y) = √ 2MPl y .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='29) The Ricci scalar of the unit ˆ M2 and the trace of the induced metric ˆΦij are ˆR(x, y) = 4ex2+y2 , ˆΦ(x, y) = 4M2 Plex2+y2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='30) Since the √ˆg = e−(x2+y2), the integral of ˆR(x, y) or ˆΦ(x, y) on the whole is divergence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Therefore the average Misner-Sharp and the average work term still lack definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' One 15 can simply propose a cut-off L for the integral region to preserve the validity of the average unified first law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is worth noting that the 2-volume Ω2 should be related to the error function erf(L) = (2/√π) � L 0 e−x2dx in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' More concretely, Ω2 = π erf(L)2, k = 16L2/(π erf(L)2), such that mMS = 16L2M2 Pl r + M + π erf(L)2Vmin r3 3 , wφ = 8M2 PlL2 π erf(L)2 r2 + Vmin , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='31) which have L2 divergence if setting L → ∞.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' To draw the embedding diagram, consider coordinates transformation x = ρ cos ϕ, y = ρ sin ϕ, the line element of unit ˆ M2 becomes e−ρ2dρ2 + e−ρ2ρ2dϕ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Then compare with the polar coordinates of a plane, identify ˆr = ρe−ρ2/2, rewritten the unit ˆ M2 line element as follows, e−ρ2dρ2 + e−ρ2ρ2dϕ2 = (2 − ρ2)ρ2e−ρ2dρ2 + (dˆr dρ)2dρ2 + ˆr2dϕ2 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='32) where the sign of function (2 − ρ2)ρ2e−ρ2 determines whether the metric can be embedded into Euclidean space or Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The plot of the function is shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The Figure 4: Function ρ e−ρ2/2 and (2 − ρ2)ρ2e−ρ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The latter one is positive in 0 < ρ < √ 2 and negative in ρ > √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' turning point is ρturn = √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The Euclidean region is 0 < ρ < √ 2 while the Minkowski region is ρ > √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The embedding diagrams are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='4 22 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='0 0 2 3 4 p0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3 (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5 pFigure 5: Embedding diagrams for the metric (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='28).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3, the left two plots show the Euclidean part and the Minkowski part separately.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The right-most plot connects them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' As ρ tends to infinity, the Minkowski part tends to the center peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2 Additional sources Cosmological constant The cosmological constant, which can be regarded as the vacuum energy density, is the simplest “matter” source.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Its Lagrangian is LΛ = −MD−2 Pl Λ , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='33) which gives the Λgµν term at the LHS of Einstein equation, or classically effective energy- momentum tensor Tµν = −MD−2 Pl Λgµν when putting at the RHS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Such an energy-momentum tensor contributes a constant energy density in space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It also shifts the value of the potential V → V + MD−2 Pl Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Thus, only turning off the spacetime dependence of scalars in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8), the energy supply vector and the work term of the cosmological constant are directly read as ψa Λ = 0 , WΛ = MD−2 Pl Λ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='34) Such a constant work term contributes the MD−2 Pl Λ ¯∇aV(D−1) term to the unified first law, and the energy supply vector vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Therefore, the cosmological constant plays the role of pressure for a thermodynamics system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' There are more interesting issues in the context of cosmology if treating Λ as a pressure [25, 26, 54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' An alternative view is that the cosmological constant should be put at the LHS, such that it provides a −MD−2 Pl Λ V(D−1) correction to the MS mass, as in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 17 Maxwell field The Lagrangian of a Maxwell field is Lem = −1 4FµνF µν .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='35) It gives the energy-momentum tensor Tµν = FµλF λ ν − 1 4gµνFλρF λρ and the source-free Maxwell equations are dF = 0 , d ∗ F = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='36) in terms of differential forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The F is the 2-form field strength, while ∗F is the Hodge dual of F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The generalized Coulomb field in higher dimension Fab = Q Ω(D−2) rD−2 ¯ϵab , Fij = 0 , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='37) has already solved the Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Obviously, dF = 0 since the factor only depends on r and any ¯ M(2) 1-form p = padua satisfies p ∧ ¯ϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The Hodge dual (D − 2)-form is ∗F = Qˆϵ/Ω(D−2), so d ∗ F ∝ dˆϵ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Thus, despite what the Iabduadub part is, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='37) must be the solution of source-free Maxwell equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Then, integrating the dual (D − 2)- form ∗F on ˆ M(D−2) gives the electric charge Q = � ˆ M(D−2) ∗F , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='38) which confirms the choice of the 1/Ω(D−2) factor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Then, the energy-momentum tensor for such Coulumb field F = Q¯ϵ/(Ω(D−2) rD−2) is decomposed as Tab = − Q2 2Ω2 (D−2) 1 r2(D−2) Iab , Tij = Q2 2Ω2 (D−2) 1 r2(D−3) ˆgij , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='39) where F 2 = FabF ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' ψa em = 0 , Wem = Q2 2Ω2 (D−2) 1 r2(D−2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='40) For the 4-dimensional spherical case, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='40) reduces to the result in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Again, the energy supply vector vanishes, and the work term only changes when varying r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' If every matter source considered in the theory contributes i) only r-dependent average work term w(i)(r);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' and ii) vanishing energy supply vector ψa (i) = 0, then the Iabduadub part of the spacetime ¯ M(2) × ˆ M(D−2) is already determined as −Irrdt2 + dr2/Irr, and the function Irr should be Irr = k − 2M2−D Pl (D − 2) M Ω(D−2) 1 rD−3 − � i 2M2−D Pl (D − 2) rD−3 � r w(i)(ξ)ξD−2dξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='41) 18 Similar tricks for constructing solutions in the previous subsection are still valid for the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The average unified first law under conditions (i) and (ii) gives ¯∇amMS = � i w(i)(r) Ω(D−2)rD−2 ¯∇ar , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='42) such that directly integrating r implies that the average MS mass is mMS(r) = M + � i Ω(D−2) � r w(i)(ξ)ξD−2dξ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='43) It constricts the function Irr taking the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='41).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Concretely, the cosmological contributes − 2Λ r2 (D−1)(D−2), while the Maxwell field contributes M2−D Pl (D−2)(D−3) Q2 Ω2 (D−2) r−2(D−3) to Irr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='42) also implies 1 2 dIrr dr = D − 3 D − 2 M2−D Pl Ω(D−2) mMS(r) rD−2 − M2−D Pl D − 2 r � i w(i)(r) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='44) The RHS coincides with the geometric surface gravity relation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Thus, in the GEF coordinates {v, r}, there is σ′ = 0 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Finally, the coordinates transformation (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='15) explicitly shows that the Iabduadub part must be −Irrdt2 +dr2/Irr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It completes the proof.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Taking the case of vacuum, both ψa and w vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Meanwhile, for spherical symmetry, the ˆgij is detemeined4 and the value of k should be 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The theorem reduces to Birkhoff’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' In such a sense, this theorem generalizes Birkhoff’s theorem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Apply the above conclusion, one obtains charged novel topological black hole solutions ds2 = − (c − M−2 Pl M Ω(2) r + M−2 Pl Q2 2Ω2 (2) r2(D−3) − Λr2 3 )dt2 + dr2 c − M−2 Pl M Ω(2) r + M−2 Pl Q2 2Ω2 (2) r2(D−3) − Λr2 3 + r2( dρ2 1 − cρ2 − 2a2 log ρ + ρ2dϕ2) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='45) and ds2 = − (−M−2 Pl M Ω(2) r + M−2 Pl Q2 2Ω2 (2) r2(D−3) − Λr2 3 )dt2 + dr2 − M−2 Pl M Ω(2) r + M−2 Pl Q2 2Ω2 (2) r2(D−3) − Λr2 3 + r2e−(x2+y2)(dx2 + dy2) , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='46) where the minimum of the potential is absorbed into the cosmological constant Λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 4In the complete proof, there is one part to proof that spherical symmetry demanding the spacetime contains an orbit of SO(3) group, such that the metric has the form of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1), see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Such a step is skipped in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 19 Irreducible mass and usable energy Suppose rH is the location of the black hole horizon, since there should be Irr(rH) = 0, the averaged MS mass at the horizon is mMS = (D − 2)Ω(D−2)MD−2 Pl krD−3 H /2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Modestly, in the case of k > 0, the MS mass mMS should not decrease from the viewpoint of the area theorem [56].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It seems reasonable to define (D − 2)Ω(D−2)MD−2 Pl krD−3 H /2 as irreducible mass mirr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is natural to further define the usable energy as musable(r) = mMS(r) − mirr outside the horizon r > rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The mass parameter does not appear in musable(r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Instead, such a new definition is determined by the difference of the work term between location r and horizon rH.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [57] discussed the possibility of a Schwarzchild black hole as a battery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' One can regard that their charging process turns the rest energy of in-falling material into the usable energy of the black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Suitable discharging will extract such energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It seems the concept of usable energy at least works in the case of the asymptotic flat RN black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Non-vanishing energy supply vector The case of ∂aφI ̸= 0 violates the condition ψa = 0 such that the above generalization of Birkhoff’s theorem becomes invalid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It explains that a static spherical solution with a non-constant σ(r) is due to r-dependent scalars [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Another situation violating condition (ii) ψa = 0 is the appearance of a Vaidya mass, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=', a null time-dependent function for the mass parameter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' More concretely, in the GEF coordinates {v, r}, the Iabduadub part satisfies − � (k − ωD M(v) rD−3 − � i 2M2−D Pl (D − 2) rD−3 � r w(i)(ξ)ξD−2dξ) � dv2 ± 2dvdr , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='47) such that the average unified first law gives ¯∇amMS = ˙M ¯∇av + � i w(i)(r) Ω(D−2)rD−2 ¯∇ar , (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='48) where ˙M is dM/dv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Thus, the function M(v) implies the non-vanishing energy supply vector ψa = ˙M ¯∇av/A(D−2) without changing contributions from other matter fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='44) is still valid if change the derivative dIrr/dr as ∂Irr/∂r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Therefore, the work term of the source of Vaidya mass must vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It hints at finding the corresponding energy-momentum tensor T vr = ψa ¯∇av = 0 , T rr = ψa ¯∇ar = ± ˙M Ω(D−2) rD−2 , T vv = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='49) The result for component T vv is obtained by using the vanishing work term W = 0 again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' At the horizon, the energy supply vector is interpreted as a heat flow [13], so it is plausible that 20 the vector ψa does not vanish in dynamical cases, like the approach of obtaining evolving spacetime by multiplying a time-dependent conformal factor to the static solution [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 4 The first law including topological charge Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [48,49] introduce the concept of topological charge for a topological RN-AdS black hole to preserve the Gibbs-Duhem-like relation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Using the notation of this article, the explicit expression of the metric of a topological RN-AdS black hole is ds2 = −fdt2 + dr2 f + r2ˆgijdxidxj , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1) where f = k − 2M2−D Pl (D − 2) M Ω(D−2) 1 rD−3 + M2−D Pl (D − 2)(D − 3) Q2 Ω2 (D−2) 1 r2(D−3) + r2 l2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2) and l is the AdS radius.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' They consider formally changing k continually, then define the topological charge as ε = Ω(D−2) � |k| D−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) Treat r = rc as a holographic screen, label f(rc), df(rc)/dr as fc, f′ c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The Brown-York tensor on the screen leads to the energy density ϵ and the pressure p as ϵ = C − D − 2 M2−D Pl √fc rc , p = D − 3 M2−D Pl √fc rc + 1 2M2−D Pl f′ c √fc − C , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='4) where C is a constant from holographic renormalization [48,49,59].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' They define the total “volume”5 and energy of the whole screen as V = Ω(D−2)rD−2 c , E = ϵV .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5) However, the entropy is chosen as horizon entropy S = 2π M2−D Pl Ω(D−2)rD−2 H , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6) in which rH satisfies f(rH) = 0, determines the location of the Killing horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Replacing M2−D Pl as 8π, the entropy reduces to the well-known one quarter of the horizon area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' After suitably defining the temperature T T = TH √fc = 1 2π√fc �rH l2 + (D − 3) 2M2−D Pl (D − 2) M Ω(D−2) 1 rD−2 H − M2−D Pl D − 2 Q2 Ω2 (D−2) 1 r2D−5 H � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='7) 5It is worth noting that the “volume” V is the (D − 2)-dimensional area A(D−2) rather than the (D − 1)- dimensional volume V(D−1) in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 21 the chemical potential µ for the electric charge Q µ = − 1 (D − 3) Ω(D−2) √fc � Q rD−3 c − Q rD−3 H � , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8) and the potential ζ for the topological charge ε ζ = sgn(k) ε 4−D D−2 Ω D−4 D−2 (D−2) M2−D Pl rD−3 H − rD−3 c √fc , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='9) on the screen6, not only the first law of thermodynamics dE + pdV = TdS + µdQ + ζdε is obtained, the Gibbs-Duhem-like relation E + pV = TS + µQ + ζε , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='10) still holds for non-planar (k ̸= 0) cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is the motivation of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [48,49] introducing the topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' This charge is suggested as the last charge of a black hole with maximal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' On the other hand, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [60] introduces the chemical potential for the number of colors, which is applied to study the phase structure of a topological black hole [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The number of colors seems to have similar behavior with the topological charge under scale transformation of ˆ M(D−2), but their connection is still unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is worth mentioning details about scaling ˆ M(D−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The power (D − 2)/2 in the definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) implies that the topological charge is also a k-normalized volume of MD−2 or the part of MD−2 in interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' For the maximally symmetric ˆ M(D−2) with a non-normalized and nonzero k, re-scale the metric ˆgij → l2 ˆgij to set |k| as 1, then the additional scale l can always be absorbed by adjusting other parameters in Irr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Therefore, the maximally symmetric ˆ M(D−2) with a non-normalized k does not distinguish it from the k-normalized situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Modestly, if introducing a topological charge for MD−2, it seems better to consider the MD−2 as an Einstein manifold without maximal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1 Topological charge as the portal of ˆ M(D−2) Nevertheless, the parameter k plays the role of a portal connecting the manifold ¯ M(2) and shape parameters of ˆ M(D−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' In higher dimensions, even if the Ricci scalar of ˆ M(D−2) is a constant, the manifold ˆ M(D−2) is not necessary to have maximal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' As an example, consider the following 4-dimensional Einstein manifold satisfied ˆRij = kˆgij, dˆs2 = ∆ˆr ˆr2 − ˆa2 cos2 ˆθ � dˆτ − ˆa sin2 ˆθ 1 − k ˆa2 3 dˆφ �2 + (ˆr2 − ˆa2 cos2 ˆθ) �dˆr2 ∆ˆr + dˆθ2 ∆ˆθ � + ∆ˆθ sin2 ˆθ ˆr2 − ˆa2 cos2 ˆθ � ˆadˆτ + ˆr2 − ˆa2 1 − k ˆa2 3 dˆφ �2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='11) 6The notation for factors of M and Q is different with Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [48,49].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 22 where ∆ˆr = (ˆr2 − ˆa2)(1 − kˆr2 3 ) − 2 ˆmˆr , ∆ˆθ = 1 − kˆa2 3 cos2 ˆθ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='12) This is the Euclideanized Kerr-(A)dS gravitational instanton [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The 4-dimensional Planck scale MPl(4) is set as 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The additional hat mark reminds us that the instanton would be regarded as the ˆ M(D−2) manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Then it is not hard to construct the following metric in 6-dimensional Einstein gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' ds2 = −(k − ω6 M r3 )dt2 + dr2 k − ω6 M r3 + r2dˆs2 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='13) where ω6 can be treated as a suitable constant in 6-dimension when the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='11) is the metric ˆgij of the ˆ M(4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' This is a vacuum solution of the 6-dimensional Einstein equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' At least the situation of M > 0 and k > 0 can describe a black hole.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The shape of its horizon is a 4-dimensional Kerr-dS instanton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is reasonable to allow continuously varying k for the Einstein manifold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The pa- rameters of the Euclidean spacetime satisfy the following Smarr-like relationship due to ∆ˆr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='7 ˆ M2 = ˆS(1 − k ˆS 3π )2 4π − π ˆJ2 ˆS + k ˆJ2 3 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='14) where ˆ M = ˆm (1 − k ˆa2 3 )2 , ˆJ = ˆmˆa (1 − k ˆa2 3 )2 , ˆS = π ˆr2 + + ˆa2 1 − k ˆa2 3 , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='15) in which the ˆr+ is the larger root of the equation ∆ˆr = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Treating ˆ M as the function of ˆS, ˆJ, k, the variation of the Smarr-like relation leads to the first law δ ˆ M = ˆTδ ˆS + ˆΩδ ˆJ + ˆΘδk , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='16) where ˆT = 12π4 ˆJ2 + 3π2 ˆS2 − 4πk ˆS3 + k2 ˆS4 24π3 ˆ M ˆS2 , ˆΩ = ˆJ(k ˆS − 3π) 3 ˆ M ˆS , ˆΘ = 6π3 ˆJ2 − 3π ˆS2 + k ˆS3 36π3 ˆ M .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='17) Thus, there should be the first law for the 6-dimensional black hole, δM = TδS + r3 H ω6 ˆΘ (δ ˆ M − ˆTδ ˆS − ˆΩHδ ˆJ) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='18) where rH = (ω6M k )1/3 , T = κ 2π , S = A 4G6 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='19) 7The J2 term in Euclidean signature has a different sign from the one in Lorentz signature of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 23 Parameters of the Einstein manifold can join the first law of black hole thermodynamics for the whole ¯ M(2) × ˆ M(D−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The δk plays the role of a portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' From such a point of view, the concrete definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) does not matter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' For convenience, this article still states that continuously varying k introduces the topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2 Unusual shapes and topological charges Reminding that the role of k for the average unified first law is similar to the role of K for the non-average first law, it may be beneficial to consider the non-average law to figure out natural interpretation of the topological charge, since changing the direction, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=', the position in ˆ M(D−2) naturally causes continually varying K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Such a consideration requires clarifying the relationship between the unified first law, and the usual first law of black hole thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The unified first law states a relation in spacetime, while the usual one deals with the variation of global parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' To handle the tension between two viewpoints of the first law, consider a finite falling energy package that goes through the trapping horizon8, shown in fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6, in such a way that the horizon begins as a Killing horizon, and finally settles down as a new Killing horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' To ensure the unified first law is valid, it is assumed that the ˆ M(D−2) part keeps unchanging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The trapping horizon only changes its size during the process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Select a vector za tangent to the ¯ M(2) and the trapping horizon, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=', (za ¯∇aIrr)H = 0, and donate f′ as (za ¯∇af)H, then according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='15), contracting za with the A(D−2)ψa generates the MD−2 Pl κgeo A′ (D−2) term, the equivalent expression of the heat flow term TdS.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The tunneling approach for Hawking radiation in dynamical spacetime confirms the relationship between κgeo at the horizon and the horizon temperature T = κH/(2π) [20–24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Remarkably, two versions of the unified first law Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='16) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='20) become M′ MS − WH V ′ (D−1) = m′ MS − wH V ′ (D−1) = κgeo A′ (D−2) M2−D Pl , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='20) which is regarded as the quasi-local first law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The average first law implies the integral mMS,f − mMS,i − � TH wH dVH = 1 M2−D Pl � TH κH dAH , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='21) where mMS,f is the final average MS mass on the trapping horizon, mMS,i is the initial one, lower subscribes H represent taking value on the horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Suppose the change is tiny 8The trapping horizon is defined as a hypersurface foliated by marginal surfaces, so it has vanishing expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The equation Irr = 0 determines the location of a trapping horizon, see appendix C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 24 Figure 6: The time coordinate v is the null retarded time, while ¯t is given by v − r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Such a sketch describes a falling process changing the size of the trapping horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The red curve represents the evolving tapping horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Gray arrows portray the in-falling energy package.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Both the initial and final moment is marked by gray dotted lines while Killing horizons are indicated by black lines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' enough such that every global parameter becomes M + δM and Q(i) + δQ(i) at the final state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Then the (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='21) reduces to δmMS(rH) − wH δVH = κH δAH M2−D Pl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='22) The term about δk is omitted since the shape of ˆ M(D−2) is fixed during the falling process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is significant that the term κH δAH appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' To indicate that the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='20) is the quasi- local viewpoint for the first law, although it will be less general, calculate δmMS(rH) of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='43).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Keeping every small change of global parameters except k, the result is δmMS(rH) = δM + Ω(D−2) � i � � rH ∂w(i)(ξ) ∂Q(i) ξD−2dξ � δQ(i) + wH δVH .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='23) Identify the corresponding potential for the charge Q(i) as Φ(i),H = −Ω(D−2) � rH ∂w(i)(ξ) ∂Q(i) ξD−2dξ , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='24) such that the relation δmMS(rH) − wH δVH = δM − � i Φ(i),HδQ(i) , (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='25) 25 2is obtained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='25) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='22) together give the usual first law formulated by variation of global parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Thus, the result (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='25) shows how to connect the quasi-local viewpoint (LHS) with the global viewpoint (RHS).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Now, back to the issue of x-dependence of K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is only the K that contributes the x-dependence of the non-average MMS by definition (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Consider a tangent vector hµ∂/∂Xµ = za∂/∂ua + yi∂/∂xi on the trapping horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Such vector not only has ua components za but also has xi component yi.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Then redefine f′ as hµ∂µf taking value on the horizon, the non-average Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='20) is modified as M′ MS − (D − 2)Ω(D−2) rD−3 H 2M2−D Pl K′ − WH V ′ (D−1) = κgeo A′ (D−2) M2−D Pl .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='26) Again, considering integrating Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='26) along h and supposing only a small change of global parameters, all derivatives f′ are replaced by δf, including K′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Keeping δk in the usual first law or the average unified first law is only a formal operation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' However, from the non-average point of view, allowing δK is naturally achieved by changing the direction in ˆ M(D−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It opens up the consideration of defining topological charge (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) by K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The K term responds to the x-dependence in MMS, relating to the LHS of the non- average unified first law (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='16).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' For the RHS side, as is discussed in section 2, the energy supply vector does not depend on x, so the only term responding to the x-dependence in MMS must appear in the work term with the correct power of r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2) helps again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It specifies that the relevant work term should be proportional to r−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Multi-scalars offer such work term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Concrete examples are novel topological black holes constructed in section 3 and global monopole solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The contribution from the r−2 work term cancels the x-dependence from K(x) in the charged solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='45) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='46).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The type II solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='46) have the same Iabduadub part with the planar black hole, while the type I solution (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='45) describes diverse situations replacing k by c in Irr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' However, it is unclear what the topological significance is if replace k with c in the definition (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' An alternative approach is to define topological charge in a concrete context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' In novel solutions and global monopole solutions in appendix D, scalar fields φI map the sub-manifold ˆ M(D−2) to the vacuum manifold of the sigma model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Thus, another choice for defining the topological charge is the degree of map for scalars, which is proportional to the integral of the volume element of the vacuum manifold deg(φ) ∝ � vac ϵIJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='K dφI ∧ dφJ ∧ · · · ∧ dφK .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='27) Here, it is assumed an appropriate compactification was chosen to ensure the integral is finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The definition includes situations discussed in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [53] in principle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Such a definition 26 is topological because the degree is homotopy invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The x-dependence K rely on the x-dependence work term with r−2 power.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' One can treat the non-average unified first law should contain both “topological charges” from curvature K and degree of map from scalars simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Two topological charges satisfy some relation through c if concerning the ˆ M(D−2) part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The x-dependence is canceled while the constant c may be retained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' In this sense, it is the c playing the role of the portal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It seems that introducing topological charges only touches the corner of an iceberg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Exact solutions (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='45) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='46) remind us that the constant term in Irr still hides rich structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 5 Summary and discussion This article re-derives the unified first law from Einstein equations in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Distinguish- ing from the original law, sub-manifold ˆ M(D−2) is no limit to the maximally symmetric space, but rather an arbitrary manifold including dependent-on-position Ricci scalar ˆR(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Such unusual shapes require two versions of MS masses and unified first laws.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The non- average law keeps x-dependence, while the average law is from integrating out positions x on ˆ M(D−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Two types of novel 4-dimensional topological black hole solutions in Einstein-scalars the- ory are constructed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The improved approach of generating solutions from thermodynamics simplifies the construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Horizon shapes of these black holes are visualized by plotting embedding diagrams in one higher dimension space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Both cannot be entirely embedded into a single Euclidean or Minkowski space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The improved approach further gives a simple method to introduce cosmological constant, electric charge, and Vaidya mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The non-constant K(x) gives natural motivation to consider topological charge since its value changes when moving the position x in ˆ M(D−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Novel solutions show that K(x) is achieved by x-dependent scalars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Contribution for Irr from K(x) and ˆΦ(x) cancel out, but they left a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The residual constant seems to challenge the original motivation for introducing the topological charge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' A modest viewpoint is viewing the residual constant as a portal to parameters about the horizon shape for the first law.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The next task is to investigate how to add angular momentum to topological black holes obtained here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It has been successfully done for planar and hyperbolic solutions in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [63–66].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' We expect a systematic method to generate spinning solutions from the solutions with the metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1), at least type I and type II solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Once corresponding spinning solutions are obtained, they are worth testing the validity of Kerr/CFT corresponding [67], 27 and may even give some new ideas about black hole evaporation and entropy of horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Recent progress in quantum gravity has calculated the entropy of horizon with arbitrary shape [68].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Solutions here offer concrete examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is unclear whether the non-asymptotic flat or dS features forbid the production of topological black holes with unusual shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is worth exploring these possibilities like Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' An alternative approach to pushing forward such an investigation is considering modified gravity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Generalizing the MS mass and the unified first law to modified gravity theories has been studied in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [44–46], but they are still only concerned about the maximal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Especially the MS mass for the Gauss-Bonet gravity has non-linear terms of about k [44], such that the relation between the average unified first law and the non- average law may be non-trivial.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Despite the difficulties, the investigation of this article suggests that the generating solutions approach for modified gravity theories [69–71] may still be worth studying.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' One can expect the corresponding improved approach benefit of finding new topological black holes with unusual horizon shapes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Novel topological black holes and the construction method may imply some unexpected application in the holographic context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' At first, the topological charge, or the color charge introduced in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [60] enriches phase structures of topological black hole [61].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Such previous researches are interested in maximal symmetry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Nevertheless, novel topological black holes imply an inhomogeneous AdS boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The entrance of parameters about the shape in the first law would enlarge the black hole phase structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It seems studying black hole phase transition would also inspire some new ideas for cosmology [54, 55].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Secondly, Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [61] emphasizes the sub-system aspect for formulating the thermodynamics when dealing with the non-compact nature of the plane and the hyperbolic surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The sub-system consideration inspires the possibility that the metric of ˆgij describes the unusual shape with richer inhomogeneous structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The whole ˆ M(D−2) may be too inhomogeneous to be described by a single coordinates frame {xi}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Instead, the entire spacetime may need many small pieces to write down the line-element (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Different patches are not the same but relate to each other by scaling r → lr if they overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Applying the AdS/CFT corresponding to the type I and type II topological AdS black holes, the boundary field theory should live in the non-Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The AdS boundary with unusual shape means that novel solutions are not asymptotic AdS spacetime in a precise sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Solutions do not reduce to asymptotically flat spacetime when turning off the cosmological constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Thus, the topological theorem in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [10] is not violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is natural to expect this is the holographic description for some boundary solidary material 28 in unusual shapes, although the part embedded in Minkowski space seems less plausible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The construction in section 3 implies inverse engineering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' For a given ˆ M(D−2), one can design an appropriate field space metric GIJ for the sigma model to achieve the metric ˆgij of ˆ M(D−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The warp ˆgij should respond to a non-vanishing stress tensor Tij, which may be similar to the previous study about holographic viscoelastic hydrodynamics [72].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Moreover, it is remarkable that the ansatz (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='25) and (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='29) for scalars is similar to the ansatz that appeared in the holographic axion model [73].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Several scalars have VEV proportional to coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Completing the inverse engineering technique may give a systematic method to find analytic models, opening up a new field of holographic solids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Appendix A: Calculate Γλ µν and Rλ ρµν This appendix shows an efficient method for calculating the Levi-Civita connection and Riemann tensor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Usually, the components of the Levi-Civita connection for a given metric, are calculated by Γλ µν = 1 2gλσ(∂gµσ ∂xν + ∂gνσ ∂xµ − ∂gµν ∂xσ ) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1) The geodesic equations give hints to finding the trick.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Considering the geodesic equations d2xλ dτ 2 + Γλ µν dxµ dτ dxν dτ = d2xλ dτ 2 + gλσ(dgσν dτ dxν dτ − 1 2 ∂gµν ∂xσ dxµ dτ dxν dτ ) = 0 , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2) in which dgσν/dτ = (∂gσν/∂xµ)(dxµ/dτ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' one can extract the following structure Γλ µνdxµdxν = gλσ(dgσνdxν − 1 2 ∂ds2 ∂xσ ) , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) which can be viewed as a coordinate-dependent rank-2 symmetric tensor, in which ∂ds2/∂xσ is the short notation of (∂gµν/∂xσ)dxµdxν.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Since the coordinates frame is fixed under a particular calculation, one can simply treat gσν as several functions of xµ and dgσν represents their differential.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The trick is to calculate the structure Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) rather than to calculate components of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1) one by one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Now we use this trick to calculate the Levi-Civita connection of the metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The components of its inverse metric are gab = Iab , gij = ˆgij r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='4) Thus, Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) gives Γa µνdxµdxν = ¯Γa bcdubduc − (r ¯∇ar)ˆgijdxidxj , Γi µνdxµdxν = 2 ¯∇ar r duadxi + ˆΓi jkdxjdxk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5) 29 Here, the property ∂r/∂ua = ¯∇ar is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Noticing that product terms like duadxi are the short notation for symmetric tensor product (duadxi ⊗ dxidua)/2, every component can be correctly read as Γa bc = ¯Γa bc , Γa ij = −rIab ∂r ∂ub ˆgij , Γi aj = 1 r ∂r ∂ua δi j , Γi jk = ˆΓi jk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6) It is worth noting that the components Γa bc and Γi jk are just the independent Levi-Civita connection of ¯ M(2) and ˆ M(D−2) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The author would like to introduce the covariant differential operator ¯∇a for ¯ M(2) and ˆ∇i for ˆ M(D−2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The areal radius r(u) can be treated as a scalar field in ¯ M(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The notation ¯∇ar also means ∂r/∂ua while ¯∇ar means Iab(∂r/∂ub).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Once the connection was obtained, the Riemann tensor can be calculated through Rλ ρµν = ∂Γλ νρ/∂xµ + Γλ µσΓσ νρ − (µ ↔ ν).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It can also be treated as several 2-forms due to the anti-symmetry of exchanging µ and ν, 1 2Rλ ρµνdxµ ∧ dxν = dΓλ νρ ∧ dxν + (Γλ µσdxµ) ∧ (Γσ νρdxν) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='7) In order to simplify the notation, label 1 2Rλ ρµνdxµ ∧ dxν as Ωλ ρ and Γλ µσdxµ as Aλ ρ , then Ωλ ρ = dAλ ρ + Aλ σ ∧ Aσ ρ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8) These 1-forms Aλ ρ can be viewed as the connection 1-forms for the coordinates tetrad (∂µ)ν = δν µ while Ωλ ρ are their curvature 2-forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Concretely, 1-forms Aλ ρ for the metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1) are Aa b = ¯Γa cb duc = ¯Aa b , Aa i = − (r ¯∇ar)ˆgijdxj , Ai a = ¯∇ar r dxi , Ai j = ¯∇ar r δi jdua + ˆΓi kjdxk = dr r δi j + ˆAi j .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='9) Then one obtains curvature 2-forms by applying Eq .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Firstly, Ωa b = dAa b + Aa c ∧ Ac b + Aa i ∧ Ai b = ¯Ωa b , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='10) since term containing ˆgijdxi ∧ dxj vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Notice that Ωai = IabΩb i = −Ωia = −r2ˆgijΩj a, calculating Ωi a can avoid dealing with dˆgij here: Ωi a = ¯∇a ¯∇br r dub ∧ dxi , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='11) therefore, Ωa i = −r ¯∇a ¯∇br ˆgij dub ∧ dxj , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='12) 30 The final 2-form Ωi j is Ωi j = ˆΩi j − Irrδi k ˆgjl dxk ∧ dxl = 1 2 � ˆRi jkl − Irr(δi k ˆgjl − δi l ˆgjk) � dxk ∧ dxl , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='13) where the term Irr is the short notation for Iab ¯∇ar ¯∇br.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Reminding dxµ ∧ dxν = dxµ ⊗ dxν − dxν ⊗ dxµ, one can read the components of Riemann tensor as Ra bcd = ¯Ra bcd , Ra ibj = −Ra ijb = −r( ¯∇a ¯∇br) ˆgij , Ri ajb = −Ri abj = − ¯∇a ¯∇br r δi j , Ri jkl = ˆRi jkl − Irr(δi kˆgjl − δi lˆgjk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='14) The same result can be found in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Contracting δi i = D −2, components of the Ricci tensor are Rab = ¯Rab − (D − 2) ¯∇a ¯∇br r , Rij = ˆRij − ˆgij � r ¯∇2r + (D − 3)Irr� , (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='15) in which the ¯∇2r is ¯∇a ¯∇ar for short.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The Ricci scalar is R = IabRab + ˆgij r2 Rij = ¯R − 2(D − 2) ¯∇2r r + ˆR r2 − (D − 2)(D − 3)Irr r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='16) Appendix B: Calculate ∗J and ∗K Then the author will calculate the Hodge dual ∗K and ∗J directly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' One can read the volume D-form ϵ from the metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1) as following ϵ = rD−2� −I ˆg du1 ∧ du2 ∧ dx1 ∧ · · · ∧ dxn−2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1) where I, ˆg is the determinate of Iab, ˆgij respectly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Meanwhile, the volume 2-form ¯ϵ for ¯ M(2) is ¯ϵ = √ −I du1 ∧ du2 , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2) and the volume (D − 2)-form ˆϵ for ˆ M(D−2) is ˆϵ = � ˆg dx1 ∧ · · · ∧ dxD−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) This notation makes the following relation obvious, ϵ = rD−2¯ϵ ∧ ˆϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='4) It is still necessary to use the notation of tensor components to calculate the Hodge dual, ϵab i1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='iD−2 = rD−2¯ϵab ˆϵi1i2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='iD−2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5) 31 Then, ∗J is given by ∗ J = rD−2Ja¯ϵabdub ∧ ˆϵ = rD−2T ac(¯ϵab¯ϵcd) ¯∇dr dub ∧ ˆϵ , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6) Noting that tensor notation of the volume 2-form ¯ϵab can be expressed as ¯ϵab = √ −I¯εab , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='7) in which the symbol ¯εab satisfies ¯ε12 = −¯ε21 = 1 and ¯ε11 = ¯ε22 = 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Then the ¯ϵab is ¯ϵab = −( √ −I)−1¯εab , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8) where the symbol ¯εab also share the same pattern with ¯εab, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=', ¯ε12 = −¯ε21 = 1 and ¯ε11 = ¯ε22 = 0, such that ¯ϵab¯ϵcd = −¯εab¯εcd = −δa cδb d + δa dδb c .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='9) Lowering the ab indexes leads to ¯ϵab¯ϵcd = −IacIbd + IadIbc .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='10) Thus, the (D − 1)-form ∗J becomes ∗J =rD−2(Tab ¯∇ar + 2W ¯∇br) dub ∧ ˆϵ , (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='11) As for ∗K, it can be obtained by simply replacing the T ab in ∗J by −Iab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Thus, ∗K = rD−2 ¯∇br dub ∧ ˆϵ = d( rD−1 D − 1) ∧ ˆϵ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='12) Appendix C: General Eddington-Finkelstein coordinates The 2-dimensional sub-spacetime must permits double null coordinates {u, v} such that the line element becomes ds2 = −Ω2(u, v)dudv + r2(u, v)ˆgij(x)dxidxj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1) In the region where ¯∇ar does not vanish, r itself can be a coordinate, such that one can change to other coordinates frame like {v, r}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Since dr = r,u du + r,v dv where r,u, r,v represents ∂r/∂u, ∂r/∂v, the line element becomes ds2 = Ω2 r,v r,u dv2 − Ω2 r,u drdv + r2ˆgij(x)dxidxj .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2) 32 The line element (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2) is still general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The coordinates frame {v, r} is called general Eddington-Finkelstein (GEF) coordinates in this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Define functions σ(v, r) = −2r,u/Ω2 and f(v, r) = −4r,ur,v/Ω2, metric components under the GEF coordinates are gvv = Ivv = − f(v, r) σ2(v, r) , grr = Irr = 0 , gvr = grv = Ivr = Irv = 1 σ(v, r) , gij = r2ˆgij , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) while inverse metric components are gvv = Ivv = 0 , grr = Irr = f(v, r) , gvr = grv = Ivr = Irv = σ(v, r) , gij = ˆgij r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='4) In general, the shape of spacetime ¯ M(2) × ˆ M(D−2) with a given metric ˆgij for the unit ˆ M(D−2) is described by two functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' In double null coordinates, they are Ω(u, v) and r(u, v), while in GEF coordinates, they are σ(v, r) and f(v, r).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The usage of function f(v, r) is convenient since it picks up the important function Irr = ∇µr∇µr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Further, the determinant of Iab in the GEF coordinates is simply I = −σ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Then √ −I equals to σ−1 up to a sign.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Therefore, the Laplacian of r in ¯ M(2) is ¯∇2r = σ∂(σ−1Ivr) ∂v + σ∂(σ−1Irr) ∂r = f′ − σ′ σ f , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5) where f′ = ∂f/∂r and σ′ = ∂σ/∂r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The result leads to a simple expression (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='4) for the geometric surface gravity (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1) in terms of f and σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The next task is to calculate expansions of null vector fields tangent to ¯ M(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Specify fields as kµ ∂ ∂Xµ = ∂ ∂v + f 2σ ∂ ∂r , lµ ∂ ∂Xµ = −σ ∂ ∂r , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6) and require the condition kµlν = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Assuming the increasing direction of v is future, kµ and lν are all future-pointed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Their expansion can be easily calculated by the trick θ(k) = (D − 2) ka ¯∇ar/r without dealing with gµν∇µkν θ(k) = (D − 2) f 2σr , θ(l) = −(D − 2) σ r .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='7) Such a method is also used in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The hypersurface Irr = f = 0 leads to θ(k) = 0, thus determining a trapping horizon, which is defined as a hypersurface foliated by marginal surfaces [12, 13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' A marginal surface is a 2-codimensional spatial surface with vanishing expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' One can further classify types of trapping horizons according to the behavior of θ(l) and Llθ(k), see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [12,13].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' 33 It seems there is a problem that the vector lµ vanishes at location σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' However, such location also implies a singular line element.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' To handle this issue, let us back to the non-singular double null coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Another choice for two null vector fields is ˜kµ ∂ ∂Xµ = √ 2 Ω ∂ ∂v , ˜lµ ∂ ∂Xµ = √ 2 Ω ∂ ∂u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8) Their expansions are θ(˜k) = √ 2(D − 2) Ω r r,v , θ(˜l) = √ 2(D − 2) Ω r r,u .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='9) Apply the coordinates transformation and carefully deal with the chain rule ∂ ∂vDN = ∂ ∂vGEF + r,v ∂ ∂r , ∂ ∂u = r,u ∂ ∂r , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='10) where vGEF donates the coordinate v in GEF frame and vDN donates the v in double null frame, one confirms the following relation ˜kµ = √ 2 Ω kµ , ˜lµ = Ω √ 2 lµ , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='11) such that θ(˜k) = √ 2 Ω θ(k) , θ(˜l) = Ω √ 2 θ(l) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='12) Since Ω ̸= 0, σ = 0 is just a coordinate singularity, implying that r,u = 0, which is another trapping horizon which satisfies θ(l) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' However, the relationship Irr = −2( r D − 2)2θ(k)θ(l) , (C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='13) is covariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The normalization condition kµlµ forces that the construction θ(k)θ(l) does not change under a regular rescaling like Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='(C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Switching the role of u and v changes the concerned null vector field from ˜k to ˜l.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Thus, the equation Irr = 0 always picks up the hypersurface with vanishing expansion (θ(k) = 0 or θ(l) = 0), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=', the trapping horizon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Appendix D: Design the field space metric GIJ This appendix gives examples of designing the field space metric GIJ for a given ˆgij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [53] modifies the field space metric to let the global monopole becomes an exact solution and further generalizes it to planar and hyperbolic situations in higher dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Such research serves as a good example of designing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' This appendix aims to repeat results in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [53] through the approach of this article.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [47] overstated the approach of generating 34 solutions from thermodynamics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It was thought that the approach cannot derive the global monopole solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is going to show the validity of the improved approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Consider a D dimensional spacetime with metric (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1) and the ansatz φI = η nI(x).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' As what has been discussed in section 3, such ansatz implies the vanishing energy supply vector and the work term depending on ua only through r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Thus, according to the theorem in subsection 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3, the line element Iabduadub must be −f(r)dt2 +dr2/f(r) with the function f(r) = c − 2M2−D Pl (D − 2) M Ω(D−2) 1 rD−3 − 2M2−D Pl Vmin (D − 1)(D − 2)r2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='1) The only problem is to seek the suitable field space metric GIJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Suppose the ˆgij part is a (D − 2)-dimensional sphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The simplest choice is GIJ = δIJ in (D − 1) dimensions where I, J = 1, 2, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' , D − 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Demanding δIJnInJ = 1 respects the symmetry of rotation, no matter in ˆ M(D−2) or the field space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It means embedding the unit sphere ˆgijdxidxj into a higher dimensional Euclidean space δIJdφIdφJ, such that δIJ ∂inI ∂jnJ = ˆgij , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='2) However, a flat field space metric δIJ does not satisfy the goal of exact solutions [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It inspires the choice of GIJ = F(X) δIJ , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='3) where X = δIJφIφJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Then the induced metric due to φI = η nI(x) is ˆΦij = F(η2) η2 ∂inI∂jnI = F(η2) η2 ˆgij .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='4) Thus, the work term of the model is Wφ = (D − 2) η2 F(η2) 2r2 + Vmin , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='5) which implies c = k − M2−D Pl η2F(η2)/(D − 3) according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='(3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='12).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The final check is the Klein-Gordon (KG) equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Since the field space metric GIJ is conformally flat, the field space connection is Γ I φ JK = 1 F dF dX (φJ δI K + φK δI J − φIδJK) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='6) Thus, the KG equations give ˆ∇2nI(x) = ˆΦ nI F(η2)( dF dX )X=η2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='7) Tricks in appendix A lead to ˆ∇2nI = −(D − 2)nI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Consider the following line element ds2 = f(ρ)dρ2 + ρ2ˆgijdxidxj , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8) 35 in which the Iab part is only one dimensional.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Since the ˆgij is supposed as a (D − 2)- dimensional metric in this article, the metric (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8) has D − 1 dimensions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Results in appendix A implies Γρ ρρ = 1 2f df dρ , Γa ij = − ρ f ˆgij , Γi aj = 1 ρδi j , Γi jk = ˆΓi jk .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='9) and Rρ iρj = − ρ 2f2 df dρ ˆgij , Ri ρjρ = 1 2ρf df dρδi j , Ri jkl = ˆRi jkl − 1 f (δi kˆgjl − δi lˆgjk) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='10) If f = 1 and the ˆgijdxidxj part is a sphere, the metric (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8) describes a Euclidean space ds2 = δIJdXIdXJ in spherical coordinates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Thus, flatness requires vanishing curvature, so there should be ˆRi jkl = δi kˆgjl−δi lˆgjk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The case for a Minkowski space ds2 = ηIJdXIdXJ has f = −1 and the ˆgijdxidxj part is a hyperbolic surface, such that ˆRi jkl = −(δi kˆgjl − δi lˆgjk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' This is a quick way to ensure the curvature of the maximum symmetric space is ˆRi jkl = k(δi kˆgjl − δi lˆgjk).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' As for the Laplacian, its expression under coordinates {ρ, xi} is ∇I∇Iφ = 1 fρD−2 ∂ ∂ρ(ρD−2 ∂φ ∂ρ) + ˆ∇2φ ρ2 , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='11) where ∇I donates the covariant derivative with respect to the metric (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='8) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' If choosing the scalar φ as one of the coordinates XI = ρnI, then ∇J∇JXI = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' In coordinates {ρ, xi}, the vanishing Laplacian in a flat space implies ˆ∇2nI = −D − 2 f nI .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='12) For sphere, f = 1, so ˆ∇2nI = −(D − 2)nI.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Therefore, to be consistent, there should be ( dF dX )X=η2 = − 1 η2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='13) For simplicity, one can further require F(η2) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' These are conditions for the field space metric F(X)δIJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Such an argument repeats the results of Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [53] for the spherical case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' For the hyper- bolic case, ˆgijdxidxj has k = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It can only be embedded into one higher dimensional Minkowski space rather than the Euclidean space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The above argument is valid by simply replacing δIJ with ηIJ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' As for the planar case k = 0, there is no need to introduce higher dimensional field space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Simply using the ansatz φI = ηxI leads to a similar result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The unified expression for the metric is ds2 = − (k − M2−D Pl η2 D − 3 − 2M2−D Pl (D − 2) M Ω(D−2) 1 rD−3 − Vminr2 3MD−2 Pl )dt2 + dr2 k − M2−D Pl η2 D−3 − 2M2−D Pl (D−2) M Ω(D−2) 1 rD−3 − Vminr2 3MD−2 Pl + r2ˆgijdxidxj , (D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content='14) 36 where k = −1, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It completes deriving solutions and finding suitable field space metrics, and obtains the same results with Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' [53].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' Acknowledgments The author thanks Hyat Huang and Hongwei Tan for their helpful and inspiring discus- sion, and the support of Mayumi Aoki and Ryoko Nishikawa regarding personnel affairs at Kanazawa University.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' It is also grateful for the invitation from Yi Wang to visit HKUST Jockey Club Institute for Advanced Study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} +page_content=' The author thanks for the useful comment from Ali Akil.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_dAzT4oBgHgl3EQfvf3k/content/2301.01709v1.pdf'} diff --git a/_tAyT4oBgHgl3EQfdveI/content/tmp_files/2301.00308v1.pdf.txt b/_tAyT4oBgHgl3EQfdveI/content/tmp_files/2301.00308v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..e4ca86c7397f9e7c2ab30705c498727692979197 --- /dev/null +++ b/_tAyT4oBgHgl3EQfdveI/content/tmp_files/2301.00308v1.pdf.txt @@ -0,0 +1,2509 @@ +High-Accuracy Absolute-Position-Aided Code Phase +Tracking Based on RTK/INS Deep Integration in +Challenging Static Scenarios +A Preprint +Yiran Luo +Department of Geomatics Engineering +University of Calgary +Calgary T2N1N4, Canada +yiran.luo@ucalgary.ca +Li-Ta Hsu +Department of Aeronautical and Aviation Engineering +The Hong Kong Polytechnic University +Hung Hom, Hong Kong SAR, China +lt.hsu@polyu.edu.hk +Yang Jiang +Department of Geomatics Engineering +University of Calgary +Calgary T2N1N4, Canada +yang.jiang1@ucalgary.ca +Baoyu Liu +Department of Geomatics Engineering +University of Calgary +Calgary T2N1N4, Canada +baoyu.liu@ucalgary.ca +Zhetao Zhang +School of Earth Sciences and Engineering +Hohai University +Nanjing 211100, China +ztzhang@hhu.edu.cn +Yan Xiang +School of Electronic Information +and Electrical Engineering +Shanghai Jiao Tong University +Shanghai 200240, China +yan.xiang@sjtu.edu.cn +Naser El-Sheimy +Department of Geomatics Engineering +University of Calgary +Calgary T2N1N4, Canada +elsheimy@ucalgary.ca +January 3, 2023 +Abstract +Many multi-sensor navigation systems urgently demand accurate positioning initialization +from global navigation satellite systems (GNSSs) in challenging static scenarios. How- +ever, ground blockages against line-of-sight (LOS) signal reception make it difficult for +GNSS users. Steering local codes in GNSS basebands is a desiring way to correct instanta- +neous signal phase misalignment, efficiently gathering useful signal power and increasing +positioning accuracy. Besides, inertial navigation systems (INSs) have been used as a well- +complementary dead reckoning (DR) sensor for GNSS receivers in kinematic scenarios +resisting various interferences since early. But little work focuses on the case of whether the +INS can improve GNSS receivers in static scenarios. Thus, this paper proposes an enhanced +navigation system deeply integrated with low-cost INS solutions and GNSS high-accuracy +carrier-based positioning. First, an absolute code phase is predicted from base station +information, and integrated solution of the INS DR and real-time kinematic (RTK) results +through an extended Kalman filter (EKF). Then, a numerically controlled oscillator (NCO) +leverages the predicted code phase to improve the alignment between instantaneous local +arXiv:2301.00308v1 [eess.SP] 31 Dec 2022 + +A Preprint +code phases and received ones. The proposed algorithm is realized in a vector-tracking +GNSS software-defined radio (SDR). Real-world experiments demonstrate the proposed +SDR regarding estimating time-of-arrival (TOA) and positioning accuracy. +Keywords GNSS baseband · code phase domain · vector tracking · vector receiver · positioning · float RTK · +multipath mitigation · deep integration · low-cost IMU +1 +Introduction +Demands for GNSS devices will keep increasing over the following decades due to the explosion of +smartphone-based navigation Sharma et al. [2020] and intelligent transportation construction Zhang et al. +[2020]. Therefore, realizing accurate positioning in challenging environments using global navigation satel- +lite system (GNSS) devices is a hot debate these days. However, current GNSS receiver techniques are uneasy +to achieve a next-generation positioning, navigation, and timing (PNT) performance due to the intrinsic +mechanism of GNSS electromagnetic waveforms. For instance, unlike LTE/5G wireless communication +signals with an orthogonal frequency division multiple access (OFDMA) and substantial transmission power +Liu et al. [2014], Cimini [1985], the GNSS signals are transmitted at the same frequency. Differently, each +signal channel is divided through the code division multiple access (CDMA). The GNSS signals are also +confronted with severe channel fading over a long-distance transmission (approximately 20,000 km for the +GNSS satellites operated in the medium Earth orbit). The former is naturally immune to the multipath +effect, while the GNSS signals are less capable of resisting these interferences in the transmission. Due to +this, recent research proposed a hybrid optical–wireless network that achieves a decimeter-level terrestrial +positioning and sub-nanosecond timing aiming to enact as a supplement or even substitute the GNSS device +in the future commercial market Koelemeij et al. [2022]. +Against this background, there are many remaining problems and vast space regarding new GNSS receiver +design, especially in challenging cases, and it is urgent to embark on renewing the current commercial +receiver architecture. The CDMA signals are sensitive to the non-line-of-sight (NLOS) ray within one chip +range causing big issues in channel estimation. In recent years, super-resolution algorithms (SRAs) emerged +in GNSS signal processing to separate line-of-sight (LOS) and NLOS signals into different orthogonal spaces +Krasner and McBurney [2022], Luo et al. [2021a], Da Rosa Zanatta et al. [2020]. Recent work presented +a graph Fourier transform (GFT) filter denoising the complex correlator outputs to replace the old GNSS +tracking loop, which can be considered as a direct way to steer the code phase in challenging cases Luo and +El-Sheimy [2022]. Except for the separation using the state-of-the-art SRAs or the GNSS antenna changes +Suzuki et al. [2020], Hong et al. [2020], Daneshmand et al. [2013], modeling the superposition signals +formed with LOS and NLOS rays is mainstream in the current GNSS community to overcome multipath +interference Lau and Cross [2007], Yan et al. [2022], Smolyakov et al. [2020]. +Early in the 1990s, research revealed the prominence of GNSS products to overcome the predicament of +tracking accurate Doppler frequency between the users’ end and the satellite——vector delay lock loop +(VDLL) Spilker Jr [1996]. The vector approach is an intelligent choice to model the LOS Doppler frequency +into a more proper shape. The basic idea of this technique is to leverage the user’s navigation estimates +to predict the Doppler frequency as compensation for the time of arrival (TOA) estimation in the GNSS +baseband processing, stepping into a more accurate single-point positioning (SPP) solution. The VDLL +allows GNSS researchers to optimize baseband signal modeling with information from multiple channels +instead of a conservative loop filter algorithm in a single channel. Later, this idea was extended to assist +the carrier phase modeling, for which it was nominated as a vector phase lock loop (VPLL) Zhodzishsky +et al. [1998]. Improved versions and more specific experimental results based on the VPLL techniques have +been presented over the following years Henkel et al. [2009], Shafaati et al. [2018]. However, the stability +and convergence of the VPLL are vulnerable to being destroyed when the biased error in the code phase +modeling cannot be well removed (meaning that a multipath effect interferes with the GNSS baseband and +its high-precision navigation solutions). It can be explained that the GNSS carrier and code signals are +synchronized and significantly interact with each other. +The wireless communication theory indicates that the fast fading Satyanarayana et al. [2012], such as the +multipath effect on carrier signals, causes a modeling error of a maximum of approximately one-fourth +of a signal wavelength, e.g., about 5 cm for global positioning system (GPS) L1 signals Kelly and Braasch +[2001]. This value is much smaller than the GNSS code phase error. For example, a typical value of the +multipath effect on the code signal is commonly at the meter level, which is two orders of the magnitude of +the carrier phase VAN NEE [1992]. Partially resorting to this mentioning, carrier-aiding is always used in +2 + +A Preprint +a traditional GNSS baseband to support the code signal estimation Kaplan and Hegarty [2017]. Similarly, +a vector delay/frequency lock loop (VDFLL) was also presented to enhance the code phase tracking by +combining the carrier-aiding and the VDLL approaches Lashley [2009]. +The conventional vector tracking techniques impose an indirect approach to improving the TOA modeling +over the code tracking process inside a GNSS receiver. The vector tracking has the potential to enable the +baseband to yield a more accurate code Doppler frequency production. Then, the improved code Doppler +replicates instantaneous code phases (i.e., TOA modeling) more precisely, contributing to a higher-quality +navigation solution. This type of vector receiver can ultimately alleviate the harmful interference to the +LOS signal estimation, especially when the user’s end is moving DIetmayer et al. [2020]. Nevertheless, +little efficiency will be available in the traditional vector tracking loop once the received LOS and NLOS +rays cannot be well discriminated in terms of the Doppler frequency feature (i.e., the frequency or Fourier +domain). Unfortunately, this case often occurs to a GNSS user today. +The authors’ recent research presented a method to optimize the TOA model in the GNSS tracking process +assisted by an absolute position solution, not simply relying on the code frequency error between the +baseband signal replica and the mapped code signal prediction from the user’s navigation solution Luo +et al. [2022a]. More specifically, we take advantage of the high-accuracy positioning solution (more accurate +than the code-based-only positioning result) to improve the TOA modeling aided by the vector tracking +technique in the code phase domain instead of the code frequency domain. By coincidence, a recent paper +using the classic SRA, i.e., root MUSIC, was also published aiming to achieve an analogous goal as stated +before Krasner and McBurney [2022]. +An inertial navigation system (INS) can resist the high-frequency random noise in the user’s navigation as it +works upon autonomous dead reckoning (DR), not relying on external information Groves [2013]. Hence, +integrating INS has been a prevalent way to rug the GNSS-based navigation system since early. To date, it has +attracted much attention in the navigation field to what extent the low-cost INS increases the GNSS-based +navigation Harke and O’Keefe [2022], Zhang et al. [2022]. However, research space remains in terms of the +influence of the low-cost INS on GNSS baseband signal processing. +Regarding these discussions, this work proposes an improved version of the authors’ previous research Luo +et al. [2022a, 2021b]. More specifically, a low-cost inertial measurement unit (IMU) is deeply integrated into +a GNSS vector delay/frequency/phase lock loop (VDFPLL) software-defined radio (SDR). Meanwhile, an +extended Kalman filter (EKF) is used to fuse float-RTK solutions and INS DR results. Hence, compared to +our previous research Luo et al. [2022a], the main contributions of this work include the following: +1. A low-cost IMU is combined with the float-RTK solutions via an EKF; integrated navigation solutions +are used to improve the GNSS code phase domain TOA modeling directly; +2. An RTK/INS VDFPLL SDR is proposed and developed where the integration of RTK solutions and +INS dead reckoning results, traditional scalar tracking loop (STL), VDFLL, and VDFPLL are realized +and combined; +3. An approach showing how the INS enhances the GNSS baseband in a static scenario is presented +based on real-world experiments, which few previous research discussed. +A diagram explaining the difference between the proposed and traditional algorithms towards the code +phase estimation in the GNSS baseband is provided in Figure 1. Before explaining Figure 1, it is worthwhile +to emphasize that the instantaneous code phase error consists of two primary parts regarding standard +GNSS baseband processing. They are an absolute one from the initial code phase error and a relative one +caused by the Doppler frequency error in which the received signal subtracts the local replica. These lead to +the following discussions. +At first, the conventional STL causes an apparent code Doppler frequency error and initial code phase error +(see Figure 1(a)) Kaplan and Hegarty [2017]; then, the traditional vector tracking compensates for parts +of the Doppler frequency error reducing the relative code phase error (see Figure 1(b)) Spilker Jr [1996], +Lashley et al. [2021]; next, when the vector tracking technique is further aided with an IMU sensor, the +GNSS baseband becomes more capable of alleviating the frequency error Lashley and Bevly [2013], but the +initial code phase error is remained (see Figure 1(c)); after that, when the RTK-based absolute-position-aided +(APA) technique is involved in tracking, the initial code phase error can be reduced (see Figure 1(d)) Luo +et al. [2022a]; finally, this work proposes a deep integration method of INS and GNSS RTK processing to +correct a more absolute code phase error in the local replica (see Figure 1(e)). +3 + +A Preprint +Figure 1: Diagrammatic sketch reflecting the actual and locally replicated code signals varying with the time +regarding different algorithms where colored curves correspond to the locally replicated code signals. (a) +code phase misalignment is caused by the code frequency error and the initial (absolute) code phase error; +(b) moderate frequency error is reduced (c) significant frequency error is reduced (d) significant frequency +error and moderate initial code phase error are reduced (e) significant frequency error and significant initial +code phase error are reduced. +4 + +0.5 +Timestamp +11 +Replicated signal +based on STL +0.5 +-1 +1 +0.5 +Replicated signal +1 +based on standalone +traditional VT +0.5 +0.5 +Replicated signal +01 +based on RTK/IMU +traditional VT +0.5 +0.5 +Replicated signal +based on RTK +VDFPLL +-0.5 +1 +1 +0.5 +Replicated signal +1 +based on proposed +RTK/IMU VDFPLL +-0.5A Preprint +Figure 2 further depicts the code phase errors at the timestamp (see the dashed red lines in Figure 1) in the +tracking process regarding the RTK-only APA and RTK/INS APA techniques. It is worth emphasizing that +the timestamp denotes the local clock count to get the TOA estimation in the GNSS baseband (i.e., the time +to extract the instantaneous GNSS measurements). Compared to our previous work Luo et al. [2022a], the +proposed algorithm can improve the code phase estimation by removing the initial code phase error related +to the multipath/NLOS effect and the carrier cycle slip (because of the involvement of the RTK-based APA +technique). However, the case always occurs in the real world: for example, the multipath interference in a +static GNSS user’s receiver will cause such an absolute code phase error issue which is challenging in the +current GNSS community. Therefore, this research comes up with a method to solve it. +Figure 2: Comparison of the code phase error at the timestamp (see the dashed red line in Figure 4) for +extracting the instantaneous GNSS measurements (e.g., pseudoranges and carrier phases). +The remaining of this paper is organized as follows: Section 2 introduces the methodology where the +proposed VDFPLL based on RTK/INS deep integration is discussed in detail; before that, the RTK-position- +aided VDFPLL is briefly introduced; then, two real-world stationary experiments are provided, and discussed +in Section 3; finally, Section 4 concludes this work. +2 +Materials and Methods +This section will investigate how the APA code phase tracking in a GNSS baseband is realized with the +proposed VDFPLL (deeply integrated with the float-RTK positioning and the INS DR). We will first recap +the VDFPLL based on standalone GNSS RTK solutions; then, its improved form, deeply integrating the INS +DR navigation solutions, will be discussed. Finally, how the proposed RTK/INS VDFPLL are combined with +the STL and the VDFLL in a GPS SDR will be elaborated on. +2.1 +RTK-position-aided VDFPLL +As mentioned earlier, the VDFPLL provides a way to directly steer the local code replica with the user’s +absolute position in the code phase domain instead of the conventional code frequency domain. Our previous +work achieved this goal by presenting a practical means in the baseband that applies the user’s RTK solution +as a source of high-accuracy code phase prediction. This technique will be briefly stated in the following for +the integrity of this work. +5 + +Codephaseerror(differencebetweenthelocallyreplicated +signal andthereceived signal)at thetimestamp +元/2 +Improvedcode +phase error (dueto +multipath/NLOS +errorandcarrier +cycle slip) +0 +Codephase +Codephaseerror +errorbased +basedonRTK +onthe +VDFPLL +RTKIIMU +(Luo et al. [2022a]) +VDFPLL +(proposed) +3元/2A Preprint +The architecture of the RTK-position-aided VDFPLL is illustrated in Figure 3, where APA and RPA corre- +spond to absolute- and relative-position-aided, respectively. It is worth noting that the RPA is achieved +with the traditional VDFLL technique. Besides, the absolute code phase is also tracked aided by the vector +tracking technique in the code phase domain. In this case, the following discriminates the entire code phase +error +∆ ˆτi +r,k = ∆ ˆτi,(S) +r,k ++ ∆ ˆτi,(RT K) +r,k +(1) +with +∆ ˆτi,(RT K) +r,k += fc +c +� +˜ρi +r,k−1 − ˆρi,(RT K) +r,k−1 +� +(2) +ˆρi,(RTK) +r,k−1 += ˆri,(RTK) +r,k−1 ++ +� ˆBr,ρ,t,k−1 + ˆBi +ρ,sys,k−1 +� +− κD ˆBr,mp,k−1 +ˆri,(RT K) +r,k−1 += +����ˆpi +k−1 − ˆp(RT K) +r,k−1 +���� +where subscript k denotes the index of tracking epochs; ∆ ˆτi,(S) +r,k +is the traditional discriminated code phase +error through an early-minus-late-envelope code discriminator; ∆ ˆτi,(RT K) +r,k +is the code phase error obtained +from the APA approach; ˜ρi +r,k−1 and ˆρi,(RTK) +r,k−1 +are the pseudoranges measured from the code tracking filter and +predicted from the float RTK solution, respectively; ˆri,(RT K) +r,k−1 +is the predicted geometry distance; ˆpi +k−1 is the +vector of satellite position; ˆp(RTK) +r,k−1 is the vector of the estimated float RTK position; +� ˆBr,ρ,t,k−1 + ˆBi +ρ,sys,k−1 +� +is +the summation of the local clock bias error estimation and systematic error estimation, and it is computed +from base station information and master satellite measurements Luo et al. [2022a]; ˆBr,mp,k−1 is the estimated +multipath delay error imposed on the absolute code phase error via a between-satellite single difference +algorithm, and κD is its tuned coefficient constant based on the involved early-late spacing Luo et al. [2021b]. +Next, the work process of the RTK-position-aided VDFPLL in the GPS SDR within the same tracking epoch +is stated as follows: +Step 1: the SDR receives the incoming intermediate frequency (IF) GPS L1 C/A data via front-end equipment; +Step 2: integration and dumping (I&D) procedures upon correlators are implemented between the local +code replica and the incoming IF GPS signals; +Step 3: the correlator output passing through the traditional code discriminator yields ∆ ˆτ(S) +r,k ; +Step 4: the bias of the discriminated code error compensated by the RTK-position aided (i.e., the APA +operation) code error estimation ∆ ˆτi,(RTK) +r,k +gives ∆ ˆτi +r,k; +Step 5: a code tracking loop filter denoises the code phase error from Step 4; +Step 6: an RPA technique (i.e., the VDFLL) is executed to alleviate the code frequency error in the code- +tracking process; +Step 7: a numerically controlled oscillator (NCO) leverages the output of Step 6 to produce the TOA +estimation (the raw output of the code loop filter aided by the Doppler prediction), the pseudorange +measurements (be de-noised by the carrier smoothing technique); +Step 8: the RTK engine leverages the pseudoranges and carrier phases (the raw output of the carrier loop +filter aided by the Doppler prediction) from all the tracking channels, navigation data, and the base station +information to compute the float RTK solutions; +Step 9: the APA code phase error ∆ ˆτi,(RTK) +r,k +is computed with the float-RTK solutions and the pseudorange +measurement by (2); +Step 10: repeating Step 2, the RTK-position-aided VDFPLL is working for the next tracking epoch. +To conclude, the work process of the VDFPLL based on the float-RTK solutions executed in a GNSS SDR has +been browsed. +6 + +A Preprint +Figure 3: Overview of the GNSS baseband architecture with the RTK-position-aided VDFPLL Luo et al. +[2022a, 2021b]. +2.2 +The proposed VDFPLL based on RTK/INS deep integration +2.2.1 +Architectures of the proposed VDFPLL SDR +The architectures of the proposed APA VDFPLL GPS SDR deeply integrated with the float RTK solutions +and INS dead reckoning results are displayed in Figure 4. It is worthwhile to mention that hybrid tracking +loops are adopted here due to the data rates discrepancy corresponded to various sources, i.e., the proposed +SDR tracking, the base station, and the IMU sensor raw data. +First, there are two procedures for updating the code tracking loop with the APA method in the SDR. On the +one hand, the TOA model is predicted from the integrated float RTK/INS EKF, as depicted in Figure 4(a). +On the other hand, the proposed VDFPLL updating rate is 5 Hz, that is higher than the RTK solution rate, +so the INS DR navigation solutions (with a rate of 50 Hz) are interpolated in the updating process when the +RTK solutions (with a rate of 1 Hz) are absent. It is worth noting that we take two samples of the IMU raw +data per update to compensate for coning and sculling errors, so the raw data rate of the IMU is 100 Hz, +while the DR navigation results are at the rate of 50 Hz. +Finally, as the tracking loop updating rate in the proposed SDR baseband is 200 Hz, much higher than the +VDFPLL rate, the traditional STLs for code and carrier tracking are interpolated across the intervals where +the VDFPLL is not activated. +As a result, the three tracking loops are jointly working in the proposed RTK/INS-based VDFPLL SDR, and +they are the VDFPLL based on the RTK/INS integrated EKF (see Figure 4(a)), the VDFPLL based on the INS +DR (see Figure 4(b)), and the traditional STL (see Figure 4(c)). +7 + +Channel 1 +Ephemeris +B +Channel L +p,t,k-1 +Clock error +BL +p,sys,k-1 +Systematic +Predicted APA code +error +phase error +Multipath +error + APA +Base station +coming +r,k +RTK +Code +signals +engine +tracking +4tL +r,k +loop filter +I&D +At(s) +r,k +Local +Code +Base +discriminator +carrier +station +RPA +replica +Local code +Code NCO +replica +APA VDFPLLA Preprint +Figure 4: Architectures of the proposed VDFPLL-enhanced GPS SDR based on the deep integration of float +RTK solutions and INS DR navigation results (detailed discussions refer to the Algorithm 1 stated later.) +8 + +Base +station +Channel 1 +1 Hz +Channel L +Pseudo +Incoming +APA +ranges +Integrated +RTK +signals +VDFPLL +EKF +engine +(Figure 3) +navigator +rier aiding +Hz +/DFLL) +Carrier +-- +1Hz +Carrier +phases +NLS +tracking +velocity +loop +estimator +Low-cost +100 Hz +50 Hz +IMU raw +Mechanization +dataChannel 1 +5 Hz +Channel L +Pseudo +Incoming +APA +ranges +signals +VDFPLL +(Figure 3) +rier aiding +C +5 Hz +/DFLL) +p +Carrier +D +z +tracking +loop +Str,k-1 +Pik'-1 or +5 Hz +PG/I|DR,'-1 +Low-cost +100 Hz +IMU raw +Mechanization +dataChannel 1 +Channel L +200 Hz +Pseudo +Incoming +Code +ranges +signals +tracking +loop +Carrier +C +200 Hz +aiding +p +Carrier +D +tracking +loopA Preprint +2.2.2 +RTK/INS EKF Navigator and INS DR +As illustrated in Figure 4, there are three types of TOA estimation formed in the proposed GNSS SDR +corresponding to Figures 4(a), 4(b), and 4(c), respectively. They will be elaborated on one by one in the +following. +At first, one is formed using the absolute position estimated from the integrated RTK/INS EKF navigator +(see Figure 4(a)). The RTK engine comes from an open-source package goGPS v0.4.3 Herrera et al. [2016]. +Then, the float RTK deeply integrated into the proposed SDR has been introduced in the authors’ previous +work Luo et al. [2022a]. Then, the SDR platform to realize the deep integration of RTK and INS has also +been built and investigated in the authors’ previous publications Luo et al. [2019a, 2021c]. +Besides, we will discuss the EKF algorithm used in this work. The state transition equation is given by +δxe +k′′ = Φe +k′′,k′′−1δxe +k′′−1 + we +k′′−1 +with +k′′ = +� +k +′/K +� +k′ = ⌊k/M⌋ +where superscript e represents the Earth-centered, Earth-fixed (ECEF) coordinate frame; subscript k′′ denotes +the epoch index of the EKF updates; K is the integer ratio of the INS DR and the EKF updating rates, where +the former and the latter are 50 Hz and 1 Hz (constrained by the rate of the base station information), +respectively; k′ is the epoch index of INS DR solutions; M is the integer ratio of the GNSS tracking rate and +the INS updating rate where the tracking rate (200 Hz) is no lower than the INS updating rate (50 Hz) here; +so it satisfies K,M ∈ Z+; Φe +k′′,k′′−1 is the transition matrix; we +k′′−1 is the process noise vector. Then, the state +vector of the EKF model in the ECEF frame is given by +δxe +k′′ = +� +(δψe)T ,(δve)T ,(δpe)T , +� +δbg +�T ,(δba)T �T +k′′ +where δψe is the attitude error vector; δve is the 3D velocity error vector; δpe is the 3D position error vector; +δbg and δba are the respective gyro and accelerometer bias error vectors. +Next, the observation equation is provided as +δzk′′ = He +k′′δxe +k′′ + vk′′ +where He +k′′ is the observation matrix; vk′′ is the observation noise vector. The observation vector, including +position errors and velocity errors, is provided as +δzk′′ = +� +�������� +� +������� +˜˙xI − ˜˙xG,NLS +˜˙yI − ˜˙yG,NLS +˜˙zI − ˜˙zG,NLS +� +������� +T +, +� +������ +˜xI − ˜xG,RT K +˜yI − ˜yG,RT K +˜zI − ˜zG,RT K +� +������ +T � +�������� +k′′ +T +where [ ˜x, ˜y, ˜z] and +� ˜˙x, ˜˙y, ˜˙z +� +correspond to the 3D positions and velocities w.r.t. the ECEF frame, respectively; +subscript I and G correspond to the solutions obtained from the INS and the GNSS, respectively; the +subscript “RTK” means that the GNSS position results are solved by the float RTK algorithm Luo et al. +[2022a], and the subscript “NLS” represents that the GNSS velocity results are calculated from the standard +non-linear squared (NLS) method Luo et al. [2019b]. How to build Φe +k′′ ,k′′−1 and He +k′′, as well as how to +form the process noise covariance matrix and the observation noise covariance matrix can refer to Luo et al. +[2019a]. +After the system model is built, the recursive estimation of the EKF algorithm is to predict and update the +state vector Faragher [2012]. Finally, upon the time epoch where base station information is available for the +RTK algorithm, the navigation solutions corresponding to the velocity, position, and attitude information +from the EKF algorithm are given by +ˆCe +G/I,b,k′′−1 ≈ +� +I3 − +� +δ ˆψe +k′′−1 +� +× +� ˆCe +I,b,k′−1 +ˆve +G/I,k′′−1 = ˆve +I,k′−1 − δˆve +k′′−1 +ˆpe +G/I,k′′−1 = ˆpe +I,k′−1 − δ ˆpe +k′′−1 +9 + +A Preprint +where (·)× denotes the skew matrix operator; I3 is the 3-order identity matrix; δˆve +k′′−1 δ ˆpe +k′′−1, and δ ˆψe +k′′−1 are +the estimated state vectors about velocity and position errors, and attitude errors, and ˆve +G/I,k′′−1, ˆpe +G/I,k′′−1 +and ˆCe +G/I,k′′−1 are the estimated navigation vectors (corresponding to the respective velocity, position, and +attitude); ˆve +I,k′−1, ˆpe +I,k′−1 and ˆCe +I,b,k′−1 are the counterparts solely upon the INS DR process, which will be +introduced subsequently. +Within the epochs where the base station information is missing, the EKF-based results at the previous +epoch can contribute to the INS DR process at the current epoch as the following +ˆve +G/I|DR,k′ = ˆve +G/I,k′′−1 + +� +C +e +G/I,k′′−1˜fb +ib,k′−1 − 2(ωie) × ˆve +G/I,k′′−1 + ge � +ˆpe +G/I,k′′−1 +�� +∆tI +ˆpe +G/I|DR,k′ = ˆpe +G/I,k′′−1 + +� +ˆve +G/I|DR,k′ + ˆve +G/I,k′′−1 +� ∆tI +2 +ˆCe +G/I|DR,b,k′ = Ce +i (∆tI) ˆCe +G/I,b,k′′−1 +� +I3 + +� +˜ωb +ib,k′−1∆tI +� +× +� +with +ωie = � 0 +0 +ωie +�T +Ce +i (∆tI) = +� +������ +cos(ωie∆tI) +sin(ωie∆tI) +0 +−sin(ωie∆tI) +cos(ωie∆tI) +0 +0 +0 +1 +� +������ +where ∆tI is the updating interval of the EKF; ˜fb +ib,k′−1 and ˜ωb +ib,k′−1 are the specific force and angular rate +measurement vectors of the body frame w.r.t. ECEF frame; ωie is the Earth rotation rate, i.e., 7.292115e-5 +rad/s and ωie is its vector form; ge � +ˆpe +G/I,k′′−1 +� +is the gravity acceleration vector function in the ECEF frame +varying with the user’s position ˆpe +G/I ,k′′−1 (see Equations (2.133) and (2.142) in Groves [2013]); Ce +i (∆tI) is +the Earth rotation matrix from the Earth-centered inertial (ECI) to the ECEF coordinate frame varying with +the updating interval ∆tI; C +e +G/I,b,k′′−1 is the averaging transformation matrix w.r.t. the body-to-ECEF-frame +coordinate obtained from ˆCe +G/I,b,k′′−1 (see Equations (5.84) and (5.85) in Groves [2013]). +Then, considering the case where the navigating solutions are derived from the INS DR process (see Figure +4(b)), the mechanization in the ECEF frame can be expressed as +ˆve +I,k′ = ˆve +G/I|DR,k′−1 + +� +C +e +I,b,k′−1˜fb +ib,k′−1 − 2(ωie) × ˆve +G/I|DR,k′−1 + ge � +ˆpe +G/I|DR,k′−1 +�� +∆tI +ˆpe +I,k′ = ˆpe +G/I|DR,k′−1 + +� +ˆve +I,k′ + ˆve +G/I|DR,k′−1 +� ∆tI +2 +ˆCe +I,b,k′ = Ce +i (∆tI) ˆCe +G/I|DR,b,k′−1 +� +I3 + +� +˜ωb +ib,k′−1∆tI +� +× +� +or +ˆve +I,k′ = ˆve +I,k′−1 + +� +C +e +I,b,k′−1˜fb +ib,k′−1 − 2(ωie) × ˆve +I,k′−1 + ge � +ˆpe +I,k′−1 +�� +∆tI +ˆpe +I,k′ = ˆpe +I,k′−1 + +� +ˆve +I,k′ + ˆve +I,k′−1 +� ∆tI +2 +ˆCe +I,b,k′ = Ce +i (∆tI) ˆCe +I,b,k′−1 +� +I3 + +� +˜ωb +ib,k′−1∆tI +� +× +� +where C +e +I,b,k′−1 is the averaging transformation matrix computed from ˆCe +I,b,k′−1. +It is worth mentioning that the tracking rate (200 Hz) is higher than the INS DR updating rate (50 Hz). +So, three out of four tracking intervals do not have an update for the INS DR. Assuming that the user’s +navigation results are not changed significantly over the time 0.02s (i.e., +1 +50Hz), when k′ = k/M, we make an +approximation that the navigation estimations in the following (k + 1)th, (k + 2)th, and (k + 3)th tracking +epochs are identical to the ones computed at the kth, to interpolate the tracking epochs without the INS +updating. +10 + +A Preprint +2.2.3 +RTK/INS APA Code Phase Tracking +The baseband TOA modeling at the start of the kth epoch aided by the absolute positions from the integrated +RTK/INS EKF and the INS DR is estimated through +� +TOA +i +k = c−1 ˜ρi +k − δˆtr,k +(3) +with the pseudorange model of +˜ρi +k ≜ ˜ρi +k−1 + cf −1 +c +�� +fc + ˆf i +code,dop,k +� +Tcoh + ∆ ˆτi+,(RT K/INS) +code,k,0 +� +(4) +where ˜ρi +k−1 and ˜ρi +k are the instantaneous pseudorange measurements at the respective previous and current +epochs; δˆtr,k is the estimated local clock bias error; fc and c are the spreading code rate and the speed of +light, respectively; Tcoh is the coherent integration time ˆf i +code,dop,k is the estimated code Doppler frequency +and ∆ ˆτi+,(RTK/INS) +code,k,0 +is the proposed initial code phase error estimate in chips at the start of the kth epoch, +and the estimation processes of them will be subsequently elaborated. +On the one hand, ˆf i +code,dop,k can be written as +ˆf i +code,dop,k = −fc +fr +˜fcarr,dop,k + T −1 +coh∆ ˆτi+ +r,k +(5) +˜f i +carr,dop,k = ∆ ˆf i,(aid) +carr,k + T −1 +coh∆ ˆϕi+ +r,k +(6) +with +∆ ˆf i,(aid) +carr,k =fr +c +� +ˆve +G/I,k′′−1 · ˆei +k +� +ˆpe +G/I,k′′−1 +� +− ˆvi +k · ˆei +k +� +ˆpe +G/I,k′′−1 +� ++ cδˆ˙tr,k − cδˆ˙t +i +k +� +(7) +or +∆ ˆf i,(aid) +carr,k =fr +c +� +ˆve +G/I|DR,k′−1 · ˆei +k +� +ˆpe +G/I|DR,k′−1 +� +− ˆvi +k · ˆei +k +� +ˆpe +G/I|DR,k′−1 +� ++ cδˆ˙tr,k − cδˆ˙t +i +k +� +(8) +∆ ˆf i,(aid) +carr,k =fr +c +� +ˆve +G/I|DR,k′ · ˆei +k +� +ˆpe +G/I|DR,k′ +� +− ˆvi +k · ˆei +k +� +ˆpe +G/I|DR,k′ +� ++ cδˆ˙tr,k − cδˆ˙t +i +k +� +(9) +∆ ˆf i,(aid) +carr,k =fr +c +� +ˆve +I,k′ · ˆei +k +� +ˆpe +I,k′ +� +− ˆvi +k · ˆei +k +� +ˆpe +I,k′ +� ++ cδˆ˙tr,k − cδˆ˙t +i +k +� +(10) +∆ ˆf i,(aid) +carr,k =fr +c +� +ˆve +I,k′−1 · ˆei +k +� +ˆpe +I,k′−1 +� +− ˆvi +k · ˆei +k +� +ˆpe +I,k′−1 +� ++ cδˆ˙tr,k − cδˆ˙t +i +k +� +(11) +where ˜fcarr,dop,k denotes the carrier Doppler frequency measurement; +� +T −1 +coh∆ ˆτi+ +r,k +� +and +� +T −1 +coh∆ ˆϕi+ +r,k +� +are the +filtered code phase error and the filtered carrier phase error through the loop filters, respectively, which have +accounted for the coherent integration interval in tracking, and its input is ∆ ˆτi +x,k which will be explained +later, with x ∈ {I,G/I,G/I|DR}; ∆ ˆf i,(aid) +carr,k is the aided Doppler frequency computed via the user’s velocity +estimation, known as a VDFLL technique Lashley et al. [2021]; ˆvx,k′ / ˆvx,k′′ and δˆ˙tr,k are the predicted user’s +velocity vector and the predicted user’s clock drift; ˆvi +k and δˆ˙t +i +k are the satellite velocity vector and the satellite +clock drift predicted with the broadcast ephemeris; ˆei +k (·) is the operator of the unit cosine vector varied with +the position estimation. +As mentioned above, ∆ ˆτi +x,k is the APA discriminated code phase error, and there are three ways to obtain +this estimate in the code tracking loop. For instance, the ones estimated via the respective RTK/INS EKF +solution, the two-consecutive-epoch INS DR, and INS DR right after the EKF are computed as +∆ ˆτi +G/I,k = ∆ ˆτi,(S) +r,k ++ ∆ ˆτi,(APA) +r,k +� +ˆri +G/I,k−1 +� +(12) +11 + +A Preprint +∆ ˆτi +I,k = ∆ ˆτi,(S) +r,k ++ ∆ ˆτi,(APA) +r,k +� +ˆri +I,k−1 +� +(13) +∆ ˆτi +G/I|DR,k = ∆ ˆτi,(S) +r,k ++ ∆ ˆτi,(APA) +r,k +� +ˆri +G/I|DR,k−1 +� +(14) +with +ˆri +G/I,k−1 = +���ˆpi +k−1 − ˆpe +G/I,k′′−1 +��� +ˆri +I,k−1 = +���ˆpi +k−1 − ˆpe +I,k′−1 +��� +ˆri +G/I|DR,k−1 = +���ˆpi +k−1 − ˆpe +G/I|DR,k′−1 +��� +where ∆ ˆτi,(S) +r,k +is the traditional discriminated code phase error as introduced earlier; ˆpi +k−1 is the satellite +position vector computed from the broadcast ephemeris; ∆ ˆτi,(APA) +x,k +(·) is the operator to obtain the absolute +code phase error with the geometry distance prediction (i.e., the APA process) and the error models, and its +analytical expression is defined as +∆ ˆτi,(APA) +r,k +� +ˆri +x,k−1 +� +≜ fc +c +� +˜ρi +r,k−1 − +� +ˆri +x,k−1 + +� ˆBr,ρ,t,k−1 + ˆBi +ρ,sys,k−1 +� +− κD ˆBr,mp,k−1 +�� +(15) +Therefore, based on these discussions, it is easy to find that the absolute code phase error estimate +∆ ˆτi+,(RTK/INS) +code,0,k +in (4) (i.e., the difference between the received initial code phase and the counterpart of +the local code replica synthesized with the NCO) can be alleviated by the proposed algorithm. +Finally, the proposed algorithm in this paper is summarized in Algorithm 1. This algorithm is realized in a +GPS SDR prototype where L1 C/A signals are used to validate the TOA and position estimation performance. +3 +Results and Discussion +The experimental equipment is set up as shown in Figure 5. Two stationary data sets were collected in +the real world to verify the proposed algorithm. A NovAtel antenna was used to receive the GPS L1 C/A +IF signals through a Fraunhofer IIS RF frond-end, where the IF sampling rate is 10.125 MHz. The IMU +raw data were collected from the Crossbow Nav 440 device, where the IMU’s gyro and accelerometer bias +stabilities are 10 deg/h and 1 mg, respectively. Besides, it is worth mentioning that two samples are taken +for updating the inertial sensor data for our navigation equation, so the updating rate of the INS DR is half +(50 Hz) of the IMU raw data rate (100 Hz). The reference positions of the two experiments are obtained by +averaging the results provided by the Crossbow Nav440 GPS/INS integration solutions (the centers of the +IMU sensor and the GNSS antenna are sufficiently close in the setup and neglected in this experiment). +The proposed algorithm is tested in a GPS SDR platform where the coherent integration time is 5 ms, the +classic discriminators are chosen as the noncoherent-early-minus-late-amplitude code discriminator and +Costas carrier discriminator, and the early-late spacing is four IF sample intervals. Five types of tracking +algorithms are compared in the same SDR conditions except for the parameter adjustment in Table 1. +First, an open sky area is chosen to carry out the experiment where the test spot in Google Map and the sky +plot of the available GPS satellites are shown in Figure 6. +We first assess the SPP results for this open-sky case. The SPP is based on the weighted NLS algorithm in the +tested SDR, which refers to an open-source package RTKLIB Takasu and Yasuda [2009]. Figure 7 depicts the +dilution of precision (DOP) results, SPP errors, and the 3D position cumulative distribution function (CDF) +curves of 3D SPP root-mean-squared errors (RMSEs). +Although the results show that the proposed algorithm does not increase the SPP accuracy compared to +the classic STL algorithm in the open sky, it proves that the APA-based VDFPLL algorithms enhance the +traditional RPA-based VDFLL in the static situation. Meanwhile, using the INS can moderately enhance the +RTK-based APA tracking process. +Then, the RTK position errors for the different SDR algorithms are compared in Figure 8, where the position +errors, horizontal position results for a Google Map show, and the CDF curves of 3D and 2D position +estimates RMSE are included. The traditional RPA VDFLL does not help the RTK position accuracy in +this open-sky and static case. The finding is that the RTK-only-based VDFPLL improves the RTK results +within a 3D range while the RTK/INS-based VDFPLL slightly makes it outperform the STL-based horizontal +12 + +A Preprint +Algorithm 1 High-accuracy APA GNSS code phase tracking based on RTK/INS deep integration +Require: +k∗ ≜ k mod KM, subject to K,M ∈ Z+ and k,k∗ ∈ N +1: +while new digital IF samples (for a coherent processing interval) are received at the kth +epoch do +2: +Synthesize the code and carrier local replicas with the code/carrier NCOs; +3: +Produce the early- prompt- and late-branch samples through the I&D using the local replicas +and the received IF samples; +4: +Discriminate the code/carrier phase errors with the outputs of the I&D (i.e., correlator outputs); +5: +if the base station information is available at the tracking epoch(s) {k∗ − 2M, ...,k∗ − M − 1} then +6: +Compensate for the discriminated code phase error with (14); +7: +else if the base station information is available at the tracking epoch(s) {k∗ − M,..., k∗ − 1} then +8: +Compensate for the discriminated code phase error with (12); +9: +else +10: +Compensate for the discriminated code phase error with (13); +11: +end if +12: +Optimize the compensated code phase error from Step 6/8/10 with a 1-Hz 2nd-order loop filter; +13: +if the vector tracking trigger (5 Hz) is activated then +14: +Optimize the discriminated carrier phase error with a 0.5-Hz 1st-order loop filter; +15: +if the RTK/INS EKF is updated at the epoch(s) {k∗,k∗ − 2M,k∗ − 3M,...,k∗ − (K − 1)M} then +16: +Predict the carrier Doppler with (10); +17: +else if the RTK/INS EKF is updated at the epoch(s) {k∗ − 2M + 1,...,k∗ − M − 1} then +18: +Predict the carrier Doppler with (8); +19: +else if the RTK/INS EKF is updated at the epoch(s) {k∗ − M} then +20: +Predict the carrier Doppler with (9); +21: +else if the RTK/INS EKF is updated at the epoch(s) {k∗ − M + 1,...,k∗ − 1} then +22: +Predict the carrier Doppler with (7); +23: +else +24: +Predict the carrier Doppler with (11); +25: +end if +26: +Compute the carrier frequency with (6) (for carrier NCO); +27: +else +28: +Optimize and predict the carrier Doppler with a 15-Hz 3rd-order loop filter (for carrier NCO); +29: +end if +30: +Compute the code frequency with (5) (for code NCO); +31: +end while +Figure 5: Setup for the stationary experiments. +13 + +GNSS Satellite +CrossbowNav440 +NovAtel +Antenna +GPSSDR +Fraunhofer IIS +RFFront-EndA Preprint +Figure 6: Open-sky test spot (Google Map show) and sky plot of available GPS satellites. +Figure 7: Single point navigation results and statistical analysis of different SDRs in the open-sky situation +where dashed lines correspond to outlier epochs. (a) DOP values (b) SPP results (c) CDF curves of 3D SPP +RMSE. +14 + +N +30 +330 +15. +300 +30 +G25 +45. +60 +7.5 +G23 +G31 +G03 +G14 +G321 +G22 +G01 +G26 +210 +150SDR:STL +SDR:RTK-basedVDFLL +Initializationof +SDR:RTK/INS-basedVDFLL +theSDR +(a) +SDR:RTK-basedVDFPLL +SDR:RTK/INS-baSedVDFPLL +3 +East +2 +3 +North +2 +2 +20 +40 +60 +80 +100wW +East [m] +50 +North [m] +10 +20 +E +21 +20 +40 +60 +80 +100 +Epoch [s]0.8 +robability +0.6 +0.4 +n +Datafrom522555s +0.2 +to522645s +10 +20 +30 +40 +50 +3D SPPRASErailNw +51.0824 +51.0822 +51.082 +0 +1A +1A +St Davids United Church +0 +51.0818 +★ +titude +51.0816 +Testspot +51.0814 +CrowchildT +Close park +51.0812 +20 m +114.125 +114.1245 +114.124 +114.1235 +-114.123 +Longitude ldegreeA Preprint +Table 1: Parameter settings of the SDR regarding the real-world experimental validation +SDR Type +Relative +position/ +velocity +aiding +RTK +absolute +position +aiding +INS deep +integration +RTK/INS +integration +navigation +solution +Tracking +loop filter +STL (traditional) +Kaplan and Hegarty [2017] +No +No +No +No +1-Hz 2nd-order DLL & +18-Hz 3rd-order PLL +RTK-based VDFLL +Lashley et al. [2021] +Yes +No +No +No +1-Hz 2nd-order DLL & +0.5-Hz 1st-order PLL & +15-Hz 3rd-order PLL +(see Algorithm 1) +RTK/INS-based VDFLL +Lashley et al. [2021] +Yes +Yes +Yes +Yes +RTK-based VDFPLL +Luo et al. [2022a] +Yes +Yes +No +No +RTK/INS-based VDFPLL +(proposed) +Yes +Yes +Yes +Yes +Figure 8: RTK position results and statistical analysis of different SDRs in the open-sky situation where +dashed lines correspond to the outlier epochs. (a) RTK position errors (b) horizontal RTK results in Google +Map (c) CDF curves of 3D RTK RMSE (d) CDF curves of horizontal (2D) RTK RMSE. +15 + +SDR:STL +SDR:RTK-basedVDFLL +Initializationof +(a) +SDR:RTK/INS-basedVDFLL +the SDR +SDR:RTK-basedVDFPLL +SDR:RTK/INS-baSedVDFPLL +East[m] +t +4 +2 +0 +LO +North [m] +5 +up [m] +5 +20 +40 +60 +80 +100 +GPStimesince522535s +Epoch[s]SDR.STL +Datafrom522555s +SDR.RTK-based VDFLL +31.0819 +to522645s +SDR.RTKINS-based VDFLL +SDR.RTK-basedVDFPLL +SDR.RTKINS-basedVDFPLL +☆ +Ground truth +5108185 +51.0818 +☆ +08173 +5 m +51.0817 +114.1238 +114.123 +114.1236 +114.1285 +-114.1234 +ongtuce0.8 +Pro bability +0.6 +0.4. +0.2 +0 +2 +80.8 +Probability +0.6 +0.4 +0.2 +8A Preprint +positioning. It can be explained that the vertical INS DR position solution would cause more code phase +errors (compared to the STL tracking in a static open-sky case) in terms of the vertical position estimation. +Next, we compare the RTK/INS integrated results of three SDRs in Figure 9, where “Ground-truth-based +VDFPLL” means that the SDR leverages the actual position coordinates instead of on-the-fly RTK or +integrated RTK/INS position estimates. It is a hardware-in-the-loop (HIL) simulation strategy to provide a +reference for the proposed algorithm under the same SDR conditions. In other words, the HIL simulation +results represent the upper bound of the performance that the used SDR platform can achieve. +Based on the CDF curves in Figure 9(d), the proposed accuracy slightly exceeds the traditional STL-based +integrated RTK/INS solution within the 78% probability. So, the large errors can be alleviated in the +integration results using the proposed algorithm in the open sky area. In contrast, the traditional VDFLL- +based integrated RTK/INS positioning accuracy drops much in this well-condition static scenario. +Figure 9: RTK position results and statistical analysis of different SDRs in the open-sky situation where +dashed lines correspond to the outlier epochs. (a) RTK position errors (b) horizontal RTK results in Google +Map (c) CDF curves of 3D RTK RMSE (d) CDF curves of horizontal (2D) RTK RMSE. +TOA curves will be evaluated in the following parts to examine the baseband processing discrepancy. The +TOA estimation from the GPS SDR is computed via (3), where it is worth noting that the ˜ρi +k here is the raw +data from the loop filter output without the carrier-smoothing algorithm. Therefore, the measurement and +reference TOA residuals in meters are computed as follows +(TOA measurement residual)i +k +∆= ˜ρi +k − c +� +TOAi +0 + δˆtr,0 +� +− c +� +−f i +d,0f −1 +r ++ δˆ˙tr,0 +� +kTcoh +(TOA reference residual)i +k +∆= cTOAi +k − cTOAi +0 − c +� +−f i +d,0f −1 +r +� +kTcoh +TOA residual error ∆= (TOA measurement residual)i +k − (TOA reference residual)i +k +where the TOA residuals exclude the initial pseudorange and initial Doppler frequency for the simplicity of +analysis; subscript 0 and k denote the initial and the kth epoch, respectively; TOAi +0 and f i +d,0 represent the +16 + +Initialization +SDR:STL,POS:RTK/IMUIntegration +SDR:RTK/INS-basedVDFLL,POS:RTK/IMUIntegration +of the SDR +SDR:RTK/INS-basedVDFPLL,POS:RTK/IMUIntegration +(a) +SDR:Ground-truth-basedYDFPLL.POS:RTK/iMUIntegratior +East [m] +2 +0 +North [m] +-2 +[] +dn +20 +40 +60 +80 +100 +GPStimesince522535s +Epoch[s]SDR:STL, POS:RTK/IMU Integration +SDR:RTK/INS-based VDFLL, POS:RTK/IMU Integration +51.0819 +SDR:RTK/INS-based VDFPLL, POS:RTK/IMU Integration +0 +SDR:Ground-truth-based VDFPLL, POS:RTK/IMU Integration +★ +ground truth +51.08185 +2 +Datafrom522555sto522645s +ep. +51.0818 +★ +51.08175 +5 m +51.0817 +-114.1238 +-114.1237 +-114.1236 +-114.1235 +114.1234 +Longitude Tdegree0.8 +Probability +0.6 +0.4 +0.20.8 +Probability +0.6 +0.4 +0.2A Preprint +TOA and Doppler frequency reference at the initial epoch, and they are computed as +T OAi +0=c−1 ����pgt,0 − pi +0 +��� + ˆBI,0 + ˆBT ,0 − cδˆti +0 +� +f i +d,0 = fr +c +� +vgt,0 · ˆei +0 − ˆvi +0 · ˆei +0 − cδˆ˙t +i +0 +� +where pgt,0 and vgt,0 are the ground truth vectors in the ECEF coordinate frame for the user’s position and +velocity in the stationary experiments, with vgt,0 = [0,0,0]T ; pi +0, vi +0, δˆti +0, and δˆ˙t +i +0 are the satellite position and +velocity vectors, satellite clock bias and drift errors, respectively, which are obtained and computed using the +broadcast ephemeris; meanwhile, ˆBI,0 and ˆBT ,0 are the ionospheric error derived from the Klobuchar model +and the tropospheric error calculated via the Saastamoninen model; ˆei +0 is the unit cosine vector computed +from pgt,0; δˆti +0 and δˆ˙t +i +0 are the estimated user’s clock bias and drift errors computed as +δˆtr,0 = averaging clock bias error − (navigation time spanning) +2 +× δˆ˙tr,0 +where δˆ˙tr,0 is the averaging value of the clock drift estimates from the “Ground-truth-based VDFPLL” SDR. +It is worthwhile to say that the given δˆ˙t +i +0 and δˆti +0 references are not sufficiently accurate, but they are +satisfactory in validating the TOA performance amid different SDR algorithms. +Figure 10 shows the error curves of the TOA residuals for the signals from the satellite of PRN1 (low +elevation angle) and PRN22 (high elevation angle). It proves that the APA-based vector tracking algorithms +perform better in interference mitigation for the signals from the low satellite. +In summary, even if the proposed algorithm does not manifest much better than the traditional STL in an +open-sky static environment, it demonstrates a significant improvement in comparison with the traditional +RPA vector tracking in the same condition. More specifically, it can be used as boosting complement for the +existing vector GNSS receivers. +Another set of data was collected under a semi-open-sky situation where the GPS antenna was receiving the +signals affected by the eastern CCIT building at the campus of the University of Calgary. The Google Map +show of the test spot and the corresponding satellite sky plot are provided in Figure 11. +In this test, where the results are displayed in Figure 12, we also give the DOP values to offer the satellite +geometry status; the SPP error curves and their 3D CDF are provided as well. +First, it can be observed that the two VDFPLLs and the STL produce more reliable solutions as the position +outliers occur to the two VDFLLs at around the 95th epoch. Then, it is evident that the proposed algorithm +embraces the highest SPP accuracy. +We also assess the RTK solution accuracy related to the different SDRs. The RTK position error curves, posi- +tion results in Google Map, and the 3D CDF curves are provided in Figure 13. After the SDR RTK solutions +become stable, the RTK accuracy from the proposed SDR solutions still shows the highest performance. The +RTK-only-based APA algorithm can reduce the random noise, but it is more biased than the traditional STL +algorithm. The two RPA-only vector tracking techniques are still less capable of offering efficient assistance +in the static test. +Next, the integrated RTK/INS solutions are compared in the semi-open-sky environment as shown in Figure +14. In this case, the proposed algorithm significantly improved the 2D positioning performance. At the same +time, the RTK/INS-based RPA method elevates the 3D positioning accuracy compared to the STL-based +integration. The RPA- and APA-based integrated navigation accuracies over the error range of approximately +55% probability outperform the traditional STL one. Furthermore, the positioning results with the proposed +algorithm perform more stable than the traditional vector tracking. Another finding is that RTK/INS-based +APA vector tracking yields a much more ideal horizontal positioning estimate than the RPA one. By contrast, +the latter is superior to the former in the vertical direction. By comparing to the upper bound (i.e., the +estimation from the ground-truth-based VDFPLL SDR), it can be explained that the fusion of the low-cost +IMU has a side effect on the vertical position solution when it is applied to the proposed RTK/INS-based +VDFPLL SDR in this stationary experiment. +Then, the raw TOA performance in terms of the signal from the high-elevation-angle satellite (PRN3) and +the one with a low elevation angle affected by the multipath interference (PRN6) are plotted in Figure 15. +17 + +A Preprint +Figure 10: Error curves of the TOA residuals for the GPS satellites PRN1 and PRN22 in the open-sky +situation. +Figure 11: Semi-open-sky test spot (Google Map show) and the sky plot of available GPS satellites. +18 + +PRN1 +350 +300 +SDR:STL +SDR:RTK-basedVDFLL +SDR:RTK/INS-basedVDFLL +[w] +250 +SDR:RTK-basedVDFPLL +SDR:RTK/INS-basedVDFPLL +200 +150 +Residuals +100 +50 +0 +Timesince522536s +-50 +0 +20 +40 +60 +80 +100 +Time [s]PRN1 +Error of Residuals of TOA [m] +-10 +20 +30 +40 +50 +-60 +40 +60 +80 +100 +Timesince522536s +Time [s]PRN22 +700 +600 +Residuals of TOA [m] +500 +400 +300 +200 +100 +0 +Timesince522536s +-100 +0 +20 +40 +60 +80 +100 +Time [s]PRN22 +-5 +-10 +Error of Residuals of TOA [m] +-15 +-20 +25 +30 +35 +40 +45 +-50 +-55 +0 +20 +40 +60 +80 +100 +Time [s]51.0812 +51.081 +Blockage +CCITbuilding +51.0808 +1 +1 +68 +51.0806 +★ +tud +51.0804 +PIN +CcIT BuIding +Test spot +EnergiSimulation/ +51.0802 +FrankandSarahMe +51.08 +20 m +114.1355 +114.135 +-114.1345 +-114.134 +334.4335 +Longitude [degreeN +30 +330 +15 +G25 +G06 +300 +30 +45 +60 +G31 +75 +G23 +G14. +G03 +E +G22 +G26 +2 +G16 +210 +150 +SA Preprint +Figure 12: Single point navigation results and statistical analysis of different SDRs in the semi-open-sky +situation where dashed lines correspond to the outlier epochs. (a) DOP values (b) SPP results (c) CDF curves +of 3D SPP RMSE. +A more significant fluctuation in the TOA curves emerges at the PRN6 in this semi-open-sky experiment +compared to the open-sky PRN1 (see Figure 10). Also, both APA-based tracking loops are more capable of +alleviating the TOA error varying with a long-term time spanning (i.e., the level of dozens of seconds) than +the RPA-based vector tracking and the STL. Nevertheless, the curves computed from the high-quality signal, +PRN3, are highly homogeneous regarding all the tested tracking loop algorithms. +The next part will quantitatively examine the exact improvement the proposed algorithm can offer for the +GNSS baseband estimation. As mentioned, an upper bound of the instantaneous integrated positioning +performance is obtained from the SDR under the HIL test using the “Ground-truth-based VDFPLL”. +Therefore, the corresponding TOA error curve representing the upper bound can also be extracted from +the tracking results. Then, we compute the TOA error of the PRN22 and PRN3 (with the highest elevation +angles during the experiments) in the open-sky and semi-open-sky cases as the respective references. The +proposed TOA error references reasonably model the remained local clock errors in meters varying with the +time where the other biased errors, like the atmospheric delay and initial TOA errors, are assumed to be +well removed by the given models. +Ultimately, the TOA curve references are derived and illustrated in Figure 16. After that, the TOA accuracy +of different satellites in the two testing situations will be analyzed via these references in the following. +Table 2 summarizes the statistical analysis of the TOA performances of different tracking algorithms where +the RMSE results are computed for the used satellites. Then, in regard to the traditional STL, the TOA +accuracy improvements of the two RPA- and APA-based vector tracking algorithms operated in the GPS +SDR are computed and depicted in Figure 17. +19 + +SDR:STL +SDR:RTK-basedVDFLL +Initializationof +SDR:RTK/INS-baSedVDFLL +SDR:RTK-basedVDFPLL +theSDR +(a) +SDR:RTK/INS-basedVDFPLL +2.5 +East +2 +1.5 +1.5 +North +3 +- +20 +40 +60 +80 +100 +GPStimesince524045sEpoch[s]Initialization of +the SDR +21 +[m +181 +20 +North [m] +40 +20 +20 +40 +60 +80 +100 +Epoch [s]0.8 +robability +0.6 +0.4 +Datafrom524055s +to524160s +0.2 +0 +10 +20 +30 +40 +5.0 +3 spprmsuA Preprint +Figure 13: RTK position results and statistical analysis of different SDRs in the semi-open-sky situation +where dashed lines correspond to the outlier epochs. (a) RTK position error (b) horizontal RTK position +results in Google Map (c) CDF curves of 3D RTK RMSE. +The curves in Figure 17 indicate that both APA tracking methods outperform two RPA ones in elevating the +TOA accuracy. Furthermore, regarding the lower-elevation satellites and the smaller-TOA-error channels, +the TOA errors induced by the navigation results through the vector feedback procedure are more likely to +drop in the proposed RTK/INS-based APA approach than in the RTK-only APA one. +Figure 18 plots the APA error curves estimated from the proposed RTK/INS-based VDFPLL modeling the +instantaneous initial/absolute code phase error in meters at each tracking epoch of which the analytical +expression is given by (15). Compared to the traditional tracking algorithms (scalar and old vector tracking +loops), the proposed algorithm can individually discriminate the absolute code phase error unrelating to +the frequency error given by the same-epoch local replica subtracting incoming signals. This operation +established through the proposed architecture is reasonable and it proves efficient. The implied information +that the RTK/INS integrated EKF navigator provides more accurate positioning than the code-based-only +SPP method can adequately explain the results. +So, the dashed black lines in Figure 18 mean that the traditional scalar and vector tracking loops have +nothing to recognize the code phase error not varying with the time spanning (the error residual remained +by the traditional code discriminating process). However, the proposed RTK/INS-based APA vector tracking +can directly estimate the absolute code phase error at every tracking epoch. The APA code discriminated +results can show how the code phases are corrected by the accurate user’s position solution, especially for +the satellites facing the deterministic biased error changing as the cycles of dozens of seconds or longer. +This phenomenon commonly occurs to the static user’s antenna receiving incoming signals affected by the +multipath effect. +20 + +SDR:STL +SDR:RTK-basedVDFLL +Initialization.of +SDR:RTK/INS-based VDFLL +theSDR +SDR:RTK-baSedVDFPLL +(a) +SDR:RTK/INS-basedVDFPLL +10 +5 +0 +10 +North [m] +5 +0 +20 +E +10 +0 +-10 +20 +40 +60 +80 +100 +GPS time since 524045s Epoch [s]51.0807 +SDR:STL +SDR:RTK-based VDFLL +SDR:RTK/INS-based VDFLL +SDR:RTK-based VDFPL +51.08065 +★ +SDR:RTK/INS-based VDFPLL +Ground truth +atitude [degree] +allegiatePINV +51.0806 +Datafrom524055sto524160s +5.08055 +51.0805 +5 m +-114.1345 +114.1344 +114.1343 +114.1342 +334.34 +Longitude [degreel0.8 +robability +0.6 +0.4 +n +0.2 +5 +10 +15 +20 +3DRKRASEA Preprint +Figure 14: RTK/INS integration position results and statistical analysis of different SDRs where dashed lines +correspond to the outlier epochs in the semi-open-sky situation. (a) RTK/INS integration position error (b) +horizontal RTK/INS integration position results in Google Map (c) CDF curves of 2D RTK/INS integration +position RMSE (d) CDF curves of 3D RTK/INS integration position RMSE. +Finally, it is worth mentioning that the proposed RTK/INS-based VDFPLL is a simplified prototype applying +the APA discriminated error relying on the INS and RTK to the GNSS baseband processing. The tracking +performance still has space to be further improved by redoing loop filter algorithms (e.g., the EKF) or other +GNSS baseband optimizing methods (e.g., snapshot processing Luo et al. [2022b], Fernández-Hernández +et al. [2022] and open-loop tracking Tsang et al. [2022], van Graas et al. [2009]). In other words, the +proposed algorithm has a broad scope of use towards the GNSS signals at all frequencies and constellations, +potentially contributing to the development of next-generation GNSS receivers and GNSS-based multi-sensor +integrating navigation systems. +4 +Conclusions +This work proposes a deep integration of RTK and INS, enhancing the instantaneous code phase tracking +performance in challenging static environments. In the presented algorithm, the navigation solutions, +especially the absolute position solution, from the integrated EKF navigator are deeply fused into the GNSS +tracking loop, forming an APA code phase discriminator. The RTK/INS-based APA discriminator combined +with the vector tracking technique realized upon a GPS L1 C/A SDR can serve for more satisfactory tracking +and positioning results than the RTK-based-only APA vector tracking approach. Two real-world stationary +experiments have verified the performance. Finally, the conclusions of this work can be drawn as follows: +21 + +SDR:STL,POS:RTK/IMUIntegration +Initialization +SDR:RTK/INS-basedVDFLL,POS:RTK/IMUIntegration +of the SDR +SDR:RTK/INS-basedVDFPLL,POS:RTK/IMUIntegration +SDR:Ground-truth-basedVDFPLL,POS:RTK/IMUIntegration +(a) +3 +East[m] +2 +North [m] +2 +0 +.2 +[u] +0 +-5 +-10 +20 +40 +60 +80 +100 +GPStimesince524045s +Epoch[s]51.0807 +SDR:STL, POS:RTK/IMU Integration +SDR:RTK/INS-based VDFLL, POS:RTK/IMU Integration +O +SDR:RTK/INS-based VDFPLL, POS:RTK/IMU Integration +SDR:Ground-truth-based VDFPLL, POS:RTK/IMU Integration +★ +51.08065 +ground truth +llegiatePINW +51.0806 +Datafrom524055sto524160s +53.08055 +51.0805 +5 m +-114.1345 +114.1344 +114.1343 +114.1342 +24.334 +Longitude [degree0.8 +0.6 +0.4 +0.2 +2 +30.8 +Probabil +0.6 +0.4 +0.2 +- +.6A Preprint +Figure 15: Error curves of the TOA residuals for the GPS satellites PRN6 and PRN3 in the semi-open-sky +situation. +Figure 16: TOA curve references (regarding the error curves of the TOA residuals) derived from the ground- +truth-based VDFPLL SDR. (a) TOA reference from PRN22 for the open-sky experiment (b) TOA reference +from PRN3 for the semi-open-sky experiment. +22 + +PRN6 +800 +700 +SDR:STL +SDR:RTK-based VDFLL +600 +SDR:RTK/INS-baSedVDFLL +[] +SDR:RTK-basedVDFPLL +500 +SDR:RTK/INS-baSedVDEPLL +TOA +Residualsof +400 +300 +200 +100 +0 +Timesince524042s +-100 +0 +20 +40 +60 +80 +100 +Time [s]PRN6 +40 +Error of Residuals of TOA [m] +20 +20 +40 +60 +Timesince524042s +-80 +0 +20 +40 +60 +80 +100 +Time [s]PRN3 +900 +800 +700 +Residuals of TOA [m] +600 +500 +400 +300 +200 +100 +0 +Timesince524042s +-100 +0 +20 +40 +60 +80 +100 +Time [s]PRN3 +- +Error of Residuals of TOA [m] +-10 +20 +30 +40 +-50 +Timesince524042s +-60 +0 +20 +40 +60 +80 +100 +Time [s]PRN22 +Error of Residuals of TOA (reference) [m] +-10 +20 +30 +Timesince522536s +40 +50 +-60 +0 +20 +40 +60 +80 +100 +Time [s]PRN3 +Error of Residuals of TOA (reference) [m] +-10 +20 +Timesince524042s +30 +40 +50 +-60 +0 +20 +40 +60 +80 +100 +Time [s]A Preprint +Table 2: RMSEs of the TOA estimates from the active satellites regarding the two stationary experiments +where the reference curves for the error of residuals of TOA refer to Figure 16 (“OS” and” SOS” correspond +to “open-sky” and “semi-open-sky” testing situations, respectively; “Averaging C/N0” is estimated from the +“Ground-truth-based VDFPLL” SDR). +PRN +numbers +Elevation +angle [◦] +Averaging +C/N0 +[dB-Hz] +PRN +numbers +of the +TOA error +reference +RMSE for the error of residuals of TOA [m] +STL +RTK +VDFLL +RTK/INS +VDFLL +RTK +VDFPLL +Proposed +RTK/INS +VDFPLL +SOS-25 +8.2 +43.4 +SOS-03 +10.92 +9.03 +9.12 +14.17 +11.85 +OS-32 +13.9 +45.2 +OS-22 +2.62 +4.21 +4.17 +1.80 +1.80 +SOS-06 +15.0 +39.6 +SOS-03 +14.49 +13.31 +13.24 +7.18 +9.05 +OS-25 +15.2 +44.8 +OS-22 +13.23 +12.12 +12.16 +14.08 +13.22 +SOS-16 +17.4 +43.5 +SOS-03 +3.89 +9.39 +9.22 +9.81 +8.08 +OS-01 +24.1 +37.4 +OS-22 +8.31 +9.18 +9.19 +4.80 +4.61 +SOS-14 +26.2 +44.2 +SOS-03 +15.18 +16.92 +16.89 +12.87 +13.67 +OS-26 +26.5 +41.0 +OS-22 +19.45 +24.68 +24.54 +18.96 +18.97 +OS-23 +32.3 +47.4 +OS-22 +19.23 +23.96 +23.91 +20.07 +19.24 +SOS-26 +36.0 +46.8 +SOS-03 +3.47 +7.66 +7.51 +6.22 +5.65 +OS-14 +36.2 +46.9 +OS-22 +4.78 +6.19 +6.17 +4.31 +4.46 +SOS-23 +42.2 +45.6 +SOS-03 +8.30 +5.06 +5.10 +2.75 +3.23 +SOS-31 +49.0 +51.4 +SOS-03 +12.64 +11.34 +11.38 +14.12 +14.17 +OS-31 +59.3 +52.9 +OS-22 +2.16 +2.49 +2.49 +1.63 +1.67 +SOS-22 +61.6 +48.2 +SOS-03 +2.54 +2.57 +2.58 +3.58 +3.54 +OS-03 +65.4 +49.8 +OS-22 +7.63 +10.23 +10.18 +7.87 +7.52 +Figure 17: Comparison of TOA accuracy improvements varying with the satellite elevation angles and the +TOA errors (positive and negative values represent the improved and reduced performance percentages, +respectively). +23 + +100 +of TOA Accuracy (RMSE) [%] +50 +50 +.100 +150 +SDR:RTK-based VDFLL +SDR:RTK/INS-based VDFLL +SDR:RTK-based VDFPLL +SDR:RTK/INS-based VDFPLL +200 +0 +10 +20 +30 +40 +50 +60 +70 +Satellite Elevation Angle [Degrees]100 +of TOA Accuracy (RMSE) [%] +50 +50 +100 +mproement +-150 +200 +2 +4 +6 +8 +10 +12 +14 +16 +18 +20 +STL sR TOA RMsE [mA Preprint +Figure 18: APA code phase errors from the proposed RTK/INS-based VDFPLL SDR where the numbers +correspond to the satellite PRN numbers and dashed black lines correspond to the estimates from the +traditional scalar and vector tracking algorithms (a) open sky (b) semi-open sky. +1. The proposed RTK/INS APA vector tracking has improved the multipath mitigation performance of +the GNSS baseband in static situations compared to the traditional scalar/vector tracking and the +RTK-aided-only APA vector tracking; +2. The deeply integrated INS in the proposed high-accuracy APA GPS SDR has enhanced the TOA +estimation accuracy more significantly regarding the satellites with low elevation angles; +3. The technique regarding the tested low-cost IMU deeply integrated into the RTK-position-aided +vector GPS has proved to be inferior in improving the vertical positioning accuracy but can efficiently +increase the horizontal positioning accuracy in challenging static environments. +Our future work will focus on base-station-free APA GNSS tracking upon INS DR and precise point +positioning (PPP) technique. +References +H. Sharma, C. A. Lichtenberger, and T. Pany. 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ISSN 19324553. doi:10.1109/JSTSP.2009.2023350. +27 + diff --git a/_tAyT4oBgHgl3EQfdveI/content/tmp_files/load_file.txt b/_tAyT4oBgHgl3EQfdveI/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f213b19a38aa03f7c0e393d7db1c58282e02ffe4 --- /dev/null +++ b/_tAyT4oBgHgl3EQfdveI/content/tmp_files/load_file.txt @@ -0,0 +1,1478 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf,len=1477 +page_content='High-Accuracy Absolute-Position-Aided Code Phase Tracking Based on RTK/INS Deep Integration in Challenging Static Scenarios A Preprint Yiran Luo Department of Geomatics Engineering University of Calgary Calgary T2N1N4, Canada yiran.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='luo@ucalgary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='ca Li-Ta Hsu Department of Aeronautical and Aviation Engineering The Hong Kong Polytechnic University Hung Hom, Hong Kong SAR, China lt.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='hsu@polyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='hk Yang Jiang Department of Geomatics Engineering University of Calgary Calgary T2N1N4, Canada yang.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='jiang1@ucalgary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='ca Baoyu Liu Department of Geomatics Engineering University of Calgary Calgary T2N1N4, Canada baoyu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='liu@ucalgary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='ca Zhetao Zhang School of Earth Sciences and Engineering Hohai University Nanjing 211100, China ztzhang@hhu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='cn Yan Xiang School of Electronic Information and Electrical Engineering Shanghai Jiao Tong University Shanghai 200240, China yan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='xiang@sjtu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='cn Naser El-Sheimy Department of Geomatics Engineering University of Calgary Calgary T2N1N4, Canada elsheimy@ucalgary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='ca January 3, 2023 Abstract Many multi-sensor navigation systems urgently demand accurate positioning initialization from global navigation satellite systems (GNSSs) in challenging static scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' How- ever, ground blockages against line-of-sight (LOS) signal reception make it difficult for GNSS users.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Steering local codes in GNSS basebands is a desiring way to correct instanta- neous signal phase misalignment, efficiently gathering useful signal power and increasing positioning accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Besides, inertial navigation systems (INSs) have been used as a well- complementary dead reckoning (DR) sensor for GNSS receivers in kinematic scenarios resisting various interferences since early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' But little work focuses on the case of whether the INS can improve GNSS receivers in static scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Thus, this paper proposes an enhanced navigation system deeply integrated with low-cost INS solutions and GNSS high-accuracy carrier-based positioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' First, an absolute code phase is predicted from base station information, and integrated solution of the INS DR and real-time kinematic (RTK) results through an extended Kalman filter (EKF).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Then, a numerically controlled oscillator (NCO) leverages the predicted code phase to improve the alignment between instantaneous local arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='00308v1 [eess.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='SP] 31 Dec 2022 A Preprint code phases and received ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The proposed algorithm is realized in a vector-tracking GNSS software-defined radio (SDR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Real-world experiments demonstrate the proposed SDR regarding estimating time-of-arrival (TOA) and positioning accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Keywords GNSS baseband · code phase domain · vector tracking · vector receiver · positioning · float RTK · multipath mitigation · deep integration · low-cost IMU 1 Introduction Demands for GNSS devices will keep increasing over the following decades due to the explosion of smartphone-based navigation Sharma et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2020] and intelligent transportation construction Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Therefore, realizing accurate positioning in challenging environments using global navigation satel- lite system (GNSS) devices is a hot debate these days.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' However, current GNSS receiver techniques are uneasy to achieve a next-generation positioning, navigation, and timing (PNT) performance due to the intrinsic mechanism of GNSS electromagnetic waveforms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' For instance, unlike LTE/5G wireless communication signals with an orthogonal frequency division multiple access (OFDMA) and substantial transmission power Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2014], Cimini [1985], the GNSS signals are transmitted at the same frequency.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Differently, each signal channel is divided through the code division multiple access (CDMA).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The GNSS signals are also confronted with severe channel fading over a long-distance transmission (approximately 20,000 km for the GNSS satellites operated in the medium Earth orbit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The former is naturally immune to the multipath effect, while the GNSS signals are less capable of resisting these interferences in the transmission.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Due to this, recent research proposed a hybrid optical–wireless network that achieves a decimeter-level terrestrial positioning and sub-nanosecond timing aiming to enact as a supplement or even substitute the GNSS device in the future commercial market Koelemeij et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Against this background, there are many remaining problems and vast space regarding new GNSS receiver design, especially in challenging cases, and it is urgent to embark on renewing the current commercial receiver architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The CDMA signals are sensitive to the non-line-of-sight (NLOS) ray within one chip range causing big issues in channel estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' In recent years, super-resolution algorithms (SRAs) emerged in GNSS signal processing to separate line-of-sight (LOS) and NLOS signals into different orthogonal spaces Krasner and McBurney [2022], Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2021a], Da Rosa Zanatta et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Recent work presented a graph Fourier transform (GFT) filter denoising the complex correlator outputs to replace the old GNSS tracking loop, which can be considered as a direct way to steer the code phase in challenging cases Luo and El-Sheimy [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Except for the separation using the state-of-the-art SRAs or the GNSS antenna changes Suzuki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2020], Hong et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2020], Daneshmand et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2013], modeling the superposition signals formed with LOS and NLOS rays is mainstream in the current GNSS community to overcome multipath interference Lau and Cross [2007], Yan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022], Smolyakov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Early in the 1990s, research revealed the prominence of GNSS products to overcome the predicament of tracking accurate Doppler frequency between the users’ end and the satellite——vector delay lock loop (VDLL) Spilker Jr [1996].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The vector approach is an intelligent choice to model the LOS Doppler frequency into a more proper shape.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The basic idea of this technique is to leverage the user’s navigation estimates to predict the Doppler frequency as compensation for the time of arrival (TOA) estimation in the GNSS baseband processing, stepping into a more accurate single-point positioning (SPP) solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The VDLL allows GNSS researchers to optimize baseband signal modeling with information from multiple channels instead of a conservative loop filter algorithm in a single channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Later, this idea was extended to assist the carrier phase modeling, for which it was nominated as a vector phase lock loop (VPLL) Zhodzishsky et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [1998].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Improved versions and more specific experimental results based on the VPLL techniques have been presented over the following years Henkel et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2009], Shafaati et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2018].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' However, the stability and convergence of the VPLL are vulnerable to being destroyed when the biased error in the code phase modeling cannot be well removed (meaning that a multipath effect interferes with the GNSS baseband and its high-precision navigation solutions).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' It can be explained that the GNSS carrier and code signals are synchronized and significantly interact with each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The wireless communication theory indicates that the fast fading Satyanarayana et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2012], such as the multipath effect on carrier signals, causes a modeling error of a maximum of approximately one-fourth of a signal wavelength, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', about 5 cm for global positioning system (GPS) L1 signals Kelly and Braasch [2001].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' This value is much smaller than the GNSS code phase error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' For example, a typical value of the multipath effect on the code signal is commonly at the meter level, which is two orders of the magnitude of the carrier phase VAN NEE [1992].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Partially resorting to this mentioning, carrier-aiding is always used in 2 A Preprint a traditional GNSS baseband to support the code signal estimation Kaplan and Hegarty [2017].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Similarly, a vector delay/frequency lock loop (VDFLL) was also presented to enhance the code phase tracking by combining the carrier-aiding and the VDLL approaches Lashley [2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The conventional vector tracking techniques impose an indirect approach to improving the TOA modeling over the code tracking process inside a GNSS receiver.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The vector tracking has the potential to enable the baseband to yield a more accurate code Doppler frequency production.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Then, the improved code Doppler replicates instantaneous code phases (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', TOA modeling) more precisely, contributing to a higher-quality navigation solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' This type of vector receiver can ultimately alleviate the harmful interference to the LOS signal estimation, especially when the user’s end is moving DIetmayer et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2020].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Nevertheless, little efficiency will be available in the traditional vector tracking loop once the received LOS and NLOS rays cannot be well discriminated in terms of the Doppler frequency feature (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', the frequency or Fourier domain).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Unfortunately, this case often occurs to a GNSS user today.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The authors’ recent research presented a method to optimize the TOA model in the GNSS tracking process assisted by an absolute position solution, not simply relying on the code frequency error between the baseband signal replica and the mapped code signal prediction from the user’s navigation solution Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' More specifically, we take advantage of the high-accuracy positioning solution (more accurate than the code-based-only positioning result) to improve the TOA modeling aided by the vector tracking technique in the code phase domain instead of the code frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' By coincidence, a recent paper using the classic SRA, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', root MUSIC, was also published aiming to achieve an analogous goal as stated before Krasner and McBurney [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' An inertial navigation system (INS) can resist the high-frequency random noise in the user’s navigation as it works upon autonomous dead reckoning (DR), not relying on external information Groves [2013].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Hence, integrating INS has been a prevalent way to rug the GNSS-based navigation system since early.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' To date, it has attracted much attention in the navigation field to what extent the low-cost INS increases the GNSS-based navigation Harke and O’Keefe [2022], Zhang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' However, research space remains in terms of the influence of the low-cost INS on GNSS baseband signal processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Regarding these discussions, this work proposes an improved version of the authors’ previous research Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022a, 2021b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' More specifically, a low-cost inertial measurement unit (IMU) is deeply integrated into a GNSS vector delay/frequency/phase lock loop (VDFPLL) software-defined radio (SDR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Meanwhile, an extended Kalman filter (EKF) is used to fuse float-RTK solutions and INS DR results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Hence, compared to our previous research Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022a], the main contributions of this work include the following: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' A low-cost IMU is combined with the float-RTK solutions via an EKF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' integrated navigation solutions are used to improve the GNSS code phase domain TOA modeling directly;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' An RTK/INS VDFPLL SDR is proposed and developed where the integration of RTK solutions and INS dead reckoning results, traditional scalar tracking loop (STL), VDFLL, and VDFPLL are realized and combined;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' An approach showing how the INS enhances the GNSS baseband in a static scenario is presented based on real-world experiments, which few previous research discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' A diagram explaining the difference between the proposed and traditional algorithms towards the code phase estimation in the GNSS baseband is provided in Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Before explaining Figure 1, it is worthwhile to emphasize that the instantaneous code phase error consists of two primary parts regarding standard GNSS baseband processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' They are an absolute one from the initial code phase error and a relative one caused by the Doppler frequency error in which the received signal subtracts the local replica.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' These lead to the following discussions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' At first, the conventional STL causes an apparent code Doppler frequency error and initial code phase error (see Figure 1(a)) Kaplan and Hegarty [2017];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' then, the traditional vector tracking compensates for parts of the Doppler frequency error reducing the relative code phase error (see Figure 1(b)) Spilker Jr [1996], Lashley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2021];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' next, when the vector tracking technique is further aided with an IMU sensor, the GNSS baseband becomes more capable of alleviating the frequency error Lashley and Bevly [2013], but the initial code phase error is remained (see Figure 1(c));' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' after that, when the RTK-based absolute-position-aided (APA) technique is involved in tracking, the initial code phase error can be reduced (see Figure 1(d)) Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022a];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' finally, this work proposes a deep integration method of INS and GNSS RTK processing to correct a more absolute code phase error in the local replica (see Figure 1(e)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 3 A Preprint Figure 1: Diagrammatic sketch reflecting the actual and locally replicated code signals varying with the time regarding different algorithms where colored curves correspond to the locally replicated code signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' (a) code phase misalignment is caused by the code frequency error and the initial (absolute) code phase error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' (b) moderate frequency error is reduced (c) significant frequency error is reduced (d) significant frequency error and moderate initial code phase error are reduced (e) significant frequency error and significant initial code phase error are reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5 Timestamp 11 Replicated signal based on STL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5 Replicated signal 1 based on standalone traditional VT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5 Replicated signal 01 based on RTK/IMU traditional VT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5 Replicated signal based on RTK VDFPLL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5 1 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5 Replicated signal 1 based on proposed RTK/IMU VDFPLL 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5A Preprint Figure 2 further depicts the code phase errors at the timestamp (see the dashed red lines in Figure 1) in the tracking process regarding the RTK-only APA and RTK/INS APA techniques.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' It is worth emphasizing that the timestamp denotes the local clock count to get the TOA estimation in the GNSS baseband (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', the time to extract the instantaneous GNSS measurements).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Compared to our previous work Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022a], the proposed algorithm can improve the code phase estimation by removing the initial code phase error related to the multipath/NLOS effect and the carrier cycle slip (because of the involvement of the RTK-based APA technique).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' However, the case always occurs in the real world: for example, the multipath interference in a static GNSS user’s receiver will cause such an absolute code phase error issue which is challenging in the current GNSS community.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Therefore, this research comes up with a method to solve it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Figure 2: Comparison of the code phase error at the timestamp (see the dashed red line in Figure 4) for extracting the instantaneous GNSS measurements (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', pseudoranges and carrier phases).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The remaining of this paper is organized as follows: Section 2 introduces the methodology where the proposed VDFPLL based on RTK/INS deep integration is discussed in detail;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' before that, the RTK-position- aided VDFPLL is briefly introduced;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' then, two real-world stationary experiments are provided, and discussed in Section 3;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' finally, Section 4 concludes this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 2 Materials and Methods This section will investigate how the APA code phase tracking in a GNSS baseband is realized with the proposed VDFPLL (deeply integrated with the float-RTK positioning and the INS DR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' We will first recap the VDFPLL based on standalone GNSS RTK solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' then, its improved form, deeply integrating the INS DR navigation solutions, will be discussed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Finally, how the proposed RTK/INS VDFPLL are combined with the STL and the VDFLL in a GPS SDR will be elaborated on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1 RTK-position-aided VDFPLL As mentioned earlier, the VDFPLL provides a way to directly steer the local code replica with the user’s absolute position in the code phase domain instead of the conventional code frequency domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Our previous work achieved this goal by presenting a practical means in the baseband that applies the user’s RTK solution as a source of high-accuracy code phase prediction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' This technique will be briefly stated in the following for the integrity of this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 5 Codephaseerror(differencebetweenthelocallyreplicated signal andthereceived signal)at thetimestamp 元/2 Improvedcode phase error (dueto multipath/NLOS errorandcarrier cycle slip) 0 Codephase Codephaseerror errorbased basedonRTK onthe VDFPLL RTKIIMU (Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022a]) VDFPLL (proposed) 3元/2A Preprint The architecture of the RTK-position-aided VDFPLL is illustrated in Figure 3, where APA and RPA corre- spond to absolute- and relative-position-aided, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' It is worth noting that the RPA is achieved with the traditional VDFLL technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Besides, the absolute code phase is also tracked aided by the vector tracking technique in the code phase domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' In this case, the following discriminates the entire code phase error ∆ ˆτi r,k = ∆ ˆτi,(S) r,k + ∆ ˆτi,(RT K) r,k (1) with ∆ ˆτi,(RT K) r,k = fc c � ˜ρi r,k−1 − ˆρi,(RT K) r,k−1 � (2) ˆρi,(RTK) r,k−1 = ˆri,(RTK) r,k−1 + � ˆBr,ρ,t,k−1 + ˆBi ρ,sys,k−1 � − κD ˆBr,mp,k−1 ˆri,(RT K) r,k−1 = ����ˆpi k−1 − ˆp(RT K) r,k−1 ���� where subscript k denotes the index of tracking epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ∆ ˆτi,(S) r,k is the traditional discriminated code phase error through an early-minus-late-envelope code discriminator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ∆ ˆτi,(RT K) r,k is the code phase error obtained from the APA approach;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ˜ρi r,k−1 and ˆρi,(RTK) r,k−1 are the pseudoranges measured from the code tracking filter and predicted from the float RTK solution, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ˆri,(RT K) r,k−1 is the predicted geometry distance;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ˆpi k−1 is the vector of satellite position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ˆp(RTK) r,k−1 is the vector of the estimated float RTK position;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' � ˆBr,ρ,t,k−1 + ˆBi ρ,sys,k−1 � is the summation of the local clock bias error estimation and systematic error estimation, and it is computed from base station information and master satellite measurements Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022a];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ˆBr,mp,k−1 is the estimated multipath delay error imposed on the absolute code phase error via a between-satellite single difference algorithm, and κD is its tuned coefficient constant based on the involved early-late spacing Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2021b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Next, the work process of the RTK-position-aided VDFPLL in the GPS SDR within the same tracking epoch is stated as follows: Step 1: the SDR receives the incoming intermediate frequency (IF) GPS L1 C/A data via front-end equipment;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Step 2: integration and dumping (I&D) procedures upon correlators are implemented between the local code replica and the incoming IF GPS signals;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Step 3: the correlator output passing through the traditional code discriminator yields ∆ ˆτ(S) r,k ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Step 4: the bias of the discriminated code error compensated by the RTK-position aided (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', the APA operation) code error estimation ∆ ˆτi,(RTK) r,k gives ∆ ˆτi r,k;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Step 5: a code tracking loop filter denoises the code phase error from Step 4;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Step 6: an RPA technique (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', the VDFLL) is executed to alleviate the code frequency error in the code- tracking process;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Step 7: a numerically controlled oscillator (NCO) leverages the output of Step 6 to produce the TOA estimation (the raw output of the code loop filter aided by the Doppler prediction), the pseudorange measurements (be de-noised by the carrier smoothing technique);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Step 8: the RTK engine leverages the pseudoranges and carrier phases (the raw output of the carrier loop filter aided by the Doppler prediction) from all the tracking channels, navigation data, and the base station information to compute the float RTK solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Step 9: the APA code phase error ∆ ˆτi,(RTK) r,k is computed with the float-RTK solutions and the pseudorange measurement by (2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Step 10: repeating Step 2, the RTK-position-aided VDFPLL is working for the next tracking epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' To conclude, the work process of the VDFPLL based on the float-RTK solutions executed in a GNSS SDR has been browsed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 6 A Preprint Figure 3: Overview of the GNSS baseband architecture with the RTK-position-aided VDFPLL Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022a, 2021b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2 The proposed VDFPLL based on RTK/INS deep integration 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1 Architectures of the proposed VDFPLL SDR The architectures of the proposed APA VDFPLL GPS SDR deeply integrated with the float RTK solutions and INS dead reckoning results are displayed in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' It is worthwhile to mention that hybrid tracking loops are adopted here due to the data rates discrepancy corresponded to various sources, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', the proposed SDR tracking, the base station, and the IMU sensor raw data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' First, there are two procedures for updating the code tracking loop with the APA method in the SDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' On the one hand, the TOA model is predicted from the integrated float RTK/INS EKF, as depicted in Figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' On the other hand, the proposed VDFPLL updating rate is 5 Hz, that is higher than the RTK solution rate, so the INS DR navigation solutions (with a rate of 50 Hz) are interpolated in the updating process when the RTK solutions (with a rate of 1 Hz) are absent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' It is worth noting that we take two samples of the IMU raw data per update to compensate for coning and sculling errors, so the raw data rate of the IMU is 100 Hz, while the DR navigation results are at the rate of 50 Hz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Finally, as the tracking loop updating rate in the proposed SDR baseband is 200 Hz, much higher than the VDFPLL rate, the traditional STLs for code and carrier tracking are interpolated across the intervals where the VDFPLL is not activated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' As a result, the three tracking loops are jointly working in the proposed RTK/INS-based VDFPLL SDR, and they are the VDFPLL based on the RTK/INS integrated EKF (see Figure 4(a)), the VDFPLL based on the INS DR (see Figure 4(b)), and the traditional STL (see Figure 4(c)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 7 Channel 1 Ephemeris B Channel L p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='t,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k-1 Clock error BL p,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='sys,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k-1 Systematic Predicted APA code error phase error Multipath error APA Base station coming r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k RTK Code signals engine tracking 4tL r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k loop filter I&D At(s) r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k Local Code Base discriminator carrier station RPA replica Local code Code NCO replica APA VDFPLLA Preprint Figure 4: Architectures of the proposed VDFPLL-enhanced GPS SDR based on the deep integration of float RTK solutions and INS DR navigation results (detailed discussions refer to the Algorithm 1 stated later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=') 8 Base station Channel 1 1 Hz Channel L Pseudo Incoming APA ranges Integrated RTK signals VDFPLL EKF engine (Figure 3) navigator rier aiding Hz /DFLL) Carrier -- 1Hz Carrier phases NLS tracking velocity loop estimator Low-cost 100 Hz 50 Hz IMU raw Mechanization dataChannel 1 5 Hz Channel L Pseudo Incoming APA ranges signals VDFPLL (Figure 3) rier aiding C 5 Hz /DFLL) p Carrier D z tracking loop Str,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content="k-1 Pik'-1 or 5 Hz PG/I|DR," metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content="'-1 Low-cost 100 Hz IMU raw Mechanization dataChannel 1 Channel L 200 Hz Pseudo Incoming Code ranges signals tracking loop Carrier C 200 Hz aiding p Carrier D tracking loopA Preprint 2." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2 RTK/INS EKF Navigator and INS DR As illustrated in Figure 4, there are three types of TOA estimation formed in the proposed GNSS SDR corresponding to Figures 4(a), 4(b), and 4(c), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' They will be elaborated on one by one in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' At first, one is formed using the absolute position estimated from the integrated RTK/INS EKF navigator (see Figure 4(a)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The RTK engine comes from an open-source package goGPS v0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='3 Herrera et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2016].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Then, the float RTK deeply integrated into the proposed SDR has been introduced in the authors’ previous work Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Then, the SDR platform to realize the deep integration of RTK and INS has also been built and investigated in the authors’ previous publications Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2019a, 2021c].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Besides, we will discuss the EKF algorithm used in this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The state transition equation is given by δxe k′′ = Φe k′′,k′′−1δxe k′′−1 + we k′′−1 with k′′ = � k ′/K � k′ = ⌊k/M⌋ where superscript e represents the Earth-centered, Earth-fixed (ECEF) coordinate frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' subscript k′′ denotes the epoch index of the EKF updates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' K is the integer ratio of the INS DR and the EKF updating rates, where the former and the latter are 50 Hz and 1 Hz (constrained by the rate of the base station information), respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' k′ is the epoch index of INS DR solutions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' M is the integer ratio of the GNSS tracking rate and the INS updating rate where the tracking rate (200 Hz) is no lower than the INS updating rate (50 Hz) here;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' so it satisfies K,M ∈ Z+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Φe k′′,k′′−1 is the transition matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' we k′′−1 is the process noise vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Then, the state vector of the EKF model in the ECEF frame is given by δxe k′′ = � (δψe)T ,(δve)T ,(δpe)T , � δbg �T ,(δba)T �T k′′ where δψe is the attitude error vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' δve is the 3D velocity error vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' δpe is the 3D position error vector;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' δbg and δba are the respective gyro and accelerometer bias error vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Next, the observation equation is provided as δzk′′ = He k′′δxe k′′ + vk′′ where He k′′ is the observation matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' vk′′ is the observation noise vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The observation vector, including position errors and velocity errors, is provided as δzk′′ = � �������� � ������� ˜˙xI − ˜˙xG,NLS ˜˙yI − ˜˙yG,NLS ˜˙zI − ˜˙zG,NLS � ������� T , � ������ ˜xI − ˜xG,RT K ˜yI − ˜yG,RT K ˜zI − ˜zG,RT K � ������ T � �������� k′′ T where [ ˜x, ˜y, ˜z] and � ˜˙x, ˜˙y, ˜˙z � correspond to the 3D positions and velocities w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' the ECEF frame, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' subscript I and G correspond to the solutions obtained from the INS and the GNSS, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' the subscript “RTK” means that the GNSS position results are solved by the float RTK algorithm Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022a], and the subscript “NLS” represents that the GNSS velocity results are calculated from the standard non-linear squared (NLS) method Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2019b].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' How to build Φe k′′ ,k′′−1 and He k′′, as well as how to form the process noise covariance matrix and the observation noise covariance matrix can refer to Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2019a].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' After the system model is built, the recursive estimation of the EKF algorithm is to predict and update the state vector Faragher [2012].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Finally, upon the time epoch where base station information is available for the RTK algorithm, the navigation solutions corresponding to the velocity, position, and attitude information from the EKF algorithm are given by ˆCe G/I,b,k′′−1 ≈ � I3 − � δ ˆψe k′′−1 � × � ˆCe I,b,k′−1 ˆve G/I,k′′−1 = ˆve I,k′−1 − δˆve k′′−1 ˆpe G/I,k′′−1 = ˆpe I,k′−1 − δ ˆpe k′′−1 9 A Preprint where (·)× denotes the skew matrix operator;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' I3 is the 3-order identity matrix;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' δˆve k′′−1 δ ˆpe k′′−1, and δ ˆψe k′′−1 are the estimated state vectors about velocity and position errors, and attitude errors, and ˆve G/I,k′′−1, ˆpe G/I,k′′−1 and ˆCe G/I,k′′−1 are the estimated navigation vectors (corresponding to the respective velocity, position, and attitude);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ˆve I,k′−1, ˆpe I,k′−1 and ˆCe I,b,k′−1 are the counterparts solely upon the INS DR process, which will be introduced subsequently.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Within the epochs where the base station information is missing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' the EKF-based results at the previous epoch can contribute to the INS DR process at the current epoch as the following ˆve G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ = ˆve G/I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′′−1 + � C e G/I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′′−1˜fb ib,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 − 2(ωie) × ˆve G/I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′′−1 + ge � ˆpe G/I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′′−1 �� ∆tI ˆpe G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ = ˆpe G/I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′′−1 + � ˆve G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ + ˆve G/I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′′−1 � ∆tI 2 ˆCe G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ = Ce i (∆tI) ˆCe G/I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′′−1 � I3 + � ˜ωb ib,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1∆tI � × � with ωie = � 0 0 ωie �T Ce i (∆tI) = � ������ cos(ωie∆tI) sin(ωie∆tI) 0 −sin(ωie∆tI) cos(ωie∆tI) 0 0 0 1 � ������ where ∆tI is the updating interval of the EKF;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ˜fb ib,k′−1 and ˜ωb ib,k′−1 are the specific force and angular rate measurement vectors of the body frame w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ECEF frame;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ωie is the Earth rotation rate, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='292115e-5 rad/s and ωie is its vector form;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ge � ˆpe G/I,k′′−1 � is the gravity acceleration vector function in the ECEF frame varying with the user’s position ˆpe G/I ,k′′−1 (see Equations (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='133) and (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='142) in Groves [2013]);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Ce i (∆tI) is the Earth rotation matrix from the Earth-centered inertial (ECI) to the ECEF coordinate frame varying with the updating interval ∆tI;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' C e G/I,b,k′′−1 is the averaging transformation matrix w.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='r.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='t.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' the body-to-ECEF-frame coordinate obtained from ˆCe G/I,b,k′′−1 (see Equations (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='84) and (5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='85) in Groves [2013]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Then,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' considering the case where the navigating solutions are derived from the INS DR process (see Figure 4(b)),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' the mechanization in the ECEF frame can be expressed as ˆve I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ = ˆve G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 + � C e I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1˜fb ib,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 − 2(ωie) × ˆve G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 + ge � ˆpe G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 �� ∆tI ˆpe I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ = ˆpe G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 + � ˆve I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ + ˆve G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 � ∆tI 2 ˆCe I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ = Ce i (∆tI) ˆCe G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 � I3 + � ˜ωb ib,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1∆tI � × � or ˆve I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ = ˆve I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 + � C e I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1˜fb ib,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 − 2(ωie) × ˆve I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 + ge � ˆpe I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 �� ∆tI ˆpe I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ = ˆpe I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 + � ˆve I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ + ˆve I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 � ∆tI 2 ˆCe I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ = Ce i (∆tI) ˆCe I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 � I3 + � ˜ωb ib,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1∆tI � × � where C e I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 is the averaging transformation matrix computed from ˆCe I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='b,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' It is worth mentioning that the tracking rate (200 Hz) is higher than the INS DR updating rate (50 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' So, three out of four tracking intervals do not have an update for the INS DR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Assuming that the user’s navigation results are not changed significantly over the time 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='02s (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', 1 50Hz), when k′ = k/M, we make an approximation that the navigation estimations in the following (k + 1)th, (k + 2)th, and (k + 3)th tracking epochs are identical to the ones computed at the kth, to interpolate the tracking epochs without the INS updating.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 10 A Preprint 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='3 RTK/INS APA Code Phase Tracking The baseband TOA modeling at the start of the kth epoch aided by the absolute positions from the integrated RTK/INS EKF and the INS DR is estimated through � TOA i k = c−1 ˜ρi k − δˆtr,k (3) with the pseudorange model of ˜ρi k ≜ ˜ρi k−1 + cf −1 c �� fc + ˆf i code,dop,k � Tcoh + ∆ ˆτi+,(RT K/INS) code,k,0 � (4) where ˜ρi k−1 and ˜ρi k are the instantaneous pseudorange measurements at the respective previous and current epochs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' δˆtr,k is the estimated local clock bias error;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' fc and c are the spreading code rate and the speed of light, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Tcoh is the coherent integration time ˆf i code,dop,k is the estimated code Doppler frequency and ∆ ˆτi+,(RTK/INS) code,k,0 is the proposed initial code phase error estimate in chips at the start of the kth epoch, and the estimation processes of them will be subsequently elaborated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' On the one hand,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ˆf i code,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='dop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k can be written as ˆf i code,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='dop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k = −fc fr ˜fcarr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='dop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k + T −1 coh∆ ˆτi+ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k (5) ˜f i carr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='dop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k = ∆ ˆf i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='(aid) carr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k + T −1 coh∆ ˆϕi+ r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k (6) with ∆ ˆf i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='(aid) carr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k =fr c � ˆve G/I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′′−1 · ˆei k � ˆpe G/I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′′−1 � − ˆvi k · ˆei k � ˆpe G/I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′′−1 � + cδˆ˙tr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k − cδˆ˙t i k � (7) or ∆ ˆf i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='(aid) carr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k =fr c � ˆve G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 · ˆei k � ˆpe G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 � − ˆvi k · ˆei k � ˆpe G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 � + cδˆ˙tr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k − cδˆ˙t i k � (8) ∆ ˆf i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='(aid) carr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k =fr c � ˆve G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ · ˆei k � ˆpe G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ � − ˆvi k · ˆei k � ˆpe G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ � + cδˆ˙tr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k − cδˆ˙t i k � (9) ∆ ˆf i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='(aid) carr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k =fr c � ˆve I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ · ˆei k � ˆpe I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ � − ˆvi k · ˆei k � ˆpe I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′ � + cδˆ˙tr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k − cδˆ˙t i k � (10) ∆ ˆf i,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='(aid) carr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k =fr c � ˆve I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 · ˆei k � ˆpe I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 � − ˆvi k · ˆei k � ˆpe I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 � + cδˆ˙tr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k − cδˆ˙t i k � (11) where ˜fcarr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='dop,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k denotes the carrier Doppler frequency measurement;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' � T −1 coh∆ ˆτi+ r,k � and � T −1 coh∆ ˆϕi+ r,k � are the filtered code phase error and the filtered carrier phase error through the loop filters, respectively, which have accounted for the coherent integration interval in tracking, and its input is ∆ ˆτi x,k which will be explained later, with x ∈ {I,G/I,G/I|DR};' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ∆ ˆf i,(aid) carr,k is the aided Doppler frequency computed via the user’s velocity estimation, known as a VDFLL technique Lashley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2021];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ˆvx,k′ / ˆvx,k′′ and δˆ˙tr,k are the predicted user’s velocity vector and the predicted user’s clock drift;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ˆvi k and δˆ˙t i k are the satellite velocity vector and the satellite clock drift predicted with the broadcast ephemeris;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ˆei k (·) is the operator of the unit cosine vector varied with the position estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' As mentioned above, ∆ ˆτi x,k is the APA discriminated code phase error, and there are three ways to obtain this estimate in the code tracking loop.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' For instance,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' the ones estimated via the respective RTK/INS EKF solution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' the two-consecutive-epoch INS DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' and INS DR right after the EKF are computed as ∆ ˆτi G/I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k = ∆ ˆτi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='(S) r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k + ∆ ˆτi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='(APA) r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k � ˆri G/I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k−1 � (12) 11 A Preprint ∆ ˆτi I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k = ∆ ˆτi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='(S) r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k + ∆ ˆτi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='(APA) r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k � ˆri I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k−1 � (13) ∆ ˆτi G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k = ∆ ˆτi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='(S) r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k + ∆ ˆτi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='(APA) r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k � ˆri G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k−1 � (14) with ˆri G/I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k−1 = ���ˆpi k−1 − ˆpe G/I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′′−1 ��� ˆri I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k−1 = ���ˆpi k−1 − ˆpe I,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 ��� ˆri G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k−1 = ���ˆpi k−1 − ˆpe G/I|DR,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k′−1 ��� where ∆ ˆτi,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='(S) r,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k is the traditional discriminated code phase error as introduced earlier;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ˆpi k−1 is the satellite position vector computed from the broadcast ephemeris;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ∆ ˆτi,(APA) x,k (·) is the operator to obtain the absolute code phase error with the geometry distance prediction (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', the APA process) and the error models, and its analytical expression is defined as ∆ ˆτi,(APA) r,k � ˆri x,k−1 � ≜ fc c � ˜ρi r,k−1 − � ˆri x,k−1 + � ˆBr,ρ,t,k−1 + ˆBi ρ,sys,k−1 � − κD ˆBr,mp,k−1 �� (15) Therefore, based on these discussions, it is easy to find that the absolute code phase error estimate ∆ ˆτi+,(RTK/INS) code,0,k in (4) (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', the difference between the received initial code phase and the counterpart of the local code replica synthesized with the NCO) can be alleviated by the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Finally, the proposed algorithm in this paper is summarized in Algorithm 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' This algorithm is realized in a GPS SDR prototype where L1 C/A signals are used to validate the TOA and position estimation performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 3 Results and Discussion The experimental equipment is set up as shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Two stationary data sets were collected in the real world to verify the proposed algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' A NovAtel antenna was used to receive the GPS L1 C/A IF signals through a Fraunhofer IIS RF frond-end, where the IF sampling rate is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='125 MHz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The IMU raw data were collected from the Crossbow Nav 440 device, where the IMU’s gyro and accelerometer bias stabilities are 10 deg/h and 1 mg, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Besides, it is worth mentioning that two samples are taken for updating the inertial sensor data for our navigation equation, so the updating rate of the INS DR is half (50 Hz) of the IMU raw data rate (100 Hz).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The reference positions of the two experiments are obtained by averaging the results provided by the Crossbow Nav440 GPS/INS integration solutions (the centers of the IMU sensor and the GNSS antenna are sufficiently close in the setup and neglected in this experiment).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The proposed algorithm is tested in a GPS SDR platform where the coherent integration time is 5 ms, the classic discriminators are chosen as the noncoherent-early-minus-late-amplitude code discriminator and Costas carrier discriminator, and the early-late spacing is four IF sample intervals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Five types of tracking algorithms are compared in the same SDR conditions except for the parameter adjustment in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' First, an open sky area is chosen to carry out the experiment where the test spot in Google Map and the sky plot of the available GPS satellites are shown in Figure 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' We first assess the SPP results for this open-sky case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The SPP is based on the weighted NLS algorithm in the tested SDR, which refers to an open-source package RTKLIB Takasu and Yasuda [2009].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Figure 7 depicts the dilution of precision (DOP) results, SPP errors, and the 3D position cumulative distribution function (CDF) curves of 3D SPP root-mean-squared errors (RMSEs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Although the results show that the proposed algorithm does not increase the SPP accuracy compared to the classic STL algorithm in the open sky, it proves that the APA-based VDFPLL algorithms enhance the traditional RPA-based VDFLL in the static situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Meanwhile, using the INS can moderately enhance the RTK-based APA tracking process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Then, the RTK position errors for the different SDR algorithms are compared in Figure 8, where the position errors, horizontal position results for a Google Map show, and the CDF curves of 3D and 2D position estimates RMSE are included.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The traditional RPA VDFLL does not help the RTK position accuracy in this open-sky and static case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The finding is that the RTK-only-based VDFPLL improves the RTK results within a 3D range while the RTK/INS-based VDFPLL slightly makes it outperform the STL-based horizontal 12 A Preprint Algorithm 1 High-accuracy APA GNSS code phase tracking based on RTK/INS deep integration Require: k∗ ≜ k mod KM,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' subject to K,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='M ∈ Z+ and k,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='k∗ ∈ N 1: while new digital IF samples (for a coherent processing interval) are received at the kth epoch do 2: Synthesize the code and carrier local replicas with the code/carrier NCOs;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 3: Produce the early- prompt- and late-branch samples through the I&D using the local replicas and the received IF samples;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 4: Discriminate the code/carrier phase errors with the outputs of the I&D (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', correlator outputs);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 5: if the base station information is available at the tracking epoch(s) {k∗ − 2M, .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=',k∗ − M − 1} then 6: Compensate for the discriminated code phase error with (14);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 7: else if the base station information is available at the tracking epoch(s) {k∗ − M,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', k∗ − 1} then 8: Compensate for the discriminated code phase error with (12);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 9: else 10: Compensate for the discriminated code phase error with (13);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 11: end if 12: Optimize the compensated code phase error from Step 6/8/10 with a 1-Hz 2nd-order loop filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 13: if the vector tracking trigger (5 Hz) is activated then 14: Optimize the discriminated carrier phase error with a 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5-Hz 1st-order loop filter;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 15: if the RTK/INS EKF is updated at the epoch(s) {k∗,k∗ − 2M,k∗ − 3M,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=',k∗ − (K − 1)M} then 16: Predict the carrier Doppler with (10);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 17: else if the RTK/INS EKF is updated at the epoch(s) {k∗ − 2M + 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=',k∗ − M − 1} then 18: Predict the carrier Doppler with (8);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 19: else if the RTK/INS EKF is updated at the epoch(s) {k∗ − M} then 20: Predict the carrier Doppler with (9);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 21: else if the RTK/INS EKF is updated at the epoch(s) {k∗ − M + 1,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='..' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=',k∗ − 1} then 22: Predict the carrier Doppler with (7);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 23: else 24: Predict the carrier Doppler with (11);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 25: end if 26: Compute the carrier frequency with (6) (for carrier NCO);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 27: else 28: Optimize and predict the carrier Doppler with a 15-Hz 3rd-order loop filter (for carrier NCO);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 29: end if 30: Compute the code frequency with (5) (for code NCO);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 31: end while Figure 5: Setup for the stationary experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 13 GNSS Satellite CrossbowNav440 NovAtel Antenna GPSSDR Fraunhofer IIS RFFront-EndA Preprint Figure 6: Open-sky test spot (Google Map show) and sky plot of available GPS satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Figure 7: Single point navigation results and statistical analysis of different SDRs in the open-sky situation where dashed lines correspond to outlier epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' (a) DOP values (b) SPP results (c) CDF curves of 3D SPP RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 14 N 30 330 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 300 30 G25 45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 60 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5 G23 G31 G03 G14 G321 G22 G01 G26 210 150SDR:STL SDR:RTK-basedVDFLL Initializationof SDR:RTK/INS-basedVDFLL theSDR (a) SDR:RTK-basedVDFPLL SDR:RTK/INS-baSedVDFPLL 3 East 2 3 North 2 2 20 40 60 80 100wW East [m] 50 North [m] 10 20 E 21 20 40 60 80 100 Epoch [s]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='8 robability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='4 n Datafrom522555s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2 to522645s 10 20 30 40 50 3D SPPRASErailNw 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0824 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0822 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='082 0 1A 1A St Davids United Church 0 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0818 ★ titude 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0816 Testspot 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0814 CrowchildT Close park 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0812 20 m 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='125 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1245 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='124 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1235 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='123 Longitude ldegreeA Preprint Table 1: Parameter settings of the SDR regarding the real-world experimental validation SDR Type Relative position/ velocity aiding RTK absolute position aiding INS deep integration RTK/INS integration navigation solution Tracking loop filter STL (traditional) Kaplan and Hegarty [2017] No No No No 1-Hz 2nd-order DLL & 18-Hz 3rd-order PLL RTK-based VDFLL Lashley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2021] Yes No No No 1-Hz 2nd-order DLL & 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5-Hz 1st-order PLL & 15-Hz 3rd-order PLL (see Algorithm 1) RTK/INS-based VDFLL Lashley et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2021] Yes Yes Yes Yes RTK-based VDFPLL Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022a] Yes Yes No No RTK/INS-based VDFPLL (proposed) Yes Yes Yes Yes Figure 8: RTK position results and statistical analysis of different SDRs in the open-sky situation where dashed lines correspond to the outlier epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' (a) RTK position errors (b) horizontal RTK results in Google Map (c) CDF curves of 3D RTK RMSE (d) CDF curves of horizontal (2D) RTK RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 15 SDR:STL SDR:RTK-basedVDFLL Initializationof (a) SDR:RTK/INS-basedVDFLL the SDR SDR:RTK-basedVDFPLL SDR:RTK/INS-baSedVDFPLL East[m] t 4 2 0 LO North [m] 5 up [m] 5 20 40 60 80 100 GPStimesince522535s Epoch[s]SDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='STL Datafrom522555s SDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='RTK-based VDFLL 31.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0819 to522645s SDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='RTKINS-based VDFLL SDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='RTK-basedVDFPLL SDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='RTKINS-basedVDFPLL ☆ Ground truth 5108185 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0818 ☆ 08173 5 m 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0817 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1238 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='123 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1236 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1285 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1234 ongtuce0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='8 Pro bability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2 0 2 80.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='8 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2 8A Preprint positioning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' It can be explained that the vertical INS DR position solution would cause more code phase errors (compared to the STL tracking in a static open-sky case) in terms of the vertical position estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Next, we compare the RTK/INS integrated results of three SDRs in Figure 9, where “Ground-truth-based VDFPLL” means that the SDR leverages the actual position coordinates instead of on-the-fly RTK or integrated RTK/INS position estimates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' It is a hardware-in-the-loop (HIL) simulation strategy to provide a reference for the proposed algorithm under the same SDR conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' In other words, the HIL simulation results represent the upper bound of the performance that the used SDR platform can achieve.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Based on the CDF curves in Figure 9(d), the proposed accuracy slightly exceeds the traditional STL-based integrated RTK/INS solution within the 78% probability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' So, the large errors can be alleviated in the integration results using the proposed algorithm in the open sky area.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' In contrast, the traditional VDFLL- based integrated RTK/INS positioning accuracy drops much in this well-condition static scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Figure 9: RTK position results and statistical analysis of different SDRs in the open-sky situation where dashed lines correspond to the outlier epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' (a) RTK position errors (b) horizontal RTK results in Google Map (c) CDF curves of 3D RTK RMSE (d) CDF curves of horizontal (2D) RTK RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' TOA curves will be evaluated in the following parts to examine the baseband processing discrepancy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The TOA estimation from the GPS SDR is computed via (3), where it is worth noting that the ˜ρi k here is the raw data from the loop filter output without the carrier-smoothing algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Therefore,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' the measurement and reference TOA residuals in meters are computed as follows (TOA measurement residual)i k ∆= ˜ρi k − c � TOAi 0 + δˆtr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0 � − c � −f i d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0f −1 r + δˆ˙tr,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0 � kTcoh (TOA reference residual)i k ∆= cTOAi k − cTOAi 0 − c � −f i d,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0f −1 r � kTcoh TOA residual error ∆= (TOA measurement residual)i k − (TOA reference residual)i k where the TOA residuals exclude the initial pseudorange and initial Doppler frequency for the simplicity of analysis;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' subscript 0 and k denote the initial and the kth epoch, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' TOAi 0 and f i d,0 represent the 16 Initialization SDR:STL,POS:RTK/IMUIntegration SDR:RTK/INS-basedVDFLL,POS:RTK/IMUIntegration of the SDR SDR:RTK/INS-basedVDFPLL,POS:RTK/IMUIntegration (a) SDR:Ground-truth-basedYDFPLL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='POS:RTK/iMUIntegratior East [m] 2 0 North [m] 2 [] dn 20 40 60 80 100 GPStimesince522535s Epoch[s]SDR:STL, POS:RTK/IMU Integration SDR:RTK/INS-based VDFLL, POS:RTK/IMU Integration 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0819 SDR:RTK/INS-based VDFPLL, POS:RTK/IMU Integration 0 SDR:Ground-truth-based VDFPLL, POS:RTK/IMU Integration ★ ground truth 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='08185 2 Datafrom522555sto522645s ep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0818 ★ 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='08175 5 m 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0817 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1238 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1237 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1236 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1235 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1234 Longitude Tdegree0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='8 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='8 Probability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2A Preprint TOA and Doppler frequency reference at the initial epoch, and they are computed as T OAi 0=c−1 ����pgt,0 − pi 0 ��� + ˆBI,0 + ˆBT ,0 − cδˆti 0 � f i d,0 = fr c � vgt,0 · ˆei 0 − ˆvi 0 · ˆei 0 − cδˆ˙t i 0 � where pgt,0 and vgt,0 are the ground truth vectors in the ECEF coordinate frame for the user’s position and velocity in the stationary experiments, with vgt,0 = [0,0,0]T ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' pi 0, vi 0, δˆti 0, and δˆ˙t i 0 are the satellite position and velocity vectors, satellite clock bias and drift errors, respectively, which are obtained and computed using the broadcast ephemeris;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' meanwhile, ˆBI,0 and ˆBT ,0 are the ionospheric error derived from the Klobuchar model and the tropospheric error calculated via the Saastamoninen model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ˆei 0 is the unit cosine vector computed from pgt,0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' δˆti 0 and δˆ˙t i 0 are the estimated user’s clock bias and drift errors computed as δˆtr,0 = averaging clock bias error − (navigation time spanning) 2 × δˆ˙tr,0 where δˆ˙tr,0 is the averaging value of the clock drift estimates from the “Ground-truth-based VDFPLL” SDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' It is worthwhile to say that the given δˆ˙t i 0 and δˆti 0 references are not sufficiently accurate, but they are satisfactory in validating the TOA performance amid different SDR algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Figure 10 shows the error curves of the TOA residuals for the signals from the satellite of PRN1 (low elevation angle) and PRN22 (high elevation angle).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' It proves that the APA-based vector tracking algorithms perform better in interference mitigation for the signals from the low satellite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' In summary, even if the proposed algorithm does not manifest much better than the traditional STL in an open-sky static environment, it demonstrates a significant improvement in comparison with the traditional RPA vector tracking in the same condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' More specifically, it can be used as boosting complement for the existing vector GNSS receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Another set of data was collected under a semi-open-sky situation where the GPS antenna was receiving the signals affected by the eastern CCIT building at the campus of the University of Calgary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The Google Map show of the test spot and the corresponding satellite sky plot are provided in Figure 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' In this test, where the results are displayed in Figure 12, we also give the DOP values to offer the satellite geometry status;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' the SPP error curves and their 3D CDF are provided as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' First, it can be observed that the two VDFPLLs and the STL produce more reliable solutions as the position outliers occur to the two VDFLLs at around the 95th epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Then, it is evident that the proposed algorithm embraces the highest SPP accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' We also assess the RTK solution accuracy related to the different SDRs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The RTK position error curves, posi- tion results in Google Map, and the 3D CDF curves are provided in Figure 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' After the SDR RTK solutions become stable, the RTK accuracy from the proposed SDR solutions still shows the highest performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The RTK-only-based APA algorithm can reduce the random noise, but it is more biased than the traditional STL algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The two RPA-only vector tracking techniques are still less capable of offering efficient assistance in the static test.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Next, the integrated RTK/INS solutions are compared in the semi-open-sky environment as shown in Figure 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' In this case, the proposed algorithm significantly improved the 2D positioning performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' At the same time, the RTK/INS-based RPA method elevates the 3D positioning accuracy compared to the STL-based integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The RPA- and APA-based integrated navigation accuracies over the error range of approximately 55% probability outperform the traditional STL one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Furthermore, the positioning results with the proposed algorithm perform more stable than the traditional vector tracking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Another finding is that RTK/INS-based APA vector tracking yields a much more ideal horizontal positioning estimate than the RPA one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' By contrast, the latter is superior to the former in the vertical direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' By comparing to the upper bound (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', the estimation from the ground-truth-based VDFPLL SDR), it can be explained that the fusion of the low-cost IMU has a side effect on the vertical position solution when it is applied to the proposed RTK/INS-based VDFPLL SDR in this stationary experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Then, the raw TOA performance in terms of the signal from the high-elevation-angle satellite (PRN3) and the one with a low elevation angle affected by the multipath interference (PRN6) are plotted in Figure 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 17 A Preprint Figure 10: Error curves of the TOA residuals for the GPS satellites PRN1 and PRN22 in the open-sky situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Figure 11: Semi-open-sky test spot (Google Map show) and the sky plot of available GPS satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='PRN1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='350 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='SDR:STL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='SDR:RTK-basedVDFLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='SDR:RTK/INS-basedVDFLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='[w] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='250 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='SDR:RTK-basedVDFPLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='SDR:RTK/INS-basedVDFPLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Residuals ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Timesince522536s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Time [s]PRN1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Error of Residuals of TOA [m] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Timesince522536s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Time [s]PRN22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Residuals of TOA [m] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Timesince522536s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Time [s]PRN22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Error of Residuals of TOA [m] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='15 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='25 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='35 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='45 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='55 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Time [s]51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0812 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='081 Blockage CCITbuilding 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0808 1 1 68 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0806 ★ tud 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0804 PIN CcIT BuIding Test spot EnergiSimulation/ 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0802 FrankandSarahMe 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='08 20 m 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1355 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='135 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1345 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='134 334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='4335 Longitude [degreeN 30 330 15 G25 G06 300 30 45 60 G31 75 G23 G14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' G03 E G22 G26 2 G16 210 150 SA Preprint Figure 12: Single point navigation results and statistical analysis of different SDRs in the semi-open-sky situation where dashed lines correspond to the outlier epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' (a) DOP values (b) SPP results (c) CDF curves of 3D SPP RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' A more significant fluctuation in the TOA curves emerges at the PRN6 in this semi-open-sky experiment compared to the open-sky PRN1 (see Figure 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Also, both APA-based tracking loops are more capable of alleviating the TOA error varying with a long-term time spanning (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', the level of dozens of seconds) than the RPA-based vector tracking and the STL.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Nevertheless, the curves computed from the high-quality signal, PRN3, are highly homogeneous regarding all the tested tracking loop algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The next part will quantitatively examine the exact improvement the proposed algorithm can offer for the GNSS baseband estimation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' As mentioned, an upper bound of the instantaneous integrated positioning performance is obtained from the SDR under the HIL test using the “Ground-truth-based VDFPLL”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Therefore, the corresponding TOA error curve representing the upper bound can also be extracted from the tracking results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Then, we compute the TOA error of the PRN22 and PRN3 (with the highest elevation angles during the experiments) in the open-sky and semi-open-sky cases as the respective references.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The proposed TOA error references reasonably model the remained local clock errors in meters varying with the time where the other biased errors, like the atmospheric delay and initial TOA errors, are assumed to be well removed by the given models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Ultimately, the TOA curve references are derived and illustrated in Figure 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' After that, the TOA accuracy of different satellites in the two testing situations will be analyzed via these references in the following.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Table 2 summarizes the statistical analysis of the TOA performances of different tracking algorithms where the RMSE results are computed for the used satellites.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Then, in regard to the traditional STL, the TOA accuracy improvements of the two RPA- and APA-based vector tracking algorithms operated in the GPS SDR are computed and depicted in Figure 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 19 SDR:STL SDR:RTK-basedVDFLL Initializationof SDR:RTK/INS-baSedVDFLL SDR:RTK-basedVDFPLL theSDR (a) SDR:RTK/INS-basedVDFPLL 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5 East 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='5 North 3 20 40 60 80 100 GPStimesince524045sEpoch[s]Initialization of the SDR 21 [m 181 20 North [m] 40 20 20 40 60 80 100 Epoch [s]0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='8 robability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='4 Datafrom524055s to524160s 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2 0 10 20 30 40 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0 3 spprmsuA Preprint Figure 13: RTK position results and statistical analysis of different SDRs in the semi-open-sky situation where dashed lines correspond to the outlier epochs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' (a) RTK position error (b) horizontal RTK position results in Google Map (c) CDF curves of 3D RTK RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The curves in Figure 17 indicate that both APA tracking methods outperform two RPA ones in elevating the TOA accuracy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Furthermore, regarding the lower-elevation satellites and the smaller-TOA-error channels, the TOA errors induced by the navigation results through the vector feedback procedure are more likely to drop in the proposed RTK/INS-based APA approach than in the RTK-only APA one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Figure 18 plots the APA error curves estimated from the proposed RTK/INS-based VDFPLL modeling the instantaneous initial/absolute code phase error in meters at each tracking epoch of which the analytical expression is given by (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Compared to the traditional tracking algorithms (scalar and old vector tracking loops), the proposed algorithm can individually discriminate the absolute code phase error unrelating to the frequency error given by the same-epoch local replica subtracting incoming signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' This operation established through the proposed architecture is reasonable and it proves efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The implied information that the RTK/INS integrated EKF navigator provides more accurate positioning than the code-based-only SPP method can adequately explain the results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' So, the dashed black lines in Figure 18 mean that the traditional scalar and vector tracking loops have nothing to recognize the code phase error not varying with the time spanning (the error residual remained by the traditional code discriminating process).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' However, the proposed RTK/INS-based APA vector tracking can directly estimate the absolute code phase error at every tracking epoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The APA code discriminated results can show how the code phases are corrected by the accurate user’s position solution, especially for the satellites facing the deterministic biased error changing as the cycles of dozens of seconds or longer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' This phenomenon commonly occurs to the static user’s antenna receiving incoming signals affected by the multipath effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 20 SDR:STL SDR:RTK-basedVDFLL Initialization.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='of SDR:RTK/INS-based VDFLL theSDR SDR:RTK-baSedVDFPLL (a) SDR:RTK/INS-basedVDFPLL 10 5 0 10 North [m] 5 0 20 E 10 0 10 20 40 60 80 100 GPS time since 524045s Epoch [s]51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0807 SDR:STL SDR:RTK-based VDFLL SDR:RTK/INS-based VDFLL SDR:RTK-based VDFPL 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='08065 ★ SDR:RTK/INS-based VDFPLL Ground truth atitude [degree] allegiatePINV 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0806 Datafrom524055sto524160s 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='08055 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0805 5 m 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1345 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1344 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1343 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1342 334.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='34 Longitude [degreel0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='8 robability 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='4 n 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2 5 10 15 20 3DRKRASEA Preprint Figure 14: RTK/INS integration position results and statistical analysis of different SDRs where dashed lines correspond to the outlier epochs in the semi-open-sky situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' (a) RTK/INS integration position error (b) horizontal RTK/INS integration position results in Google Map (c) CDF curves of 2D RTK/INS integration position RMSE (d) CDF curves of 3D RTK/INS integration position RMSE.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Finally, it is worth mentioning that the proposed RTK/INS-based VDFPLL is a simplified prototype applying the APA discriminated error relying on the INS and RTK to the GNSS baseband processing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The tracking performance still has space to be further improved by redoing loop filter algorithms (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', the EKF) or other GNSS baseband optimizing methods (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=', snapshot processing Luo et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022b], Fernández-Hernández et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022] and open-loop tracking Tsang et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2022], van Graas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' [2009]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' In other words, the proposed algorithm has a broad scope of use towards the GNSS signals at all frequencies and constellations, potentially contributing to the development of next-generation GNSS receivers and GNSS-based multi-sensor integrating navigation systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 4 Conclusions This work proposes a deep integration of RTK and INS, enhancing the instantaneous code phase tracking performance in challenging static environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' In the presented algorithm, the navigation solutions, especially the absolute position solution, from the integrated EKF navigator are deeply fused into the GNSS tracking loop, forming an APA code phase discriminator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The RTK/INS-based APA discriminator combined with the vector tracking technique realized upon a GPS L1 C/A SDR can serve for more satisfactory tracking and positioning results than the RTK-based-only APA vector tracking approach.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Two real-world stationary experiments have verified the performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Finally, the conclusions of this work can be drawn as follows: 21 SDR:STL,POS:RTK/IMUIntegration Initialization SDR:RTK/INS-basedVDFLL,POS:RTK/IMUIntegration of the SDR SDR:RTK/INS-basedVDFPLL,POS:RTK/IMUIntegration SDR:Ground-truth-basedVDFPLL,POS:RTK/IMUIntegration (a) 3 East[m] 2 North [m] 2 0 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2 [u] 0 5 10 20 40 60 80 100 GPStimesince524045s Epoch[s]51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0807 SDR:STL, POS:RTK/IMU Integration SDR:RTK/INS-based VDFLL, POS:RTK/IMU Integration O SDR:RTK/INS-based VDFPLL, POS:RTK/IMU Integration SDR:Ground-truth-based VDFPLL, POS:RTK/IMU Integration ★ 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='08065 ground truth llegiatePINW 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0806 Datafrom524055sto524160s 53.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='08055 51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0805 5 m 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1345 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1344 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1343 114.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1342 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='334 Longitude [degree0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2 2 30.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='8 Probabil 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='6A Preprint Figure 15: Error curves of the TOA residuals for the GPS satellites PRN6 and PRN3 in the semi-open-sky situation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Figure 16: TOA curve references (regarding the error curves of the TOA residuals) derived from the ground- truth-based VDFPLL SDR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' (a) TOA reference from PRN22 for the open-sky experiment (b) TOA reference from PRN3 for the semi-open-sky experiment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='PRN6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='SDR:STL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='SDR:RTK-based VDFLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='SDR:RTK/INS-baSedVDFLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='[] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='SDR:RTK-basedVDFPLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='SDR:RTK/INS-baSedVDEPLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='TOA ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Residualsof ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Timesince524042s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Time [s]PRN6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Error of Residuals of TOA [m] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Timesince524042s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Time [s]PRN3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='900 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='800 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='700 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Residuals of TOA [m] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='600 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='500 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='400 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='300 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='200 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Time [s]PRN3 Error of Residuals of TOA [m] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='30 ' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Time [s]PRN22 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Error of Residuals of TOA (reference) [m] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Timesince522536s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Time [s]PRN3 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Error of Residuals of TOA (reference) [m] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Timesince524042s ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='80 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Time [s]A Preprint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Table 2: RMSEs of the TOA estimates from the active satellites regarding the two stationary experiments ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='where the reference curves for the error of residuals of TOA refer to Figure 16 (“OS” and” SOS” correspond ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='to “open-sky” and “semi-open-sky” testing situations,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' “Averaging C/N0” is estimated from the “Ground-truth-based VDFPLL” SDR).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' PRN numbers Elevation angle [◦] Averaging C/N0 [dB-Hz] PRN numbers of the TOA error reference RMSE for the error of residuals of TOA [m] STL RTK VDFLL RTK/INS VDFLL RTK VDFPLL Proposed RTK/INS VDFPLL SOS-25 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2 43.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='4 SOS-03 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='92 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='03 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='12 14.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='58 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='54 OS-03 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='4 49.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='8 OS-22 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='63 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='23 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='18 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='87 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='52 Figure 17: Comparison of TOA accuracy improvements varying with the satellite elevation angles and the TOA errors (positive and negative values represent the improved and reduced performance percentages, respectively).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 23 100 of TOA Accuracy (RMSE) [%] 50 50 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='SDR:RTK-based VDFLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='SDR:RTK/INS-based VDFLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='SDR:RTK-based VDFPLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='SDR:RTK/INS-based VDFPLL ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='0 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='30 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='40 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='60 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='70 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Satellite Elevation Angle [Degrees]100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='of TOA Accuracy (RMSE) [%] ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='50 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='100 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='mproement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='150 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='200 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='6 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='8 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='10 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='14 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='16 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='18 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='20 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='STL sR TOA RMsE [mA Preprint ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='Figure 18: APA code phase errors from the proposed RTK/INS-based VDFPLL SDR where the numbers ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='correspond to the satellite PRN numbers and dashed black lines correspond to the estimates from the ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='traditional scalar and vector tracking algorithms (a) open sky (b) semi-open sky.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The proposed RTK/INS APA vector tracking has improved the multipath mitigation performance of the GNSS baseband in static situations compared to the traditional scalar/vector tracking and the RTK-aided-only APA vector tracking;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The deeply integrated INS in the proposed high-accuracy APA GPS SDR has enhanced the TOA estimation accuracy more significantly regarding the satellites with low elevation angles;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' The technique regarding the tested low-cost IMU deeply integrated into the RTK-position-aided vector GPS has proved to be inferior in improving the vertical positioning accuracy but can efficiently increase the horizontal positioning accuracy in challenging static environments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Our future work will focus on base-station-free APA GNSS tracking upon INS DR and precise point positioning (PPP) technique.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' References H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Sharma, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Lichtenberger, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Pany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Multipath Error Modelling and Position Error Over-bounding for Precise RTK Positioning using GNSS Raw Measurements from Smartphone for Automotive Navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' In Proceedings of the 33rd International Technical Meeting of the Satellite Division of The Institute of Navigation (ION GNSS+ 2020), pages 1902–1924, oct 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ISBN 0936406267.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='33012/2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='17627.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='org/publications/abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='cfm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='articleID=17627.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Guohao Zhang, Weisong Wen, Bing Xu, and Li-ta Hsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Extending Shadow Matching to Tightly-Coupled GNSS/INS Integration System.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' IEEE Transactions on Vehicular Technology, 69(5):4979–4991, may 2020.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ISSN 0018-9545.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1109/TVT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2020.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' GPS Solutions, 18(2):163–175, 2014.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ISSN 15211886.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1007/s10291-014-0368-6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Cimini.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Analysis and Simulation of a Digital Mobile Channel Using Orthogonal Frequency Divi- sion Multiplexing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' IEEE Transactions on Communications, 33(7):665–675, jul 1985.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ISSN 0096-2244.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1109/TCOM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1985.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='org/document/9173292/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Yiran Luo, You Li, Jin Wang, and Naser El-Sheimy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Supporting GNSS Baseband Using Smartphone IMU and Ultra-Tight Integration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' arXiv preprint arXiv:2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='02613, 2021c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' URL http://arxiv.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='org/abs/2111.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 02613.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Yiran Luo, Jian Li, Chunyang Yu, Zhitao Lyu, Zhe Yue, and Naser El-Sheimy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' A GNSS software- defined receiver with vector tracking techniques for land vehicle navigation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' In Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ION 2019 Pacific PNT Meeting, Honolulu, Hawaii, USA, April 8-11, volume 2019-April, pages 713–727, 2019b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ISBN 0936406224.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='33012/2019.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='16834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' URL https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='ion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='org/publications/abstract.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='cfm?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' articleID=16834.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Ramsey Faragher.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Understanding the Basis of the Kalman Filter Via a Simple and Intuitive Deriva- tion [Lecture Notes].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' IEEE Signal Processing Magazine, 29(5):128–132, sep 2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ISSN 1053-5888.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1109/MSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2012.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2203621.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' URL http://ieeexplore.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='ieee.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='org/document/6279585/.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 26 A Preprint Tomoji Takasu and Akio Yasuda.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Development of the low-cost RTK-GPS receiver with an open source program package RTKLIB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' In Proceedings of the International symposium on GPS/GNSS, Jeju, Korea, pages 4–6, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Yiran Luo, Li-Ta Hsu, and Naser El-Sheimy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' A Baseband MLE for Snapshot GNSS Receiver Using Super- Long-Coherent Correlation in a Fractional Fourier Domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' unpublished, 2022b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Ignacio Fernández-Hernández, José A López-Salcedo, and Gonzalo Seco-Granados.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 9 Snapshot Re- ceivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' In Kai Borre, Ignacio Fernández-Hernández, José A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' López-Salcedo, and M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Zahidul H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Bhuiyan, editors, GNSS Software Receivers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Cambridge University Press, oct 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ISBN 9781108934176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1017/9781108934176.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Chin Lok Tsang, Yiran Luo, and Li-Ta Hsu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Long Coherent Open-Loop GPS L5Q Signal Positioning: A Case Study for an Urban Area in Hong Kong.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' In Proceedings of the ION GNSS+ 2022, Denver, Colorado, USA, September 19-23, pages 2025–2041, oct 2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='33012/2022.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='18326.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Frank van Graas, Andrey Soloviev, Maarten Uijt de Haag, and Sanjeev Gunawardena.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' Closed-loop sequential signal processing and open-loop batch processing approaches for GNSS receiver design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' IEEE Journal on Selected Topics in Signal Processing, 3(4):571–586, 2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' ISSN 19324553.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' doi:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='1109/JSTSP.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2009.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content='2023350.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/_tAyT4oBgHgl3EQfdveI/content/2301.00308v1.pdf'} +page_content=' 27' metadata={'source': 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100644 index 0000000000000000000000000000000000000000..2d30182ee80a7c10c80bd1f7be42ffd4e310a16f --- /dev/null +++ b/bdFAT4oBgHgl3EQf5B6R/content/tmp_files/2301.08731v1.pdf.txt @@ -0,0 +1,1188 @@ +Can Peanuts Fall in Love with Distributional Semantics? +James A. Michaelov (j1michae@ucsd.edu) +Seana Coulson (scoulson@ucsd.edu) +Benjamin K. Bergen (bkbergen@ucsd.edu) +Department of Cognitive Science, University of California, San Diego +9500 Gilman Dr, La Jolla, CA 92093, USA +Abstract +The context in which a sentence appears can drastically alter +our expectations about upcoming words—for example, follow- +ing a short story involving an anthropomorphic peanut, exper- +imental participants are more likely to expect the sentence the +peanut was in love than the peanut was salted, as indexed by +N400 amplitude (Nieuwland & van Berkum, 2006). This rapid +and dynamic updating of comprehenders’ expectations about +the kind of events that a peanut may take part in based on +context has been explained using the construct of Situation +Models—updated mental representations of key elements of +an event under discussion, in this case, the peanut protago- +nist. However, recent work showing that N400 amplitude can +be predicted based on distributional information alone raises +the question whether situation models are in fact necessary for +the kinds of contextual effects observed in previous work. To +investigate this question, we attempt to model the results of +Nieuwland and van Berkum (2006) using six computational +language models and three sets of word vectors, none of which +have explicit situation models or semantic grounding. We find +that the effect found by Nieuwland and van Berkum (2006) +can be fully modeled by two language models and two sets of +word vectors, with others showing a reduced effect. Thus, at +least some processing effects normally explained through situ- +ation models may not in fact require explicit situation models. +Keywords: psycholinguistics; human language comprehen- +sion; event-related brain potentials; N400; natural language +processing; deep learning; language models; word vectors +Introduction +It is widely believed that prediction plays a key role in lan- +guage processing, with more predictable words being pro- +cessed more easily (Fischler & Bloom, 1979; Kutas & Hill- +yard, 1984; Levy, 2008; Kutas, DeLong, & Smith, 2011; +Van Petten & Luka, 2012; DeLong, Troyer, & Kutas, 2014; +Luke & Christianson, 2016; Kuperberg, Brothers, & Wlotko, +2020). Perhaps the strongest evidence for this comes from the +N400, a neural signal of processing difficulty that is highly +correlated with lexical probability—contextually probable +words elicit an N400 response of smaller (less negative) am- +plitude than contextually improbable words, whether pre- +dictability is determined based on human judgements (Kutas +& Hillyard, 1984; for review see Van Petten & Luka, 2012) +or a corpus (Parviz, Johnson, Johnson, & Brock, 2011; Frank, +Otten, Galli, & Vigliocco, 2015; Aurnhammer & Frank, +2019b; Merkx & Frank, 2021; Szewczyk & Federmeier, +2022; Michaelov, Coulson, & Bergen, 2022). +A striking feature of the predictions indexed by the N400 +is how flexible they can be. Under normal circumstances, a +sentence such as the peanut was in love would be highly im- +probable, much more so than the peanut was salted. Follow- +ing the short story in (1), however, this changes (Nieuwland +& van Berkum, 2006). +(1) A woman saw a dancing peanut who had a big smile on +his face. The peanut was singing about a girl he had just +met. And judging from the song, the peanut was totally +crazy about her. The woman thought it was really cute to +see the peanut singing and dancing like that. +In fact, Nieuwland and van Berkum (2006), who tested this +in Dutch, found that in the context of (1), the last word of +de pinda was verliefd (‘the peanut was in love’) elicited a +smaller N400 than de pinda was gezouten (‘the peanut was +salted’). How does such a dramatic reversal occur? +One possibility, that put forward by Nieuwland and van +Berkum (2006), is that while reading the context, the reader’s +mental representation of the peanut is altered such that it is +treated as an animate entity. +This, as Nieuwland and van +Berkum (2006) note, is in line with theories of situation mod- +els, which argue that we keep track of the entities under +discussion, as well as their properties and relations, among +other things. Such accounts generally involve explicitly struc- +tures or schemata, grounding in world knowledge or expe- +rience, extraction of propositional information, or a com- +bination of these (see, e.g., Bransford, Barclay, & Franks, +1972; Kintsch & van Dijk, 1978; Johnson-Laird, 1980; Gar- +nham, 1981; Johnson-Laird, 1983; van Dijk & Kintsch, 1983; +Kintsch, 1988; Zwaan, Langston, & Graesser, 1995; Zwaan, +Magliano, & Graesser, 1995; Radvansky, Zwaan, Federico, +& Franklin, 1998; Kintsch, 1998; Zwaan & Radvansky, +1998; Zwaan & Madden, 2004; Kintsch, 2005; Van Berkum, +Koornneef, Otten, & Nieuwland, 2007; Kintsch & Man- +galath, 2011; Butcher & Kintsch, 2012; Zwaan, 2014, 2016; +Zacks & Ferstl, 2016; Kintsch, 2018; Hoeben Mannaert & +Dijkstra, 2021). Under a situation model account, the reader +alters their semantic representation of the peanut such that it +has animate features in accordance with the information that +it can sing, dance, and show emotions, facilitating the pro- +cessing of in love. +While this hypothesis—that structured or grounded situ- +ation models explain N400 effects such as those found by +Nieuwland and van Berkum (2006)—is generally accepted +(e.g., by Hagoort & van Berkum, 2007; Filik & Leuthold, +arXiv:2301.08731v1 [cs.CL] 20 Jan 2023 + +2008; Warren, McConnell, & Rayner, 2008; Rosenbach, +2008; Ferguson & Sanford, 2008; Ferguson, Sanford, & +Leuthold, 2008; Menenti, Petersson, Scheeringa, & Hagoort, +2009; Bicknell, Elman, Hare, McRae, & Kutas, 2010; de +Groot, 2011; Metusalem et al., 2012; Aravena et al., 2014; +Zwaan, 2014; Xiang & Kuperberg, 2015; Kuperberg et al., +2020) and has been shown to be viable using computational +modeling (Venhuizen, Crocker, & Brouwer, 2019), there are +also alternative explanations. +The possibility that we explore in the present study is that +the effect is explained by lexical prediction based on distri- +butional linguistic knowledge. A number of researchers have +found that modeling the amplitude of the N400 based on the +statistics of language can both be used to model N400 effects +(Ettinger, Feldman, Resnik, & Phillips, 2016; Michaelov & +Bergen, 2020; Michaelov, Bardolph, Coulson, & Bergen, +2021; Michaelov & Bergen, 2022; Uchida, Lair, Ishiguro, & +Dominey, 2021) and to predict single-trial N400 amplitude +(Chwilla & Kolk, 2005; Parviz et al., 2011; Van Petten, 2014; +Frank et al., 2015; Aurnhammer & Frank, 2019a, 2019b; +Merkx & Frank, 2021; Michaelov et al., 2021; Michaelov +& Bergen, 2022; Szewczyk & Federmeier, 2022). +Specifically, we look at two possible ways in which this +might arise. One, which we refer to as event-level priming, +refers to the idea that a word associated with a previously- +discussed event may be more likely to be predicted by virtue +of this. This is something that has been previously reported in +the N400—Metusalem et al. (2012), for example, found that +that merely being related to the event under discussion leads +to a smaller N400 response to a word even when that word is +inappropriate. Michaelov and Bergen (2022) model this with +transformer language models—systems trained to calculate +the probability of a word given its context based on the statis- +tics of language alone—showing that this effect is explainable +with distributional information. Thus, it may be the case that +the fact that in love is related to, for example, being crazy +about someone that leads to it being predicted to be more +likely than salted. Following Michaelov and Bergen (2022), +we investigate this using 6 Dutch transformer language mod- +els (Havinga, 2021, 2022a, 2022b, 2022c; de Vries et al., +2019; Delobelle, Winters, & Berendt, 2020), testing whether +they show the same effect as humans—that is, whether they +predict the canonical sentence the peanut was salted to be less +likely than the noncanonical sentence the peanut was in love. +An alternative possibility is what we refer to as lexical +priming. More simply than in the case of event-level prim- +ing, it may be the case that the preceding context involving +words such as dancing, smile, singing, crazy, and cute ex- +erts a stronger pressure on prediction of in love than peanut +does on salted. Because language models appear to exhibit +lexical priming effects (Kassner & Sch¨utze, 2020; Misra, Et- +tinger, & Rayz, 2020), we instead to turn to word vectors to +test this. Word vectors are representations of words derived +from their co-occurrence statistics, either directly or based +on word embeddings learned by neural networks (see, e.g., +Dumais, Furnas, Landauer, Deerwester, & Harshman, 1988; +Landauer, Foltz, & Laham, 1998; Mikolov, Sutskever, Chen, +Corrado, & Dean, 2013; Pennington, Socher, & Manning, +2014; Mikolov, Grave, Bojanowski, Puhrsch, & Joulin, 2018; +Tulkens, Emmery, & Daelemans, 2016; Grave, Bojanowski, +Gupta, Joulin, & Mikolov, 2018). The cosine distance be- +tween the vector of each critical word (e.g. in love or salted) +and the mean of the vectors of the words in the preceding con- +text can therefore be used to test how similar the critical word +is to the words preceding it (Ettinger et al., 2016; Uchida et +al., 2021), and thus, the extent to which it is predicted based +on them. To do this this we used three sets of Dutch word +vectors (from Tulkens et al., 2016; and Grave et al., 2018). +Background +A number of researchers have attempted to model the N400 +computationally, including using language models (Parviz +et al., 2011; Frank et al., 2015; Aurnhammer & Frank, +2019b; Michaelov & Bergen, 2020; Merkx & Frank, 2021; +Michaelov et al., 2021, 2022; Szewczyk & Federmeier, 2022) +and the distances between vector representations of words +(Parviz et al., 2011; Van Petten, 2014; Ettinger et al., 2016; +Uchida et al., 2021). There have also been several attempts +to computationally model whether the amplitude of the N400 +response is impacted by situation models (Uchida et al., 2021; +Venhuizen et al., 2019) and thematic roles (Brouwer, Crocker, +Venhuizen, & Hoeks, 2017; Fitz & Chang, 2019; Rabovsky, +Hansen, & McClelland, 2018). +To the best of our knowledge, only one previous study +(Uchida et al., 2021) has directly attempted to model the dis- +course effect found by Nieuwland and van Berkum (2006), +and it does not rely on purely distributional linguistic infor- +mation. Specifically, Uchida et al. (2021) base their model on +Wikipedia2Vec (Yamada et al., 2020) vectors—while these +include distributional information derived from the surface- +level statistics of language, they also include information +about hyperlinks between Wikipedia pages, and thus struc- +tured semantic relations based on human judgements of rel- +evance and importance (Yamada et al., 2020). Additionally, +Uchida et al. (2021) only look at the English-translated ver- +sion of the single stimulus item presented in (1), and thus, it +is unclear whether the results generalize to all the stimuli in +the original study. The current study overcomes these infer- +ential limitations by using the original Dutch stimuli and by +using neural language models and word vectors trained only +on natural language input. +The present study +We investigate the adequacy of distributional knowledge to +explain the human N400 effect found by Nieuwland and van +Berkum (2006) using predictions of neural network language +models and the distance between the word vectors of the criti- +cal words and their context. Specifically, we ask this question +for two possible variants of the effect found by Nieuwland +and van Berkum (2006). + +Reduction effect +Nieuwland and van Berkum (2006) presented experimental +participants with short stories such as those in (1) including +“canonical” sentences like the peanut was salted or “non- +canonical” ones like the peanut was in love. One approach +to whether language models and human show the same pre- +diction patterns (taken by Uchida et al., 2021) is to compare +the statistical metrics the critical words elicit in the context of +the full story versus in isolation. Without preceding context, +these sentences should produce values that match the canon- +icality of the sentence, but the difference should attenuate or +reverse following the story context. +Thus, we ran a statistical analysis testing for an interac- +tion between stimulus length (full story or only the last sen- +tence) and canonicality (canonical or noncanonical). Such +an interaction would reveal a context-dependent difference in +the effect of canonicality on our statistical metrics; and thus +would replicate in neural language models the effect found by +Nieuwland and van Berkum (2006). +However, an interaction between stimulus length and +canonicality in this direction could result from either a rever- +sal or a decrease in the magnitude of the canonicality effect. +Canonical stimuli might elicit lower surprisals or smaller co- +sine distances in both context conditions, but of different +magnitudes. For this reason, we label the effect measured by +an interaction (in the expected direction) a reduction effect. +Reversal effect +Nieuwland and van Berkum (2006) did not employ the 2 x +2 design that would allow them to detect an interaction— +they compared the N400 in context only, finding that canon- +ical stimuli actually elicited larger N400 responses than non- +canonical stimuli. To replicate this finding, we test whether +the canonical full-length stimuli elicit higher surprisals or +greater cosine distances than the noncanonical full-length +stimuli, a reversal effect. +Implications +If either language models or word vectors can successfully +model the reversal effect, this would suggest that distribu- +tional information is sufficient to explain the data reported +by Nieuwland and van Berkum (2006). Thus, while situation +models and extralinguistic information may be involved in the +neurocognitive system underlying the N400, additional evi- +dence is required to prove this. If neither can model either ef- +fect, this would undermine the claim that distributional infor- +mation is sufficient to explain the effect found by Nieuwland +and van Berkum (2006). Finally, if either language models +or word vectors can successfully model the reduction effect +but not the full reversal effect, this may support the idea that +distributional information could be used as part of the neu- +rocognitive system underlying the N400 response, that it is +not sufficient to yield the dynamic contextual sensitivity hu- +mans display. Situation models and other sources of informa- +tion might explain the remainder. +Method and approach +Original experiment and stimuli +Stimuli were used from the original experiment, and are pro- +vided online1 by the authors (Nieuwland & van Berkum, +2006). We compared the effect of experimental condition on +the N400 and on neural network surprisal (as in Michaelov +& Bergen, 2020) and the cosine similarity between the word +vector of the critical word and the mean of the word vectors +in its context (as in Ettinger et al., 2016). +The stimuli use 60 full-length story frames, each of which +has either a canonical or noncanonical predicate, for 120 +unique stories. As the aim is to model human online compre- +hension processes, the models only used the text before the +critical words (e.g., in love or salted) to predict the critical +words, so stories were truncated after the critical word. For +the critical sentence stimuli, we isolated the last sentence of +these truncated stories, including and up to the critical word +in each story (e.g., The peanut was in love). This produced +240 stimuli, as shown in Table 1. +Predicate Type +Stimulus Length +Count +Canonical +Full-length +60 +Canonical +critical sentence +60 +Noncanonical +Full-length +60 +Noncanonical +critical sentence +60 +Table 1: Experimental stimuli derived from Nieuwland and +van Berkum (2006). +Experiment 1: Language Models +Language models +In this study, we use six pretrained models available through +the transformers package (Wolf et al., 2020). Four of these +models—Dutch versions of the medium (Havinga, 2021) and +large (Havinga, 2022a) GPT-2 models (Radford et al., 2019) +and Dutch versions of the 125 million parameter (Havinga, +2022b) and 1.3 billion parameter (Havinga, 2022c) GPT-Neo +models (Black, Gao, Wang, Leahy, & Biderman, 2021)— +were autoregressive, meaning that they are trained to predict +a word based only on its preceding context. +The remain- +ing two models—BERTje (de Vries et al., 2019) and Rob- +BERT v2 (Delobelle et al., 2020), based on BERT (Devlin, +Chang, Lee, & Toutanova, 2019) and RoBERTa (Liu et al., +2019), respectively—are masked language models, meaning +that they are also trained to predict a word based on the text +following the critical word. However, as stated, in the present +study, all models were only provided with the context preced- +ing the critical words. We ran the stimuli through each lan- +guage model, calculating the surprisal of each critical word +that was in the model’s vocabulary. To do this, we calculated +the negative of the logarithm of the probabilities provided +1https://www.researchgate.net/publication/268208198 + +for each critical word by each of the language models. We +then tested for the reduction and reversal effects with these +surprisal values. The language models were run in Python +(Van Rossum & Drake, 2009), using the PyTorch (Paszke et +al., 2019) implementation of each model, as provided by the +transformers package (Wolf et al., 2020). +Statistical analysis and data manipulation were carried out +in R (R Core Team, 2020) using Rstudio (RStudio Team, +2020) and the tidyverse (Wickham et al., 2019) and lme4 +(Bates, M¨achler, Bolker, & Walker, 2015) packages. +Reduction effect +In order to test the reduction effect, we constructed linear +mixed-effects regression models, with the surprisal calculated +from each language model as the dependent variable. In each +model, predicate type (canonical or noncanonical) and stimu- +lus length (full-length or critical sentence) were fixed effects +and story frame (each of the 60) was a random intercept. For +the regressions with the autoregressive models and BERTje +surprisal as their dependent variables, we then constructed +regressions also including an interaction between predicate +type and stimulus length. Using likelihood ratio tests, we +found that these regressions including the interaction fit the +data significantly better than those without the interaction +(GPT-2 Medium: χ2(1) = 116.0, p < 0.001; GPT-2 Large: +χ2(1) = 107.3, p < 0.001; GPT-Neo 125M: χ2(1) = 61.7, p < +0.001; GPT-Neo 1.3B: χ2(1) = 65.3, p < 0.001; BERTje: +χ2(1) = 44.4, p < 0.001), indicating a significant interaction +between predicate type and stimulus length. The regression +with RobBERT surprisal as its dependent variable and no in- +teraction had a singular fit, but the regression with the inter- +action did not. Thus, instead of running a likelihood ratio +test to investigate whether there was a significant interaction, +we used a a Type III ANOVA with Satterthwaite’s method +for estimating degrees of freedom (Kuznetsova, Brockhoff, +& Christensen, 2017) on the regression with the interaction, +finding it to be a significant predictor of RobBERT surprisal +(F(1,71.2) = 81.0, p < 0.001). +Note that all reported p- +values are corrected for multiple comparisons based on false +discovery rate (Benjamini & Yekutieli, 2001). +For all language models, there was a significant interac- +tion between predicate type and stimulus length. Further in- +spection of the regressions showed that in all cases, the in- +teraction was in the expected direction. +Thus, all models +displayed the reduction effect. This can be seen visually in +Figure 1—in all models, when only the critical sentence was +presented, the mean surprisal for critical words in canonical +sentences is lower than for critical words in noncanonical sen- +tences. Conversely, when the full-length story is presented to +the language models, the critical words in the noncanonical +sentences elicit a lower or roughly-equal surprisal than the +critical words in the canonical sentences. +Reversal effect +To test for which models this latter finding was statistically +significant, we initially attempted to fit linear mixed-effects +regression models for each the full-length and critical sen- +tence stimulus results for each language model; however, this +led to several models with singular fits. +Instead, we car- +ried out pairwise two-tailed t-tests, comparing the surprisal of +canonical and noncanonical stimuli for full-length and critical +sentence stimuli for each language model. +First, we test whether canonical critical sentence (only) +stimuli elicit lower surprisals than noncanonical critical sen- +tence stimuli. +For all language models, the decontextual- +ized canonical critical sentence stimuli elicit significantly +lower surprisals than noncanonical critical sentence stimuli +after correction for multiple comparisons (GPT-2 Medium: +t(88.9) = −9.65, p < 0.001; GPT-2 Large: t(89.0) = −9.59, +p < 0.001; GPT-Neo 125M: t(88.6) = −9.20, p < 0.001 +; GPT-Neo 1.3B: t(89.0) = −9.72, p < 0.001; BERTje: +t(48.4) = −5.99, p < 0.001; RobBERT: t(55.1) = −7.67, +p < 0.001). +Next, in order to investigate the reversal effect, we test +whether canonical full-length stimuli elicit lower surprisals +than noncanonical full-length stimuli. After correction for +multiple comparisons, only the Dutch GPT-2 models suc- +cessfully model the reversal effect—they are the only mod- +els for which canonical full-length stimuli elicit signifi- +cantly higher surprisals than noncanonical full-length stimuli +(GPT-2 Medium: t(86.3) = 5.61, p < 0.001; GPT-2 Large: +t(88.5) = 5.34, p < 0.001). +With the other models, while the difference was in the cor- +rect direction, it was not significant after correction for mul- +tiple comparisons (GPT-Neo 125M: t(89.0) = −0.57, p = +1.000 ; GPT-Neo 1.3B: t(89.0) = 0.71, p = 1.000; BERTje: +t(51.5) = 0.79, p = 1.000; RobBERT: t(46.6) = 2.31, p = +0.120). +However, it is worth noting that the contrast between the +two sets of results (critical sentence only vs. full stimulus) +means that significant canonicality effects for the critical sen- +tence stimuli disappear in the full-length stimuli, underscor- +ing the presence of a reduction effect in the Dutch GPT-Neo +models, BERTje, and RobBERT. +Discussion +Nieuwland and van Berkum (2006) found that in a suitably +supportive context, noncanonical stimuli like de pinda was +verliefd (‘the peanut was in love’) elicit smaller N400 re- +sponses than canonical stimuli such as de pinda was gezouten +(‘the peanut was salted’)—context not only mitigated but re- +verse the effect of animacy violation. +We find that two language models also display this rever- +sal effect: Dutch GPT-2 Medium (Havinga, 2021) and Dutch +GPT-2 Large (Havinga, 2022a). When these models are pre- +sented with the same contexts, the surprisal of critical words +in the noncanonical condition is lower than that elicited by +those in the canonical condition. +This is not the case for the remaining four language mod- +els: Dutch GPT-Neo 125M (Havinga, 2022b), Dutch GPT- +Neo 1.3B (Havinga, 2022c), BERTje (de Vries et al., 2019), +and RobBERT (Delobelle et al., 2020). However, these mod- + +GPT−2 Large +GPT−Neo 1.3B +RobBERT +GPT−2 Medium +GPT−Neo 125M +BERTje +Target Sentence +Full Story +Target Sentence +Full Story +Target Sentence +Full Story +0 +5 +10 +15 +20 +0 +5 +10 +15 +20 +Stimulus Length +Surprisal +Predicate Type +Canonical +Uncanonical +Figure 1: Surprisal elicited by critical words for each predicate type and stimulus length. +els do display the weaker reduction effect, and further, the ab- +sence of a significant difference between conditions for these +models when presented with the full stories shows that the +difference between canonical and noncanonical critical sen- +tence stimuli is not just reduced, but disappears entirely. +It may be tempting to infer that the architecture of au- +toregressive transformers, and in particular, those based on +the GPT-2 architecture, leads to success capturing the ef- +fect. However, it should be noted that before correction for +multiple comparisons, RobBERT also successfully displays +the reversal effect. In addition, not all language models had +the same vocabulary, and thus, a different number of items +were analyzed across models. For these reasons, and because +these models are all of various sizes and trained on several +different datasets, we believe it would be premature to draw +conclusions about how language model architecture impacts +whether a model displays the reversal effect. +Experiment 2: Word Embeddings +Cosine Distance +In this study, we used 3 sets of pretrained word vectors: +the 300-dimensional Dutch fastText vectors (Grave et al., +2018) trained on Dutch text from Wikipedia2 and Common +Crawl3 and two 320-dimensional Dutch word vectors re- +leased by Tulkens et al. (2016)—one trained on COW (COr- +pora from the Web; Sch¨afer & Bildhauer, 2012) and one +trained on a Combined corpus made up of the SoNaR cor- +pus (Oostdijk, Reynaert, Hoste, & Schuurman, 2013) and text +from Wikipedia and Roularta4. To calculate cosine distance, +2https://nl.wikipedia.org/ +3https://commoncrawl.org/ +4https://www.roularta.be +we calculated the cosine distance (using SciPy; Virtanen et +al., 2020) between the mean of the word embeddings for all +words in the preceding context and the word embedding for +the critical word. Only critical words that were present in +the pretrained embeddings were included, and words in the +context that were not present in the pretrained embeddings +were ignored when calculating cosine distance. The cosine +distances for critical words in each condition are shown in +Figure 2. +Reduction effect +As with language model surprisal, we constructed linear +mixed-effects regressions with predicate type and stimulus +length as fixed effects and story from as a random intercept. +With these models, the cosine distance calculated using each +set of pretrained embeddings was the dependent variable. The +interaction between predicate type and stimulus length was +significant for all embeddings after correcting for multiple +comparisons (fastText: χ2(1) = 12.0, p = 0.003; Combined: +χ2(1) = 40.8, p < 0.001; COW: χ2(1) = 66.4, p < 0.001). +Reversal effect +When comparing the cosine distances calculated between the +embedding of the critical words and the preceding words +of the critical sentence using two-tailed t-tests as with sur- +prisal, there was a significant difference between canonincal +and noncanonical critical words for Combined and COW em- +beddings (Combined: t(116.9) = −3.48, p = 0.004; COW: +t(116.5) = −4.45, p < 0.001), but not fastText embeddings +(fastText: t(118.0) = −1.96, p = 0.237). +Similarly, when comparing the cosine distances between +the critical word and the preceding words of the full story, +there was a significant difference between canonical and + +FastText +Combined +COW +Target Sentence +Full Story +Target Sentence +Full Story +Target Sentence +Full Story +0.0 +0.2 +0.4 +0.6 +0.8 +Stimulus Length +CosineDistance +Predicate Type +Canonical +Uncanonical +Figure 2: Cosine distance elicited by critical words for each predicate type and stimulus length. +noncanonical critical words for Combined and COW em- +beddings (Combined: t(117.0) = 4.82, p < 0.001; COW: +t(117.0) = 6.78, p < 0.001), but not fastText embeddings +(fastText: t(117.4) = 1.68, p = 0.418). +Discussion +The cosine distances calculated from all three sets of word +vectors displayed the reduction effect, and two out of three +displayed the reversal effect. Thus, the results suggest that +the N400 effect reported by Nieuwland and van Berkum +(2006) can be explained on the basis of distributional linguis- +tic knowledge alone. +The present study corroborates the finding of Uchida et al. +(2021), and expands upon it in several ways. First, it explic- +itly tests for the reversal effect—not just whether canonical +and noncanonical stimuli differ depending on whether there +is a preceding story or not, but also whether the noncanon- +ical sentence is more expected than the canonical when the +story is present. Second, we found that language model sur- +prisal can model the effect for multiple stimuli, not just the +peanut was in love example. Third, we found that the effect +can be modeled in Dutch, the language in which the human +study was carried out. And finally, we found that embeddings +trained on text data only (i.e., without additional information) +are able to model the effect. +General Discussion +Human comprehenders use context to update expectations +about upcoming words, making a sentence that would be +highly unlikely on its own more predictable than a sentence +that would be relatively likely on its own. More strikingly, +humans do this even when the event described is implausible, +violating the constraint, for instance, that only animate, con- +scious entities can fall in love. The human comprehension +system is quite flexible if it can update expectations about +what peanuts, for example, can do based only a story that in- +directly implies the animacy of a fictional peanut. +It has often been assumed that this flexibility requires sit- +uation models that are explicitly structured (Venhuizen et al., +2019) or involve non-linguistic world knowledge or experi- +ence (Uchida et al., 2021). However, the present findings +show that it is possible for a purely linguistic model with +no direct experiential grounding to update its expectations +based on the linguistic context and its extant knowledge of the +statistics of language. One possibility suggested by this re- +sult is that that the dynamics of lexical prediction in language +models may in some implicit, unspecified way approximate +situation models even though they only learn from distribu- +tional information. +But there may be an even simpler explanation. We com- +puted the similarity between critical words and their contexts. +Within final sentences alone, canonical critical words were +more similar to their contexts than noncanonical words, but +when we include the full story context, it is the noncanoni- +cal critical words that are more similar to their contexts. This +could explain both the human results and the language model +results. The amplitude of the N400 to a given word is reduced +when it is semantically related to a previously-seen word +(Bentin, McCarthy, & Wood, 1985; Rugg, 1985; Van Petten +& Kutas, 1988; Kutas & Hillyard, 1989; Holcomb, 1988; Ku- +tas, 1993; Lau, Holcomb, & Kuperberg, 2013); and similarly, +words that are semantically related to previously-mentioned +words are more strongly predicted by language models than +words that are not (Misra et al., 2020; Michaelov et al., 2021). +Overall, then, our results show that in principle, it is possible +that the pattern in the N400 responses reported by Nieuwland +and van Berkum (2006) may not rely on situation models at +all, but rather reflect some form of lexical priming. +It may still be the case that humans use structured or +semantically-rich situation models in online language com- +prehension (see, e.g., Kuperberg et al., 2020). 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A. +(1998). +Situation +models in language comprehension and memory. +Psy- +chological Bulletin, 123(2), 162–185. doi: 10.1037/0033- +2909.123.2.162 + diff --git a/bdFAT4oBgHgl3EQf5B6R/content/tmp_files/load_file.txt b/bdFAT4oBgHgl3EQf5B6R/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f8b41939a6fb6e148d551ab75526498141cd7979 --- /dev/null +++ b/bdFAT4oBgHgl3EQf5B6R/content/tmp_files/load_file.txt @@ -0,0 +1,1408 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf,len=1407 +page_content='Can Peanuts Fall in Love with Distributional Semantics?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' James A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Michaelov (j1michae@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='edu) Seana Coulson (scoulson@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='edu) Benjamin K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Bergen (bkbergen@ucsd.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='edu) Department of Cognitive Science, University of California, San Diego 9500 Gilman Dr, La Jolla, CA 92093, USA Abstract The context in which a sentence appears can drastically alter our expectations about upcoming words—for example, follow- ing a short story involving an anthropomorphic peanut, exper- imental participants are more likely to expect the sentence the peanut was in love than the peanut was salted, as indexed by N400 amplitude (Nieuwland & van Berkum, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' This rapid and dynamic updating of comprehenders’ expectations about the kind of events that a peanut may take part in based on context has been explained using the construct of Situation Models—updated mental representations of key elements of an event under discussion, in this case, the peanut protago- nist.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' However, recent work showing that N400 amplitude can be predicted based on distributional information alone raises the question whether situation models are in fact necessary for the kinds of contextual effects observed in previous work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' To investigate this question, we attempt to model the results of Nieuwland and van Berkum (2006) using six computational language models and three sets of word vectors, none of which have explicit situation models or semantic grounding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' We find that the effect found by Nieuwland and van Berkum (2006) can be fully modeled by two language models and two sets of word vectors, with others showing a reduced effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Thus, at least some processing effects normally explained through situ- ation models may not in fact require explicit situation models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Keywords: psycholinguistics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' human language comprehen- sion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' event-related brain potentials;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' N400;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' natural language processing;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' deep learning;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' language models;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' word vectors Introduction It is widely believed that prediction plays a key role in lan- guage processing, with more predictable words being pro- cessed more easily (Fischler & Bloom, 1979;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Kutas & Hill- yard, 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Levy, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Kutas, DeLong, & Smith, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Van Petten & Luka, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' DeLong, Troyer, & Kutas, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Luke & Christianson, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Kuperberg, Brothers, & Wlotko, 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Perhaps the strongest evidence for this comes from the N400, a neural signal of processing difficulty that is highly correlated with lexical probability—contextually probable words elicit an N400 response of smaller (less negative) am- plitude than contextually improbable words, whether pre- dictability is determined based on human judgements (Kutas & Hillyard, 1984;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' for review see Van Petten & Luka, 2012) or a corpus (Parviz, Johnson, Johnson, & Brock, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Frank, Otten, Galli, & Vigliocco, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Aurnhammer & Frank, 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Merkx & Frank, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Szewczyk & Federmeier, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Michaelov, Coulson, & Bergen, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' A striking feature of the predictions indexed by the N400 is how flexible they can be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Under normal circumstances, a sentence such as the peanut was in love would be highly im- probable, much more so than the peanut was salted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Follow- ing the short story in (1), however, this changes (Nieuwland & van Berkum, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' (1) A woman saw a dancing peanut who had a big smile on his face.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' The peanut was singing about a girl he had just met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' And judging from the song, the peanut was totally crazy about her.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' The woman thought it was really cute to see the peanut singing and dancing like that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' In fact, Nieuwland and van Berkum (2006), who tested this in Dutch, found that in the context of (1), the last word of de pinda was verliefd (‘the peanut was in love’) elicited a smaller N400 than de pinda was gezouten (‘the peanut was salted’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' How does such a dramatic reversal occur?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' One possibility, that put forward by Nieuwland and van Berkum (2006), is that while reading the context, the reader’s mental representation of the peanut is altered such that it is treated as an animate entity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' This, as Nieuwland and van Berkum (2006) note, is in line with theories of situation mod- els, which argue that we keep track of the entities under discussion, as well as their properties and relations, among other things.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Such accounts generally involve explicitly struc- tures or schemata, grounding in world knowledge or expe- rience, extraction of propositional information, or a com- bination of these (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', Bransford, Barclay, & Franks, 1972;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Kintsch & van Dijk, 1978;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Johnson-Laird, 1980;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Gar- nham, 1981;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Johnson-Laird, 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' van Dijk & Kintsch, 1983;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Kintsch, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Zwaan, Langston, & Graesser, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Zwaan, Magliano, & Graesser, 1995;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Radvansky, Zwaan, Federico, & Franklin, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Kintsch, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Zwaan & Radvansky, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Zwaan & Madden, 2004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Kintsch, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Van Berkum, Koornneef, Otten, & Nieuwland, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Kintsch & Man- galath, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Butcher & Kintsch, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Zwaan, 2014, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Zacks & Ferstl, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Kintsch, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Hoeben Mannaert & Dijkstra, 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Under a situation model account, the reader alters their semantic representation of the peanut such that it has animate features in accordance with the information that it can sing, dance, and show emotions, facilitating the pro- cessing of in love.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' While this hypothesis—that structured or grounded situ- ation models explain N400 effects such as those found by Nieuwland and van Berkum (2006)—is generally accepted (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', by Hagoort & van Berkum, 2007;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Filik & Leuthold, arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='08731v1 [cs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='CL] 20 Jan 2023 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Warren, McConnell, & Rayner, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Rosenbach, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Ferguson & Sanford, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Ferguson, Sanford, & Leuthold, 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Menenti, Petersson, Scheeringa, & Hagoort, 2009;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Bicknell, Elman, Hare, McRae, & Kutas, 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' de Groot, 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Metusalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Aravena et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Zwaan, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Xiang & Kuperberg, 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Kuperberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2020) and has been shown to be viable using computational modeling (Venhuizen, Crocker, & Brouwer, 2019), there are also alternative explanations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' The possibility that we explore in the present study is that the effect is explained by lexical prediction based on distri- butional linguistic knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' A number of researchers have found that modeling the amplitude of the N400 based on the statistics of language can both be used to model N400 effects (Ettinger, Feldman, Resnik, & Phillips, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Michaelov & Bergen, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Michaelov, Bardolph, Coulson, & Bergen, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Michaelov & Bergen, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Uchida, Lair, Ishiguro, & Dominey, 2021) and to predict single-trial N400 amplitude (Chwilla & Kolk, 2005;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Parviz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Van Petten, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Frank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Aurnhammer & Frank, 2019a, 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Merkx & Frank, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Michaelov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Michaelov & Bergen, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Szewczyk & Federmeier, 2022).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Specifically, we look at two possible ways in which this might arise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' One, which we refer to as event-level priming, refers to the idea that a word associated with a previously- discussed event may be more likely to be predicted by virtue of this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' This is something that has been previously reported in the N400—Metusalem et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' (2012), for example, found that that merely being related to the event under discussion leads to a smaller N400 response to a word even when that word is inappropriate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Michaelov and Bergen (2022) model this with transformer language models—systems trained to calculate the probability of a word given its context based on the statis- tics of language alone—showing that this effect is explainable with distributional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Thus, it may be the case that the fact that in love is related to, for example, being crazy about someone that leads to it being predicted to be more likely than salted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Following Michaelov and Bergen (2022), we investigate this using 6 Dutch transformer language mod- els (Havinga, 2021, 2022a, 2022b, 2022c;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' de Vries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Delobelle, Winters, & Berendt, 2020), testing whether they show the same effect as humans—that is, whether they predict the canonical sentence the peanut was salted to be less likely than the noncanonical sentence the peanut was in love.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' An alternative possibility is what we refer to as lexical priming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' More simply than in the case of event-level prim- ing, it may be the case that the preceding context involving words such as dancing, smile, singing, crazy, and cute ex- erts a stronger pressure on prediction of in love than peanut does on salted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Because language models appear to exhibit lexical priming effects (Kassner & Sch¨utze, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Misra, Et- tinger, & Rayz, 2020), we instead to turn to word vectors to test this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Word vectors are representations of words derived from their co-occurrence statistics, either directly or based on word embeddings learned by neural networks (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', Dumais, Furnas, Landauer, Deerwester, & Harshman, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Landauer, Foltz, & Laham, 1998;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Mikolov, Sutskever, Chen, Corrado, & Dean, 2013;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Pennington, Socher, & Manning, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Mikolov, Grave, Bojanowski, Puhrsch, & Joulin, 2018;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Tulkens, Emmery, & Daelemans, 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Grave, Bojanowski, Gupta, Joulin, & Mikolov, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' The cosine distance be- tween the vector of each critical word (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' in love or salted) and the mean of the vectors of the words in the preceding con- text can therefore be used to test how similar the critical word is to the words preceding it (Ettinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Uchida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2021), and thus, the extent to which it is predicted based on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' To do this this we used three sets of Dutch word vectors (from Tulkens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' and Grave et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Background A number of researchers have attempted to model the N400 computationally, including using language models (Parviz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Frank et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Aurnhammer & Frank, 2019b;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Michaelov & Bergen, 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Merkx & Frank, 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Michaelov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2021, 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Szewczyk & Federmeier, 2022) and the distances between vector representations of words (Parviz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2011;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Van Petten, 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Ettinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Uchida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' There have also been several attempts to computationally model whether the amplitude of the N400 response is impacted by situation models (Uchida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Venhuizen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2019) and thematic roles (Brouwer, Crocker, Venhuizen, & Hoeks, 2017;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Fitz & Chang, 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Rabovsky, Hansen, & McClelland, 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' To the best of our knowledge, only one previous study (Uchida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2021) has directly attempted to model the dis- course effect found by Nieuwland and van Berkum (2006), and it does not rely on purely distributional linguistic infor- mation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Specifically, Uchida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' (2021) base their model on Wikipedia2Vec (Yamada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2020) vectors—while these include distributional information derived from the surface- level statistics of language, they also include information about hyperlinks between Wikipedia pages, and thus struc- tured semantic relations based on human judgements of rel- evance and importance (Yamada et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Additionally, Uchida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' (2021) only look at the English-translated ver- sion of the single stimulus item presented in (1), and thus, it is unclear whether the results generalize to all the stimuli in the original study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' The current study overcomes these infer- ential limitations by using the original Dutch stimuli and by using neural language models and word vectors trained only on natural language input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' The present study We investigate the adequacy of distributional knowledge to explain the human N400 effect found by Nieuwland and van Berkum (2006) using predictions of neural network language models and the distance between the word vectors of the criti- cal words and their context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Specifically, we ask this question for two possible variants of the effect found by Nieuwland and van Berkum (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Reduction effect Nieuwland and van Berkum (2006) presented experimental participants with short stories such as those in (1) including “canonical” sentences like the peanut was salted or “non- canonical” ones like the peanut was in love.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' One approach to whether language models and human show the same pre- diction patterns (taken by Uchida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2021) is to compare the statistical metrics the critical words elicit in the context of the full story versus in isolation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Without preceding context, these sentences should produce values that match the canon- icality of the sentence, but the difference should attenuate or reverse following the story context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Thus, we ran a statistical analysis testing for an interac- tion between stimulus length (full story or only the last sen- tence) and canonicality (canonical or noncanonical).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Such an interaction would reveal a context-dependent difference in the effect of canonicality on our statistical metrics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' and thus would replicate in neural language models the effect found by Nieuwland and van Berkum (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' However, an interaction between stimulus length and canonicality in this direction could result from either a rever- sal or a decrease in the magnitude of the canonicality effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Canonical stimuli might elicit lower surprisals or smaller co- sine distances in both context conditions, but of different magnitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' For this reason, we label the effect measured by an interaction (in the expected direction) a reduction effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Reversal effect Nieuwland and van Berkum (2006) did not employ the 2 x 2 design that would allow them to detect an interaction— they compared the N400 in context only, finding that canon- ical stimuli actually elicited larger N400 responses than non- canonical stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' To replicate this finding, we test whether the canonical full-length stimuli elicit higher surprisals or greater cosine distances than the noncanonical full-length stimuli, a reversal effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Implications If either language models or word vectors can successfully model the reversal effect, this would suggest that distribu- tional information is sufficient to explain the data reported by Nieuwland and van Berkum (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Thus, while situation models and extralinguistic information may be involved in the neurocognitive system underlying the N400, additional evi- dence is required to prove this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' If neither can model either ef- fect, this would undermine the claim that distributional infor- mation is sufficient to explain the effect found by Nieuwland and van Berkum (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Finally, if either language models or word vectors can successfully model the reduction effect but not the full reversal effect, this may support the idea that distributional information could be used as part of the neu- rocognitive system underlying the N400 response, that it is not sufficient to yield the dynamic contextual sensitivity hu- mans display.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Situation models and other sources of informa- tion might explain the remainder.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Method and approach Original experiment and stimuli Stimuli were used from the original experiment, and are pro- vided online1 by the authors (Nieuwland & van Berkum, 2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' We compared the effect of experimental condition on the N400 and on neural network surprisal (as in Michaelov & Bergen, 2020) and the cosine similarity between the word vector of the critical word and the mean of the word vectors in its context (as in Ettinger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' The stimuli use 60 full-length story frames, each of which has either a canonical or noncanonical predicate, for 120 unique stories.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' As the aim is to model human online compre- hension processes, the models only used the text before the critical words (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', in love or salted) to predict the critical words, so stories were truncated after the critical word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' For the critical sentence stimuli, we isolated the last sentence of these truncated stories, including and up to the critical word in each story (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', The peanut was in love).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' This produced 240 stimuli, as shown in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Predicate Type Stimulus Length Count Canonical Full-length 60 Canonical critical sentence 60 Noncanonical Full-length 60 Noncanonical critical sentence 60 Table 1: Experimental stimuli derived from Nieuwland and van Berkum (2006).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Experiment 1: Language Models Language models In this study, we use six pretrained models available through the transformers package (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Four of these models—Dutch versions of the medium (Havinga, 2021) and large (Havinga, 2022a) GPT-2 models (Radford et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2019) and Dutch versions of the 125 million parameter (Havinga, 2022b) and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='3 billion parameter (Havinga, 2022c) GPT-Neo models (Black, Gao, Wang, Leahy, & Biderman, 2021)— were autoregressive, meaning that they are trained to predict a word based only on its preceding context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' The remain- ing two models—BERTje (de Vries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2019) and Rob- BERT v2 (Delobelle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2020), based on BERT (Devlin, Chang, Lee, & Toutanova, 2019) and RoBERTa (Liu et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2019), respectively—are masked language models, meaning that they are also trained to predict a word based on the text following the critical word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' However, as stated, in the present study, all models were only provided with the context preced- ing the critical words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' We ran the stimuli through each lan- guage model, calculating the surprisal of each critical word that was in the model’s vocabulary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' To do this, we calculated the negative of the logarithm of the probabilities provided 1https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='researchgate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='net/publication/268208198 for each critical word by each of the language models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' We then tested for the reduction and reversal effects with these surprisal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' The language models were run in Python (Van Rossum & Drake, 2009), using the PyTorch (Paszke et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2019) implementation of each model, as provided by the transformers package (Wolf et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Statistical analysis and data manipulation were carried out in R (R Core Team, 2020) using Rstudio (RStudio Team, 2020) and the tidyverse (Wickham et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2019) and lme4 (Bates, M¨achler, Bolker, & Walker, 2015) packages.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Reduction effect In order to test the reduction effect, we constructed linear mixed-effects regression models, with the surprisal calculated from each language model as the dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' In each model, predicate type (canonical or noncanonical) and stimu- lus length (full-length or critical sentence) were fixed effects and story frame (each of the 60) was a random intercept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' For the regressions with the autoregressive models and BERTje surprisal as their dependent variables, we then constructed regressions also including an interaction between predicate type and stimulus length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Using likelihood ratio tests, we found that these regressions including the interaction fit the data significantly better than those without the interaction (GPT-2 Medium: χ2(1) = 116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='0, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' GPT-2 Large: χ2(1) = 107.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='3, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' GPT-Neo 125M: χ2(1) = 61.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='7, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' GPT-Neo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='3B: χ2(1) = 65.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='3, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' BERTje: χ2(1) = 44.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='4, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001), indicating a significant interaction between predicate type and stimulus length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' The regression with RobBERT surprisal as its dependent variable and no in- teraction had a singular fit, but the regression with the inter- action did not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Thus, instead of running a likelihood ratio test to investigate whether there was a significant interaction, we used a a Type III ANOVA with Satterthwaite’s method for estimating degrees of freedom (Kuznetsova, Brockhoff, & Christensen, 2017) on the regression with the interaction, finding it to be a significant predictor of RobBERT surprisal (F(1,71.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='2) = 81.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='0, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Note that all reported p- values are corrected for multiple comparisons based on false discovery rate (Benjamini & Yekutieli, 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' For all language models, there was a significant interac- tion between predicate type and stimulus length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Further in- spection of the regressions showed that in all cases, the in- teraction was in the expected direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Thus, all models displayed the reduction effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' This can be seen visually in Figure 1—in all models, when only the critical sentence was presented, the mean surprisal for critical words in canonical sentences is lower than for critical words in noncanonical sen- tences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Conversely, when the full-length story is presented to the language models, the critical words in the noncanonical sentences elicit a lower or roughly-equal surprisal than the critical words in the canonical sentences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Reversal effect To test for which models this latter finding was statistically significant, we initially attempted to fit linear mixed-effects regression models for each the full-length and critical sen- tence stimulus results for each language model;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' however, this led to several models with singular fits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Instead, we car- ried out pairwise two-tailed t-tests, comparing the surprisal of canonical and noncanonical stimuli for full-length and critical sentence stimuli for each language model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' First, we test whether canonical critical sentence (only) stimuli elicit lower surprisals than noncanonical critical sen- tence stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' For all language models, the decontextual- ized canonical critical sentence stimuli elicit significantly lower surprisals than noncanonical critical sentence stimuli after correction for multiple comparisons (GPT-2 Medium: t(88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='9) = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='65, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' GPT-2 Large: t(89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='0) = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='59, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' GPT-Neo 125M: t(88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='6) = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='20, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' GPT-Neo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='3B: t(89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='0) = −9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='72, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' BERTje: t(48.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='4) = −5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='99, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' RobBERT: t(55.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='1) = −7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='67, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Next, in order to investigate the reversal effect, we test whether canonical full-length stimuli elicit lower surprisals than noncanonical full-length stimuli.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' After correction for multiple comparisons, only the Dutch GPT-2 models suc- cessfully model the reversal effect—they are the only mod- els for which canonical full-length stimuli elicit signifi- cantly higher surprisals than noncanonical full-length stimuli (GPT-2 Medium: t(86.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='3) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='61, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' GPT-2 Large: t(88.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='5) = 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='34, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' With the other models, while the difference was in the cor- rect direction, it was not significant after correction for mul- tiple comparisons (GPT-Neo 125M: t(89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='0) = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='57, p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='000 ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' GPT-Neo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='3B: t(89.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='71, p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' BERTje: t(51.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='5) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='79, p = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='000;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' RobBERT: t(46.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='6) = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='31, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='120).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' However, it is worth noting that the contrast between the two sets of results (critical sentence only vs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' full stimulus) means that significant canonicality effects for the critical sen- tence stimuli disappear in the full-length stimuli, underscor- ing the presence of a reduction effect in the Dutch GPT-Neo models, BERTje, and RobBERT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Discussion Nieuwland and van Berkum (2006) found that in a suitably supportive context, noncanonical stimuli like de pinda was verliefd (‘the peanut was in love’) elicit smaller N400 re- sponses than canonical stimuli such as de pinda was gezouten (‘the peanut was salted’)—context not only mitigated but re- verse the effect of animacy violation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' We find that two language models also display this rever- sal effect: Dutch GPT-2 Medium (Havinga, 2021) and Dutch GPT-2 Large (Havinga, 2022a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' When these models are pre- sented with the same contexts, the surprisal of critical words in the noncanonical condition is lower than that elicited by those in the canonical condition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' This is not the case for the remaining four language mod- els: Dutch GPT-Neo 125M (Havinga, 2022b), Dutch GPT- Neo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='3B (Havinga, 2022c), BERTje (de Vries et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2019), and RobBERT (Delobelle et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' However, these mod- GPT−2 Large GPT−Neo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='3B RobBERT GPT−2 Medium GPT−Neo 125M BERTje Target Sentence Full Story Target Sentence Full Story Target Sentence Full Story 0 5 10 15 20 0 5 10 15 20 Stimulus Length Surprisal Predicate Type Canonical Uncanonical Figure 1: Surprisal elicited by critical words for each predicate type and stimulus length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' els do display the weaker reduction effect, and further, the ab- sence of a significant difference between conditions for these models when presented with the full stories shows that the difference between canonical and noncanonical critical sen- tence stimuli is not just reduced, but disappears entirely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' It may be tempting to infer that the architecture of au- toregressive transformers, and in particular, those based on the GPT-2 architecture, leads to success capturing the ef- fect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' However, it should be noted that before correction for multiple comparisons, RobBERT also successfully displays the reversal effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' In addition, not all language models had the same vocabulary, and thus, a different number of items were analyzed across models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' For these reasons, and because these models are all of various sizes and trained on several different datasets, we believe it would be premature to draw conclusions about how language model architecture impacts whether a model displays the reversal effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Experiment 2: Word Embeddings Cosine Distance In this study, we used 3 sets of pretrained word vectors: the 300-dimensional Dutch fastText vectors (Grave et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2018) trained on Dutch text from Wikipedia2 and Common Crawl3 and two 320-dimensional Dutch word vectors re- leased by Tulkens et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' (2016)—one trained on COW (COr- pora from the Web;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Sch¨afer & Bildhauer, 2012) and one trained on a Combined corpus made up of the SoNaR cor- pus (Oostdijk, Reynaert, Hoste, & Schuurman, 2013) and text from Wikipedia and Roularta4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' To calculate cosine distance, 2https://nl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='wikipedia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='org/ 3https://commoncrawl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='org/ 4https://www.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='roularta.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='be we calculated the cosine distance (using SciPy;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Virtanen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2020) between the mean of the word embeddings for all words in the preceding context and the word embedding for the critical word.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Only critical words that were present in the pretrained embeddings were included, and words in the context that were not present in the pretrained embeddings were ignored when calculating cosine distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' The cosine distances for critical words in each condition are shown in Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Reduction effect As with language model surprisal, we constructed linear mixed-effects regressions with predicate type and stimulus length as fixed effects and story from as a random intercept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' With these models, the cosine distance calculated using each set of pretrained embeddings was the dependent variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' The interaction between predicate type and stimulus length was significant for all embeddings after correcting for multiple comparisons (fastText: χ2(1) = 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='0, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='003;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Combined: χ2(1) = 40.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='8, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' COW: χ2(1) = 66.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='4, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Reversal effect When comparing the cosine distances calculated between the embedding of the critical words and the preceding words of the critical sentence using two-tailed t-tests as with sur- prisal, there was a significant difference between canonincal and noncanonical critical words for Combined and COW em- beddings (Combined: t(116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='9) = −3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='48, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='004;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' COW: t(116.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='5) = −4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='45, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001), but not fastText embeddings (fastText: t(118.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='0) = −1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='96, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='237).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Similarly, when comparing the cosine distances between the critical word and the preceding words of the full story, there was a significant difference between canonical and FastText Combined COW Target Sentence Full Story Target Sentence Full Story Target Sentence Full Story 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='8 Stimulus Length CosineDistance Predicate Type Canonical Uncanonical Figure 2: Cosine distance elicited by critical words for each predicate type and stimulus length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' noncanonical critical words for Combined and COW em- beddings (Combined: t(117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='0) = 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='82, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' COW: t(117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='0) = 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='78, p < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='001), but not fastText embeddings (fastText: t(117.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='4) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='68, p = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='418).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Discussion The cosine distances calculated from all three sets of word vectors displayed the reduction effect, and two out of three displayed the reversal effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Thus, the results suggest that the N400 effect reported by Nieuwland and van Berkum (2006) can be explained on the basis of distributional linguis- tic knowledge alone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' The present study corroborates the finding of Uchida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' (2021), and expands upon it in several ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' First, it explic- itly tests for the reversal effect—not just whether canonical and noncanonical stimuli differ depending on whether there is a preceding story or not, but also whether the noncanon- ical sentence is more expected than the canonical when the story is present.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Second, we found that language model sur- prisal can model the effect for multiple stimuli, not just the peanut was in love example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Third, we found that the effect can be modeled in Dutch, the language in which the human study was carried out.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' And finally, we found that embeddings trained on text data only (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', without additional information) are able to model the effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' General Discussion Human comprehenders use context to update expectations about upcoming words, making a sentence that would be highly unlikely on its own more predictable than a sentence that would be relatively likely on its own.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' More strikingly, humans do this even when the event described is implausible, violating the constraint, for instance, that only animate, con- scious entities can fall in love.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' The human comprehension system is quite flexible if it can update expectations about what peanuts, for example, can do based only a story that in- directly implies the animacy of a fictional peanut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' It has often been assumed that this flexibility requires sit- uation models that are explicitly structured (Venhuizen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2019) or involve non-linguistic world knowledge or experi- ence (Uchida et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' However, the present findings show that it is possible for a purely linguistic model with no direct experiential grounding to update its expectations based on the linguistic context and its extant knowledge of the statistics of language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' One possibility suggested by this re- sult is that that the dynamics of lexical prediction in language models may in some implicit, unspecified way approximate situation models even though they only learn from distribu- tional information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' But there may be an even simpler explanation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' We com- puted the similarity between critical words and their contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Within final sentences alone, canonical critical words were more similar to their contexts than noncanonical words, but when we include the full story context, it is the noncanoni- cal critical words that are more similar to their contexts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' This could explain both the human results and the language model results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' The amplitude of the N400 to a given word is reduced when it is semantically related to a previously-seen word (Bentin, McCarthy, & Wood, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Rugg, 1985;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Van Petten & Kutas, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Kutas & Hillyard, 1989;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Holcomb, 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Ku- tas, 1993;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Lau, Holcomb, & Kuperberg, 2013);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' and similarly, words that are semantically related to previously-mentioned words are more strongly predicted by language models than words that are not (Misra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2020;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Michaelov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Overall, then, our results show that in principle, it is possible that the pattern in the N400 responses reported by Nieuwland and van Berkum (2006) may not rely on situation models at all, but rather reflect some form of lexical priming.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' It may still be the case that humans use structured or semantically-rich situation models in online language com- prehension (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', Kuperberg et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=', 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' However, the results of the study carried out by Nieuwland and van Berkum (2006) appear to provide weaker evidence for this than previ- ously believed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/bdFAT4oBgHgl3EQf5B6R/content/2301.08731v1.pdf'} +page_content=' Language model predictions or even lexical priming based on language statistics appear to be sufficient.' metadata={'source': 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Mahapatra2023 +©ESO 2023 +January 27, 2023 +From exo-Earths to exo-Venuses +Flux and Polarization Signatures of Reflected Light +G. Mahapatra,1,⋆, F. Abiad1, L. Rossi2, and D.M.Stam1 +1 Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands +e-mail: g.mahapatra@tudelft.nl +2 CNRS/INSU, LATMOS-IPSL, Guyancourt, France +Accepted 11 January, 2023 +ABSTRACT +Context. Terrestrial-type exoplanets in or near stellar habitable zones appear to be ubiquitous. It +is, however, unknown which of these planets have temperate, Earth-like climates or e.g. extreme, +Venus-like climates. +Aims. Technical tools to distinguish different types of terrestrial-type planets are crucial for deter- +mining whether a planet could be habitable or incompatible with life as we know it. We investigate +the potential of spectropolarimetry for distinguishing exo-Earths from exo-Venuses. +Methods. We present numerically computed fluxes and degrees of linear polarization of starlight +that is reflected by exoplanets with atmospheres in evolutionary states ranging from similar to the +current Earth to similar to the current Venus, with cloud compositions ranging from pure water +to 75% sulfuric acid solution, for wavelengths between 0.3 and 2.5 µm. We also present flux and +polarization signals of such planets in stable but spatially unresolved orbits around the star Alpha +Centauri A. +Results. The degree of polarization of the reflected starlight shows larger variations with the +planetary phase angle and wavelength than the total flux. Across the visible, the largest degree of +polarization is reached for an Earth-like atmosphere with water clouds, due to Rayleigh scattering +above the clouds and the rainbow feature at phase angles near 40◦. At near-infrared wavelengths, +the planet with a Venus-like CO2 atmosphere and thin water clouds shows the most prominent +polarization features due to Rayleigh-like scattering by the small cloud droplets. A planet in +a stable orbit around Alpha Centauri A would leave temporal variations on the order of 10−13 +W/m3 in the total reflected flux and 10−11 in the total degree of polarization as the planet orbits +the star and assuming a spatially unresolved star-planet system. Star-planet contrasts are in the +order of 10−10 and vary proportionally with planetary flux. +⋆ Now at SRON Netherlands Institute for Space Research, Leiden, The Netherlands +Article number, page 1 of 24 +arXiv:2301.11314v1 [astro-ph.EP] 26 Jan 2023 + +Mahapatra et al.: From exo-Earths to exo-Venuses +Conclusions. Current polarimeters appear to be incapable to distinguish between the possible +evolutionary phases of spatially unresolved terrestrial exo-planets, as a sensitivity close to 10−10 +would be required to discern the planetary signal given the background of unpolarized starlight. +A telescope/instrument capable of achieving planet-star contrasts lower than 10−9 should be able +to observe the large variation of the planet’s resolved degree of polarization as a function of its +phase angle and thus be able to discern an exo-Earth from an exo-Venus based on its clouds’ +unique polarization signatures. +Key words. Exo-planets – Venus – Radiative transfer – Polarimetry +1. Introduction +Despite having similar sizes, being formed around the same time and from similar materials, it is +clear that the Earth and Venus have evolved into dramatically different worlds. While it is generally +acknowledged that Venus once had much larger amounts of water than today, it is still debated +whether Venus was once more Earth-like with oceans of water before the runaway-greenhouse- +effect took off (Donahue et al. 1982), or whether the atmospheric water vapour never actually +condensed on the surface (Turbet et al. 2021). Bullock & Grinspoon (2001) conducted a detailed +study of the possible evolution of Venus’s climate over long time periods starting with a water +vapour enriched atmosphere. Terrestrial-type exoplanets are also expected to harbour a wide vari- +ety of atmospheric compositions with maybe only a few planets hospitable to life as we know it. +Various climate models suggest that the likelihood of a planetary atmosphere exhibiting a Venus- +like runaway-greenhouse-effect is higher than that of an atmosphere in an Earth-like, N2-dominated +state (Lincowski et al. 2018, and references therein). A study by Kane et al. (2020) even shows that +Jupiter’s migration might have stimulated the runaway-greenhouse-effect on Venus, suggesting that +there could be more Venus-analogs than Earth-analogs in planetary systems with Jupiter-like plan- +ets. +As planned powerful telescopes and dedicated, sensitive detection techniques will allow us to +characterize smaller exoplanets in the near-future, it will become possible to probe terrestrial-type +planets in and near the habitable zones of solar-type stars and to find out whether they resemble +Earth or Venus, or something else all together. The high-altitude cloud deck on an exo-Venus would +make it difficult to use a technique like transit spectroscopy for the characterization of the planet +as the clouds themselves would block the transmission of the starlight and apart from a spectral +dependence of the cloud optical thickness which could leave a wavelength dependent transmission +through the cloud tops, the microphysical properties of the cloud particles, such as their composi- +tion, shape and size distribution would remain a mystery. Also, the clouds would inhibit measuring +trace gas column densities as they would block the planet’s lower atmosphere and only allow tran- +sit spectroscopy of the highest regions of the atmosphere (see, e.g. Lustig-Yaeger et al. 2019a, and +references therein). Indeed, a Venus-like ubiquitous cloud deck could possibly be mistaken for the +Article number, page 2 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +planet’s surface, as one would only measure the transmittance through the gaseous atmosphere +above the clouds, possibly inferring the atmosphere to be thin and eroding (Lustig-Yaeger et al. +2019b). +Jordan et al. (2021) modelled the photochemistry of some of the primary sulphuric chemical +species that should be responsible for the formation and sustenance of Venus’s sulfuric acid solution +clouds, such as SO2, OCS and H2S, and found that the abundances of such species above the cloud +deck would depend heavily on the effective temperature and distance to the parent star, with their +abundances decreasing with increasing temperature and being depleted, as we see on Venus, in +the presence of a star like the Sun. Thus it would be challenging to rule out the possibility of +an exoplanet being a Venus analog solely on the basis of the detection of such chemical species +in transmission spectra (Jordan et al. 2021). Indeed, the full characterization of rocky exoplanets +and their classification appears to require the direct imaging of starlight that is reflected by such +planets and/or the thermal radiation that is emitted by them. While telescopes able to perform such +measurements are not yet available, plans are underway for their development and deployment +(Keller et al. 2010; The Luvoir Team 2019). +While most of telescopes and instruments are designed for only measuring total fluxes of exo- +planets, including (spectro)polarimetry is also being considered. The main reason to include (spec- +tro)polarimetry (see e.g. Rossi et al. 2021, and references therein) is that it increases the contrast +between star and planet, as the stellar flux will be mostly unpolarized when integrated over the disk +(Kemp et al. 1987), while the flux of the reflected starlight will usually be (linearly) polarized. And +in addition, (spectro)polarimetry can be used for the characterization of planetary atmospheres and +surfaces. As a classic example of the latter, Hansen & Hovenier (1974) used Earth-based measure- +ments of the disk-integrated degree of polarization of sunlight that was reflected by Venus in three +spectral bands and across a broad phase angle range, to deduce that the particles forming Venus’s +main cloud deck consist of 75% sulfuric acid solution, that the effective radius of their size dis- +tribution is 1.05 µm, and that the effective width of the distribution is 0.07. They also derived the +cloud top altitude (at 50 mbars) by determining the amount of Rayleigh scattering in the gas above +the cloud tops at a wavelength of 0.365 µm. This was later confirmed by the Pioneer Venus mission +which performed in-situ measurements using a nephelometer on a probe that descended through +the clouds (Knollenberg & Hunten 1980). +Polarimetry proved to be an effective technique for disentangling Venus’s cloud properties be- +cause the scattering particles leave a unique angular polarization pattern in the reflected sunlight +depending on the particles’ micro- and macro-physical properties (for an extensive explanation +of the application of polarimetry for the characterization of planetary atmospheres, see Hansen +& Travis 1974). While multiple scattered light usually has a low degree of polarization, and thus +dilutes the angular polarization patterns of the singly scattered light, the angles where the abso- +lute degree of polarization reaches a local maximum and/or where it is zero (the so-called ’neutral +points’) are preserved and thus still allow for the characterization of the particles. +Article number, page 3 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +Another factor in the successful application of (spectro)polarimetry for the characterization of +Venus’s clouds and hazes is that with Earth-based telescopes, inner planet Venus can be observed +at a wide range of phase angles, thus allowing observations of the angular variation of the degree +of polarization due to the light that has been singly scattered by the atmospheric constituents. In +our solar system, only Venus, Mercury, and the Moon can be observed at a large phase angle range +with Earth-based telescopes (ignoring the proximity of Mercury to the Sun). To effectively apply +polarimetry to the outer planets in the solar system, a polarimeter onboard a space mission would be +needed. An example of such an instrument was OCCP onboard NASA’s Galileo mission (Russell +et al. 1992) that orbited Jupiter. Regarding exoplanets, however, the range of observable phase +angles depends on the inclination angle of the planetary orbits: for a face-on orbit, the planet’s +phase angle will be 90◦ everywhere along the orbit, while for an edge-on orbit, the phase angle will +range from close to 0◦ (when the planetary disk is fully illuminated) to 180◦ (when the night-side +of the planet is in view). The precise range of accessible phase angles would of course depend on +the observational technique and e.g. the use of a coronagraph or star-shade. +Here we investigate the total flux and degree of polarization of starlight that is reflected by +terrestrial-type exoplanets, focusing on the possible evolutionary stages of Venus as described by +Bullock & Grinspoon (2001). Our goal is to identify characteristic signatures that could help to +identify the properties of exo-Venuses, thus to guide the design of future telescope instruments. +We compute the disk-integrated total and polarized fluxes of light reflected between wavelengths +of 0.3 to 2.5 µm. First, we study the single scattering properties of spherical cloud droplets of pure +water (H2O) or 75% sulphuric acid (H2SO4) in order to identify potentially distinct signatures for +each particle type as a function of wavelength and planetary phase angle. Second, we compute the +multiple scattered flux and polarization signals that are integrated over the planet’s illuminated disk +as functions of the planet’s phase angle. Third, we compute the signals of the planets in the four +evolutionary phases in stable orbits around the nearby solar-type Alpha Centauri A, simulating the +observations of such planets if they are spatially unresolved from their parent star. +The outline of this paper is as follows. In Sect. 2, we define the fluxes and polarization of +planets, and we describe our numerical algorithm and the four model planets in the evolutionary +phases as described by Bullock & Grinspoon (2001). In Sect. 3, we present the total and polarized +fluxes as computed for planets that are spatially resolved from their star and for planets that are +spatially unresolved. In the latter case, the planet’s signal is thus combined with the stellar light. +We specifically assume that our model planet orbits the solar-type star Alpha Centauri A. In Sect. 4, +we summarize our results and present our conclusions. +2. Numerical method +2.1. Flux and polarization definitions +In this paper, we present the flux and polarization signals of starlight that is reflected by potentially +habitable exoplanets that orbit solar-type stars, and in particular, Alpha Centauri A. Because these +Article number, page 4 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +planets will be very close in angular distance to their parent star, they will usually be spatially unre- +solved, i.e. it will not be possible to spatially separate the planet’s signal from that of its parent star. +The flux vector Fu (u= ’unresolved’) that describes the light of the star and its spatially unresolved +planet, and that arrives at a distant observer is then written as +Fu(λ, α) = Fs(λ) + Fp(λ, α), +(1) +with Fs the star’s flux vector and Fp that of the planet. Furthermore, λ is the wavelength (or wave- +length band), and α is the planetary phase angle, i.e. the angle between the star and the observer +as measured from the center of the planet. We assume that the light of the star is captured together +with the starlight that is reflected by the planet. A telescope with a coronagraph or star-shade would +of course limit the amount of captured direct starlight, depending on its design and the angular dis- +tance between the star and the planet. +A flux (column) vector is given by (Hansen & Travis 1974) +F = [F, Q, U, V], +(2) +with F the total flux, Q and U the linearly polarized fluxes, and V the circularly polarized flux. The +dimensions of F, Q, U, and V are W m−2, or W m−3 when defined per wavelength. +Measurements of FGK-stars, such as the Sun and Alpha Centauri A, indicate that their (disk- +integrated) polarized fluxes are virtually negligible (Kemp et al. 1987; Cotton et al. 2017), thus we +describe the star’s flux (column) vector that arrives at the observer located at a distance D as +Fs(λ) = Fs(λ) 1 = R2 +s +D2 πB(λ, Ts) 1, +(3) +with πB the stellar surface flux, Ts the star’s effective temperature, Rs the stellar radius, and 1 the +unit (column) vector. The parameter values that we adopt for the Alpha Centauri A system are +listed in Table 1. +Because of the huge distances to stars and their planets, flux vector Fp of the starlight that is +reflected by an exoplanet pertains to the planet as a whole, thus integrated across the illuminated +and visible part of the planetary disk. It is given by (see e.g. Rossi et al. 2018) +Fp(λ, α) += +AG(λ) Rp(λ, α) +r2 +p +D2 +R2 +s +d2 πB(λ, Ts) 1 +(4) += +AG(λ) R1p(λ, α) +r2 +p +D2 +R2 +s +d2 πB(λ, Ts). +(5) +Here, AG is the planet’s geometric albedo, Rp the matrix describing the reflection by the planet +and R1p its first column, rp is the planet’s radius, d the distance between the star and the planet, +and D the distance to the observer. The planet’s reflection is normalized such that planetary phase +function R1p, which is the first element of R1p, equals 1.0 at α = 0◦. +Article number, page 5 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +The contrast C between the total flux of the planet and the total flux of the star is then given by +C(λ, α) = Fp(λ, α) +Fs(λ) += AG(λ) R1p(λ, α) +r2 +p +d2 , +(6) +with Fp the first element of the planetary flux vector Fp. Using the parameters from Table 1, the con- +trast C between a planet with the radius of Venus at a Venus-like distance from Alpha Centauri A +equals about 2·10−9AG at α = 0◦ (at this phase angle, the planet would actually be precisely behind +the star with respect to the observer and thus out of sight). +The degree of polarization of the spatially resolved planet (without including any direct light +of the star) is defined as +Pp = +� +Q2p + U2p +Fp +, +(7) +where we ignore the planet’s circularly polarized flux Vp as it is expected to be very small com- +pared to the linearly polarized fluxes (Rossi & Stam 2018). We also ignore the circularly polarized +fluxes in our radiative transfer computations, as this saves significant amounts of computing time +without introducing significant errors in the computed total and linearly polarized fluxes (see Stam +& Hovenier 2005). +Fluxes Qp and Up are defined with respect to the planetary scattering plane, which is the plane +through the planet, the star and the observer. In case the planet is mirror-symmetric with respect to +the planetary scattering plane, linearly polarized flux Up equals zero and we can use an alternative +definition of the degree of polarization that includes the polarization direction as follows +Pp = −Qp +Fp +. +(8) +If Pp > 0 (Pp < 0), the light is polarized perpendicular (parallel) to the reference plane. +In case a planet is not completely spatially resolved from its parent star, and the background of +the planet on the sky is thus filled with (unpolarized) starlight, the observable degree of polarization +Pu can be written as (cf. Eqs. 6-7) +Pu = +� +Q2p + U2p +Fp + xFs += +Fp +Fp + xFs +Pp = +C +C + xPp, +(9) +with x the fraction of the stellar flux that is in the background, which will depend on the angular +distance between the star and the planet, on the starlight suppressing techniques that are employed, +such as a coronagraph or star-shade, and on the spatial resolution of the telescope at the wavelength +under consideration. This equation also holds for the signed degree of polarization as given by +Eq. 8. If x = 1, the planetary and the stellar flux are measured together. In that case, +Pu = +C +C + 1Pp ≈ CPp. +(10) +Article number, page 6 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +Table 1. The values of the parameters describing the planetary system of Alpha Centauri A used in our +numerical modelling1. +Parameter (unit) +Symbol +Value +Stellar radius (RSun) +Rs +1.2234 +Stellar effective temperature (K) +Ts +5790 +Planet radius (km) +rp +6052 +Planet orbital distance (AU) +d +0.86 +Planet orbital period (yr) +P +0.76 +Distance to the system (ly) +D +4.2 +Angular separation (arcsecs) +S +0.67 +Here we used the fact that the contrast C will usually be very small (on the order of 10−9 as shown +earlier). +The planet’s degree of polarization Pp and the contrast C both depend on λ and α, but generally +in a different way. The dependence of Pu on λ and α will thus generally differ from that of either +Pp or C. +2.2. Our radiative transfer algorithm +Our procedure to compute the flux vector Fp (Eq. 5) of the starlight that is reflected by the planet, +is described in Rossi et al. (2018). The radiative transfer algorithm is based on an efficient adding- +doubling algorithm (de Haan et al. 1987) and fully includes polarization for all orders of scattering. +With this algorithm, through the use of a Fourier-series expansion of the planetary reflection matrix +Rp, the reflected flux vector can be computed for any planetary phase angle α. +Our model planetary atmospheres consist of horizontally homogeneous layers. For each layer, +we prescribe the total optical thickness b, the single-scattering albedo a, and the single-scattering +matrix P. Our layered model atmosphere is bounded below by a Lambertian reflecting surface (i.e. +the light is reflected isotropically and unpolarized) with an albedo asurf. +A layer’s optical thickness b at a wavelength λ is the sum of the optical thicknesses of the gas +molecules, bm, and, if present, the cloud particles, bc. We ignore other atmospheric particles, such +as haze particles. The single-scattering matrix P of a mixture of gas molecules and cloud particles +in a layer is given by +P(Θ, λ) = bm +sca(λ) Pm(Θ, λ) + bc +sca(λ) Pc(Θ, λ) +bmsca(λ) + bcsca(λ) +, +(11) +with subscript ‘sca’ referring to ‘scattering’, thus bsca = ab, with a the single scattering albedo. +Furthermore, Pm is the single-scattering matrix of the gas molecules, and Pc that of the cloud +particles. Θ is the single scattering angle: Θ = 180◦ − α. +We use two types of model atmospheres to study the influence of an exoplanet’s atmospheric +evolution on the reflected light signals: an Earth-like and a Venus-like atmosphere. For our Earth- +like atmosphere, we define the pressure and temperature across 17 layers, representing a mid- +1 The orbital distance d of the planet has been chosen such that it receives the same stellar flux as Venus +receives from the Sun, and in accordance with the orbit stability requirements for a planet around Alpha +Centauri A predicted by Quarles & Lissauer (2016). For the radius of the Sun, RSun, we use 695,700 km. +Article number, page 7 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +Fig. 1. The four evolutionary phases of the model planets (Bullock & Grinspoon 2001). In Phases 1 and 2, +the clouds consist of liquid water droplets, and in Phases 3 and 4, of liquid sulfuric acid solution droplets. +The cloud optical thickness is indicated by bc and the cloud particle effective radius by reff. For the effective +variance veff of the size distributions in Phases 1-3, we use 0.1, and for Phase 4, veff = 0.07. +latitude summer profile (following Stam 2008). For our Venus-like atmosphere, we use 71 layers +with pressure and temperature profiles from the Venus International Reference Atmosphere (VIRA) +(Kliore et al. 1985), representing a mid-latitude afternoon profile. With these vertical profiles, and +assuming anisotropic Rayleigh scattering (Hansen & Travis 1974), we compute each layer’s single +scattering matrix Pm and the scattering optical thickness bm +sca. We neglect absorption, thus bm = bm +sca. +The depolarization factor for computing Pm and bm +sca for anisotropic Rayleigh scattering depends on +the atmospheric composition. For the Earth-like atmosphere, we use a (wavelength independent) +depolarization factor of 0.03, which is representative for dry air, and for the Venus-like atmosphere, +we use 0.09, which is representative for a pure CO2 atmosphere (Hansen & Travis 1974). We +use wavelength-independent refractive indices of 1.00044 and 1.00027 for the Venus-like and the +Earth-like model atmospheres, respectively; note that this assumption has a negligible effect on the +reflected total and polarized fluxes. +The cloud particles in our model atmospheres are spherical and distributed in size according +to a two-parameter gamma size distribution (see Hansen & Travis 1974) that is described by an +effective radius reff and an effective variance veff. The terrestrial clouds are located between 1 and +3 km altitude, and the Venusian clouds, depending on their evolutionary phase, between 47 and +80 km. The cloud optical thickness has a uniform vertical distribution through the altitude range +(see Fig. 1). +The single-scattering properties of the cloud particles are computed using Mie-theory (De Rooij +& Van der Stap 1984), as these particles are expected to be spherical. For these computations we +specify the wavelength λ and nr, the refractive index of the cloud particles. The cloud particles are +composed of either pure water or a sulphuric acid solution with varying concentration. We use the +refractive index of water from Hale & Querry (1973) and that of sulphuric acid with 75 % acid +concentration from Palmer & Williams (1975). We use a negligible value for the imaginary part of +the particles’ refractive indices, ni = 10−8. +Article number, page 8 of 24 + +Cloud +thickness +Ititude +A +Cloud top = 3 km +Cloud top = 80 km +Cloud top = 65 km +Cloud top = 65 km +bc = 2 +bc = 4 +bc = 120 +bc = 30 +reff = 10.0 μm +reff = 0.5 μm +reff = 2.0 μm +reff = 1.0 μm +Phase 1 +Phase 2 +Phase 3 +Phase 4 +Current Earth +Thin clouds Venus +Thick clouds Venus +Current VenusMahapatra et al.: From exo-Earths to exo-Venuses +2.3. Cloud properties through the planet’s evolution +It is suspected that early Venus had a thin, Earth-like atmosphere and (possibly) an Earth-like ocean +that was later lost due to the runaway greenhouse effect (Donahue et al. 1982; Kasting 1988; Way & +Del Genio 2020). As the planet’s surface heated up, the water would have evaporated and enriched +the atmosphere with water vapor. The macroscopic cloud properties for the 4 evolutionary phases +that we will use are illustrated in Fig. 1. +We start our evolutionary model of Venus assuming Earth-like conditions (Phase 1), i.e. an +atmosphere consisting of 78% N2 and 22% O2. Model simulations showed that the actual depen- +dence of the total and polarized flux signals on the percentage of oxygen appeared to be negligible. +Hence we used the present day Earth atmosphere as the Earth-like atmosphere model while the ac- +tual percentage of oxygen on an exo-planet could be different. The cloud particles have an effective +radius reff of 10 µm in agreement with ISCCP (Tselioudis 2001) and an effective variance veff of +0.1. The total cloud optical thickness bc is 10.0 at λ = 0.55 µm and the cloud layer extends from +2 to 4 km. +The next evolutionary phases are also inspired by the Venus climate model of Bullock & Grin- +spoon (2001). In Phase 2, the atmosphere is Venus-like as it consists of pure CO2 gas, and has +relatively thin liquid water clouds with bc = 4, and with the cloud tops at 80 km. For this phase, +we use reff of 0.5 µm, which is smaller than the present day value, because the atmosphere is ex- +pected to be too hot for strong condensation to take place thus preventing the particles to grow +larger. In Phase 3, the clouds are thick sulphuric-acid solution clouds, with bc = 120 and the cloud +tops at 65 km, because the atmosphere is cool enough to allow condensation and/or coalescence +of saturated vapour over a large altitude range. Since the region of condensation covers a large +altitude range, the particles can grow large until they evaporate. In this phase, reff = 2 µm, which is +twice the effective radius of the present day Venus cloud particles. For both phases 2 and 3, we use +veff = 0.1. In Phase 4, the clouds have the present-day properties of Venus’s clouds with bc = 30 +and the cloud tops at 65 km (Rossi et al. 2015; Ragent et al. 1985). For the cloud particle sizes +in this phase, we use reff = 1.05 µm and veff = 0.07 following the values derived by Hansen & +Hovenier (1974). We ignore the absorption by cloud particles in the UV in all of our Venus-like +clouds to avoid adding complexity and because the exact nature and location of the UV-absorption +is still under debate (Titov et al. 2018). +Figure 2 shows the phase function (i.e. single scattering matrix element P11) and the degree +of linear polarization for unpolarized incident light (the ratio of single scattering matrix elements +−P21/P11) that has been singly scattered by the four different types of cloud particles as functions +of α (i.e. 180◦ - Θ), for a range of wavelengths λ. +As can be seen in Fig. 2, the phase functions show strong forward scattering peaks (near α = +180◦, thus when the night-side of the planet would be turned towards the observer) that decrease +with increasing λ, thus with decreasing effective particle size parameter xeff = 2πreff/λ (for the +large H2O cloud particles, with reff = 10 µm, this decrease is not readily apparent from the figure). +Article number, page 9 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +Fig. 2. The total flux and degree of polarization of incident unpolarized light that has been singly scattered by +four different types of cloud particles, as functions of the phase angle α and the wavelength λ. Left column: the +total flux or phase function (single scattering matrix element P11). Right column: the degree of polarization +(−P21/P11). First row: H2O particles with reff = 10 µm and veff = 0.1 (belonging to the Phase 1 model planet: +’Current Earth’); Second row: H2O particles with reff = 0.5 µm and veff = 0.1 (Phase 2: ’Thin clouds Venus’); +Third row: 75% H2SO4 particles with reff = 2.0 µm and veff = 0.1 (Phase 3: ’Thick clouds Venus’); Fourth +row: 75% H2SO4 particles with reff = 1.05 µm and veff = 0.07 (Phase 4: ’Current Venus’). +The H2O particles with reff = 10 µm show a moderate local maximum in the phase function around +α = 40◦, which is usually referred to as the primary rainbow (see e.g. Hansen & Travis 1974). The +large H2O and the H2SO4 particles also produce higher fluxes towards α = 0◦ that are referred +to as the glory (Laven 2008; García-Muñoz et al. 2014; Markiewicz et al. 2014; Rossi et al. 2015; +Markiewicz et al. 2018). For the small H2O particles and the H2SO4 particles at larger wavelengths, +the phase functions become more isotropic and the glory and other angular features disappear. +Figure 2 also shows the degree of linear polarization of the singly scattered light. This degree +of polarization appears to be more sensitive to the particle composition than the scattered flux, +especially for λ between 0.5 and 2 µm, where H2O particles yield relatively high positive degrees +of polarization (perpendicular to the scattering plane) between phase angles of about 20◦ and 100◦, +whereas the H2SO4 particles impart a mostly negative degree of polarization through a broad range +of phase angles, except for narrow regions around α = 20◦ and 80◦. The tiny, reff = 0.5 µm, water +droplets have a strong, broad positive polarization region for λ ≥ 1 µm, where they are so small +with respect to the wavelength that they scatter like Rayleigh scatterers. +As mentioned before (see e.g. Hansen & Travis 1974; Hansen & Hovenier 1974), patterns in +the single scattering degree of polarization are generally preserved when multiple scattered light is +added, as the latter usually has a low degree of polarization, and thus adds mostly total flux, which +Article number, page 10 of 24 + +H20,reff:10.0μm +H20,reff:10.0μm +2.5 +0.5 +2.5 +Wavelength (μm) +Wavelength (μm) +0.500 +0.6 +2.0 +102 +2.0 +0.000 +Phase function +0.4 +1.5 +0.037 +101 +1.5 +0.2 +0.000 +1.0 +0.139 +1.931 +0.0 +0.139 +7.197~ +100 +0.000 +0.500 +0.2 +0.5 +0.5 +10~1 +.82 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +H20, reff: 0.5 μm +H20,reff: 0.5μm +2.5 +2.5 +Wavelength (μm) +Wavelength (μum) +0.6 +2.0 +102 +Phase function +0.200 +0.800 + 0.4 +1.5 +0.518 +101 +1.5 +0.2 +1.931 +0.400 +1.0 +1.0 +0.200 +0.0 +100 +0.2 +0.5 +0.139 +0.5 +0.000 +0.000 +0200 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +75%H2SO4,reff:2.0μm +75%H2SO4,reff:2.0μm +2.5 +2.512 +2.5 +Wavelength (um) +Wavelength (μm) +0.000 +0.6 +2.0 +102 +2.0 +Phase function +0.4 +1.5 +101 +1.5 +0.2 +1.0 +1.0 +0.0 +100 +0.398 +-0.2 +0.5 +0.5 +0.000 +0.200 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +75%H2SO4,reff:1.05μm +75%H2S04,reff:1.05μm +2.5 +1668- +0.077 +E +0.25 +Wavelength (um) +Wavelength (μm) +0.6 +2.0 +102 +Phase function +2.0 +0.4 +1.5 +1.5 +0.000 +101 +0.2 +4.642- +1.0 +1.0 +0.0 +100 +-0.2 +0.5 +10-1 +0.5 +0.028 +0.000 +0.250 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Phase angle (deg) +Phaseangle (deg)Mahapatra et al.: From exo-Earths to exo-Venuses +subdues angular features, but does not change the angular pattern (local maxima, minima, neutral +points) itself. The single scattering angular features in the polarization will thus also show up in +the polarization signature of a planet as a whole, and can be used for characterisation of the cloud +particle properties and thus possibly of various phases in the evolution of a Venus-like exoplanet. +This will be investigated in the next section. +3. Results +Here we present the disk-integrated total flux and degree of polarization of incident unpolarized +starlight that is reflected by the model planets at different wavelengths λ and for phase angles α +ranging from 0◦ to 180◦. The actual range of phase angles at which an exoplanet can be observed +depends on the inclination angle i of the planet’s orbit (the angle between the normal on the orbital +plane and the direction towards the observer): α ranges between 90◦ − i to 90◦ + i. Obviously, +at α = 0◦, the planet would be precisely behind its star, and at 180◦ it would be precisely in +front of its star (in transit). Other phase angles might be inaccessible due to restrictions of inner +working angles of telescopes and/or instruments. For completeness, we include all phase angles in +our computations. +Section 3.1 shows results for spatially resolved planets and Sect. 3.2 for planets that are spatially +unresolved from their star. In particular, we show these latter results for a model planet orbiting the +star Alpha Centauri A at a distance where the incident stellar flux is similar to the solar flux that +reaches Venus. Because our model planets are all mirror-symmetric with respect to the reference +plane, their linearly polarized flux Up equals zero and will not be discussed further. +3.1. Flux and polarization of spatially resolved planets +Figure 3 shows the total flux Fp (the planetary phase function) and degree of polarization Pp as +functions of α and λ for the four evolutionary phases illustrated in Fig. 1. The total fluxes are +normalized such that at α = 0◦, they equal the planet’s geometric albedo AG (see Eq. 5). Figure 4 +shows AG of the planets in the four evolutionary phases as functions of the wavelength λ. Table 2 +lists the geometric albedo’s at 0.5, 1.0, 1.5 and 2.0 µm. The ’Current Earth’ (Phase 1) shows very +little variation in AG, and the ’Thin clouds Venus’ (Phase 2) has the lowest albedo because of +the small cloud particles and the small cloud optical thickness. The geometric albedo’s of the +’Thick clouds Venus’ (Phase 3) and the ’Current Venus’ (Phase 4) are very similar. Thus across +the wavelength region investigated in this paper, the ’Current Earth’ (Phase 1) has the highest +geometric albedo. +Table 2. The model planets’ geometric albedo’s AG for the four evolutionary phases at four wavelengths. +λ (µm) +0.5 +1.0 +1.5 +2.0 +Current Earth +0.757 +0.752 +0.740 +0.739 +Thin clouds Venus +0.186 +0.184 +0.240 +0.310 +Thick clouds Venus +0.727 +0.627 +0.580 +0.564 +Current Venus +0.726 +0.573 +0.500 +0.484 +Article number, page 11 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +Fig. 3. Left column: The total flux (or phase function); and right column: The degree of polarization of +incident unpolarized starlight that is reflected by the model planets in the 4 evolutionary phases as functions +of α and λ. First row: Phase 1 (’Current Earth’); Second row: Phase 2 (’Thin clouds Venus’); Third row: Phase +3 (’Thick clouds Venus’); Fourth row: Phase 4 (’Current Venus’). The phase functions are normalised such +that at α = 0◦, they equal the planet’s geometric albedo AG. +For each model planet, the total flux Fp decreases with increasing α mostly because less of +the planet’s observable disc is illuminated. The planet with thin H2O clouds (Phase 2) is very dark +over all α’s because the cloud optical thickness bc is small and the surface is black. The total fluxes +show vague similarities with the single scattering phase functions of the cloud particles (Fig. 2). In +particular, for the ’Current Earth’ (Phase 1) with large H2O particles, Fp increases slightly around +α = 40◦, the rainbow angle. Also, the decrease of Fp with λ is stronger for the Venus-type planets +with H2SO4 clouds (Phases 3 and 4) than for the planet with the large H2O particles (’Current +Earth’, Phase 1), because the single scattering phase function of the sulfuric acid particles decreases +stronger with λ than that of the water droplets (see Fig. 2). +0.5 +1.0 +1.5 +2.0 +2.5 +( m) +0.2 +0.4 +0.6 +0.8 +1.0 +Geometric albedo +Phase 1 (Current Earth) +Phase 2 (Thin clouds Venus) +Phase 3 (Thick clouds Venus) +Phase 4 (Current Venus) +Fig. 4. The planets’ geometric albedo’s AG as functions of the wavelength λ for the four evolutionary phases. +Article number, page 12 of 24 + +Current Earth +Current Earth +2.5 +2.5 +0.3 +Degree of polarization +2.0 +0.6 +2.0 +0.00 +000:0 +0.000 +入(μm) +0.2 +1.5 +1.5 +1.0 +0.600 +0.300 +1.0 +0.1 +0.000 + 0.2 +0.5 +0.150 +0.5 +100 +0.459 +0.0 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Thin clouds Venus +Thin clouds Venus +2.5 +Degree of linear polarization +2.5 +0.3 + 0.6 +0.300 +2.0 +2.0 +0.150 +0.000 +0.2 +入(μm) +1.5 +0.200 +0.4 +1.5 +1.0 +1.0 +0.1 +0.2 +0.5 +0.5 +0.0 +0.000 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Thick clouds Venus +Thick clouds Venus +2.5 +2.5 +0.050 +Degree of polarization +2.0 +0.8 +2.0 +0.10 +(ur) +1.5 +0.6 +1.5 +0.05 +1.0 +1.0 +0.00 +0.5 +.500 +p.250 +0.2 +0.5 +D +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Current Venus +CurrentVenus +2.5 +2.5 +Degree of polarization +2.0 +2.0 +0.20 +0.6 +0.000 +0.15 +(ur)x +1.5 +1.5 +0.10 +1.0 +0.400 +0.200 +1.0 +0.05 +0.600 +0.2 +0.5 +0.5 +0.000 +0.00 +0.05 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Phase angle (deg) +Phase angle (deg)Mahapatra et al.: From exo-Earths to exo-Venuses +Unlike Fp, the degree of polarization Pp of each of the model planets, shows angular and +spectral features that depend strongly on the cloud properties and should thus allow distinguishing +between the different evolutionary phases. In Phase 1 (’Current Earth’), Pp is high and positive up +till λ = 0.5 µm and around α = 90◦, which is due to Rayleigh scattering by the gas above the +clouds. Starting at the shortest λ, Pp increases slightly with λ before decreasing. This is due to the +slightly larger contribution of multiple scattered light, with a lower degree of polarization, at the +shortest wavelengths. A Rayleigh-scattering peak is also seen for Phase 4 (’Current Venus’), except +there, the peak decreases more rapidly with λ because the clouds are higher in the atmosphere and +there is thus less gas above them. In Phase 3 (’Thick clouds Venus’), the Rayleigh-scattering peak +is suppressed by the contribution of low polarized light that is reflected by the thicker clouds below +the gas. In Phase 2 (’Thin clouds Venus’), the relatively thin clouds are higher in the atmosphere +than in Phase 1 (’Current Earth’), which is why the Rayleigh-scattering peak only occurs at the +very shortest wavelengths (the peak is hardly visible in Fig. 3). Because the Phase 2 cloud particles +are small (reff = 0.5 µm), they themselves give rise to a Rayleigh-scattering peak at λ ≥ 1.0 µm. +The two model planets with the H2O cloud particles (Phases 1 and 2) show a narrow region of +positive polarization between 30◦ and 40◦, which is the rainbow peak (see Fig. 2). On exoplanets, +this local maximum in Pp could be used to detect liquid water clouds on exoplanets (Karalidi et al. +2011, 2012; Bailey 2007). In Phase 1 (’Current Earth’), the rainbow region starts near the Rayleigh +scattering peak of the gas and extends towards the largest wavelengths. In Phase 2 (’Thin clouds +Venus’), with the small water droplets, the rainbow only occurs at the shortest wavelengths. With +increasing wavelength, it broadens and disappears into the cloud particles’ Rayleigh scattering +peak. +The H2SO4 cloud particles (Phases 3 and 4) have their own specific polarization patterns, such +as the broad negative polarization region at α ⪆ 80◦, which can be traced back to their single +scattering patterns (Fig. 2). In Phase 3 (’Thick clouds Venus’), the cloud particles give rise to a +sharp positive polarization peak at the shortest wavelengths and for 20◦ ≤ α ≤ 30◦. In Phase 4 +(’Current Venus’), there is a broader, lower, positive polarization branch across this phase angle +range, which resembles the positive polarization branch of the tiny H2O droplets in Phase 2 (’Thin +clouds Venus’). However, at the longer wavelengths, the phase angle dependence of the polarization +of the latter planet is very different which should help to distinguish between such planets. This +emphasizes the need for measurements at a wide range of wavelengths and especially phase angles +(if the planet’s orbital inclination angle allows this). +3.2. Flux and polarization of spatially unresolved planets +In the previous section, we showed the signals of spatially resolved planets, thus without back- +ground starlight. When observing an exoplanet in the habitable zone of a solar-type star, it will +be difficult to avoid the starlight. Here, we show the total flux of the planet Fp, the star-planet +contrast C (see Eq. 6), the spatially resolved degree of polarization of the planet Pp (thus without +Article number, page 13 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +Fig. 5. A sketch of the geometries within the Alpha Centauri system: the orbital plane of the stars Alpha +Centauri A and B is inclined by about 80◦ with respect to the observer on Earth. Our model planet (the blue +dot) orbits Alpha Centauri A. In this sketch, the line of nodes of the planet’s orbit was chosen to coincide with +that of the stellar orbits. The inclination angle im of the planet’s orbit with respect to the stellar orbital plane is +45◦, and the inclination angle of the planet’s orbit with respect to the observer is 80◦ − 45◦ = 35◦. The phase +angles of the planet in this sketch would range from 90◦ − 35◦ = 55◦ to 90◦ + 35◦ = 125◦. +the starlight) and the spatially unresolved degree of polarization of the combined star-planet signal +Pu (thus including the starlight). While the total planet fluxes shown in Fig. 3 were normalized +at α = 0◦ to the planets’ geometric albedo’s AG, here they are computed according to Eq. 5, and +thus depend on the parameters of the planet-star system. We assume our model planets orbit Alpha +Centauri A. +The solar-type star Alpha Centauri A is part of a double star system, and the orbital parameters +of ourplanets are chosen based on the stable planet orbital distances and orbital inclination angles +around this star as predicted by Quarles & Lissauer (2016). Figure 5 shows a sketch of the system. +We use a planetary orbital distance d of 0.86 AU, such that each model planet receives a stellar +flux similar to the solar flux received by Venus. Additional system parameter values are listed in +Table 1. According to Quarles & Lissauer (2016), stable orbits around Alpha Centauri A can be +found for a range of angles between the planetary orbital plane and the plane in which the two stars +orbit, and thus for a range of inclination angles i of the planetary orbit. +Figure 6 shows the variation of the planetary phase angle α along a planetary orbit for two +values of the longitude of the orbit’s ascending node Ω: for Ω = 0◦ (the line connecting the planet’s +ascending and descending nodes is perpendicular to the line to the observer) and for Ω = 205◦ +which represents the configuration of Earth with Alpha Centauri A. The orbital phase of the planet +is defined such that at an orbital phase angle of 0◦, α = 180◦. The inclination angle im is the +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Alpha (deg) + = 0 +im = -10 +im = 80 +im = 0 +im = 35 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Alpha (deg) + = 205 +im = -10 +im = 80 +im = 0 +im = 35 +Fig. 6. Variation of the planet’s phase angle α along the planet’s orbit around Alpha Centauri A for different +mutual inclination angles im of the planetary orbit with respect to the orbital plane of the two stars. Top: Ω, +the longitude of the ascending node of the planet’s orbit, is 0◦. For im = −10◦, the planet is then in a face-on +orbit (i = 0◦), while for im = 80◦, it is in an edge-on orbit (i = 90◦). Bottom: Ω = 205◦, and the planetary orbit +is aligned with the node of the stellar orbital plane. +Article number, page 14 of 24 + += 800 +Alpha Centauri B +i= 35o +to the observer +Alpha Centauri AMahapatra et al.: From exo-Earths to exo-Venuses +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +10 +20 +30 +40 +Fp (10 +12 W m +2m +1) +im = -10 +im = 80 +im = 0 +im = 35 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +C (10 +9) +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.01 +0.00 +0.01 +0.02 +0.03 +0.04 +Pp +im = -10 +im = 80 +im = 0 +im = 35 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.01 +0.00 +0.01 +0.02 +0.03 +0.04 +0.05 +Pu (10 +9) +im = -10 +im = 80 +im = 0 +im = 35 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +2 +4 +6 +8 +10 +12 +14 +16 +Fp (10 +12 W m +2m +1) +im = -10 +im = 80 +im = 0 +im = 35 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +C (10 +9) +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.04 +0.03 +0.02 +0.01 +0.00 +0.01 +0.02 +0.03 +Pp +im = -10 +im = 80 +im = 0 +im = 35 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.01 +0.00 +0.01 +0.02 +0.03 +Pu (10 +9) +im = -10 +im = 80 +im = 0 +im = 35 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +1 +2 +3 +4 +5 +Fp (10 +12 W m +2m +1) +im = -10 +im = 80 +im = 0 +im = 35 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +1.0 +C (10 +9) +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.03 +0.02 +0.01 +0.00 +0.01 +0.02 +0.03 +Pp +im = -10 +im = 80 +im = 0 +im = 35 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.005 +0.000 +0.005 +0.010 +0.015 +0.020 +0.025 +0.030 +Pu (10 +9) +im = -10 +im = 80 +im = 0 +im = 35 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +1.8 +Fp (10 +12 W m +2m +1) +im = -10 +im = 80 +im = 0 +im = 35 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +0.9 +C (10 +9) +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.02 +0.01 +0.00 +0.01 +0.02 +Pp +im = -10 +im = 80 +im = 0 +im = 35 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.005 +0.000 +0.005 +0.010 +0.015 +0.020 +Pu (10 +9) +im = -10 +im = 80 +im = 0 +im = 35 +Fig. 7. The total planetary flux Fp (in W m−3) and the star-planet contrast C, the degree of polarization Pp of +the spatially resolved planet, and Pu, the degree of polarization of the star and the spatially unresolved planet, +all for a ’Current Venus’ model planet (Phase 4) as functions of the planet’s orbital phase for four mutual +inclination angles im and four wavelengths λ: 0.5 µm (row 1); 1.0 µm (row 2); 1.5 µm (row 3); and 2.0 µm +(row 4). The longitude of the ascending node of the planetary orbit, Ω, is 205◦. +angle between the plane in which the stars move and the planetary orbital plane. For Ω = 0◦, +im = −10◦ would yield a face-on planetary orbit (i = 0◦) with α = 90◦ everywhere along the orbit. +For im = 80◦, the orbit is edge-on (i = 90◦) and α varies between 0◦ and 180◦. Figure 6 also shows +the range of α for im = 0◦ and 35◦. According to Quarles & Lissauer (2016), the latter is the most +probable orientation of a stable planetary orbit. For these two cases, the accessible phase angles +range from 80◦ to 100◦, and from 45◦ to 135◦, respectively. For Ω = 205◦, the maximum range of +α would be from 20◦ to 160◦, depending on im. +Figure 7 shows Fp, Pp and Pu (the spatially unresolved planet, thus with starlight included) for +Phase 4 (’Current Venus’) as functions of the planet’s orbital phase for Ω = 205◦, four values of +im (-10◦, 80◦, 0◦, and 35◦), and four wavelengths (0.5, 1.0, 1.5, and 2.0 µm). The plots for Fp also +show the contrast C. Because C is the ratio of the planetary flux Fp to the stellar flux Fs (Eq. 6), its +variation with the orbital phase is proportional to that of Fp. +At the two orbital phases in each plot where all the lines cross, the planetary phase angles +α are the same (see Fig. 6) and thus all Fp and Pp are the same. The plots for Fp appear to be +very similar for the different wavelengths, apart from a difference in magnitude which is mainly +due to the decrease of the stellar flux that is incident on the planet with increasing wavelength, +although the planetary albedo AG and phase function R1p also decrease with increasing λ as can +be seen in Fig. 3. This wavelength dependence of Fp also causes the decrease of the contrast C +Article number, page 15 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +with increasing wavelength (i.e. the planet darkens with increasing λ), as C is independent of the +wavelength dependence of the stellar flux (see Eq. 6). The shape differences between the Fp (and +C) curves are due to the wavelength dependence of the planetary flux that can also be seen in Fig. 3. +At each wavelength λ, the largest variation in Fp with the orbital phase is seen for im = 80◦, +because for that configuration the variation of α along the orbit is largest (see Fig. 6). The degree of +polarization Pp of the planet shows significant variation with the orbital phase at all wavelengths. +A particularly striking feature for the geometry with im = 80◦ is the double peak close to the orbital +phases of 150◦ and 200◦. As can be seen in Fig. 6 for Ω = 205◦ and im = 80◦, α decreases from +about 160◦ at an orbital phase of 0◦, to 20◦ at an orbital phase around 175◦, to then increase again +with increasing orbital phase. Tracing this path of α through the Pp panel in the bottom row of +Fig. 3 explains the double peaked behaviour of Pp and its wavelength dependence as shown in +Fig. 7. For the other values of im, the phase angle range that is covered along the orbit is smaller, +and therefore the variation in Pp is also smaller. +The degree of polarization of the spatially unresolved planet, Pu, shows similar variations along +the orbital phase as Pp, except that most features are flattened out because of the addition of the +unpolarized stellar flux, which is independent of the orbital phase angle. The double peaked feature +for im = 80◦ remains strong, however, as at those orbital phase angles, the contrast C is relatively +large and thus the influence of the added stellar flux relatively small. The variation in the polarisa- +tion of the unresolved system due to the orbiting planet is on the order of 10−11. +Figure 8 is similar to Fig. 7, except for the four model planets in the different evolutionary +phases and all for im = 80◦ and Ω = 0◦. Because here the planetary orbits are seen edge-on +(i = 90◦), the full range of phase angles is covered, which makes it possible to explore the full +extent of variation of flux and polarization signals. Because of this large phase angle range, Fp +varies strongly with the orbital phase. The wavelength dependence of the total flux can be traced +back to Fig. 3, where in particular the Phase 2 planet (’Thin cloud Venus’) is dark at all wavelengths, +but relatively bright at the longest wavelengths and small phase angles. As was the case in Fig. 7, +the variation of C is the same as that of Fp, except for the off-set due to the stellar flux. The largest +values of C (about 1.6×10−9), are found for λ = 0.5 µm and around the orbital phase of 180◦ (at +180◦, the planets would actually be behind the star). +Furthermore in Fig. 8, Pp depends strongly on λ and the planet’s evolutionary phase. At 0.5 µm, +the Phase 1 planet (’Current Earth’) shows the largest values of Pp due to the Rayleigh scattering +gas above the low altitude clouds. At the longer wavelengths, where the Rayleigh scattering is less +prominent, the curves for the Phase 1 planet clearly show the positive polarization of the rainbow +around the orbital phases of 140◦ and 220◦ (see Fig. 6). For the Phase 2 planet (’Thin clouds +Venus’) and 0.5 µm, the small cloud particles cause positive polarization around 150◦ and 210◦, +which connect the rainbow and the Rayleigh scattering maximum in Fig. 3. At longer wavelengths, +the broad positive polarization signature of Rayleigh scattering by the cloud particles dominates the +curves, while the curves for the Phase 3 and 4 planets show mostly negative polarization apart from +Article number, page 16 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0 +10 +20 +30 +40 +50 +60 +Fp (10 +12 W m +2 m +1) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0.00 +0.25 +0.50 +0.75 +1.00 +1.25 +1.50 +1.75 +C (10 +9) +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +Pp +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.02 +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +Pu (10 +9) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0 +5 +10 +15 +20 +Fp (10 +12 W m +2 m +1) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +C (10 +9) +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 +Pp +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.02 +0.00 +0.02 +0.04 +0.06 +Pu (10 +9) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0 +2 +4 +6 +8 +Fp (10 +12 W m +2 m +1) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +C (10 +9) +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.05 +0.00 +0.05 +0.10 +0.15 +Pp +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.02 +0.00 +0.02 +0.04 +0.06 +Pu (10 +9) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.0 +0.5 +1.0 +1.5 +2.0 +2.5 +3.0 +Fp (10 +12 W m +2 m +1) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +C (10 +9) +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Pp +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.02 +0.00 +0.02 +0.04 +0.06 +0.08 +Pu (10 +9) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +Fig. 8. Similar to Fig. 7 except for the model planets in the four evolutionary phases and Ω = 0◦ and im = 80◦ +(thus for an edge–on orbit, i = 90◦): Phase 1 (’Current Earth’), Phase 2 (’Thin clouds Venus’), Phase 3 (’Thick +clouds Venus’), Phase 4 (’Current Venus’). The wavelengths λ are like before: 0.5 µm (row 1); 1.0 µm (row +2); 1.5 µm (row 3); and 2.0 µm (row 4). +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +10 +15 +20 +25 +Fp (10 +12 W m +2 m +1) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +0.8 +C (10 +9) +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +Pp +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.00 +0.02 +0.04 +0.06 +0.08 +0.10 +0.12 +Pu (10 +9) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +4 +6 +8 +10 +Fp (10 +12 W m +2 m +1) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +C (10 +9) +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.04 +0.02 +0.00 +0.02 +0.04 +0.06 +Pp +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.015 +0.010 +0.005 +0.000 +0.005 +0.010 +0.015 +0.020 +Pu (10 +9) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +1.5 +2.0 +2.5 +3.0 +3.5 +Fp (10 +12 W m +2 m +1) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +C (10 +9) +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.05 +0.00 +0.05 +0.10 +0.15 +Pp +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.02 +0.01 +0.00 +0.01 +0.02 +0.03 +0.04 +Pu (10 +9) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.6 +0.8 +1.0 +1.2 +1.4 +1.6 +Fp (10 +12 W m +2 m +1) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0.3 +0.4 +0.5 +0.6 +0.7 +C (10 +9) +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.05 +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +Pp +Phase 1 +Phase 2 +Phase 3 +Phase 4 +0 +50 +100 +150 +200 +250 +300 +350 +Orbital phase (deg) +0.00 +0.02 +0.04 +0.06 +0.08 +Pu (10 +9) +Phase 1 +Phase 2 +Phase 3 +Phase 4 +Fig. 9. Similar to Fig. 8 except for the most probable, stable orbit around Alpha Centauri A, i.e. for Ω = 205◦ +and im= 35◦ (Quarles & Lissauer 2016). The wavelengths are like before: 0.5 µm (row 1); 1.0 µm (row 2); +1.5 µm (row 3); and 2.0 µm (row 4). +Article number, page 17 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +the orbital phases around 180◦. When adding the starlight, the angular features of the polarization +Pu are suppressed along those parts of the orbits where C is smallest, thus away from the orbital +phase angle of 180◦. In particular within 180◦ ±40◦, Pu still shows distinguishing features, although +they are very small in absolute sense (smaller than 10−10). +Figure 9 is similar to Fig. 8 except here the model planets are in the most probable stable orbit +around Alpha Centauri A as predicted by Quarles & Lissauer (2016), namely with Ω = 205◦ and +im = 35◦. As can be seen in Fig. 6, for this geometry α varies between about 60◦ and 120◦. In +Fig. 9, Fp shows a similar variation as the curves in Fig. 8, although less prominent, as the planets +do not reach a ’full’ phase (where α = 0◦) nor the full night phase (α = 180◦) along their orbit. +The flux curves in Fig. 9 also miss small angular features that appear in the single scattering phase +functions of the cloud particles (see Fig. 2), such as the glory, again because the planets do not go +through the related phase angles. +In this particular orbital geometry, Pp shows less pronounced angular features than for the +same model planets in edge-on orbits (Fig. 8) because of the more limited phase angle range. For +example, the ’Current Earth’ (Phase 1) shows no rainbow despite the H2O clouds, because the +phase angle of about 40◦ is not reached. In the visible (λ = 0.5 µm), Pp reaches the largest values +for the ’Current Earth’ (Phase 1). At longer wavelengths, Pp of the ’Thin clouds Venus’ (Phase 2) +strongly dominates because of the Rayleigh scattering by the small cloud particles. The ’Thick +clouds Venus’ (Phase 3) shows predominant negative polarization at all wavelengths and across the +whole orbital phase angle range except at λ = 0.5 µm around an orbital phase angle of 20◦. +The polarization of the spatially unresolved planets, Pu, clearly shows the suppression of the +polarization features due to the added starlight towards the smaller and larger orbital phase angles, +where the planets are darker. While the Phase 2 planet (’Thin clouds Venus’) is relatively dark (C is +very small), its Rayleigh scattering polarization signal is so strong that its unresolved polarization +signal is larger than that of the other planets, except the Phase 1 planet (’Current Earth’) at 1.0 µm +and orbital phase angles close to 180◦. +3.3. Evolutionary phases across α and λ +In Fig. 10, we show which evolutionary phase has the highest values of |Pp| across all phase angles +α and wavelengths λ. We find that |Pp| of the Phase 1 planet (’Current Earth’) dominates between +30◦ and 150◦, and mostly for λ < 1.0 µm. In particular, around α = 40◦ and up to λ = 2.0 µm, +the polarization signal of the rainbow produced by the large water cloud particles is about 0.1 (see +the bottom plot of Fig. 10). For λ > 1.0 µm, the Phase 2 planet (’Thin clouds Venus’) shows +the strongest polarization due to the Rayleigh scattering by the small H2O cloud particles, as can +clearly be seen in the bottom plot. The small patches where the strongest polarization signal is from +the ’Thick clouds Venus’ (Phase 3), for example near α = 20◦ and λ < 1.0 µm, or from the ’Current +Venus’ (Phase 4) are due to the single scattering polarization features of the H2SO4 cloud particles, +as can be seen in Fig. 2. +Article number, page 18 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +Fig. 10. Top: The planet models that yield the largest absolute degree of polarization |Pp| over all phase angles +α and wavelengths λ: Phase 1 - ’Current Earth’ (blue); Phase 2 - ’Thin clouds Venus’ (light orange); Phase 3 +- ’Thick clouds Venus’ (dark orange); and Phase 4 - ’Current Venus’ (brown). Bottom: The maximum values +of |Pp| of the four model planets as functions of α and λ. +The accessible phase angle range for direct observations of such exoplanets obviously depends +on the actual orientation of the planetary orbits and cannot be optimized by the observer. Precisely +because of that, Fig. 10 clearly indicates that measurements should be performed across a broad +wavelength range, including wavelengths below 1 µm to allow distinguishing between Earth-like +and Venus-like planets in various evolutionary phases. +4. Summary and conclusions +We presented the total flux and linear polarization of starlight that is reflected by model planets +of various atmospheric types to investigate whether different phases in the evolution of planets +like the Earth and Venus can be distinguished from each other. We have used four planet models +to represent possible evolutionary phases. Phase 1 (’Current Earth’) has an Earth-like atmosphere +and liquid water clouds; Phase 2 (’Thin clouds Venus’) has a Venus-like CO2 atmosphere and thin +water clouds; Phase 3 (’Thick clouds Venus’) has a Venus-like CO2 atmosphere and thick sulphuric +acid solution clouds; and Phase 4 (’Current Venus’) has a CO2 atmosphere and thin sulphuric acid +solution clouds. We have computed the total flux and polarization signals specifically for model +Article number, page 19 of 24 + +2.5 +Earth +Thin clouds Venus +ThickcloudsVenus +Wavelength (microns) +2.0 +Current Venus +1.5 +1.0 +0.5 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Alpha (deg)2.5 +0.300 +Wavelength (microns) +2.0 +0.200 +0.010 +0.050 +0.100 +0.010 +1.5 +1.0 +0.05b +0.5 +0.100 +0.300 +0 +0.200 +100 +0 +20 +40 +60 +80 +100 +120 +140 +160 +180 +Alpha (deg)Mahapatra et al.: From exo-Earths to exo-Venuses +planets orbiting our neighbouring solar-type star Alpha Centauri A using predicted stable orbits +(Quarles & Lissauer 2016) in its habitable zone. +We have computed the reflected starlight for wavelengths λ ranging from 0.3 µm to 2.5 µm and +for planetary phase angles α from 0◦ to 180◦. We not only present the fluxes and polarization of +spatially resolved model planets (thus without background starlight) but also of spatially unresolved +planets, thus the combined signal of the planet and the star. For the latter cases, we also computed +the planet-star contrast C as a function of α and λ to determine what would technically be required +to detect the planetary signals upon the background starlight. +The range of planetary phase angles α at which a planet can be observed (spatially resolved or +unresolved) depends on the inclination of the planetary orbit with respect to the observer. We have +specifically studied the reflected light signals of planets orbiting solar-type star Alpha Centauri A. +This star is part of a double star system with solar-type star Alpha Centauri B. The distance between +the two stars varies from 35.6 to 11.2 AU (M-dwarf Alpha Centauri C or Proxima Centauri orbits +the pair at a distance of about 13,000 AU). Dynamical computations (Quarles & Lissauer 2016) +predict stable planetary orbits around Alpha Centauri A in a narrow range of mutual inclination +angles im between the orbital planes of the two stars and that of the planet. In particular, the most +stable orbit has im = 35◦, which provides an α range from 60◦ to 120◦. We find that with this orbital +geometry the degree of polarization of the planet would be largest for the ’Current Earth’ (Phase 1) +across the visible (λ < 1.0 µm) due to Rayleigh scattering by the gas above the clouds. At near +infrared wavelengths (1.0 µm < λ < 2.5 µm), the polarization of the ’Thin clouds Venus’ (Phase 2) +is highest, because this planet has small cloud droplets that scatter like Rayleigh scatterers at the +longer wavelengths. +The well-known advantage of measuring the degree of polarization for the characterization of +(exo)planets is that the angular features in the signal of the planet as a whole are similar to the +angular features in the light that has been singly scattered by the gas molecules and cloud particles, +which are very sensitive to the microphysical properties (such as the size distribution, composi- +tion, shape) of the scattering molecules and cloud particles and to the atmosphere’s macrophysical +properties (such as cloud altitude and thickness). The reflected total flux is much less sensitive to +the atmospheric properties than the degree and direction of polarization (see e.g. Hansen & Travis +1974, for several examples). Indeed, the variations of the planetary flux Fp along a planet’s or- +bit appear to be mostly due to the change of the fraction of the planetary disk that is illuminated +and visible to the observer. They provide limited information on the planet’s atmospheric charac- +teristics, especially if one takes into account that with real observations, the planet radius will be +unknown unless the planet happens to transit its star. The variation of the planetary flux Fp with +phase angle α and wavelength λ is similar that of planet-star contrast C, the ratio of Fp to the stellar +flux Fs. For a Venus-like planet orbiting Alpha Centauri A, C is on the order of 10−9. +Article number, page 20 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +Our numerical simulations show that variations in Pp, the degree of polarization of the spatially +resolved planets (thus without any starlight), with α combined with variations with λ could be used +to distinguish the planetary evolutionary phases explored in this paper: +– Phase 1 planets (’Current Earth’) show strong positive (perpendicular to the reference plane +through the star, planet, and observer) polarization around α = 40◦ due to scattering by large +water cloud droplets (the rainbow) and also higher polarization for λ < 0.5 µm and α ≈ 90◦ +due to Rayleigh scattering by the gas above the clouds. +– Phase 2 planets (’Thin clouds Venus’) polarize light negatively (parallel to the reference plane) +across most phase angles and at visible wavelengths. At near-infrared wavelengths, they have +strong positive polarization around α = 90◦ due to Rayleigh scattering by the small cloud +droplets, and a ’bridge’ of higher polarization from the Rayleigh maximum to the rainbow +angle (α ≈ 40◦) with decreasing λ. +– Phase 3 planets (’Thick clouds Venus’) have predominantly negative polarization from the vis- +ible to the near-infrared, with small regions of positive polarization for λ < 1.0 µm and for +20◦ ≤ α ≤ 30◦, that are characteristic for the 75% H2SO4 cloud particles with reff = 2.0 µm. +– Phase 4 planets (’Current Venus’) yield similar polarization patterns as the Phase 3 planets, ex- +cept with more prominent negative polarization for 10◦ ≤ α ≤ 30◦ and for 0.5 µm ≤ λ ≤ 2 µm. +Rayleigh scattering by the small cloud particles produces a maximum of positive polarization +around α = 90◦ and for λ > 2 µm. +Our simulations of the planetary polarisation Pp do not include any background starlight, and +can thus reach several percent to even 20% for the ’Current Earth’ (Phase 1) planet in an edge-on +orbit (Fig. 8). Whether or not such polarization variations could be measured depends strongly on +the techniques used to suppress the light of the parent star. If the background of the planet signal +contains a fraction x of the flux of the star, the degree of polarization of the light gathered by the +detector pixel that contains the planet will equal (C/(C + x))Pp ≈ (C/x)Pp with C the contrast +between the total fluxes of the planet and the star (Eq. 9). For a Venus-like planet orbiting Alpha +Centauri A, C is on the order of 10−9, thus x should be as small as 10−4 to get a polarisation signal +on the order of 10−6, assuming Pp is about 0.1. Not only excellent direct starlight suppression +techniques, but also a very high spatial resolution would help to decrease x. +Our simulations further show that temporal variations in the total flux Fu of Alpha Centauri A +with a spatially unresolved terrestrial-type planet orbiting in its habitable zone would be less than +10−12 W/m3. The degree of polarization Pu of the combined star and planet signals, would show +variations smaller than 0.05 ppb. To identify this planetary flux on top of the stellar flux, a very +high-sensitivity instrument would be required and an even higher sensitivity would be required to +subsequently characterize the planetary atmosphere. Recall that the orbital period of such a planet, +and with that the period of the signal variation and presumably the stability requirements of an +instrument, would be about 0.76 years. Bailey et al. (2018) computed the polarization signal of +spatially unresolved, hot, cloudy, Jupiter-like planet HD 189733b to be on the order of ∼20 ppm. +Article number, page 21 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +Because of their large size, Jupiter-like planets, and in particular those in close-in orbits that receive +large stellar fluxes, would clearly be less challenging observing targets than terrestrial-type planets. +The HARPS instrument on ESO’s 3.6 m telescope includes polarimetric observations with a +polarimetric sensitivity of 10−5 (?). Planetpol on the 4.2 m William Herschel Telescope (WHT) +on La Palma achieved a sensitivity to fractional linear polarization (the ratio of linearly polar- +ized flux to the total flux) of 10−6. While PlanetPol did not succeed in detecting exoplanets, it +did provide upper limits on the albedo’s of a number of exoplanets (Lucas et al. 2009). The Ex- +treme Polarimeter (ExPo), that was also mounted on the WHT, was designed to target young stars +embedded in protoplanetary disks and evolved stars surrounded by dusty envelopes, with a polari- +metric sensitivity better than 10−4 (Rodenhuis et al. 2012). The HIPPI-2 instrument uses repeated +observations of bright stars in the SDSS g’ band for achieving better than 3.5 ppm accuracy on +the 3.9-m Anglo-Australian Telescope and better than 11 ppm on the 60-cm Western Sydney Uni- +versity’s telescope (Bailey et al. 2020). The POLLUX-instrument on the LUVOIR space telescope +concept aims at high-resolution (R∼120,000) spectropolarimetric observations across ultra-violet +and visible wavelengths (100-400 nm) to characterize atmospheres of terrestrial-type exoplanets +(The Luvoir Team 2019; Rossi et al. 2021). The EPICS instrument planned to be mounted on the +ELT telescope, is designed to achieve a contrast of 10−10 depending the angular separation of the +objects (Kasper et al. 2010). +In our simulations, we have neglected absorption by atmospheric gases. Including such absorp- +tion would yield lower total fluxes in specific spectral regions, depending on the type and amount +of absorbing gas, its vertical distribution, and on the altitude and microphysical properties of the +clouds and hazes. Including absorption by atmospheric gases could increase or decrease the degree +of polarization, depending on the amount and vertical distribution of the absorbing gas and on the +microphysical properties of the scattering particles at various altitudes (see e.g. Trees & Stam 2022; +Stam 2008, for examples of polarization spectra of Earth-like planets). While measuring total and +polarized fluxes of reflected starlight across gaseous absorption bands is of obvious interest for +the characterization of planets and their atmospheres, the small numbers of photons inside gaseous +absorption bands would make such observations extremely challenging. +We have also neglected any intrinsic polarization of Alpha Centauri A. Measurements of the +degree of linear polarization of FKG stars indicate that active stars like Alpha Centauri A have a +typical mean polarization of 28.5 ± 2.2 ppm (Cotton et al. 2017). This could add to the challenges +in distinguishing the degree of polarization of the planet from that of the star if the planet is spa- +tially unresolved, although the phase angle variation of the planetary signal and the direction of +polarization of the planet signal (i.e. usually either perpendicular or parallel to the plane through +the star, the planet, and the observer) could be helpful provided of course that the instrument that is +used for the observations has the capability to measure the extremely small variations in the signal +as the planet orbits its star (on the order of 10−9). +Article number, page 22 of 24 + +Mahapatra et al.: From exo-Earths to exo-Venuses +State-of-the-art instruments with sensitivity to polarization signals down to 10−6 (i.e. 1000 ppb) +are still a few orders of magnitude away from detecting variations in polarization signals from +spatially unresolved exo-Earths or exo-Venuses around nearby solar-type stars such as Alpha Cen- +tauri A. To be able to distinguish between the different planetary evolutionary phases explored in +this paper, e.g. between water clouds or sulphuric acid clouds, variations on the order of 10−9 +and hence significant improvements in sensitivity would be needed if the planets are spatially +unresolved. The variation in the degree of polarization of spatially resolved planets along their +orbital phase should be detectable by instruments capable of achieving star-planet contrasts of +10−9 and that would allow to distinguish between water clouds or sulphuric acid clouds. Current +high-contrast imaging instruments manage to directly image self-luminous objects such as young +exoplanets and brown dwarfs in NIR total fluxes at contrasts of 10−2-10−6 (Bowler 2016; Nielsen +et al. 2019; Langlois et al. 2021; van Holstein 2021). Further, instruments such as EPICS on ELT +and concepts for instruments on future space observatories such as HabEx (Gaudi et al. 2020) and +LUVOIR (The Luvoir Team 2019) hold the promise for attaining contrasts of ∼10−10. Reaching +such extreme contrasts would make it possible to directly detect terrestrial-type planets and to use +polarimetry to differentiate between exo-Earths and exo-Venuses. +Acknowledgements. This work was supported by the Netherlands Organisation for Scientific Research (NWO) through the +User Support Programme Space Research, project number ALW GO 15-37. We thank the referee for the valuable feedback +that improved this paper. +References +Bailey, J. 2007, Astrobiology, 7, 320 +Bailey, J., Cotton, D. V., Kedziora-Chudczer, L., De Horta, A., & Maybour, D. 2020, Publications of the Astronomical +Society of Australia, 37 +Bailey, J., Kedziora-Chudczer, L., & Bott, K. 2018, Monthly Notices of the Royal Astronomical Society, 480, 1613 +Bowler, B. 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D. 2020, Journal of Geophysical Research: Planets, 125, e2019JE006276 +Article number, page 24 of 24 + diff --git a/ctFIT4oBgHgl3EQfnitG/content/tmp_files/load_file.txt b/ctFIT4oBgHgl3EQfnitG/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..0299d818421404c013f53c4466478053bdf13271 --- /dev/null +++ b/ctFIT4oBgHgl3EQfnitG/content/tmp_files/load_file.txt @@ -0,0 +1,1427 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf,len=1426 +page_content='Astronomy & Astrophysics manuscript no.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Mahapatra2023 ©ESO 2023 January 27, 2023 From exo-Earths to exo-Venuses Flux and Polarization Signatures of Reflected Light G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Mahapatra,1,⋆, F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Abiad1, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Rossi2, and D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='Stam1 1 Faculty of Aerospace Engineering, Delft University of Technology, Delft, The Netherlands e-mail: g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='mahapatra@tudelft.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='nl 2 CNRS/INSU, LATMOS-IPSL, Guyancourt, France Accepted 11 January, 2023 ABSTRACT Context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Terrestrial-type exoplanets in or near stellar habitable zones appear to be ubiquitous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' It is, however, unknown which of these planets have temperate, Earth-like climates or e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' extreme, Venus-like climates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Aims.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Technical tools to distinguish different types of terrestrial-type planets are crucial for deter- mining whether a planet could be habitable or incompatible with life as we know it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We investigate the potential of spectropolarimetry for distinguishing exo-Earths from exo-Venuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We present numerically computed fluxes and degrees of linear polarization of starlight that is reflected by exoplanets with atmospheres in evolutionary states ranging from similar to the current Earth to similar to the current Venus, with cloud compositions ranging from pure water to 75% sulfuric acid solution, for wavelengths between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='3 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We also present flux and polarization signals of such planets in stable but spatially unresolved orbits around the star Alpha Centauri A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Results.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The degree of polarization of the reflected starlight shows larger variations with the planetary phase angle and wavelength than the total flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Across the visible, the largest degree of polarization is reached for an Earth-like atmosphere with water clouds, due to Rayleigh scattering above the clouds and the rainbow feature at phase angles near 40◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' At near-infrared wavelengths, the planet with a Venus-like CO2 atmosphere and thin water clouds shows the most prominent polarization features due to Rayleigh-like scattering by the small cloud droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' A planet in a stable orbit around Alpha Centauri A would leave temporal variations on the order of 10−13 W/m3 in the total reflected flux and 10−11 in the total degree of polarization as the planet orbits the star and assuming a spatially unresolved star-planet system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Star-planet contrasts are in the order of 10−10 and vary proportionally with planetary flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' ⋆ Now at SRON Netherlands Institute for Space Research, Leiden, The Netherlands Article number, page 1 of 24 arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='11314v1 [astro-ph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='EP] 26 Jan 2023 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses Conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Current polarimeters appear to be incapable to distinguish between the possible evolutionary phases of spatially unresolved terrestrial exo-planets, as a sensitivity close to 10−10 would be required to discern the planetary signal given the background of unpolarized starlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' A telescope/instrument capable of achieving planet-star contrasts lower than 10−9 should be able to observe the large variation of the planet’s resolved degree of polarization as a function of its phase angle and thus be able to discern an exo-Earth from an exo-Venus based on its clouds’ unique polarization signatures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Key words.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Exo-planets – Venus – Radiative transfer – Polarimetry 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Introduction Despite having similar sizes, being formed around the same time and from similar materials, it is clear that the Earth and Venus have evolved into dramatically different worlds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' While it is generally acknowledged that Venus once had much larger amounts of water than today, it is still debated whether Venus was once more Earth-like with oceans of water before the runaway-greenhouse- effect took off (Donahue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1982), or whether the atmospheric water vapour never actually condensed on the surface (Turbet et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Bullock & Grinspoon (2001) conducted a detailed study of the possible evolution of Venus’s climate over long time periods starting with a water vapour enriched atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Terrestrial-type exoplanets are also expected to harbour a wide vari- ety of atmospheric compositions with maybe only a few planets hospitable to life as we know it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Various climate models suggest that the likelihood of a planetary atmosphere exhibiting a Venus- like runaway-greenhouse-effect is higher than that of an atmosphere in an Earth-like, N2-dominated state (Lincowski et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2018, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' A study by Kane et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' (2020) even shows that Jupiter’s migration might have stimulated the runaway-greenhouse-effect on Venus, suggesting that there could be more Venus-analogs than Earth-analogs in planetary systems with Jupiter-like plan- ets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' As planned powerful telescopes and dedicated, sensitive detection techniques will allow us to characterize smaller exoplanets in the near-future, it will become possible to probe terrestrial-type planets in and near the habitable zones of solar-type stars and to find out whether they resemble Earth or Venus, or something else all together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The high-altitude cloud deck on an exo-Venus would make it difficult to use a technique like transit spectroscopy for the characterization of the planet as the clouds themselves would block the transmission of the starlight and apart from a spectral dependence of the cloud optical thickness which could leave a wavelength dependent transmission through the cloud tops,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' the microphysical properties of the cloud particles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' such as their composi- tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' shape and size distribution would remain a mystery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Also, the clouds would inhibit measuring trace gas column densities as they would block the planet’s lower atmosphere and only allow tran- sit spectroscopy of the highest regions of the atmosphere (see, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Lustig-Yaeger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2019a, and references therein).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Indeed, a Venus-like ubiquitous cloud deck could possibly be mistaken for the Article number, page 2 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses planet’s surface, as one would only measure the transmittance through the gaseous atmosphere above the clouds, possibly inferring the atmosphere to be thin and eroding (Lustig-Yaeger et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2019b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Jordan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' (2021) modelled the photochemistry of some of the primary sulphuric chemical species that should be responsible for the formation and sustenance of Venus’s sulfuric acid solution clouds,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' such as SO2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' OCS and H2S,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' and found that the abundances of such species above the cloud deck would depend heavily on the effective temperature and distance to the parent star,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' with their abundances decreasing with increasing temperature and being depleted,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' as we see on Venus,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' in the presence of a star like the Sun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Thus it would be challenging to rule out the possibility of an exoplanet being a Venus analog solely on the basis of the detection of such chemical species in transmission spectra (Jordan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Indeed, the full characterization of rocky exoplanets and their classification appears to require the direct imaging of starlight that is reflected by such planets and/or the thermal radiation that is emitted by them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' While telescopes able to perform such measurements are not yet available, plans are underway for their development and deployment (Keller et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2010;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The Luvoir Team 2019).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' While most of telescopes and instruments are designed for only measuring total fluxes of exo- planets, including (spectro)polarimetry is also being considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The main reason to include (spec- tro)polarimetry (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2021, and references therein) is that it increases the contrast between star and planet, as the stellar flux will be mostly unpolarized when integrated over the disk (Kemp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1987), while the flux of the reflected starlight will usually be (linearly) polarized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' And in addition, (spectro)polarimetry can be used for the characterization of planetary atmospheres and surfaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' As a classic example of the latter, Hansen & Hovenier (1974) used Earth-based measure- ments of the disk-integrated degree of polarization of sunlight that was reflected by Venus in three spectral bands and across a broad phase angle range, to deduce that the particles forming Venus’s main cloud deck consist of 75% sulfuric acid solution, that the effective radius of their size dis- tribution is 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 µm, and that the effective width of the distribution is 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' They also derived the cloud top altitude (at 50 mbars) by determining the amount of Rayleigh scattering in the gas above the cloud tops at a wavelength of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='365 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' This was later confirmed by the Pioneer Venus mission which performed in-situ measurements using a nephelometer on a probe that descended through the clouds (Knollenberg & Hunten 1980).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Polarimetry proved to be an effective technique for disentangling Venus’s cloud properties be- cause the scattering particles leave a unique angular polarization pattern in the reflected sunlight depending on the particles’ micro- and macro-physical properties (for an extensive explanation of the application of polarimetry for the characterization of planetary atmospheres, see Hansen & Travis 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' While multiple scattered light usually has a low degree of polarization, and thus dilutes the angular polarization patterns of the singly scattered light, the angles where the abso- lute degree of polarization reaches a local maximum and/or where it is zero (the so-called ’neutral points’) are preserved and thus still allow for the characterization of the particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Article number, page 3 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses Another factor in the successful application of (spectro)polarimetry for the characterization of Venus’s clouds and hazes is that with Earth-based telescopes, inner planet Venus can be observed at a wide range of phase angles, thus allowing observations of the angular variation of the degree of polarization due to the light that has been singly scattered by the atmospheric constituents.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In our solar system, only Venus, Mercury, and the Moon can be observed at a large phase angle range with Earth-based telescopes (ignoring the proximity of Mercury to the Sun).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' To effectively apply polarimetry to the outer planets in the solar system, a polarimeter onboard a space mission would be needed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' An example of such an instrument was OCCP onboard NASA’s Galileo mission (Russell et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1992) that orbited Jupiter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Regarding exoplanets, however, the range of observable phase angles depends on the inclination angle of the planetary orbits: for a face-on orbit, the planet’s phase angle will be 90◦ everywhere along the orbit, while for an edge-on orbit, the phase angle will range from close to 0◦ (when the planetary disk is fully illuminated) to 180◦ (when the night-side of the planet is in view).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The precise range of accessible phase angles would of course depend on the observational technique and e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' the use of a coronagraph or star-shade.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Here we investigate the total flux and degree of polarization of starlight that is reflected by terrestrial-type exoplanets, focusing on the possible evolutionary stages of Venus as described by Bullock & Grinspoon (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Our goal is to identify characteristic signatures that could help to identify the properties of exo-Venuses, thus to guide the design of future telescope instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We compute the disk-integrated total and polarized fluxes of light reflected between wavelengths of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='3 to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' First, we study the single scattering properties of spherical cloud droplets of pure water (H2O) or 75% sulphuric acid (H2SO4) in order to identify potentially distinct signatures for each particle type as a function of wavelength and planetary phase angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Second, we compute the multiple scattered flux and polarization signals that are integrated over the planet’s illuminated disk as functions of the planet’s phase angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Third, we compute the signals of the planets in the four evolutionary phases in stable orbits around the nearby solar-type Alpha Centauri A, simulating the observations of such planets if they are spatially unresolved from their parent star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The outline of this paper is as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2, we define the fluxes and polarization of planets, and we describe our numerical algorithm and the four model planets in the evolutionary phases as described by Bullock & Grinspoon (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 3, we present the total and polarized fluxes as computed for planets that are spatially resolved from their star and for planets that are spatially unresolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In the latter case, the planet’s signal is thus combined with the stellar light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We specifically assume that our model planet orbits the solar-type star Alpha Centauri A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 4, we summarize our results and present our conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Numerical method 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Flux and polarization definitions In this paper, we present the flux and polarization signals of starlight that is reflected by potentially habitable exoplanets that orbit solar-type stars, and in particular, Alpha Centauri A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Because these Article number, page 4 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses planets will be very close in angular distance to their parent star, they will usually be spatially unre- solved, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' it will not be possible to spatially separate the planet’s signal from that of its parent star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The flux vector Fu (u= ’unresolved’) that describes the light of the star and its spatially unresolved planet, and that arrives at a distant observer is then written as Fu(λ, α) = Fs(λ) + Fp(λ, α), (1) with Fs the star’s flux vector and Fp that of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Furthermore, λ is the wavelength (or wave- length band), and α is the planetary phase angle, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' the angle between the star and the observer as measured from the center of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We assume that the light of the star is captured together with the starlight that is reflected by the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' A telescope with a coronagraph or star-shade would of course limit the amount of captured direct starlight, depending on its design and the angular dis- tance between the star and the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' A flux (column) vector is given by (Hansen & Travis 1974) F = [F, Q, U, V], (2) with F the total flux, Q and U the linearly polarized fluxes, and V the circularly polarized flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The dimensions of F, Q, U, and V are W m−2, or W m−3 when defined per wavelength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Measurements of FGK-stars, such as the Sun and Alpha Centauri A, indicate that their (disk- integrated) polarized fluxes are virtually negligible (Kemp et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1987;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Cotton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2017), thus we describe the star’s flux (column) vector that arrives at the observer located at a distance D as Fs(λ) = Fs(λ) 1 = R2 s D2 πB(λ, Ts) 1, (3) with πB the stellar surface flux, Ts the star’s effective temperature, Rs the stellar radius, and 1 the unit (column) vector.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The parameter values that we adopt for the Alpha Centauri A system are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Because of the huge distances to stars and their planets, flux vector Fp of the starlight that is reflected by an exoplanet pertains to the planet as a whole, thus integrated across the illuminated and visible part of the planetary disk.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' It is given by (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2018) Fp(λ, α) = AG(λ) Rp(λ, α) r2 p D2 R2 s d2 πB(λ, Ts) 1 (4) = AG(λ) R1p(λ, α) r2 p D2 R2 s d2 πB(λ, Ts).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' (5) Here, AG is the planet’s geometric albedo, Rp the matrix describing the reflection by the planet and R1p its first column, rp is the planet’s radius, d the distance between the star and the planet, and D the distance to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The planet’s reflection is normalized such that planetary phase function R1p, which is the first element of R1p, equals 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 at α = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Article number, page 5 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses The contrast C between the total flux of the planet and the total flux of the star is then given by C(λ, α) = Fp(λ, α) Fs(λ) = AG(λ) R1p(λ, α) r2 p d2 , (6) with Fp the first element of the planetary flux vector Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Using the parameters from Table 1, the con- trast C between a planet with the radius of Venus at a Venus-like distance from Alpha Centauri A equals about 2·10−9AG at α = 0◦ (at this phase angle, the planet would actually be precisely behind the star with respect to the observer and thus out of sight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The degree of polarization of the spatially resolved planet (without including any direct light of the star) is defined as Pp = � Q2p + U2p Fp , (7) where we ignore the planet’s circularly polarized flux Vp as it is expected to be very small com- pared to the linearly polarized fluxes (Rossi & Stam 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We also ignore the circularly polarized fluxes in our radiative transfer computations, as this saves significant amounts of computing time without introducing significant errors in the computed total and linearly polarized fluxes (see Stam & Hovenier 2005).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Fluxes Qp and Up are defined with respect to the planetary scattering plane, which is the plane through the planet, the star and the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In case the planet is mirror-symmetric with respect to the planetary scattering plane, linearly polarized flux Up equals zero and we can use an alternative definition of the degree of polarization that includes the polarization direction as follows Pp = −Qp Fp .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' (8) If Pp > 0 (Pp < 0), the light is polarized perpendicular (parallel) to the reference plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In case a planet is not completely spatially resolved from its parent star, and the background of the planet on the sky is thus filled with (unpolarized) starlight, the observable degree of polarization Pu can be written as (cf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Eqs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 6-7) Pu = � Q2p + U2p Fp + xFs = Fp Fp + xFs Pp = C C + xPp, (9) with x the fraction of the stellar flux that is in the background, which will depend on the angular distance between the star and the planet, on the starlight suppressing techniques that are employed, such as a coronagraph or star-shade, and on the spatial resolution of the telescope at the wavelength under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' This equation also holds for the signed degree of polarization as given by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' If x = 1, the planetary and the stellar flux are measured together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In that case, Pu = C C + 1Pp ≈ CPp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' (10) Article number, page 6 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The values of the parameters describing the planetary system of Alpha Centauri A used in our numerical modelling1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Parameter (unit) Symbol Value Stellar radius (RSun) Rs 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2234 Stellar effective temperature (K) Ts 5790 Planet radius (km) rp 6052 Planet orbital distance (AU) d 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='86 Planet orbital period (yr) P 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='76 Distance to the system (ly) D 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 Angular separation (arcsecs) S 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='67 Here we used the fact that the contrast C will usually be very small (on the order of 10−9 as shown earlier).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The planet’s degree of polarization Pp and the contrast C both depend on λ and α, but generally in a different way.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The dependence of Pu on λ and α will thus generally differ from that of either Pp or C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Our radiative transfer algorithm Our procedure to compute the flux vector Fp (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 5) of the starlight that is reflected by the planet, is described in Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' (2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The radiative transfer algorithm is based on an efficient adding- doubling algorithm (de Haan et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1987) and fully includes polarization for all orders of scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' With this algorithm, through the use of a Fourier-series expansion of the planetary reflection matrix Rp, the reflected flux vector can be computed for any planetary phase angle α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Our model planetary atmospheres consist of horizontally homogeneous layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For each layer, we prescribe the total optical thickness b, the single-scattering albedo a, and the single-scattering matrix P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Our layered model atmosphere is bounded below by a Lambertian reflecting surface (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' the light is reflected isotropically and unpolarized) with an albedo asurf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' A layer’s optical thickness b at a wavelength λ is the sum of the optical thicknesses of the gas molecules, bm, and, if present, the cloud particles, bc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We ignore other atmospheric particles, such as haze particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The single-scattering matrix P of a mixture of gas molecules and cloud particles in a layer is given by P(Θ, λ) = bm sca(λ) Pm(Θ, λ) + bc sca(λ) Pc(Θ, λ) bmsca(λ) + bcsca(λ) , (11) with subscript ‘sca’ referring to ‘scattering’, thus bsca = ab, with a the single scattering albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Furthermore, Pm is the single-scattering matrix of the gas molecules, and Pc that of the cloud particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Θ is the single scattering angle: Θ = 180◦ − α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We use two types of model atmospheres to study the influence of an exoplanet’s atmospheric evolution on the reflected light signals: an Earth-like and a Venus-like atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For our Earth- like atmosphere, we define the pressure and temperature across 17 layers, representing a mid- 1 The orbital distance d of the planet has been chosen such that it receives the same stellar flux as Venus receives from the Sun, and in accordance with the orbit stability requirements for a planet around Alpha Centauri A predicted by Quarles & Lissauer (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For the radius of the Sun, RSun, we use 695,700 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Article number, page 7 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The four evolutionary phases of the model planets (Bullock & Grinspoon 2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In Phases 1 and 2, the clouds consist of liquid water droplets, and in Phases 3 and 4, of liquid sulfuric acid solution droplets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The cloud optical thickness is indicated by bc and the cloud particle effective radius by reff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For the effective variance veff of the size distributions in Phases 1-3, we use 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='1, and for Phase 4, veff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='07.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' latitude summer profile (following Stam 2008).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For our Venus-like atmosphere, we use 71 layers with pressure and temperature profiles from the Venus International Reference Atmosphere (VIRA) (Kliore et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1985), representing a mid-latitude afternoon profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' With these vertical profiles, and assuming anisotropic Rayleigh scattering (Hansen & Travis 1974), we compute each layer’s single scattering matrix Pm and the scattering optical thickness bm sca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We neglect absorption, thus bm = bm sca.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The depolarization factor for computing Pm and bm sca for anisotropic Rayleigh scattering depends on the atmospheric composition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For the Earth-like atmosphere, we use a (wavelength independent) depolarization factor of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='03, which is representative for dry air, and for the Venus-like atmosphere, we use 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='09, which is representative for a pure CO2 atmosphere (Hansen & Travis 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We use wavelength-independent refractive indices of 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00044 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00027 for the Venus-like and the Earth-like model atmospheres, respectively;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' note that this assumption has a negligible effect on the reflected total and polarized fluxes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The cloud particles in our model atmospheres are spherical and distributed in size according to a two-parameter gamma size distribution (see Hansen & Travis 1974) that is described by an effective radius reff and an effective variance veff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The terrestrial clouds are located between 1 and 3 km altitude, and the Venusian clouds, depending on their evolutionary phase, between 47 and 80 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The cloud optical thickness has a uniform vertical distribution through the altitude range (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The single-scattering properties of the cloud particles are computed using Mie-theory (De Rooij & Van der Stap 1984), as these particles are expected to be spherical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For these computations we specify the wavelength λ and nr, the refractive index of the cloud particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The cloud particles are composed of either pure water or a sulphuric acid solution with varying concentration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We use the refractive index of water from Hale & Querry (1973) and that of sulphuric acid with 75 % acid concentration from Palmer & Williams (1975).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We use a negligible value for the imaginary part of the particles’ refractive indices, ni = 10−8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Article number, page 8 of 24 Cloud thickness Ititude A Cloud top = 3 km Cloud top = 80 km Cloud top = 65 km Cloud top = 65 km bc = 2 bc = 4 bc = 120 bc = 30 reff = 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 μm reff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 μm reff = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 μm reff = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 μm Phase 1 Phase 2 Phase 3 Phase 4 Current Earth Thin clouds Venus Thick clouds Venus Current VenusMahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Cloud properties through the planet’s evolution It is suspected that early Venus had a thin, Earth-like atmosphere and (possibly) an Earth-like ocean that was later lost due to the runaway greenhouse effect (Donahue et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1982;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Kasting 1988;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Way & Del Genio 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' As the planet’s surface heated up, the water would have evaporated and enriched the atmosphere with water vapor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The macroscopic cloud properties for the 4 evolutionary phases that we will use are illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We start our evolutionary model of Venus assuming Earth-like conditions (Phase 1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' an atmosphere consisting of 78% N2 and 22% O2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Model simulations showed that the actual depen- dence of the total and polarized flux signals on the percentage of oxygen appeared to be negligible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Hence we used the present day Earth atmosphere as the Earth-like atmosphere model while the ac- tual percentage of oxygen on an exo-planet could be different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The cloud particles have an effective radius reff of 10 µm in agreement with ISCCP (Tselioudis 2001) and an effective variance veff of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The total cloud optical thickness bc is 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 at λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='55 µm and the cloud layer extends from 2 to 4 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The next evolutionary phases are also inspired by the Venus climate model of Bullock & Grin- spoon (2001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In Phase 2, the atmosphere is Venus-like as it consists of pure CO2 gas, and has relatively thin liquid water clouds with bc = 4, and with the cloud tops at 80 km.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For this phase, we use reff of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm, which is smaller than the present day value, because the atmosphere is ex- pected to be too hot for strong condensation to take place thus preventing the particles to grow larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In Phase 3, the clouds are thick sulphuric-acid solution clouds, with bc = 120 and the cloud tops at 65 km, because the atmosphere is cool enough to allow condensation and/or coalescence of saturated vapour over a large altitude range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Since the region of condensation covers a large altitude range, the particles can grow large until they evaporate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In this phase, reff = 2 µm, which is twice the effective radius of the present day Venus cloud particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For both phases 2 and 3, we use veff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In Phase 4, the clouds have the present-day properties of Venus’s clouds with bc = 30 and the cloud tops at 65 km (Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Ragent et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1985).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For the cloud particle sizes in this phase, we use reff = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 µm and veff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='07 following the values derived by Hansen & Hovenier (1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We ignore the absorption by cloud particles in the UV in all of our Venus-like clouds to avoid adding complexity and because the exact nature and location of the UV-absorption is still under debate (Titov et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Figure 2 shows the phase function (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' single scattering matrix element P11) and the degree of linear polarization for unpolarized incident light (the ratio of single scattering matrix elements −P21/P11) that has been singly scattered by the four different types of cloud particles as functions of α (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 180◦ - Θ), for a range of wavelengths λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2, the phase functions show strong forward scattering peaks (near α = 180◦, thus when the night-side of the planet would be turned towards the observer) that decrease with increasing λ, thus with decreasing effective particle size parameter xeff = 2πreff/λ (for the large H2O cloud particles, with reff = 10 µm, this decrease is not readily apparent from the figure).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Article number, page 9 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The total flux and degree of polarization of incident unpolarized light that has been singly scattered by four different types of cloud particles, as functions of the phase angle α and the wavelength λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Left column: the total flux or phase function (single scattering matrix element P11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Right column: the degree of polarization (−P21/P11).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' First row: H2O particles with reff = 10 µm and veff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='1 (belonging to the Phase 1 model planet: ’Current Earth’);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Second row: H2O particles with reff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm and veff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='1 (Phase 2: ’Thin clouds Venus’);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Third row: 75% H2SO4 particles with reff = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm and veff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='1 (Phase 3: ’Thick clouds Venus’);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Fourth row: 75% H2SO4 particles with reff = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 µm and veff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='07 (Phase 4: ’Current Venus’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The H2O particles with reff = 10 µm show a moderate local maximum in the phase function around α = 40◦, which is usually referred to as the primary rainbow (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Hansen & Travis 1974).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The large H2O and the H2SO4 particles also produce higher fluxes towards α = 0◦ that are referred to as the glory (Laven 2008;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' García-Muñoz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Markiewicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2014;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2015;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Markiewicz et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2018).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For the small H2O particles and the H2SO4 particles at larger wavelengths, the phase functions become more isotropic and the glory and other angular features disappear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Figure 2 also shows the degree of linear polarization of the singly scattered light.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' This degree of polarization appears to be more sensitive to the particle composition than the scattered flux, especially for λ between 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 and 2 µm, where H2O particles yield relatively high positive degrees of polarization (perpendicular to the scattering plane) between phase angles of about 20◦ and 100◦, whereas the H2SO4 particles impart a mostly negative degree of polarization through a broad range of phase angles, except for narrow regions around α = 20◦ and 80◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The tiny, reff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm, water droplets have a strong, broad positive polarization region for λ ≥ 1 µm, where they are so small with respect to the wavelength that they scatter like Rayleigh scatterers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' As mentioned before (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Hansen & Travis 1974;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Hansen & Hovenier 1974), patterns in the single scattering degree of polarization are generally preserved when multiple scattered light is added, as the latter usually has a low degree of polarization, and thus adds mostly total flux, which Article number, page 10 of 24 H20,reff:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0μm H20,reff:10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0μm 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 Wavelength (μm) Wavelength (μm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='500 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='028 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='250 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 Phase angle (deg) Phaseangle (deg)Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses subdues angular features, but does not change the angular pattern (local maxima, minima, neutral points) itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The single scattering angular features in the polarization will thus also show up in the polarization signature of a planet as a whole, and can be used for characterisation of the cloud particle properties and thus possibly of various phases in the evolution of a Venus-like exoplanet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' This will be investigated in the next section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Results Here we present the disk-integrated total flux and degree of polarization of incident unpolarized starlight that is reflected by the model planets at different wavelengths λ and for phase angles α ranging from 0◦ to 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The actual range of phase angles at which an exoplanet can be observed depends on the inclination angle i of the planet’s orbit (the angle between the normal on the orbital plane and the direction towards the observer): α ranges between 90◦ − i to 90◦ + i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Obviously, at α = 0◦, the planet would be precisely behind its star, and at 180◦ it would be precisely in front of its star (in transit).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Other phase angles might be inaccessible due to restrictions of inner working angles of telescopes and/or instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For completeness, we include all phase angles in our computations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='1 shows results for spatially resolved planets and Sect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 for planets that are spatially unresolved from their star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In particular, we show these latter results for a model planet orbiting the star Alpha Centauri A at a distance where the incident stellar flux is similar to the solar flux that reaches Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Because our model planets are all mirror-symmetric with respect to the reference plane, their linearly polarized flux Up equals zero and will not be discussed further.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Flux and polarization of spatially resolved planets Figure 3 shows the total flux Fp (the planetary phase function) and degree of polarization Pp as functions of α and λ for the four evolutionary phases illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The total fluxes are normalized such that at α = 0◦, they equal the planet’s geometric albedo AG (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Figure 4 shows AG of the planets in the four evolutionary phases as functions of the wavelength λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Table 2 lists the geometric albedo’s at 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The ’Current Earth’ (Phase 1) shows very little variation in AG, and the ’Thin clouds Venus’ (Phase 2) has the lowest albedo because of the small cloud particles and the small cloud optical thickness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The geometric albedo’s of the ’Thick clouds Venus’ (Phase 3) and the ’Current Venus’ (Phase 4) are very similar.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Thus across the wavelength region investigated in this paper, the ’Current Earth’ (Phase 1) has the highest geometric albedo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The model planets’ geometric albedo’s AG for the four evolutionary phases at four wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' λ (µm) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 Current Earth 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='757 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='752 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='740 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='739 Thin clouds Venus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='186 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='184 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='240 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='310 Thick clouds Venus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='727 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='627 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='580 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='564 Current Venus 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='726 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='573 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='500 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='484 Article number, page 11 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Left column: The total flux (or phase function);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' and right column: The degree of polarization of incident unpolarized starlight that is reflected by the model planets in the 4 evolutionary phases as functions of α and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' First row: Phase 1 (’Current Earth’);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Second row: Phase 2 (’Thin clouds Venus’);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Third row: Phase 3 (’Thick clouds Venus’);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Fourth row: Phase 4 (’Current Venus’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The phase functions are normalised such that at α = 0◦, they equal the planet’s geometric albedo AG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For each model planet, the total flux Fp decreases with increasing α mostly because less of the planet’s observable disc is illuminated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The planet with thin H2O clouds (Phase 2) is very dark over all α’s because the cloud optical thickness bc is small and the surface is black.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The total fluxes show vague similarities with the single scattering phase functions of the cloud particles (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In particular, for the ’Current Earth’ (Phase 1) with large H2O particles, Fp increases slightly around α = 40◦, the rainbow angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Also, the decrease of Fp with λ is stronger for the Venus-type planets with H2SO4 clouds (Phases 3 and 4) than for the planet with the large H2O particles (’Current Earth’, Phase 1), because the single scattering phase function of the sulfuric acid particles decreases stronger with λ than that of the water droplets (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 1.' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 0 20 40 60 80 100 120 140 160 180 0 20 40 60 80 100 120 140 160 180 Phase angle (deg) Phase angle (deg)Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses Unlike Fp, the degree of polarization Pp of each of the model planets, shows angular and spectral features that depend strongly on the cloud properties and should thus allow distinguishing between the different evolutionary phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In Phase 1 (’Current Earth’), Pp is high and positive up till λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm and around α = 90◦, which is due to Rayleigh scattering by the gas above the clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Starting at the shortest λ, Pp increases slightly with λ before decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' This is due to the slightly larger contribution of multiple scattered light, with a lower degree of polarization, at the shortest wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' A Rayleigh-scattering peak is also seen for Phase 4 (’Current Venus’), except there, the peak decreases more rapidly with λ because the clouds are higher in the atmosphere and there is thus less gas above them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In Phase 3 (’Thick clouds Venus’), the Rayleigh-scattering peak is suppressed by the contribution of low polarized light that is reflected by the thicker clouds below the gas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In Phase 2 (’Thin clouds Venus’), the relatively thin clouds are higher in the atmosphere than in Phase 1 (’Current Earth’), which is why the Rayleigh-scattering peak only occurs at the very shortest wavelengths (the peak is hardly visible in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Because the Phase 2 cloud particles are small (reff = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm), they themselves give rise to a Rayleigh-scattering peak at λ ≥ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The two model planets with the H2O cloud particles (Phases 1 and 2) show a narrow region of positive polarization between 30◦ and 40◦, which is the rainbow peak (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' On exoplanets, this local maximum in Pp could be used to detect liquid water clouds on exoplanets (Karalidi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2011, 2012;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Bailey 2007).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In Phase 1 (’Current Earth’), the rainbow region starts near the Rayleigh scattering peak of the gas and extends towards the largest wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In Phase 2 (’Thin clouds Venus’), with the small water droplets, the rainbow only occurs at the shortest wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' With increasing wavelength, it broadens and disappears into the cloud particles’ Rayleigh scattering peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The H2SO4 cloud particles (Phases 3 and 4) have their own specific polarization patterns, such as the broad negative polarization region at α ⪆ 80◦, which can be traced back to their single scattering patterns (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In Phase 3 (’Thick clouds Venus’), the cloud particles give rise to a sharp positive polarization peak at the shortest wavelengths and for 20◦ ≤ α ≤ 30◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In Phase 4 (’Current Venus’), there is a broader, lower, positive polarization branch across this phase angle range, which resembles the positive polarization branch of the tiny H2O droplets in Phase 2 (’Thin clouds Venus’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' However, at the longer wavelengths, the phase angle dependence of the polarization of the latter planet is very different which should help to distinguish between such planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' This emphasizes the need for measurements at a wide range of wavelengths and especially phase angles (if the planet’s orbital inclination angle allows this).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Flux and polarization of spatially unresolved planets In the previous section, we showed the signals of spatially resolved planets, thus without back- ground starlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' When observing an exoplanet in the habitable zone of a solar-type star, it will be difficult to avoid the starlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Here, we show the total flux of the planet Fp, the star-planet contrast C (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 6), the spatially resolved degree of polarization of the planet Pp (thus without Article number, page 13 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' A sketch of the geometries within the Alpha Centauri system: the orbital plane of the stars Alpha Centauri A and B is inclined by about 80◦ with respect to the observer on Earth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Our model planet (the blue dot) orbits Alpha Centauri A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In this sketch, the line of nodes of the planet’s orbit was chosen to coincide with that of the stellar orbits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The inclination angle im of the planet’s orbit with respect to the stellar orbital plane is 45◦, and the inclination angle of the planet’s orbit with respect to the observer is 80◦ − 45◦ = 35◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The phase angles of the planet in this sketch would range from 90◦ − 35◦ = 55◦ to 90◦ + 35◦ = 125◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' the starlight) and the spatially unresolved degree of polarization of the combined star-planet signal Pu (thus including the starlight).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' While the total planet fluxes shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 3 were normalized at α = 0◦ to the planets’ geometric albedo’s AG, here they are computed according to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 5, and thus depend on the parameters of the planet-star system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We assume our model planets orbit Alpha Centauri A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The solar-type star Alpha Centauri A is part of a double star system, and the orbital parameters of ourplanets are chosen based on the stable planet orbital distances and orbital inclination angles around this star as predicted by Quarles & Lissauer (2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Figure 5 shows a sketch of the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We use a planetary orbital distance d of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='86 AU, such that each model planet receives a stellar flux similar to the solar flux received by Venus.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Additional system parameter values are listed in Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' According to Quarles & Lissauer (2016), stable orbits around Alpha Centauri A can be found for a range of angles between the planetary orbital plane and the plane in which the two stars orbit, and thus for a range of inclination angles i of the planetary orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Figure 6 shows the variation of the planetary phase angle α along a planetary orbit for two values of the longitude of the orbit’s ascending node Ω: for Ω = 0◦ (the line connecting the planet’s ascending and descending nodes is perpendicular to the line to the observer) and for Ω = 205◦ which represents the configuration of Earth with Alpha Centauri A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The orbital phase of the planet is defined such that at an orbital phase angle of 0◦, α = 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The inclination angle im is the 0 50 100 150 200 250 300 350 Orbital phase (deg) 0 20 40 60 80 100 120 140 160 180 Alpha (deg) = 0 im = -10 im = 80 im = 0 im = 35 0 50 100 150 200 250 300 350 Orbital phase (deg) 0 20 40 60 80 100 120 140 160 180 Alpha (deg) = 205 im = -10 im = 80 im = 0 im = 35 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Variation of the planet’s phase angle α along the planet’s orbit around Alpha Centauri A for different mutual inclination angles im of the planetary orbit with respect to the orbital plane of the two stars.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Top: Ω, the longitude of the ascending node of the planet’s orbit, is 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For im = −10◦, the planet is then in a face-on orbit (i = 0◦), while for im = 80◦, it is in an edge-on orbit (i = 90◦).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Bottom: Ω = 205◦, and the planetary orbit is aligned with the node of the stellar orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Article number, page 14 of 24 = 800 Alpha Centauri B i= 35o to the observer Alpha Centauri AMahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses 0 50 100 150 200 250 300 350 Orbital phase (deg) 10 20 30 40 Fp (10 12 W m 2m 1) im = -10 im = 80 im = 0 im = 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 C (10 9) 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='04 Pp im = -10 im = 80 im = 0 im = 35 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 Pu (10 9) im = -10 im = 80 im = 0 im = 35 0 50 100 150 200 250 300 350 Orbital phase (deg) 2 4 6 8 10 12 14 16 Fp (10 12 W m 2m 1) im = -10 im = 80 im = 0 im = 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 C (10 9) 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='03 Pp im = -10 im = 80 im = 0 im = 35 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='03 Pu (10 9) im = -10 im = 80 im = 0 im = 35 0 50 100 150 200 250 300 350 Orbital phase (deg) 1 2 3 4 5 Fp (10 12 W m 2m 1) im = -10 im = 80 im = 0 im = 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='9 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 C (10 9) 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='03 Pp im = -10 im = 80 im = 0 im = 35 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='030 Pu (10 9) im = -10 im = 80 im = 0 im = 35 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='8 Fp (10 12 W m 2m 1) im = -10 im = 80 im = 0 im = 35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='9 C (10 9) 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 Pp im = -10 im = 80 im = 0 im = 35 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='020 Pu (10 9) im = -10 im = 80 im = 0 im = 35 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The total planetary flux Fp (in W m−3) and the star-planet contrast C, the degree of polarization Pp of the spatially resolved planet, and Pu, the degree of polarization of the star and the spatially unresolved planet, all for a ’Current Venus’ model planet (Phase 4) as functions of the planet’s orbital phase for four mutual inclination angles im and four wavelengths λ: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm (row 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm (row 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm (row 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm (row 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The longitude of the ascending node of the planetary orbit, Ω, is 205◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' angle between the plane in which the stars move and the planetary orbital plane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For Ω = 0◦, im = −10◦ would yield a face-on planetary orbit (i = 0◦) with α = 90◦ everywhere along the orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For im = 80◦, the orbit is edge-on (i = 90◦) and α varies between 0◦ and 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Figure 6 also shows the range of α for im = 0◦ and 35◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' According to Quarles & Lissauer (2016), the latter is the most probable orientation of a stable planetary orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For these two cases, the accessible phase angles range from 80◦ to 100◦, and from 45◦ to 135◦, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For Ω = 205◦, the maximum range of α would be from 20◦ to 160◦, depending on im.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Figure 7 shows Fp, Pp and Pu (the spatially unresolved planet, thus with starlight included) for Phase 4 (’Current Venus’) as functions of the planet’s orbital phase for Ω = 205◦, four values of im (-10◦, 80◦, 0◦, and 35◦), and four wavelengths (0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5, and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The plots for Fp also show the contrast C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Because C is the ratio of the planetary flux Fp to the stellar flux Fs (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 6), its variation with the orbital phase is proportional to that of Fp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' At the two orbital phases in each plot where all the lines cross, the planetary phase angles α are the same (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 6) and thus all Fp and Pp are the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The plots for Fp appear to be very similar for the different wavelengths, apart from a difference in magnitude which is mainly due to the decrease of the stellar flux that is incident on the planet with increasing wavelength, although the planetary albedo AG and phase function R1p also decrease with increasing λ as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' This wavelength dependence of Fp also causes the decrease of the contrast C Article number, page 15 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses with increasing wavelength (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' the planet darkens with increasing λ), as C is independent of the wavelength dependence of the stellar flux (see Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The shape differences between the Fp (and C) curves are due to the wavelength dependence of the planetary flux that can also be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' At each wavelength λ, the largest variation in Fp with the orbital phase is seen for im = 80◦, because for that configuration the variation of α along the orbit is largest (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The degree of polarization Pp of the planet shows significant variation with the orbital phase at all wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' A particularly striking feature for the geometry with im = 80◦ is the double peak close to the orbital phases of 150◦ and 200◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 6 for Ω = 205◦ and im = 80◦, α decreases from about 160◦ at an orbital phase of 0◦, to 20◦ at an orbital phase around 175◦, to then increase again with increasing orbital phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Tracing this path of α through the Pp panel in the bottom row of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 3 explains the double peaked behaviour of Pp and its wavelength dependence as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For the other values of im, the phase angle range that is covered along the orbit is smaller, and therefore the variation in Pp is also smaller.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The degree of polarization of the spatially unresolved planet, Pu, shows similar variations along the orbital phase as Pp, except that most features are flattened out because of the addition of the unpolarized stellar flux, which is independent of the orbital phase angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The double peaked feature for im = 80◦ remains strong, however, as at those orbital phase angles, the contrast C is relatively large and thus the influence of the added stellar flux relatively small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The variation in the polarisa- tion of the unresolved system due to the orbiting planet is on the order of 10−11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Figure 8 is similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 7, except for the four model planets in the different evolutionary phases and all for im = 80◦ and Ω = 0◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Because here the planetary orbits are seen edge-on (i = 90◦), the full range of phase angles is covered, which makes it possible to explore the full extent of variation of flux and polarization signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Because of this large phase angle range, Fp varies strongly with the orbital phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The wavelength dependence of the total flux can be traced back to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 3, where in particular the Phase 2 planet (’Thin cloud Venus’) is dark at all wavelengths, but relatively bright at the longest wavelengths and small phase angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' As was the case in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 7, the variation of C is the same as that of Fp, except for the off-set due to the stellar flux.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The largest values of C (about 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6×10−9), are found for λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm and around the orbital phase of 180◦ (at 180◦, the planets would actually be behind the star).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Furthermore in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 8, Pp depends strongly on λ and the planet’s evolutionary phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' At 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm, the Phase 1 planet (’Current Earth’) shows the largest values of Pp due to the Rayleigh scattering gas above the low altitude clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' At the longer wavelengths, where the Rayleigh scattering is less prominent, the curves for the Phase 1 planet clearly show the positive polarization of the rainbow around the orbital phases of 140◦ and 220◦ (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For the Phase 2 planet (’Thin clouds Venus’) and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm, the small cloud particles cause positive polarization around 150◦ and 210◦, which connect the rainbow and the Rayleigh scattering maximum in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' At longer wavelengths, the broad positive polarization signature of Rayleigh scattering by the cloud particles dominates the curves, while the curves for the Phase 3 and 4 planets show mostly negative polarization apart from Article number, page 16 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses 0 50 100 150 200 250 300 350 Orbital phase (deg) 0 10 20 30 40 50 60 Fp (10 12 W m 2 m 1) Phase 1 Phase 2 Phase 3 Phase 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='75 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='25 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='50 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='75 C (10 9) 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='20 Pp Phase 1 Phase 2 Phase 3 Phase 4 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='12 Pu (10 9) Phase 1 Phase 2 Phase 3 Phase 4 0 50 100 150 200 250 300 350 Orbital phase (deg) 0 5 10 15 20 Fp (10 12 W m 2 m 1) Phase 1 Phase 2 Phase 3 Phase 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 C (10 9) 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='06 Pp Phase 1 Phase 2 Phase 3 Phase 4 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='06 Pu (10 9) Phase 1 Phase 2 Phase 3 Phase 4 0 50 100 150 200 250 300 350 Orbital phase (deg) 0 2 4 6 8 Fp (10 12 W m 2 m 1) Phase 1 Phase 2 Phase 3 Phase 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 C (10 9) 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='15 Pp Phase 1 Phase 2 Phase 3 Phase 4 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='06 Pu (10 9) Phase 1 Phase 2 Phase 3 Phase 4 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 Fp (10 12 W m 2 m 1) Phase 1 Phase 2 Phase 3 Phase 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 C (10 9) 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='25 Pp Phase 1 Phase 2 Phase 3 Phase 4 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='08 Pu (10 9) Phase 1 Phase 2 Phase 3 Phase 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 7 except for the model planets in the four evolutionary phases and Ω = 0◦ and im = 80◦ (thus for an edge–on orbit, i = 90◦): Phase 1 (’Current Earth’), Phase 2 (’Thin clouds Venus’), Phase 3 (’Thick clouds Venus’), Phase 4 (’Current Venus’).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The wavelengths λ are like before: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm (row 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm (row 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm (row 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm (row 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 0 50 100 150 200 250 300 350 Orbital phase (deg) 10 15 20 25 Fp (10 12 W m 2 m 1) Phase 1 Phase 2 Phase 3 Phase 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='7 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='8 C (10 9) 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='20 Pp Phase 1 Phase 2 Phase 3 Phase 4 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='12 Pu (10 9) Phase 1 Phase 2 Phase 3 Phase 4 0 50 100 150 200 250 300 350 Orbital phase (deg) 4 6 8 10 Fp (10 12 W m 2 m 1) Phase 1 Phase 2 Phase 3 Phase 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='7 C (10 9) 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='06 Pp Phase 1 Phase 2 Phase 3 Phase 4 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='020 Pu (10 9) Phase 1 Phase 2 Phase 3 Phase 4 0 50 100 150 200 250 300 350 Orbital phase (deg) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 Fp (10 12 W m 2 m 1) Phase 1 Phase 2 Phase 3 Phase 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='7 C (10 9) 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='15 Pp Phase 1 Phase 2 Phase 3 Phase 4 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='04 Pu (10 9) Phase 1 Phase 2 Phase 3 Phase 4 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='8 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 Fp (10 12 W m 2 m 1) Phase 1 Phase 2 Phase 3 Phase 4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='7 C (10 9) 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='25 Pp Phase 1 Phase 2 Phase 3 Phase 4 0 50 100 150 200 250 300 350 Orbital phase (deg) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='08 Pu (10 9) Phase 1 Phase 2 Phase 3 Phase 4 Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 8 except for the most probable, stable orbit around Alpha Centauri A, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' for Ω = 205◦ and im= 35◦ (Quarles & Lissauer 2016).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The wavelengths are like before: 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm (row 1);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm (row 2);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm (row 3);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' and 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm (row 4).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Article number, page 17 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses the orbital phases around 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' When adding the starlight, the angular features of the polarization Pu are suppressed along those parts of the orbits where C is smallest, thus away from the orbital phase angle of 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In particular within 180◦ ±40◦, Pu still shows distinguishing features, although they are very small in absolute sense (smaller than 10−10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Figure 9 is similar to Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 8 except here the model planets are in the most probable stable orbit around Alpha Centauri A as predicted by Quarles & Lissauer (2016), namely with Ω = 205◦ and im = 35◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' As can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 6, for this geometry α varies between about 60◦ and 120◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 9, Fp shows a similar variation as the curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 8, although less prominent, as the planets do not reach a ’full’ phase (where α = 0◦) nor the full night phase (α = 180◦) along their orbit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The flux curves in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 9 also miss small angular features that appear in the single scattering phase functions of the cloud particles (see Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2), such as the glory, again because the planets do not go through the related phase angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In this particular orbital geometry, Pp shows less pronounced angular features than for the same model planets in edge-on orbits (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 8) because of the more limited phase angle range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For example, the ’Current Earth’ (Phase 1) shows no rainbow despite the H2O clouds, because the phase angle of about 40◦ is not reached.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In the visible (λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm), Pp reaches the largest values for the ’Current Earth’ (Phase 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' At longer wavelengths, Pp of the ’Thin clouds Venus’ (Phase 2) strongly dominates because of the Rayleigh scattering by the small cloud particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The ’Thick clouds Venus’ (Phase 3) shows predominant negative polarization at all wavelengths and across the whole orbital phase angle range except at λ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm around an orbital phase angle of 20◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The polarization of the spatially unresolved planets, Pu, clearly shows the suppression of the polarization features due to the added starlight towards the smaller and larger orbital phase angles, where the planets are darker.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' While the Phase 2 planet (’Thin clouds Venus’) is relatively dark (C is very small), its Rayleigh scattering polarization signal is so strong that its unresolved polarization signal is larger than that of the other planets, except the Phase 1 planet (’Current Earth’) at 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm and orbital phase angles close to 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Evolutionary phases across α and λ In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 10, we show which evolutionary phase has the highest values of |Pp| across all phase angles α and wavelengths λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We find that |Pp| of the Phase 1 planet (’Current Earth’) dominates between 30◦ and 150◦, and mostly for λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In particular, around α = 40◦ and up to λ = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm, the polarization signal of the rainbow produced by the large water cloud particles is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='1 (see the bottom plot of Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For λ > 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm, the Phase 2 planet (’Thin clouds Venus’) shows the strongest polarization due to the Rayleigh scattering by the small H2O cloud particles, as can clearly be seen in the bottom plot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The small patches where the strongest polarization signal is from the ’Thick clouds Venus’ (Phase 3), for example near α = 20◦ and λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm, or from the ’Current Venus’ (Phase 4) are due to the single scattering polarization features of the H2SO4 cloud particles, as can be seen in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Article number, page 18 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Top: The planet models that yield the largest absolute degree of polarization |Pp| over all phase angles α and wavelengths λ: Phase 1 - ’Current Earth’ (blue);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Phase 2 - ’Thin clouds Venus’ (light orange);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Phase 3 ’Thick clouds Venus’ (dark orange);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' and Phase 4 - ’Current Venus’ (brown).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Bottom: The maximum values of |Pp| of the four model planets as functions of α and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The accessible phase angle range for direct observations of such exoplanets obviously depends on the actual orientation of the planetary orbits and cannot be optimized by the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Precisely because of that, Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 10 clearly indicates that measurements should be performed across a broad wavelength range, including wavelengths below 1 µm to allow distinguishing between Earth-like and Venus-like planets in various evolutionary phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Summary and conclusions We presented the total flux and linear polarization of starlight that is reflected by model planets of various atmospheric types to investigate whether different phases in the evolution of planets like the Earth and Venus can be distinguished from each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We have used four planet models to represent possible evolutionary phases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Phase 1 (’Current Earth’) has an Earth-like atmosphere and liquid water clouds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Phase 2 (’Thin clouds Venus’) has a Venus-like CO2 atmosphere and thin water clouds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Phase 3 (’Thick clouds Venus’) has a Venus-like CO2 atmosphere and thick sulphuric acid solution clouds;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' and Phase 4 (’Current Venus’) has a CO2 atmosphere and thin sulphuric acid solution clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We have computed the total flux and polarization signals specifically for model Article number, page 19 of 24 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 Earth Thin clouds Venus ThickcloudsVenus Wavelength (microns) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 Current Venus 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 0 20 40 60 80 100 120 140 160 180 Alpha (deg)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='300 Wavelength (microns) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='200 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='050 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='010 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='100 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='300 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='200 100 0 20 40 60 80 100 120 140 160 180 Alpha (deg)Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses planets orbiting our neighbouring solar-type star Alpha Centauri A using predicted stable orbits (Quarles & Lissauer 2016) in its habitable zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We have computed the reflected starlight for wavelengths λ ranging from 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='3 µm to 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm and for planetary phase angles α from 0◦ to 180◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We not only present the fluxes and polarization of spatially resolved model planets (thus without background starlight) but also of spatially unresolved planets, thus the combined signal of the planet and the star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For the latter cases, we also computed the planet-star contrast C as a function of α and λ to determine what would technically be required to detect the planetary signals upon the background starlight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The range of planetary phase angles α at which a planet can be observed (spatially resolved or unresolved) depends on the inclination of the planetary orbit with respect to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We have specifically studied the reflected light signals of planets orbiting solar-type star Alpha Centauri A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' This star is part of a double star system with solar-type star Alpha Centauri B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The distance between the two stars varies from 35.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 to 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 AU (M-dwarf Alpha Centauri C or Proxima Centauri orbits the pair at a distance of about 13,000 AU).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Dynamical computations (Quarles & Lissauer 2016) predict stable planetary orbits around Alpha Centauri A in a narrow range of mutual inclination angles im between the orbital planes of the two stars and that of the planet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In particular, the most stable orbit has im = 35◦, which provides an α range from 60◦ to 120◦.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We find that with this orbital geometry the degree of polarization of the planet would be largest for the ’Current Earth’ (Phase 1) across the visible (λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm) due to Rayleigh scattering by the gas above the clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' At near infrared wavelengths (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm < λ < 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm), the polarization of the ’Thin clouds Venus’ (Phase 2) is highest, because this planet has small cloud droplets that scatter like Rayleigh scatterers at the longer wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The well-known advantage of measuring the degree of polarization for the characterization of (exo)planets is that the angular features in the signal of the planet as a whole are similar to the angular features in the light that has been singly scattered by the gas molecules and cloud particles,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' which are very sensitive to the microphysical properties (such as the size distribution,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' composi- tion,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' shape) of the scattering molecules and cloud particles and to the atmosphere’s macrophysical properties (such as cloud altitude and thickness).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The reflected total flux is much less sensitive to the atmospheric properties than the degree and direction of polarization (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Hansen & Travis 1974, for several examples).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Indeed, the variations of the planetary flux Fp along a planet’s or- bit appear to be mostly due to the change of the fraction of the planetary disk that is illuminated and visible to the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' They provide limited information on the planet’s atmospheric charac- teristics, especially if one takes into account that with real observations, the planet radius will be unknown unless the planet happens to transit its star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The variation of the planetary flux Fp with phase angle α and wavelength λ is similar that of planet-star contrast C, the ratio of Fp to the stellar flux Fs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For a Venus-like planet orbiting Alpha Centauri A, C is on the order of 10−9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Article number, page 20 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses Our numerical simulations show that variations in Pp,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' the degree of polarization of the spatially resolved planets (thus without any starlight),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' with α combined with variations with λ could be used to distinguish the planetary evolutionary phases explored in this paper: – Phase 1 planets (’Current Earth’) show strong positive (perpendicular to the reference plane through the star,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' planet,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' and observer) polarization around α = 40◦ due to scattering by large water cloud droplets (the rainbow) and also higher polarization for λ < 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm and α ≈ 90◦ due to Rayleigh scattering by the gas above the clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' – Phase 2 planets (’Thin clouds Venus’) polarize light negatively (parallel to the reference plane) across most phase angles and at visible wavelengths.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' At near-infrared wavelengths, they have strong positive polarization around α = 90◦ due to Rayleigh scattering by the small cloud droplets, and a ’bridge’ of higher polarization from the Rayleigh maximum to the rainbow angle (α ≈ 40◦) with decreasing λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' – Phase 3 planets (’Thick clouds Venus’) have predominantly negative polarization from the vis- ible to the near-infrared, with small regions of positive polarization for λ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm and for 20◦ ≤ α ≤ 30◦, that are characteristic for the 75% H2SO4 cloud particles with reff = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='0 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' – Phase 4 planets (’Current Venus’) yield similar polarization patterns as the Phase 3 planets, ex- cept with more prominent negative polarization for 10◦ ≤ α ≤ 30◦ and for 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 µm ≤ λ ≤ 2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Rayleigh scattering by the small cloud particles produces a maximum of positive polarization around α = 90◦ and for λ > 2 µm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Our simulations of the planetary polarisation Pp do not include any background starlight, and can thus reach several percent to even 20% for the ’Current Earth’ (Phase 1) planet in an edge-on orbit (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Whether or not such polarization variations could be measured depends strongly on the techniques used to suppress the light of the parent star.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' If the background of the planet signal contains a fraction x of the flux of the star, the degree of polarization of the light gathered by the detector pixel that contains the planet will equal (C/(C + x))Pp ≈ (C/x)Pp with C the contrast between the total fluxes of the planet and the star (Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' For a Venus-like planet orbiting Alpha Centauri A, C is on the order of 10−9, thus x should be as small as 10−4 to get a polarisation signal on the order of 10−6, assuming Pp is about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Not only excellent direct starlight suppression techniques, but also a very high spatial resolution would help to decrease x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Our simulations further show that temporal variations in the total flux Fu of Alpha Centauri A with a spatially unresolved terrestrial-type planet orbiting in its habitable zone would be less than 10−12 W/m3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The degree of polarization Pu of the combined star and planet signals, would show variations smaller than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='05 ppb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' To identify this planetary flux on top of the stellar flux, a very high-sensitivity instrument would be required and an even higher sensitivity would be required to subsequently characterize the planetary atmosphere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Recall that the orbital period of such a planet, and with that the period of the signal variation and presumably the stability requirements of an instrument, would be about 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='76 years.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Bailey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' (2018) computed the polarization signal of spatially unresolved, hot, cloudy, Jupiter-like planet HD 189733b to be on the order of ∼20 ppm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Article number, page 21 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses Because of their large size, Jupiter-like planets, and in particular those in close-in orbits that receive large stellar fluxes, would clearly be less challenging observing targets than terrestrial-type planets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The HARPS instrument on ESO’s 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='6 m telescope includes polarimetric observations with a polarimetric sensitivity of 10−5 (?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=').' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Planetpol on the 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 m William Herschel Telescope (WHT) on La Palma achieved a sensitivity to fractional linear polarization (the ratio of linearly polar- ized flux to the total flux) of 10−6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' While PlanetPol did not succeed in detecting exoplanets, it did provide upper limits on the albedo’s of a number of exoplanets (Lucas et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The Ex- treme Polarimeter (ExPo), that was also mounted on the WHT, was designed to target young stars embedded in protoplanetary disks and evolved stars surrounded by dusty envelopes, with a polari- metric sensitivity better than 10−4 (Rodenhuis et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2012).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The HIPPI-2 instrument uses repeated observations of bright stars in the SDSS g’ band for achieving better than 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 ppm accuracy on the 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='9-m Anglo-Australian Telescope and better than 11 ppm on the 60-cm Western Sydney Uni- versity’s telescope (Bailey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The POLLUX-instrument on the LUVOIR space telescope concept aims at high-resolution (R∼120,000) spectropolarimetric observations across ultra-violet and visible wavelengths (100-400 nm) to characterize atmospheres of terrestrial-type exoplanets (The Luvoir Team 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Rossi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The EPICS instrument planned to be mounted on the ELT telescope, is designed to achieve a contrast of 10−10 depending the angular separation of the objects (Kasper et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2010).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' In our simulations, we have neglected absorption by atmospheric gases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Including such absorp- tion would yield lower total fluxes in specific spectral regions, depending on the type and amount of absorbing gas, its vertical distribution, and on the altitude and microphysical properties of the clouds and hazes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Including absorption by atmospheric gases could increase or decrease the degree of polarization, depending on the amount and vertical distribution of the absorbing gas and on the microphysical properties of the scattering particles at various altitudes (see e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Trees & Stam 2022;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Stam 2008, for examples of polarization spectra of Earth-like planets).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' While measuring total and polarized fluxes of reflected starlight across gaseous absorption bands is of obvious interest for the characterization of planets and their atmospheres, the small numbers of photons inside gaseous absorption bands would make such observations extremely challenging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We have also neglected any intrinsic polarization of Alpha Centauri A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Measurements of the degree of linear polarization of FKG stars indicate that active stars like Alpha Centauri A have a typical mean polarization of 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='5 ± 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='2 ppm (Cotton et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2017).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' This could add to the challenges in distinguishing the degree of polarization of the planet from that of the star if the planet is spa- tially unresolved, although the phase angle variation of the planetary signal and the direction of polarization of the planet signal (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' usually either perpendicular or parallel to the plane through the star, the planet, and the observer) could be helpful provided of course that the instrument that is used for the observations has the capability to measure the extremely small variations in the signal as the planet orbits its star (on the order of 10−9).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Article number, page 22 of 24 Mahapatra et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' : From exo-Earths to exo-Venuses State-of-the-art instruments with sensitivity to polarization signals down to 10−6 (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 1000 ppb) are still a few orders of magnitude away from detecting variations in polarization signals from spatially unresolved exo-Earths or exo-Venuses around nearby solar-type stars such as Alpha Cen- tauri A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' To be able to distinguish between the different planetary evolutionary phases explored in this paper, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' between water clouds or sulphuric acid clouds, variations on the order of 10−9 and hence significant improvements in sensitivity would be needed if the planets are spatially unresolved.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' The variation in the degree of polarization of spatially resolved planets along their orbital phase should be detectable by instruments capable of achieving star-planet contrasts of 10−9 and that would allow to distinguish between water clouds or sulphuric acid clouds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Current high-contrast imaging instruments manage to directly image self-luminous objects such as young exoplanets and brown dwarfs in NIR total fluxes at contrasts of 10−2-10−6 (Bowler 2016;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Nielsen et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2019;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Langlois et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2021;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' van Holstein 2021).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Further, instruments such as EPICS on ELT and concepts for instruments on future space observatories such as HabEx (Gaudi et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2020) and LUVOIR (The Luvoir Team 2019) hold the promise for attaining contrasts of ∼10−10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Reaching such extreme contrasts would make it possible to directly detect terrestrial-type planets and to use polarimetry to differentiate between exo-Earths and exo-Venuses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' Acknowledgements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' This work was supported by the Netherlands Organisation for Scientific Research (NWO) through the User Support Programme Space Research, project number ALW GO 15-37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' We thank the referee for the valuable feedback that improved this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' References Bailey, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2007, Astrobiology, 7, 320 Bailey, J.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} +page_content=' 2020, Journal of Geophysical Research: Planets, 125, e2019JE006276 Article number, page 24 of 24' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ctFIT4oBgHgl3EQfnitG/content/2301.11314v1.pdf'} diff --git a/ddAzT4oBgHgl3EQf3f4p/content/tmp_files/2301.01829v1.pdf.txt b/ddAzT4oBgHgl3EQf3f4p/content/tmp_files/2301.01829v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..8ad0b88827b0223da9701bf9f0b8c4602940ee32 --- /dev/null +++ b/ddAzT4oBgHgl3EQf3f4p/content/tmp_files/2301.01829v1.pdf.txt @@ -0,0 +1,1283 @@ +1 + +Fragment-based t-SMILES for de novo molecular generation +Juan-Ni Wu, Tong Wang, Yue Chen, Li-Juan Tang, Hai-Long Wu*, Ru-Qin Yu* +State Key Laboratory of Chemo/Biosensing and Chemometrics, College of Chemistry and Chemical Engineering, +Hunan University, Changsha 410082, People's Republic of China + +ABSTRACT +At present, sequence-based and graph-based models are two of popular used molecular generative +models. In this study, we introduce a general-purposed, fragment-based, hierarchical molecular +representation named t-SMILES (tree-based SMILES) which describes molecules using a +SMILES-type string obtained by doing breadth first search (BFS) on full binary molecular tree +formed from fragmented molecular graph. The proposed t-SMILES combines the advantages of +graph model paying more attention to molecular topology structure and language model possessing +powerful learning ability. Experiments with feature tree rooted JTVAE and chemical reaction- +based BRICS molecular decomposing algorithms using sequence-based autoregressive generation +models on three popular molecule datasets including Zinc, QM9 and ChEMBL datasets indicate +that t-SMILES based models significantly outperform previously proposed fragment-based +models and being competitive with classical SMILES based and graph-based approaches. Most +importantly, we proposed a new perspective for fragment based molecular designing. Hence, +SOTA powerful sequence-based solutions could be easily applied for fragment based molecular +tasks. +Keywords: Fragment-based drug discovery, Tree-based SMILES, Sequence-based De novo +design +1 Introduction +The starting point for molecular drug discovery programs is to identify initial ‘‘hit’’ compounds +that bind to the target and have the potential for optimization of clinical candidates with the desired +therapeutic effect. Fragment-based drug discovery (FBDD) builds drugs from small molecular +pieces. Since the pioneering work by Fesik and coworkers in 1996[1], FBDD has become a + +2 + +recognized technology in the pharmaceutical industry and within academia. More than 30 drug +candidates derived from fragments have been reported to enter the clinic[2]. Compared with atom- +based techniques, the size of the search space is greatly reduced by the use of the fragment strategy. +In addition, fragments could provide fundamental insights into molecular recognition between +proteins and ligands. As a consequence, there is a higher probability of finding molecules that +match the known targets. +The early FBDD was generally implemented based on fragment library using virtual screening and +other methods. Although deep learning[3] has been widely used in molecular generation tasks[4,5] +with fragment-based method as a research topic, the method of fragmenting molecules and coding +molecular substructures in the form of a string-type sequence like SMILES[6] to finally generate +molecules has not yet been fully explored. +In recent years, various deep generative models for the task of automatically generate molecules +have been proposed. Among deep learning-based methods, models with sequence representations +[7-9] such as SMILES and 2D representations such as graphs [10-14] are most popular, while +recently a plethora of models generating molecules in 3D[15] also starts to attract attention. +As a more natural representation of molecules, generally speaking, graph neural network could +generate 100% valid molecules as it can easily implement valence bond constraints and other +verification rules. However, it has been shown that the expressive power of standard GNNs is +bounded by Weisfeiler-Leman (WL) graph isomorphism phenomenon, the lack of ways to model +long-range interactions and higher-order structures limited the use of GNNs[16], though some +recent studies have proposed new methods such as subgraph isomorphism[17], message-passing +simple networks[18] and many others techniques to improve the expressive power of standard +GNNs[19]. + +3 + +From the perspective of graph-based computational procedure, SMILES is a linear string obtained +by performing depth first search (DFS) on molecular graph, which is more like human natural +language. When generating SMILES, the chemical graph is firstly trimmed to remove hydrogen +atoms and cycles are broken turning them into a spanning tree. Where cycles have been broken, +numeric suffix labels are included to indicate the connected nodes. Parentheses are used to indicate +points of branching on the tree. The generation algorithm of classical SMILES directly breaks +down the most common ring structures in molecules. As a consequence, some elements of +SMILES syntax must occur in pairs with deep nesting to represent molecular topological structure. +Without discussing what chemical information could be learned, models trained on SMILES +somehow generate part chemical invalid strings, particularly when trained on small datasets, which +some have identified as a limitation need to be addressed[20]. Two alternative solutions to the +classical SMILES have been proposed[20]. The DeepSMILES[21] aims to remove long term +dependencies associated with the representation of rings and branches from the SMILES syntax to +finally increase the proportion of valid molecules generated. Self-referencing embedded strings +(SELFIES)[22] are an entirely different molecular representation based on a Chomsky type-2 +grammar, in which every SELFIES string specifies a valid chemical graph. +In addition, almost all substructure (motif or fragment) based methods published to date rely on a +substructure dictionary (motif vocabulary dictionary or fragment library) of candidate +fragments[13,23-28]. In molecules, a small group of fragments being used frequently, while most +of fragments are rarely used. The differences among dictionary-based approaches arise solely from +how the database is searched, or the contents of the database itself. As a result, these methods are +inherently constrained to a set of predetermined rules or examples, limiting the exploration of +chemical space and the learning ability of the models to a certain extent. + +4 + +Recently, attention-based Transformer[29] pre-trained models have been proved to enable text +generation with human-like capabilities, including texts with specific properties such as style or +subject, if trained on enough data. With the rapid development of natural language processing(NLP) +technology, and the increasing interest in larger and more complex molecules for treatment, +language models may show a better ability to learn complex molecules than most graph generation +models[30]. +Motivated by the success of NLP and the strategies of FBDD, we hope to adopt sequence-based +models to handle fragment-based molecular generation tasks. So that, we propose a new molecule +code t-SMILES based on fragmented molecule, which describes a molecule with classical +SMILES-type string and takes language model as the main generation model. t-SMILES based +model combines the advantages of graph model paying attention to molecular topology structure +and language model possessing powerful NLP learning ability. +The t-SMILES method firstly generates an acyclic molecular tree (AMT) whose role is to represent +fragmented molecules. Fragments in this tree are chemically valid substructures automatically +extracted from molecules using a molecular disconnection algorithm. In the second stage, the AMT +is transformed to a full binary tree (FBT). Finally, breadth-first traversal of the FBT yields a t- +SMILES string. +Our proposed method is a general framework that does not limit the substructure scheme as long +as it could be used to generate chemically valid fragments and construct valid AMT. t-SMILES +string could be directly applied to sequence-based machine learning models without adjusting the +model architecture in specific fields. In addition, due to the hierarchical representation of +molecules, our model can clearly learn the high-level topology structural information of molecules +while processing the detailed substructure information. Most importantly, by exploiting NLP + +5 + +methods, t-SMILES based models opens an exploration door for fragment-based molecule tasks. +Therefore, SOTA sequence-based language models could be easily used for fragment-based +molecular tasks. +To evaluate the feasibility and adaptability of t-SMILES, we break down molecules using two +different strategies BRICS[31] based and feature tree rooted JTVAE[12] based algorithms +respectively, and then use Transformer-decoder based autoregressive models to generate +molecules on three popular molecular datasets Zinc[32], QM9[33] and ChEMBL[34]. Finally, we +compare t-SMILES based models with the most popular graph neural network models, SMILES- +based models and other baselines. +Five metrics including validity, uniqueness, novelty, KL divergence(KLD)[35]and Fréchet +ChemNet distance score(FCD)[36] introduced in GuacaMol[37] benchmark are used to evaluate +the general performance of the models. In addition, we use logP, Penalized logP(plogP)[38], SA +score(SAS)[39], BertzCT[40], QED[41], TPSA, NP Score(NPS)[42], and FractionCSP3 to +evaluate whether the generation model could effectively learn the physical and chemical properties +of molecules. +The t-SMILES construction philosophy and detailed comparative experiments show that the +validity of generated molecules by t-SMILES models could be greater than 99%. At the same time, +using 5 layers mini GPT2[43](mGPT2) model, t-SMILES could well capture the physicochemical +properties of molecules to maintain the similarity of the generated molecules to the distribution of +dataset molecules, which make t-SMILES based models significantly outperforming previously +proposed fragment-based models and being competitive with classical SMILES and graph-based +approaches. In addition, compared with classical SMILES which is relatively difficult to be + +6 + +augmented[20], t-SMILES is easily to be expanded to explore different chemical spaces by using +different molecular fragmentation algorithms. +Although current de novo molecular generation approaches have made impressive advances, the +sequence-based perspectives create new opportunities to advance fragment-based molecular +design tasks. The t-SMILES solution demonstrates that de novo generation of molecules from +fragmented SMILES is possible. This solution challenges the current research paradigm used for +FBDD. +2 Methods +In order to support t-SMILES algorithm, on the basis of the classic SMILES, we introduce a new +character '&' to represent the end of the sub-branch, and another new character '^' to separate the +two adjacent SMILES substructure segments. In this section, we firstly introduce the general idea +of t-SMILES, and then introduce FBT and AMT which are the core parts of t-SMILES algorithm, +followed with molecular fragmentation algorithms and finally discuss the molecular reconstruction +strategy. +2.1 t-SMILES Algorithm Overview +In t-SMILES algorithm, molecular graph is firstly divided into chemical valid fragments (or +substructures, clusters, subgroups, subgraphs) using a specified or more disconnection methods to +obtain its AMT shown in the middle of Figure 1. Following with the AMT being transformed into +a FBT shown in the right of Figure 1, and finally the FBT is traversed in breadth first search (BFS) +to obtain the t-SMILES string. During the reconstruction, the reverse process is used, and finally +the molecular fragments are assembled into the chemical correct molecular graph. + +7 + + +Figure 1. Overview of t-SMILES algorithm: A molecular graph G is first decomposed into its reduced graph, where +each colored node in the tree represents a substructure in the molecule. We then generate an AMT based on reduced +graph, following with trasnformation of AMT to FBT. Finally, the FBT is traversed in BFS to obtain its t-SMILES +code. To reconstruct the molecule, we first rebuild FBT from t-SMILES string, and then transform FBT to AMT, +finally assemble nodes in the tree back to the original molecule. +We follow below steps to build t-SMILES: +Algorithm steps to construct t-SMILES from molecule +Step 1: Break down molecule according to the selected molecular fragmentation algorith to build +AMT; +Step 2: Convert the AMT to a FBT through algorithm; +Step 3: Traverse the FBT with BFS algorithm to get t-SMILES . +BFS algorithm for the tree is a level order traversal of tree. For any node w in the BFS tree rooted +at v, the tree path from v to w corresponds to a shortest path from v to w in the corresponding +graph. +In the following, the terms nodes and subtrees are used for describing ATM and FBT, the terms +fragment, substructure and subgroup are used to describe a part of the molecule. Subtrees, + +Acyclic molecular tree +Full binary tree +H2N +CHe +Decompose +Decompose +Decompose +Reohstrue +ecost +H,S +H3C +CI +CHa8 + +fragments, substructures or subgroups are always assumed to be connected parts. FBT has two +children named left subtree and right subtree. +2.2 Full Binary Tree +Tree is the core concept in proposed t-SMILES algorithm. A tree is a special type of graph in +which there is just a single path connecting each pair of vertices, that is, there are no cycles or +rings within the graph. The root node of a tree is the starting point while the other vertices are +either branch nodes or leaf nodes. +A FBT is a special type of binary tree in which every parent node/internal node has either two or +no children. As the most trivial tree, FBT’s structure is regular and easy to calculate. The reason +for using FBT with some redundant nodes instead of complete binary tree or other trees is that its +algorithm and structure being easy to learn by deep learning models, and the redundant nodes +could be used as global marker nodes. In this work, the character '&'(tree node marked as '&’) +marking the end of the tree branch in the FBT could be regarded as the global structural +information describing the molecular topology in t-SMILES string. +With chemical meaningful molecular fragmentation representation using FBT, t-SMILES +effectively reduces the nesting depth of brackets in classical SMILES codes, weakens the long- +term dependencies in sequences, and fundamentally reduces the difficulty of learning molecular +information for using sequence-based deep learning models. In t-SMILES algorithm, except for +the extra two characters ‘&’ and ‘^’, no more symbols are introduced, nor are recursion and other +sophisticated calculations with high computational complexity introduced. + +9 + +2.3 Molecule Decomposition +The key and first step in t-SMILES is to decompose molecules into chemical valid fragments +according to a specified disconnection algorithm. According to Lounkine et al.[44], there exists +four major strategies to fragment designing: knowledge-based, synthetically oriented, random, and +systematic or hierarchical. The open-source molecular toolkit RDKit[45] has implemented some +molecular fragmentation methods, such as RECAP[46] and BRICS[31] etc. +RECAP and BRICS both disconnect a molecule to fragments on bonds based on chemical reaction +rules. In the RECAP method, molecules are cleaved at 11 chemical bond types that correspond to +common chemical reactions, while BRICS attempts to improve RECAP for molecule +fragmentation by using a more elaborate set of 16 rules and additional pre- and postfilters. In this +study, considering that the training datasets are mainly small molecules, and the molecular +fragments segmented by BRICS are large or molecules could not be broken down, say on the +subset of Zinc and ChEMBL, if necessary, we could further cut off the branch structures connected +to ring structures on the basis of BRICS, and then cut off the bridge bonds between two rings. +BRICS algorithm is not used on QM9 in this study as molecules are relatively rather small to be +broken down. We generate reduced graph according to the cutting off logic of BRICS and then +calculate its spanning tree as AMT. +Another molecular decomposition algorithm in our study could find its root in feature tree[47] +published for molecular similarity algorithm by Rarey et al. in 1998 and JTVAE[12] proposed +later by Jin et al. Given a graph G, we first find all its simple cycles, and its edges not belonging +to any cycles. Two simple rings are merged together if they have more than two overlapping atoms, +as they constitute a specific structure called bridged compounds [48]. Each of those cycles or edges + +10 + +is considered as a cluster. Next, a cluster graph is constructed by adding edges between all +intersecting clusters. Finally, we select one of its spanning trees as the AMT of G. +The t-SMILES algorithm is a general and fragment based molecular representation framework +which does not limit the choice of molecular decomposing algorithms. Different molecular +segmentation algorithm may require different fragment assembling algorithm to complete the +reconstruction of the generated molecules. +2.4 Acyclic Molecular Tree +The idea of using tree as the base data structure of algorithms to address molecular related tasks +has been long established in cheminformatics. In early study of molecular descriptor and similarity +analysis, algorithms such as reduced graph[49], feature tree[47,50] not only had shown potential +power to improve the similarity search but also being capable of retrieving more diverse active +compounds than using Daylight fingerprints[51].Some recent works[52-55] proposed to +incorporate tree-based deep learning models into molecular generation and synthesis tasks as well. +AMT being capable to describe the molecule at various levels of resolution, reduced graph[56] +provides summary representations of chemical structures by collapsing groups of connected atoms +into single nodes while preserving the topology of the original structures. In reduced graph, the +nodes represent groups of atoms and the edges represent the physical bonds between the groups. +Constructing reduced graph in this way forms a hierarchical graph, whose top layer being the +molecular topology representing global information, and the bottom layer representing molecular +fragments of detailed information. Groups of atoms are clustered into a node in the reduced graph +approach, which could be done based on fragmentation algorithms. The feature tree[47] is a +representation of a molecule similar to a reduced graph. The vertices of the feature tree are +molecular fragments and edges connect vertices that represent fragments connected in the simple + +11 + +molecular graph. In t-SMILES algorithm, the minimum spanning tree of the reduced graph and +the concept of feature tree could be regarded as an AMT in the intermediate step, and then the next +encoding algorithm is done based on this AMT. +Specific to our experiments, one approach is to generate AMT based on the tree logic fragmented +by the BRICS algorithm. Another method uses junction tree[57] introduced by Jin et al. in JTVAE +as the AMT. In the case of a junction tree, each node in the tree corresponds to a subset (subgraph, +group, cluster or clique) in the original graph and edge is used to connect two clusters. In this study, +nodes of Junction tree represent rings, bonds, bridged compounds, or singletons in the original +molecular graph which are generated by fragment decomposing algorithm and edges represent the +physical bonds between groups. +2.5 Molecular Reconstruction and Optimization +In the process of reconstructing the generated molecules from t-SMILES strings, we follow below +steps: +Algorithm steps to reconstruct molecule from t-SMILES +Step 1: Decompose t-SMILES to reconstruct the FBT; +Step 2: Convert the FBT to the AMT; +Step 3: Assemble molecular fragments according to the selected algorithm to generate the correct +molecular graph and then optimize it. +During reconstruction, one key problem is how to assemble the molecular fragments together to +get a ‘chemical correct’ molecule. Ideally, the assemble algorithm should be selected to match the +molecular fragmentation algorithm. In this study, for efficiency reasons, we assemble molecular +graph one neighborhood at a time, following the order in which the tree itself was generated. In +other words, we start from the root node of AMT and its neighbors, then we proceed to assemble + +12 + +the neighbors and their associated clusters, and so on. If there is more than one candidate when +assembling two pieces, we simply select one randomly. +The overall performance of t-SMILES based generative model is controlled by two main factors. +The first one is whether t-SMILES can be learned and generated efficiently, and the other one is +the reconstruction and molecular optimization algorithm. We evaluate the assembling algorithms +by directly reconstructing molecules from training set data. Detailed metrics are shown in Table 1 +and Figure 2. +Table 1. Distributional results by directly reconstructing molecules from the training dataset. +Dataset +Model +Valid(↑) +Uniq(↑) +Novel(↑) +KLD(↑) +FCD(↑) +Zinc +JTVAE +1.000 +0.770 +0.662 +0.982 +0.811 +BRICS +1.000 +0.780 +0.681 +0.986 +0.849 +QM9 +JTVAE +1.000 +0.741 +0.245 +0.976 +0.968 +ChEMBL +JTVAE +1.000 +0.786 +0.677 +0.969 +0.694 +BRICS +1.000 +0.787 +0.672 +0.977 +0.792 +Data in Table 1 show that FCD scores on ChEMBL are the lowest ones while it is the highest on +QM9. +Figure 2. Distributions of physiochemical properties (logP, plogP, SAS, BertzCT, QED, TPSA, NPS and +FractionCSP3) of reconstructed molecules directly from Zinc training set. Wasserstein distance is used in this figure +and later. + +13 + +From Figure 2, it can be seen that SAS and NPS are the two most variable metrics when +reconstituting molecules. +If we output all possible assembly results, we could get a set of molecules which come from the +same fragments group with different structures. From this point of view, generating new molecules +with the desired properties (desired structure) rather than duplicating the training set is exactly the +potential goal of the molecule generation task and not a negative aspect. MOG[58] argued that the +common pitfall of existing molecule generation models based on distributional learning is that the +exploration is confined to the training distribution, and the generated molecules exhibit “the +striking similarity” with known molecules included in the training set. Models that do not require +training molecules are free from this problem, but they introduce other problems such as long +training time, the sensitivity of balance between exploration and exploitation, large variance, and +importantly, a lack of information about the known distribution. Based on t-SMILES, it’s possible +to select appropriate optimization algorithm to control how the fragments are assembled, thereby +controlling the properties of the output molecules. Molecule optimization is another challenging +topic which is beyond the scope of this study. Using more complex optimization algorithms instead +of stochastic method to select target molecule would be an option for further study. +3 Results and Discussion +To evaluate t-SMILES and generatation models on different datasets, we firstly compare two t- +SMILES based mGPT2 models with fragment-based baseline models and classical SMILES based +mGPT2 models on Zinc. And then we compare t-SMILES based models with SOTA SMILES and +GNNs baseline models on QM9 and ChEMBL datasets. Metrics used in this study are still based +on distributional learning. + +14 + +3.1 Comparison with fragment-based and SMILES-based models on Zinc +The validity, uniqueness, novelty, KLD and FCD scores on Zinc are summarized in Table 2 and +the distributions of eight physiochemical properties are shown in Figure 3. +Table 2. Distributional results on Zinc, we train or retrained all these five models, t-SMILES_J_mGPT2 breaks +molecules using the same fragmentation algorithm as JTVAE[27], t-Smiles_B_mGPT2 breaks molecules using +BRICS, mGP2 means five layers mini GPT2 model is used. +Model +Valid(↑) +Uniq(↑) +Novel(↑) +KLD(↑) +FCD(↑) +SMILES_mGPT2(Ours) +0.853 +0.674 +0.672 +0.960 +0.830 +JTVAE[12] +0.997 +0.989 +0.989 +0.870 +0.439 +FragDgm[27] +1.000 +0.423 +0.422 +0.835 +0.303 +t-SMILES_J_mGPT2(Ours) +>0.99 +0.775 +0.774 +0.970 +0.773 +t-Smiles_B_mGPT2(Ours) +>0.99 +0.783 +0.782 +0.963 +0.790 +Classical SMILES based mGPT2 model is trained for reference in this study to evaluate t-SMILES +based models. Compared with t-SMILES models, classical SMILES based model gets lower +novelty and uniqueness scores. It means that t-SMILES-based models, with almost 100% validity +and relatively high FCD scores, could improve novelty to explore a wider molecular space. +JTVAE[12] is one key baseline model for this study, which splits molecule using a tree base data +struct. So far as the validity is concerned, both t-SMILES and JTVAE models could generate near +100% valid molecules. However, t-SMILES exhibits significantly higher KLD and FCD scores +with reasonably slight lower novelty and uniqueness scores. +t-SMILES based models significantly outperforms another fragment dictionary-based model, +FragDgm[27], which splits molecule in a linear mode as a sequence of fragment IDs, on all five +distribution parameters. Despite FragDgm adopting a segmented mode and based on distributional +learning, its FCD value is the lowest among the five models. +When more and more validity of models can reach above 0.9, it becomes more important to test +whether the generative model can effectively learn the physicochemical properties of molecules. + +15 + +Detailed distributions of eight physiochemical properties logP, plogP, SAS, BertzCT, QED, TPSA, +NPS and FractionCSP3 on Zin are presented in Figure 3. + +Figure 3. Distributions of physiochemical properties (logP, plogP, SAS, BertzCT, QED, TPSA, NPS and +FractionCSP3) of random selected 10K molecules from Zinc data sets, JTVAE[12], FragDgm[27] + and molecules generated by SMILES and two t-SMILES based mGPT2 models respectively +It can be seen from Figures 3 that the distribution of t-SMILES and classical SMILES based +mGPT2 models are closer to training data set on all eight physiochemical properties than other +two fragment-based models. The distribution of generated molecules by FragDgm is far from the +training set on six physiochemical properties except NPS and FractionCSP3. +Compared with JTVAE, besides TPSA and SAS which are comparable, two t-SMILES based +mGPT2 models get significantly lower scores on all eight metrics, especially on FractionCSP3 and +BertzCT. Compared with FragDgm, except NPS which are comparable, t-SMILES based models +get significantly favorite scores in other seven physicochemical properties. +Considering novelty, the distribution of the eight physicochemical properties suggests that the +generated molecules by t-SMILES-based models are better fit to the training dataset, which +obviously implies relatively low novelty. + +16 + +From the comparative analysis of these models, two t-SMILES based models in this study +significantly outperform the fragment-based baseline models, and have similar performance to the +SMILES-based model in physicochemical properties, but validity scores could be greater than +99%, that is to say, t-SMILES could be an effective molecular presentation for fragment based +molecular tasks on Zinc. +3.2 Comparison with baseline models on QM9 +The validity, uniqueness, novelty, KLD and FCD scores on QM9 are summarized in Table 3 and +the distributions of eight physiochemical properties are shown in Figure 4. +Table 3. Distributional results on QM9. The results of CharacterVAE[7], are taken from MolGAN[60], Transformer +Reg[11], GraphVAE[59,11], MolGAN[60] and MGM[11] are taken from O. Mahmood et al.[11] +Input +Model +Valid(↑) +Uniq(↑) +Novel(↑) +KLD(↑) +FCD(↑) +SMILES +CharacterVAE[7,60] +0.103 +0.675 +0.900 +N/A +N/A +Transformer Reg[11] +0.965 +0.957 +0.183 +0.994 +0.958 +SMILES_mGPT2(ours) +0.949 +0.728 +0.172 +0.992 +0.970 +Graph +GraphVAE[59,11] +0.557 +0.760 +0.616 +N/A +N/A +MolGAN[60] +0.981 +0.104 +0.942 +N/A +N/A +MGM[11] +0.886 +0.978 +0.518 +0.966 +0.842 +t-SMILES +t-SMILES_J_mGPT2(ours) +>0.99 +0.720 +0.289 +0.976 +0.953 +On QM9, the score of validity of t-SMILES based mGPT2 model could be greater than 0.99, which +is comparable to the performance of GNNs and superior to the most SOTA SMILES based models. +FCD score of t-SMILES based mGPT2 model is one of the highest ones in all seven models, which +is greater than 0.95. +From the viewpoint of sequence-based models, our approach performs similarly to or better than +existing SMILES based approaches. Our approach shows higher validity and uniqueness scores +compared to CharacterVAE, while having a reasonable lower novelty score. Compared to the +Transformer Reg model, t-SMILES based model has higher score in novelty, lower score in +uniqueness, and comparable scores in KLD and FCD. + +17 + +Compared to the graph-based models, our approach outperforms the existing baseline approaches. +t-SMILES based model has comparable uniqueness score compared with respect to GraphVAE, +and significantly outperforms MolGAN, with reasonable lower novelty score. KLD and FCD +scores are not provided for these two models. t-SMILES based mGPT2 model has good +performance against SOTA method MGM in validity, KLD and FCD scores, while having slightly +lower scores in uniqueness and novelty. +Figure 4. Distributions of physiochemical properties (logP, plogP, SAS, BertzCT, QED, TPSA, NPS and +FractionCSP3) of random selected 10K molecules from generated molecules on QM9 data sets by SMILES and t- +SMILES based mGPT2 models respectively +In general, on QM9 data set, both mGPT2 models based on t-SMILES and classical SMILES get +high KLD and FCD scores, all greater than 0.95, that means they could maintain physiochemical +similarity to the training distribution. However, the novelty scores of the models are relatively low, +which could be interpreted by that the generated molecules based on distributional learning exhibit +“the striking similarity” with known molecules included in the training set. It also indicates that +novelty is inversely correlated with the KLD and FCD scores. +3.3 Comparison with baseline models on ChEMBL +The validity, uniqueness, novelty, KLD and FCD scores on ChEMBL are summarized in Table 4 +and the distributions of eight physiochemical properties are shown in Figure 5. + +18 + +Table 4 Distributional results on ChEMBL, The results of ORGAN[8], LSTM[37], CharacterVAE[7], AAE[61], +Transformer Reg[11], Graph MCTS[14], and MGM[11] are taken from O. Mahmood et al.[11] and Guacamol[37], +the results of MolGPT is taken from Bagal et al.[9], the results of hgraph2graph[13] is calculated by us. +Input +Model +Valid(↑) +Uniq(↑) +Novel(↑) +KLD(↑) +FCD(↑) +SMILES +ORGAN[8,37] +0.379 +0.841 +0.687 +0.267 +0.000 +LSTM[37] +0.959 +1 +0.912 +0.991 +0.913 +Character VAE[7,37] +0.870 +0.999 +0.974 +0.982 +0.863 +AAE[61,37] +0.822 +1 +0.998 +0.886 +0.529 +MolGPT[9] +0.981 +0.998 +1 +0.992 +0.907 +Transformer Reg[11] +0.961 +1.000 +0.846 +0.977 +0.883 +SMILES_mGPT2(Ours) +0.850 +0.670 +0.641 +0.972 +0.809 +Graph +Graph MCTS[14, 37] +1.000 +1.000 +0.994 +0.522 +0.015 +MGM[11] +0.849 +1.000 +0.722 +0.987 +0.845 +hgraph2graph[13] +1.000 +0.994 +0.940 +0.870 +0.485 +t-SMILES +t-SMILES_J_mGPT2(Ours) +>0.99 +0.781 +0.765 +0.913 +0.564 +t-SMILES_B_mGPT2(Ours) +>0.99 +0.781 +0.769 +0.935 +0.575 +On ChEMBL, two t-SMILES based mGPT2 models outperform graph-based baseline methods +Graph MCTS and hgraph2graph, while the performance is comparable to SOTA graph mode +MGM. Compared to Graph MCTS, t-SMILES based mGPT2 models show lower novelty scores +while having significantly higher KLD and FCD scores. It seems difficult for this graph-based +baseline model to capture the properties of the dataset distributions as shown by their low KLD +scores and almost-zero FCD scores, but it gets the highest novelty score. +Compared to hgraph2graph which is an advanced model based on JTVAE and aims to solve larger +molecular problems with motif-based method, t-SMILES based mGPT2 modes have higher KLD +and FCD scores and lower uniqueness and novelty scores. +Compared to SOTA graph-based model MGM, t-SMILES based mGPT2 models have higher score +in validity, but lower scores in uniqueness, KLD and FCD and similar score in novelty. +In the realm of sequence-based models, the proposed t-SMILES based mGPT2 models are +competitive with the classical SMILES based models, and firstly, our models outperform all listed +classical SMILES based models in validity. Compared with the GAN-based model (ORGAN), t- +SMILES based mGPT2 models have significantly higher scores in validity, KLD and FCD and +novelty scores, while having a slightly lower score in uniqueness. Compared with AAE, t-SMILES + +19 + +modes get low unique and novelty scores and higher KLD and FCD scores. Similar to Graph +MCTS, ORGAN also get an almost-zero FCD score. Our t-SMILES based approach results in +lower scores across most of the metrics when compared to Transformer Reg, MolGPT, LSTM, +VAE models besides validity score. In addition, statistics data shows that SMILES-mGPT2 model +gets a slightly lower scores compared with Transformer Reg and MolGPT, which indicates that +the t-SMILES mGPT2 models could be optimized to get high scores as well. Such an issue could +be severed as a starting point for further research. +Figure 5. Distributions of physiochemical properties (logP, plogP, SAS, BertzCT, QED, TPSA, NPS and +FractionCSP3) of random selected 10K molecules from generated molecules on ChEMBL data sets, +hgraph2graph(hG2G)[13] and molecules generated by SMILES and t-SMILES based mGPT2 models respectively +Compared with hgraph2graph in Figure 5, t-SMILES based models get slightly higher scores on +BertzCT, logP, QED and TPSA, while having comparable scores on SAS, NPS, plogP and +FractionCSP3. Compared with classical SMILES based mGPT2 model, t-SMILES based mGPT2 +models have lower scores on seven metrics besides QED. +On ChEMBL, one possible reason of the proposed t-SMILES models getting a relatively little +lower scores might be that t-SMILES based mGPT2 model is not hyper-parametrically optimized +well. + +20 + +3.4 Ablation Studies +Compared to classical SMILES, only two more characters ‘&’ and ‘^’ are introduced for t-SMILES +to encode multiscale molecular topological structure. Therefore, we adopt the same network +structure to conduct ablation studies on one LSTM model and two Transformer-based models +using three downstream datasets to investigate the performance of generative models. As shown +in Table 5, different generative models get slightly different distributional results for the same +dataset. However, compared to SMILES based models, t-SMILES based models get higher +validity, uniqueness and novelty scores generally. This evidence also supports one of our claims +that t-SMILES make it possible to generate molecules fragment-by-fragment using sequence- +based models easily. +Table 5 Distributional results on Zinc, QM9 and ChEMBL datasets. The results of LSTM[37], MolGPT[9] and +mGPT2 are all trained and calculated by us. + +Dataset +Model +Valid(↑) +Uniq(↑) +Novel(↑) +KLD(↑) +FCD(↑) +Zinc +t-SMILES_J_LSTM +>0.99 +0.782 +0.781 +0.944 +0.670 +t-SMILES_J_MolGPT +>0.99 +0.783 +0.780 +0.984 +0.823 +t-SMILES_J_mGPT2 +>0.99 +0.775 +0.774 +0.970 +0.773 +QM9 +t-SMILES_J_LSTM +>0.99 +0.736 +0.325 +0.966 +0.950 +t-SMILES_J_MolGPT +>0.99 +0.746 +0.305 +0.972 +0.964 +t-SMILES_J_mGPT2 +>0.99 +0.720 +0.289 +0.976 +0.953 +ChEMBL +t-SMILES_J_LSTM +>0.99 +0.782 +0.781 +0.944 +0.670 +t-SMILES_J_MolGPT +>0.99 +0.780 +0.762 +0.964 +0.695 +t-SMILES_J_mGPT2 +>0.99 +0.781 +0.765 +0.913 +0.564 +Zinc +SMILES_LSTM +0.850 +0.667 +0.666 +0.966 +0.831 +SMILES_MolGPT +0.980 +0.770 +0.766 +0.989 +0.923 +SMILES_mGPT2 +0.853 +0.674 +0.672 +0.960 +0.830 +QM9 +SMILES_LSTM +0.899 +0.690 +0.182 +0.992 +0.965 +SMILES_MolGPT +0.965 +0.729 +0.150 +0.986 +0.907 +SMILES_mGPT2 +0.949 +0.728 +0.172 +0.992 +0.970 +ChEMBL +SMILES_LSTM +0.774 +0.609 +0.598 +0.943 +0.725 +SMILES_MolGPT +0.976 +0.768 +0.711 +0.992 +0.807 +SMILES_mGPT2 +0.850 +0.670 +0.641 +0.972 +0.809 +4 Experiments +In this section, we briefly introduce the proposed generation models which are used to evaluate t- +SMILES. This is then followed by the overview of three commonly used public data sets. Finally, + +21 + +the details of the experiments and the metrics used for the evaluation of different models are +provided. +4.1 Generation models and hyperparametric optimization +Language modelling uses probability and statistical techniques to calculate the possibility of word +sequences in sentences and then do estimation[62]. Originally, recurrent neural network (RNN) is +designed to address this kind of sequence problems. To address the limitation of RNN that lead to +gradient disappearance and explosion for long sequences, cyclic models such as LSTM and GRU +are proposed. However, it has been shown that the power of LSTMs is insufficient when the +information has ultra-long dependencies. In 2017, the attention-based Transformer architecture +broke through the limitations of LSTM and quickly achieved SOTA scores on multiple metrics in +NLP and computer vision. After that, the improved models such as GPT1-3[63,43,64]and +BERT[65] are introduced to build pre-trained model on large datasets. Although these large-size +models have achieved unprecedented performances, they come at high computational costs. +Consequently, some of the recent NLP architectures have utilized concepts of transfer learning, +pruning, quantization, and knowledge distillation to achieve moderate model sizes while keeping +nearly similar performances as achieved by their predecessors. The recent developments in the +NLP field have a great potential to be adapted to molecular de novo generation research. Bagal et +al.[9] has proposed a Transformer-decoder model with 10 layers to molecular generation task and +proved it works well. +In this study, we mainly adopt Transformer-decoder based autoregressive generation models to +evaluate our proposed t-SMILES. For the sake of data comparability, we select one model as the +basis using a Bayesian optimizer SigOpt[66] to optimize the hyperparameters related to the neural +network topology in ChEMBL dataset using classical SMILES as input. This selected model is a + +22 + +5 layers mini version of GPT2(mGPT2) with 512 hidden layer and 8 attention headers. All other +mGPT2 models in different dataset with either t-SMILES or classical SMILES use the same high- +level neural network architecture. Finally, we train all models in detail using the Adam optimizer. +4.2 Datasets +We evaluate t-SMILES on three commonly used public data sets including subset of Zinc[32], +QM9[33] and ChEMBL-21[34]. +The Zinc subset used in our experiments is the same as the subset in JTVAE which contains +approximately 250K drug-like molecules extracted from the Zinc15 database that span a wide +range of chemistry. The molecules in this Zinc subset are composed of 10 atoms including Br， +C，Cl，F，H，I，N，O，P and S. +QM9 dataset contains up to 9 heavy (non hydrogen) atoms. Molecules in the subset consist of 5 +atoms including C, F, N, H and O atoms. In all, this results in about 134k druglike organic +molecules. +For the sake of universality, we select a subset from ChEMBL-21 with SMILES character length +less than 120. Apart from this very simple rule, no other preprocessing is done. Molecules in +ChEMBL subset are composed of 11 atoms including B, Br, C, Cl, F, H, I, N, O, P and S. +4.3 Evaluation Metrics +Despite a large number of metrics have been proposed for the evaluation of generative models of +molecules, there is little consensus on which should be used. It is often biased to use simple indexes +to evaluate the performance of different models [67]. From the perspective of optimization, it could +be considered that the task can be solved once the molecules with a high score of a certain index +are generated, but these generated molecules may not be useful. Therefore, we use five benchmarks + +23 + +proposed by GuacaMol: validity (↑), uniqueness (↑), novelty (↑), KL divergence (KLD) (↑)[35]and +Fréchet ChemNet Distance(FCD)[36] score (↑) to evaluate the general performance of the model. +GuacaMol calculate every score using 10K randomly sampled molecules. Validity measures the +ratio of valid molecules which could be correctly parsed by RDKit[45]; Uniqueness is the +percentage of valid molecules that are unique; Novelty is the percentage of valid unique molecules +that are not included in the training data set; KLD[35] score compares the distribution of a variety +physicochemical descriptors of the training set and generated molecules. FCD[36] score measures +the proximity of the distribution of generated molecules to the distribution of the dataset molecules +according to the Fréchet Distance in the hidden representation space of ChemNet[68], which is +trained to predict the chemical properties of small molecules. The values of these five parameters +are between 0 and 1. The larger the value, the ‘better’ the model. While KLD and FCD are both +measure of the similarity between generated molecules and molecules from the training data set. +They are highly correlated with each other and inversely correlated with novelty[11]. +In addition, we use logP, plogP[38], SAS[39], BertzCT[40], QED[41], TPSA, NPS[42], and +FractionCSP3 to evaluate whether the generative models could effectively learn the physical and +chemical properties of the molecules in the training set, thereby comprehensively evaluating the +performance of the generative model from the perspective of distributed learning knowledge. +• logP: The logarithm of the partition coefficient. If one of the solvents is water and the other is +a nonpolar solvent, then logP is a measure of hydrophobicity. +• Penalized logP(plogP)[38]is the logarithm of the partition ratio of solute between octanol and +water subtracted by synthetic accessibility score and long cycles. It has a range of (−∞,∞) +• Synthetic Accessibility score (SAS)[39]: Measurement of the difficulty of synthesizing a +compound. It is a score between 0 (difficult) and 1(easy). + +24 + +• BertzCT[40]: A topological index meant to quantify “complexity” of molecules. It consists of +a sum of two terms, one representing the complexity of the bonding, the other representing the +complexity of the distribution of heteroatoms. +• Quantitative Estimate of Drug-likeness (QED)[41]: This quantifies drug-likeness by +considering the main molecular properties. It ranges from 0 (all properties unfavorable) to 1 +(all properties favorable). +• Topological Polar Surface Area (TPSA): The sum of surface area over all polar atoms. It +measures the drug’s ability to permeate cell membranes. Molecules with a TPSA greater than +140 Å2 tend to be poor in permeating cell membranes. +• Natural Product-likeness score (NPS)[42]: Score of similarity degree of structural space +covered by molecules and natural products. It has a range of (0,1) +• FractionCSP3: The fraction of C atoms that are SP3 hybridized. +In general, we randomly select 10K molecules from generated ones to calculate five metrics from +GuacaMol[37] and the distribution of eight physiochemical properties. +4.4 Details of baseline models +We train classical SMILES based mGPT2 models as relatively baseline on three datasets, which +share the same neural network architecture with t-SMILES. If there are some SMILS-based models +that get better scores than this SMILES-mGPT2, it means that it’s possible to build t-SMILES +based model to get better scores. +In ablation studies, we retrain the LSTM[37] model with 3 layers of hidden size 768, dropout of +0.2 using the Adam optimizer with learning rate 0.001, and MolGPT[9] model with 8 layers, 8 + +25 + +headers with hidden size 256 using AdamW optimizer with learning rate 0.001 for both SMILES +and t-SMILES on all three datasets. +We retrain JTVAE[12] on Zinc using the publicly available codebase provided by the paper’s +authors and then generate 20K molecules. +We calculate metrics based on the molecules provided by Fragment-base-DGM[27] on Zinc. +We generate 20K molecules using the pre-trained model provided by hgraph2graph[13] on +ChEMBL and calculate metrics for evaluation. +We do not train the rest of the baseline models by ourselves. For QM9 and ChEMBL, we take +some results of baseline models from Bagal et al.[9], Cao and Kipf [60], O. Mahmood et al.[11] +and GuacaMol[37]. +Finally, we follow the open-source implementation of the GuacaMol[37] benchmark baselines to +calculate metrics for all baseline and new designed models. +5 Conclusion and Outlook +One of the challenging problems in designing drug molecules is the vast chemical space to be +explored. Fragment-based approaches identify a subset of chemical moieties responsible for key +molecular recognitions early on and this allows scientists to devote their time in a much reduced +and relevant chemical space. +In general, if atoms are analogized to characters in natural language processing, fragments could +be analogized to words or phrases as functional "units", then molecule could be a fragment based +discrete structured data. To train a general-purpose string generative mode to learn the knowledge +from discrete structured data, we need to prepare large amount of valid combinations of the +structures, which is time consuming and will also face the challenge of insufficient effective data + +26 + +practical in domains like target-oriented drug discovery. Our proposed t-SMILES uses SMILES +instead of fragment dictionaries id, re-encodes fragment-based molecular tree through a full binary +tree and generates molecular sequence with multiscale structural information, thus providing a new +idea for fragment-based molecular design. In this way, powerful and rapidly developing sequence- +based solutions can be applied to fragment-based molecular tasks in the same way as classical +SMILES. +In addition, compared with classical SMILES which is relatively difficult to be augmented, by +using different fragmentation algorithms, the training dataset is easier and more efficiently to be +expanded on t-SMILES to explore different chemical spaces without having to change anything +of the architecture of generation model. +Scalability Our proposed t-SMILES solution supports any effective substructures types and +patterns as long as they could be used to obtain chemically valid molecular fragments and +ultimately construct valid acyclic molecular trees. So that, with the invaluable accumulation of +drug fragments by experienced chemists, it’s possible that the t-SMILES algorithm combines this +experience with a powerful sequence based deep neural network model to help chemists better +explore chemical space. If in some special cases, only one specific fragment space needs to be +explored, and expansion is not required, then in t-SMILES algorithm, tree nodes could be easily +encoded by dictionary id instead of SMILES. Or we could replace the newly generated fragments +with the fragments from the training data set according to specified rules. The encoding logic and +algorithm flow of t-SMILES remain unchanged. +To be improved Our experiments show that how to segment, assemble molecular fragments and +how to optimize molecular are also key steps controlling the quality of generated molecules. +Therefore, this challenging topic could serve as a starting point for further research. + +27 + +With the rapid development of NLP technology, Transform-based models have been proved to +enable text generation with human-like capabilities if trained on enough data. SMILES-based +models have proved that the sequence-based NLP models are powerful tools to generate molecules. +Then, t-SMILES makes it easily to use sequence-based NLP models to generate molecules +fragment-by-fragment. Therefore, the investigation of t-SMILES based generation models or other +fragment-based tasks would be served as interesting topic for further study. +List of abbreviations +SOTA: State-of-the-art +FBDD: Fragment-based drug discovery +AMT: Acyclic Molecular Tree +FBT: Full Binary Tree +BFS: Breadth First Search +DFS: Depth First Search +SMILES: Simplified Molecular Input Line Entry Specification +plogP: Penalized logP +SAS: Synthetic Accessibility score +QED: Quantitative Estimate of Drug-likeness +NPS: Natural Product-likeness score +FCD: ChemNet distance score +KLD: KL divergence + +Conflicts of Interest +The authors declare that they have no competing interests. + +28 + +Author Contributions +Ruqin Yu and Juanni Wu designed the study and manuscript. Juanni Wu conceived the project, +constructed the algorithms and Python script, performed the experiments, informatics analyses, +and wrote the draft manuscript. Tong Wang and Yue Chen participated in the discussion and +experiments. Lijuan Tang and Hailong Wu participated in the discussion and funding acquisition. +All authors contributed to manuscript editing, revising and have approved the final version of the +manuscript. +Acknowledgements +This research was funded by the National Natural Science Foundation of China including No. +21874040 and No. 22174036. +Data availability +The datasets used in this study are publicly available. They are referenced in the Datasets and +Evaluation Metrics part of the Experiments section. The processed data used in this study can be +found at: https://github.com/juanniwu/t-SMILES/ +Code availability +Code, pretrained t-SMILES models, training and generation scripts for this work and lists of +generated molecules can be found at: https://github.com/juanniwu/t-SMILES/ +The code of baseline models used to in this work are publicly available. We are gratefully +acknowledging all authors of these researches. +[1] MolGPT[9]: https://github.com/devalab/molgpt +[2] MGM[11]：https://github.com/nyu-dl/dl4chem-mgm + +29 + +[3] JTVAE[12]：https://github.com/wengon-jin/icml18-jtnn +[4] hgraph2graph[13]: https://github.com/wengong-jin/hgraph2graph +[5] FragDGM[27]: https://github.com/marcopodda/fragment-based-dgm +[6] Guacamol[37]：https://github.com/BenevolentAI/guacamol_baselines +[7] GPT2: https://github.com/samwisegamjeee/pytorch-transformers + +Reference +1. +Shuker SB, Hajduk P, Meadows R, Fesik S (1996) Discovering High-Affinity Ligands for +Proteins: SAR by NMR. 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Drug Discov Today Technol 32–33:55–63. +https://doi.org/10.1016/j.ddtec.2020.09.003 +68. +Goh GB, Siegel C, Vishnu A, Hodas NO (2017) ChemNet: A Transferable and + +37 + +Generalizable Deep Neural Network for Small-Molecule Property Prediction. arXiv +abs/1712.0:1712.02734 + diff --git a/ddAzT4oBgHgl3EQf3f4p/content/tmp_files/load_file.txt b/ddAzT4oBgHgl3EQf3f4p/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..b87d7732cb433991b063f648109da7fa4ba29b56 --- /dev/null +++ b/ddAzT4oBgHgl3EQf3f4p/content/tmp_files/load_file.txt @@ -0,0 +1,886 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf,len=885 +page_content="1 Fragment-based t-SMILES for de novo molecular generation Juan-Ni Wu, Tong Wang, Yue Chen, Li-Juan Tang, Hai-Long Wu*, Ru-Qin Yu* State Key Laboratory of Chemo/Biosensing and Chemometrics, College of Chemistry and Chemical Engineering, Hunan University, Changsha 410082, People's Republic of China ABSTRACT At present, sequence-based and graph-based models are two of popular used molecular generative models." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In this study, we introduce a general-purposed, fragment-based, hierarchical molecular representation named t-SMILES (tree-based SMILES) which describes molecules using a SMILES-type string obtained by doing breadth first search (BFS) on full binary molecular tree formed from fragmented molecular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The proposed t-SMILES combines the advantages of graph model paying more attention to molecular topology structure and language model possessing powerful learning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Experiments with feature tree rooted JTVAE and chemical reaction- based BRICS molecular decomposing algorithms using sequence-based autoregressive generation models on three popular molecule datasets including Zinc, QM9 and ChEMBL datasets indicate that t-SMILES based models significantly outperform previously proposed fragment-based models and being competitive with classical SMILES based and graph-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Most importantly, we proposed a new perspective for fragment based molecular designing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Hence, SOTA powerful sequence-based solutions could be easily applied for fragment based molecular tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Keywords: Fragment-based drug discovery, Tree-based SMILES, Sequence-based De novo design 1 Introduction The starting point for molecular drug discovery programs is to identify initial ‘‘hit’’ compounds that bind to the target and have the potential for optimization of clinical candidates with the desired therapeutic effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Fragment-based drug discovery (FBDD) builds drugs from small molecular pieces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Since the pioneering work by Fesik and coworkers in 1996[1], FBDD has become a 2 recognized technology in the pharmaceutical industry and within academia.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' More than 30 drug candidates derived from fragments have been reported to enter the clinic[2].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Compared with atom- based techniques, the size of the search space is greatly reduced by the use of the fragment strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In addition, fragments could provide fundamental insights into molecular recognition between proteins and ligands.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' As a consequence, there is a higher probability of finding molecules that match the known targets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The early FBDD was generally implemented based on fragment library using virtual screening and other methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Although deep learning[3] has been widely used in molecular generation tasks[4,5] with fragment-based method as a research topic, the method of fragmenting molecules and coding molecular substructures in the form of a string-type sequence like SMILES[6] to finally generate molecules has not yet been fully explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In recent years, various deep generative models for the task of automatically generate molecules have been proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Among deep learning-based methods, models with sequence representations [7-9] such as SMILES and 2D representations such as graphs [10-14] are most popular, while recently a plethora of models generating molecules in 3D[15] also starts to attract attention.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' As a more natural representation of molecules, generally speaking, graph neural network could generate 100% valid molecules as it can easily implement valence bond constraints and other verification rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' However, it has been shown that the expressive power of standard GNNs is bounded by Weisfeiler-Leman (WL) graph isomorphism phenomenon, the lack of ways to model long-range interactions and higher-order structures limited the use of GNNs[16], though some recent studies have proposed new methods such as subgraph isomorphism[17], message-passing simple networks[18] and many others techniques to improve the expressive power of standard GNNs[19].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 3 From the perspective of graph-based computational procedure, SMILES is a linear string obtained by performing depth first search (DFS) on molecular graph, which is more like human natural language.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' When generating SMILES, the chemical graph is firstly trimmed to remove hydrogen atoms and cycles are broken turning them into a spanning tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Where cycles have been broken, numeric suffix labels are included to indicate the connected nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Parentheses are used to indicate points of branching on the tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The generation algorithm of classical SMILES directly breaks down the most common ring structures in molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' As a consequence, some elements of SMILES syntax must occur in pairs with deep nesting to represent molecular topological structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Without discussing what chemical information could be learned, models trained on SMILES somehow generate part chemical invalid strings, particularly when trained on small datasets, which some have identified as a limitation need to be addressed[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Two alternative solutions to the classical SMILES have been proposed[20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The DeepSMILES[21] aims to remove long term dependencies associated with the representation of rings and branches from the SMILES syntax to finally increase the proportion of valid molecules generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Self-referencing embedded strings (SELFIES)[22] are an entirely different molecular representation based on a Chomsky type-2 grammar, in which every SELFIES string specifies a valid chemical graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In addition, almost all substructure (motif or fragment) based methods published to date rely on a substructure dictionary (motif vocabulary dictionary or fragment library) of candidate fragments[13,23-28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In molecules, a small group of fragments being used frequently, while most of fragments are rarely used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The differences among dictionary-based approaches arise solely from how the database is searched, or the contents of the database itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' As a result, these methods are inherently constrained to a set of predetermined rules or examples, limiting the exploration of chemical space and the learning ability of the models to a certain extent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 4 Recently, attention-based Transformer[29] pre-trained models have been proved to enable text generation with human-like capabilities, including texts with specific properties such as style or subject, if trained on enough data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' With the rapid development of natural language processing(NLP) technology, and the increasing interest in larger and more complex molecules for treatment, language models may show a better ability to learn complex molecules than most graph generation models[30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Motivated by the success of NLP and the strategies of FBDD, we hope to adopt sequence-based models to handle fragment-based molecular generation tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' So that, we propose a new molecule code t-SMILES based on fragmented molecule, which describes a molecule with classical SMILES-type string and takes language model as the main generation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' t-SMILES based model combines the advantages of graph model paying attention to molecular topology structure and language model possessing powerful NLP learning ability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The t-SMILES method firstly generates an acyclic molecular tree (AMT) whose role is to represent fragmented molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Fragments in this tree are chemically valid substructures automatically extracted from molecules using a molecular disconnection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In the second stage, the AMT is transformed to a full binary tree (FBT).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Finally, breadth-first traversal of the FBT yields a t- SMILES string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Our proposed method is a general framework that does not limit the substructure scheme as long as it could be used to generate chemically valid fragments and construct valid AMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' t-SMILES string could be directly applied to sequence-based machine learning models without adjusting the model architecture in specific fields.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In addition, due to the hierarchical representation of molecules, our model can clearly learn the high-level topology structural information of molecules while processing the detailed substructure information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Most importantly, by exploiting NLP 5 methods, t-SMILES based models opens an exploration door for fragment-based molecule tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Therefore, SOTA sequence-based language models could be easily used for fragment-based molecular tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' To evaluate the feasibility and adaptability of t-SMILES, we break down molecules using two different strategies BRICS[31] based and feature tree rooted JTVAE[12] based algorithms respectively, and then use Transformer-decoder based autoregressive models to generate molecules on three popular molecular datasets Zinc[32], QM9[33] and ChEMBL[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Finally, we compare t-SMILES based models with the most popular graph neural network models, SMILES- based models and other baselines.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Five metrics including validity, uniqueness, novelty, KL divergence(KLD)[35]and Fréchet ChemNet distance score(FCD)[36] introduced in GuacaMol[37] benchmark are used to evaluate the general performance of the models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In addition, we use logP, Penalized logP(plogP)[38], SA score(SAS)[39], BertzCT[40], QED[41], TPSA, NP Score(NPS)[42], and FractionCSP3 to evaluate whether the generation model could effectively learn the physical and chemical properties of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The t-SMILES construction philosophy and detailed comparative experiments show that the validity of generated molecules by t-SMILES models could be greater than 99%.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' At the same time, using 5 layers mini GPT2[43](mGPT2) model, t-SMILES could well capture the physicochemical properties of molecules to maintain the similarity of the generated molecules to the distribution of dataset molecules, which make t-SMILES based models significantly outperforming previously proposed fragment-based models and being competitive with classical SMILES and graph-based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In addition, compared with classical SMILES which is relatively difficult to be 6 augmented[20], t-SMILES is easily to be expanded to explore different chemical spaces by using different molecular fragmentation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Although current de novo molecular generation approaches have made impressive advances, the sequence-based perspectives create new opportunities to advance fragment-based molecular design tasks.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The t-SMILES solution demonstrates that de novo generation of molecules from fragmented SMILES is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' This solution challenges the current research paradigm used for FBDD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=" 2 Methods In order to support t-SMILES algorithm, on the basis of the classic SMILES, we introduce a new character '&' to represent the end of the sub-branch, and another new character '^' to separate the two adjacent SMILES substructure segments." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In this section, we firstly introduce the general idea of t-SMILES, and then introduce FBT and AMT which are the core parts of t-SMILES algorithm, followed with molecular fragmentation algorithms and finally discuss the molecular reconstruction strategy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='1 t-SMILES Algorithm Overview In t-SMILES algorithm, molecular graph is firstly divided into chemical valid fragments (or substructures, clusters, subgroups, subgraphs) using a specified or more disconnection methods to obtain its AMT shown in the middle of Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Following with the AMT being transformed into a FBT shown in the right of Figure 1, and finally the FBT is traversed in breadth first search (BFS) to obtain the t-SMILES string.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' During the reconstruction, the reverse process is used, and finally the molecular fragments are assembled into the chemical correct molecular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 7 Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Overview of t-SMILES algorithm: A molecular graph G is first decomposed into its reduced graph, where each colored node in the tree represents a substructure in the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' We then generate an AMT based on reduced graph, following with trasnformation of AMT to FBT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Finally, the FBT is traversed in BFS to obtain its t-SMILES code.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' To reconstruct the molecule, we first rebuild FBT from t-SMILES string, and then transform FBT to AMT, finally assemble nodes in the tree back to the original molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' We follow below steps to build t-SMILES: Algorithm steps to construct t-SMILES from molecule Step 1: Break down molecule according to the selected molecular fragmentation algorith to build AMT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Step 2: Convert the AMT to a FBT through algorithm;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Step 3: Traverse the FBT with BFS algorithm to get t-SMILES .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' BFS algorithm for the tree is a level order traversal of tree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' For any node w in the BFS tree rooted at v, the tree path from v to w corresponds to a shortest path from v to w in the corresponding graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In the following, the terms nodes and subtrees are used for describing ATM and FBT, the terms fragment, substructure and subgroup are used to describe a part of the molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Subtrees, Acyclic molecular tree Full binary tree H2N CHe Decompose Decompose Decompose Reohstrue ecost H,S H3C CI CHa8 fragments, substructures or subgroups are always assumed to be connected parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' FBT has two children named left subtree and right subtree.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='2 Full Binary Tree Tree is the core concept in proposed t-SMILES algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' A tree is a special type of graph in which there is just a single path connecting each pair of vertices, that is, there are no cycles or rings within the graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The root node of a tree is the starting point while the other vertices are either branch nodes or leaf nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' A FBT is a special type of binary tree in which every parent node/internal node has either two or no children.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' As the most trivial tree, FBT’s structure is regular and easy to calculate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The reason for using FBT with some redundant nodes instead of complete binary tree or other trees is that its algorithm and structure being easy to learn by deep learning models, and the redundant nodes could be used as global marker nodes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=" In this work, the character '&'(tree node marked as '&’) marking the end of the tree branch in the FBT could be regarded as the global structural information describing the molecular topology in t-SMILES string." metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' With chemical meaningful molecular fragmentation representation using FBT, t-SMILES effectively reduces the nesting depth of brackets in classical SMILES codes, weakens the long- term dependencies in sequences, and fundamentally reduces the difficulty of learning molecular information for using sequence-based deep learning models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In t-SMILES algorithm, except for the extra two characters ‘&’ and ‘^’, no more symbols are introduced, nor are recursion and other sophisticated calculations with high computational complexity introduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 9 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='3 Molecule Decomposition The key and first step in t-SMILES is to decompose molecules into chemical valid fragments according to a specified disconnection algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' According to Lounkine et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' [44], there exists four major strategies to fragment designing: knowledge-based, synthetically oriented, random, and systematic or hierarchical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The open-source molecular toolkit RDKit[45] has implemented some molecular fragmentation methods, such as RECAP[46] and BRICS[31] etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' RECAP and BRICS both disconnect a molecule to fragments on bonds based on chemical reaction rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In the RECAP method, molecules are cleaved at 11 chemical bond types that correspond to common chemical reactions, while BRICS attempts to improve RECAP for molecule fragmentation by using a more elaborate set of 16 rules and additional pre- and postfilters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In this study, considering that the training datasets are mainly small molecules, and the molecular fragments segmented by BRICS are large or molecules could not be broken down, say on the subset of Zinc and ChEMBL, if necessary, we could further cut off the branch structures connected to ring structures on the basis of BRICS, and then cut off the bridge bonds between two rings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' BRICS algorithm is not used on QM9 in this study as molecules are relatively rather small to be broken down.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' We generate reduced graph according to the cutting off logic of BRICS and then calculate its spanning tree as AMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Another molecular decomposition algorithm in our study could find its root in feature tree[47] published for molecular similarity algorithm by Rarey et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' in 1998 and JTVAE[12] proposed later by Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Given a graph G, we first find all its simple cycles, and its edges not belonging to any cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Two simple rings are merged together if they have more than two overlapping atoms, as they constitute a specific structure called bridged compounds [48].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Each of those cycles or edges 10 is considered as a cluster.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Next, a cluster graph is constructed by adding edges between all intersecting clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Finally, we select one of its spanning trees as the AMT of G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The t-SMILES algorithm is a general and fragment based molecular representation framework which does not limit the choice of molecular decomposing algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Different molecular segmentation algorithm may require different fragment assembling algorithm to complete the reconstruction of the generated molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='4 Acyclic Molecular Tree The idea of using tree as the base data structure of algorithms to address molecular related tasks has been long established in cheminformatics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In early study of molecular descriptor and similarity analysis, algorithms such as reduced graph[49], feature tree[47,50] not only had shown potential power to improve the similarity search but also being capable of retrieving more diverse active compounds than using Daylight fingerprints[51].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='Some recent works[52-55] proposed to incorporate tree-based deep learning models into molecular generation and synthesis tasks as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' AMT being capable to describe the molecule at various levels of resolution, reduced graph[56] provides summary representations of chemical structures by collapsing groups of connected atoms into single nodes while preserving the topology of the original structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In reduced graph, the nodes represent groups of atoms and the edges represent the physical bonds between the groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Constructing reduced graph in this way forms a hierarchical graph, whose top layer being the molecular topology representing global information, and the bottom layer representing molecular fragments of detailed information.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Groups of atoms are clustered into a node in the reduced graph approach, which could be done based on fragmentation algorithms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The feature tree[47] is a representation of a molecule similar to a reduced graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The vertices of the feature tree are molecular fragments and edges connect vertices that represent fragments connected in the simple 11 molecular graph.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In t-SMILES algorithm, the minimum spanning tree of the reduced graph and the concept of feature tree could be regarded as an AMT in the intermediate step, and then the next encoding algorithm is done based on this AMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Specific to our experiments, one approach is to generate AMT based on the tree logic fragmented by the BRICS algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Another method uses junction tree[57] introduced by Jin et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' in JTVAE as the AMT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In the case of a junction tree, each node in the tree corresponds to a subset (subgraph, group, cluster or clique) in the original graph and edge is used to connect two clusters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In this study, nodes of Junction tree represent rings, bonds, bridged compounds, or singletons in the original molecular graph which are generated by fragment decomposing algorithm and edges represent the physical bonds between groups.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='5 Molecular Reconstruction and Optimization In the process of reconstructing the generated molecules from t-SMILES strings, we follow below steps: Algorithm steps to reconstruct molecule from t-SMILES Step 1: Decompose t-SMILES to reconstruct the FBT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Step 2: Convert the FBT to the AMT;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Step 3: Assemble molecular fragments according to the selected algorithm to generate the correct molecular graph and then optimize it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' During reconstruction, one key problem is how to assemble the molecular fragments together to get a ‘chemical correct’ molecule.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Ideally, the assemble algorithm should be selected to match the molecular fragmentation algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In this study, for efficiency reasons, we assemble molecular graph one neighborhood at a time, following the order in which the tree itself was generated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In other words, we start from the root node of AMT and its neighbors, then we proceed to assemble 12 the neighbors and their associated clusters, and so on.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' If there is more than one candidate when assembling two pieces, we simply select one randomly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The overall performance of t-SMILES based generative model is controlled by two main factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The first one is whether t-SMILES can be learned and generated efficiently, and the other one is the reconstruction and molecular optimization algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' We evaluate the assembling algorithms by directly reconstructing molecules from training set data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Detailed metrics are shown in Table 1 and Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Table 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Distributional results by directly reconstructing molecules from the training dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Dataset Model Valid(↑) Uniq(↑) Novel(↑) KLD(↑) FCD(↑) Zinc JTVAE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='770 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='662 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='982 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='811 BRICS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='780 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='681 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='849 QM9 JTVAE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='741 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='245 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='968 ChEMBL JTVAE 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='786 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='677 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='969 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='694 BRICS 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='787 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='977 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='792 Data in Table 1 show that FCD scores on ChEMBL are the lowest ones while it is the highest on QM9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Distributions of physiochemical properties (logP, plogP, SAS, BertzCT, QED, TPSA, NPS and FractionCSP3) of reconstructed molecules directly from Zinc training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Wasserstein distance is used in this figure and later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 13 From Figure 2, it can be seen that SAS and NPS are the two most variable metrics when reconstituting molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' If we output all possible assembly results, we could get a set of molecules which come from the same fragments group with different structures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' From this point of view, generating new molecules with the desired properties (desired structure) rather than duplicating the training set is exactly the potential goal of the molecule generation task and not a negative aspect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' MOG[58] argued that the common pitfall of existing molecule generation models based on distributional learning is that the exploration is confined to the training distribution, and the generated molecules exhibit “the striking similarity” with known molecules included in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Models that do not require training molecules are free from this problem, but they introduce other problems such as long training time, the sensitivity of balance between exploration and exploitation, large variance, and importantly, a lack of information about the known distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Based on t-SMILES, it’s possible to select appropriate optimization algorithm to control how the fragments are assembled, thereby controlling the properties of the output molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Molecule optimization is another challenging topic which is beyond the scope of this study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Using more complex optimization algorithms instead of stochastic method to select target molecule would be an option for further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 3 Results and Discussion To evaluate t-SMILES and generatation models on different datasets, we firstly compare two t- SMILES based mGPT2 models with fragment-based baseline models and classical SMILES based mGPT2 models on Zinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' And then we compare t-SMILES based models with SOTA SMILES and GNNs baseline models on QM9 and ChEMBL datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Metrics used in this study are still based on distributional learning.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='1 Comparison with fragment-based and SMILES-based models on Zinc The validity, uniqueness, novelty, KLD and FCD scores on Zinc are summarized in Table 2 and the distributions of eight physiochemical properties are shown in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Table 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Distributional results on Zinc, we train or retrained all these five models, t-SMILES_J_mGPT2 breaks molecules using the same fragmentation algorithm as JTVAE[27], t-Smiles_B_mGPT2 breaks molecules using BRICS, mGP2 means five layers mini GPT2 model is used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Model Valid(↑) Uniq(↑) Novel(↑) KLD(↑) FCD(↑) SMILES_mGPT2(Ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='853 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='674 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='672 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='960 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='830 JTVAE[12] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='997 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='870 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='439 FragDgm[27] 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='000 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='423 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='422 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='835 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='303 t-SMILES_J_mGPT2(Ours) >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='775 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='970 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='773 t-Smiles_B_mGPT2(Ours) >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='783 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='782 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='963 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='790 Classical SMILES based mGPT2 model is trained for reference in this study to evaluate t-SMILES based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Compared with t-SMILES models, classical SMILES based model gets lower novelty and uniqueness scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' It means that t-SMILES-based models, with almost 100% validity and relatively high FCD scores, could improve novelty to explore a wider molecular space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' JTVAE[12] is one key baseline model for this study, which splits molecule using a tree base data struct.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' So far as the validity is concerned, both t-SMILES and JTVAE models could generate near 100% valid molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' However, t-SMILES exhibits significantly higher KLD and FCD scores with reasonably slight lower novelty and uniqueness scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' t-SMILES based models significantly outperforms another fragment dictionary-based model, FragDgm[27], which splits molecule in a linear mode as a sequence of fragment IDs, on all five distribution parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Despite FragDgm adopting a segmented mode and based on distributional learning, its FCD value is the lowest among the five models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' When more and more validity of models can reach above 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='9, it becomes more important to test whether the generative model can effectively learn the physicochemical properties of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 15 Detailed distributions of eight physiochemical properties logP, plogP, SAS, BertzCT, QED, TPSA, NPS and FractionCSP3 on Zin are presented in Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Distributions of physiochemical properties (logP, plogP, SAS, BertzCT, QED, TPSA, NPS and FractionCSP3) of random selected 10K molecules from Zinc data sets, JTVAE[12], FragDgm[27] and molecules generated by SMILES and two t-SMILES based mGPT2 models respectively It can be seen from Figures 3 that the distribution of t-SMILES and classical SMILES based mGPT2 models are closer to training data set on all eight physiochemical properties than other two fragment-based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The distribution of generated molecules by FragDgm is far from the training set on six physiochemical properties except NPS and FractionCSP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Compared with JTVAE, besides TPSA and SAS which are comparable, two t-SMILES based mGPT2 models get significantly lower scores on all eight metrics, especially on FractionCSP3 and BertzCT.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Compared with FragDgm, except NPS which are comparable, t-SMILES based models get significantly favorite scores in other seven physicochemical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Considering novelty, the distribution of the eight physicochemical properties suggests that the generated molecules by t-SMILES-based models are better fit to the training dataset, which obviously implies relatively low novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 16 From the comparative analysis of these models, two t-SMILES based models in this study significantly outperform the fragment-based baseline models, and have similar performance to the SMILES-based model in physicochemical properties, but validity scores could be greater than 99%, that is to say, t-SMILES could be an effective molecular presentation for fragment based molecular tasks on Zinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='2 Comparison with baseline models on QM9 The validity, uniqueness, novelty, KLD and FCD scores on QM9 are summarized in Table 3 and the distributions of eight physiochemical properties are shown in Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Table 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Distributional results on QM9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The results of CharacterVAE[7], are taken from MolGAN[60], Transformer Reg[11], GraphVAE[59,11], MolGAN[60] and MGM[11] are taken from O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Mahmood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' [11] Input Model Valid(↑) Uniq(↑) Novel(↑) KLD(↑) FCD(↑) SMILES CharacterVAE[7,60] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='103 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='675 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='900 N/A N/A Transformer Reg[11] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='957 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='183 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='994 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='958 SMILES_mGPT2(ours) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='970 Graph GraphVAE[59,11] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='557 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='760 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='616 N/A N/A MolGAN[60] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='981 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='104 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='942 N/A N/A MGM[11] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='886 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='978 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='518 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='842 t-SMILES t-SMILES_J_mGPT2(ours) >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='720 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='289 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='953 On QM9, the score of validity of t-SMILES based mGPT2 model could be greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='99, which is comparable to the performance of GNNs and superior to the most SOTA SMILES based models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' FCD score of t-SMILES based mGPT2 model is one of the highest ones in all seven models, which is greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='95.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' From the viewpoint of sequence-based models, our approach performs similarly to or better than existing SMILES based approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Our approach shows higher validity and uniqueness scores compared to CharacterVAE, while having a reasonable lower novelty score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Compared to the Transformer Reg model, t-SMILES based model has higher score in novelty, lower score in uniqueness, and comparable scores in KLD and FCD.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 17 Compared to the graph-based models, our approach outperforms the existing baseline approaches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' t-SMILES based model has comparable uniqueness score compared with respect to GraphVAE, and significantly outperforms MolGAN, with reasonable lower novelty score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' KLD and FCD scores are not provided for these two models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' t-SMILES based mGPT2 model has good performance against SOTA method MGM in validity, KLD and FCD scores, while having slightly lower scores in uniqueness and novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Distributions of physiochemical properties (logP, plogP, SAS, BertzCT, QED, TPSA, NPS and FractionCSP3) of random selected 10K molecules from generated molecules on QM9 data sets by SMILES and t- SMILES based mGPT2 models respectively In general, on QM9 data set, both mGPT2 models based on t-SMILES and classical SMILES get high KLD and FCD scores, all greater than 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='95, that means they could maintain physiochemical similarity to the training distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' However, the novelty scores of the models are relatively low, which could be interpreted by that the generated molecules based on distributional learning exhibit “the striking similarity” with known molecules included in the training set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' It also indicates that novelty is inversely correlated with the KLD and FCD scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='3 Comparison with baseline models on ChEMBL The validity, uniqueness, novelty, KLD and FCD scores on ChEMBL are summarized in Table 4 and the distributions of eight physiochemical properties are shown in Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 18 Table 4 Distributional results on ChEMBL, The results of ORGAN[8], LSTM[37], CharacterVAE[7], AAE[61], Transformer Reg[11], Graph MCTS[14], and MGM[11] are taken from O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Mahmood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' [11] and Guacamol[37], the results of MolGPT is taken from Bagal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' [9], the results of hgraph2graph[13] is calculated by us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Input Model Valid(↑) Uniq(↑) Novel(↑) KLD(↑) FCD(↑) SMILES ORGAN[8,37] 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='379 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='841 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='687 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='267 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='870 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='485 t-SMILES t-SMILES_J_mGPT2(Ours) >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='781 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='765 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='913 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='564 t-SMILES_B_mGPT2(Ours) >0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='99 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='781 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='769 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='935 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='575 On ChEMBL, two t-SMILES based mGPT2 models outperform graph-based baseline methods Graph MCTS and hgraph2graph, while the performance is comparable to SOTA graph mode MGM.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Compared to Graph MCTS, t-SMILES based mGPT2 models show lower novelty scores while having significantly higher KLD and FCD scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' It seems difficult for this graph-based baseline model to capture the properties of the dataset distributions as shown by their low KLD scores and almost-zero FCD scores, but it gets the highest novelty score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Compared to hgraph2graph which is an advanced model based on JTVAE and aims to solve larger molecular problems with motif-based method, t-SMILES based mGPT2 modes have higher KLD and FCD scores and lower uniqueness and novelty scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Compared to SOTA graph-based model MGM, t-SMILES based mGPT2 models have higher score in validity, but lower scores in uniqueness, KLD and FCD and similar score in novelty.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In the realm of sequence-based models, the proposed t-SMILES based mGPT2 models are competitive with the classical SMILES based models, and firstly, our models outperform all listed classical SMILES based models in validity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Compared with the GAN-based model (ORGAN), t- SMILES based mGPT2 models have significantly higher scores in validity, KLD and FCD and novelty scores, while having a slightly lower score in uniqueness.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Compared with AAE, t-SMILES 19 modes get low unique and novelty scores and higher KLD and FCD scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Similar to Graph MCTS, ORGAN also get an almost-zero FCD score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Our t-SMILES based approach results in lower scores across most of the metrics when compared to Transformer Reg, MolGPT, LSTM, VAE models besides validity score.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In addition, statistics data shows that SMILES-mGPT2 model gets a slightly lower scores compared with Transformer Reg and MolGPT, which indicates that the t-SMILES mGPT2 models could be optimized to get high scores as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Such an issue could be severed as a starting point for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Distributions of physiochemical properties (logP, plogP, SAS, BertzCT, QED, TPSA, NPS and FractionCSP3) of random selected 10K molecules from generated molecules on ChEMBL data sets, hgraph2graph(hG2G)[13] and molecules generated by SMILES and t-SMILES based mGPT2 models respectively Compared with hgraph2graph in Figure 5, t-SMILES based models get slightly higher scores on BertzCT, logP, QED and TPSA, while having comparable scores on SAS, NPS, plogP and FractionCSP3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Compared with classical SMILES based mGPT2 model, t-SMILES based mGPT2 models have lower scores on seven metrics besides QED.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' On ChEMBL, one possible reason of the proposed t-SMILES models getting a relatively little lower scores might be that t-SMILES based mGPT2 model is not hyper-parametrically optimized well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 20 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='4 Ablation Studies Compared to classical SMILES, only two more characters ‘&’ and ‘^’ are introduced for t-SMILES to encode multiscale molecular topological structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Therefore, we adopt the same network structure to conduct ablation studies on one LSTM model and two Transformer-based models using three downstream datasets to investigate the performance of generative models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' As shown in Table 5, different generative models get slightly different distributional results for the same dataset.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' However, compared to SMILES based models, t-SMILES based models get higher validity, uniqueness and novelty scores generally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' This evidence also supports one of our claims that t-SMILES make it possible to generate molecules fragment-by-fragment using sequence- based models easily.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Table 5 Distributional results on Zinc, QM9 and ChEMBL datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The results of LSTM[37], MolGPT[9] and mGPT2 are all trained and calculated by us.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Dataset Model Valid(↑) Uniq(↑) Novel(↑) KLD(↑) FCD(↑) Zinc t-SMILES_J_LSTM >0.' metadata={'source': 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+page_content='966 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='831 SMILES_MolGPT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='980 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='770 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='766 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='989 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='923 SMILES_mGPT2 0.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='182 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='965 SMILES_MolGPT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='965 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='729 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='150 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='986 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='907 SMILES_mGPT2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='949 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='728 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='172 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='970 ChEMBL SMILES_LSTM 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='774 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='609 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='598 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='943 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='725 SMILES_MolGPT 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='976 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='768 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='711 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='992 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='807 SMILES_mGPT2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='850 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='670 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='641 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='972 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='809 4 Experiments In this section, we briefly introduce the proposed generation models which are used to evaluate t- SMILES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' This is then followed by the overview of three commonly used public data sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Finally, 21 the details of the experiments and the metrics used for the evaluation of different models are provided.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='1 Generation models and hyperparametric optimization Language modelling uses probability and statistical techniques to calculate the possibility of word sequences in sentences and then do estimation[62].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Originally, recurrent neural network (RNN) is designed to address this kind of sequence problems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' To address the limitation of RNN that lead to gradient disappearance and explosion for long sequences, cyclic models such as LSTM and GRU are proposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' However, it has been shown that the power of LSTMs is insufficient when the information has ultra-long dependencies.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In 2017, the attention-based Transformer architecture broke through the limitations of LSTM and quickly achieved SOTA scores on multiple metrics in NLP and computer vision.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' After that, the improved models such as GPT1-3[63,43,64]and BERT[65] are introduced to build pre-trained model on large datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Although these large-size models have achieved unprecedented performances, they come at high computational costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Consequently, some of the recent NLP architectures have utilized concepts of transfer learning, pruning, quantization, and knowledge distillation to achieve moderate model sizes while keeping nearly similar performances as achieved by their predecessors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The recent developments in the NLP field have a great potential to be adapted to molecular de novo generation research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Bagal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' [9] has proposed a Transformer-decoder model with 10 layers to molecular generation task and proved it works well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In this study, we mainly adopt Transformer-decoder based autoregressive generation models to evaluate our proposed t-SMILES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' For the sake of data comparability, we select one model as the basis using a Bayesian optimizer SigOpt[66] to optimize the hyperparameters related to the neural network topology in ChEMBL dataset using classical SMILES as input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' This selected model is a 22 5 layers mini version of GPT2(mGPT2) with 512 hidden layer and 8 attention headers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' All other mGPT2 models in different dataset with either t-SMILES or classical SMILES use the same high- level neural network architecture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Finally, we train all models in detail using the Adam optimizer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='2 Datasets We evaluate t-SMILES on three commonly used public data sets including subset of Zinc[32], QM9[33] and ChEMBL-21[34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The Zinc subset used in our experiments is the same as the subset in JTVAE which contains approximately 250K drug-like molecules extracted from the Zinc15 database that span a wide range of chemistry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The molecules in this Zinc subset are composed of 10 atoms including Br， C，Cl，F，H，I，N，O，P and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' QM9 dataset contains up to 9 heavy (non hydrogen) atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Molecules in the subset consist of 5 atoms including C, F, N, H and O atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In all, this results in about 134k druglike organic molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' For the sake of universality, we select a subset from ChEMBL-21 with SMILES character length less than 120.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Apart from this very simple rule, no other preprocessing is done.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Molecules in ChEMBL subset are composed of 11 atoms including B, Br, C, Cl, F, H, I, N, O, P and S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='3 Evaluation Metrics Despite a large number of metrics have been proposed for the evaluation of generative models of molecules, there is little consensus on which should be used.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' It is often biased to use simple indexes to evaluate the performance of different models [67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' From the perspective of optimization, it could be considered that the task can be solved once the molecules with a high score of a certain index are generated, but these generated molecules may not be useful.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Therefore, we use five benchmarks 23 proposed by GuacaMol: validity (↑), uniqueness (↑), novelty (↑), KL divergence (KLD) (↑)[35]and Fréchet ChemNet Distance(FCD)[36] score (↑) to evaluate the general performance of the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' GuacaMol calculate every score using 10K randomly sampled molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Validity measures the ratio of valid molecules which could be correctly parsed by RDKit[45];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Uniqueness is the percentage of valid molecules that are unique;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Novelty is the percentage of valid unique molecules that are not included in the training data set;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' KLD[35] score compares the distribution of a variety physicochemical descriptors of the training set and generated molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' FCD[36] score measures the proximity of the distribution of generated molecules to the distribution of the dataset molecules according to the Fréchet Distance in the hidden representation space of ChemNet[68], which is trained to predict the chemical properties of small molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The values of these five parameters are between 0 and 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The larger the value, the ‘better’ the model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' While KLD and FCD are both measure of the similarity between generated molecules and molecules from the training data set.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' They are highly correlated with each other and inversely correlated with novelty[11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In addition, we use logP, plogP[38], SAS[39], BertzCT[40], QED[41], TPSA, NPS[42], and FractionCSP3 to evaluate whether the generative models could effectively learn the physical and chemical properties of the molecules in the training set, thereby comprehensively evaluating the performance of the generative model from the perspective of distributed learning knowledge.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' logP: The logarithm of the partition coefficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' If one of the solvents is water and the other is a nonpolar solvent, then logP is a measure of hydrophobicity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Penalized logP(plogP)[38]is the logarithm of the partition ratio of solute between octanol and water subtracted by synthetic accessibility score and long cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' It has a range of (−∞,∞) Synthetic Accessibility score (SAS)[39]: Measurement of the difficulty of synthesizing a compound.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' It is a score between 0 (difficult) and 1(easy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 24 BertzCT[40]: A topological index meant to quantify “complexity” of molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' It consists of a sum of two terms, one representing the complexity of the bonding, the other representing the complexity of the distribution of heteroatoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Quantitative Estimate of Drug-likeness (QED)[41]: This quantifies drug-likeness by considering the main molecular properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' It ranges from 0 (all properties unfavorable) to 1 (all properties favorable).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Topological Polar Surface Area (TPSA): The sum of surface area over all polar atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' It measures the drug’s ability to permeate cell membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Molecules with a TPSA greater than 140 Å2 tend to be poor in permeating cell membranes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Natural Product-likeness score (NPS)[42]: Score of similarity degree of structural space covered by molecules and natural products.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' It has a range of (0,1) FractionCSP3: The fraction of C atoms that are SP3 hybridized.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In general, we randomly select 10K molecules from generated ones to calculate five metrics from GuacaMol[37] and the distribution of eight physiochemical properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='4 Details of baseline models We train classical SMILES based mGPT2 models as relatively baseline on three datasets, which share the same neural network architecture with t-SMILES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' If there are some SMILS-based models that get better scores than this SMILES-mGPT2, it means that it’s possible to build t-SMILES based model to get better scores.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In ablation studies, we retrain the LSTM[37] model with 3 layers of hidden size 768, dropout of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='2 using the Adam optimizer with learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='001, and MolGPT[9] model with 8 layers, 8 25 headers with hidden size 256 using AdamW optimizer with learning rate 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='001 for both SMILES and t-SMILES on all three datasets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' We retrain JTVAE[12] on Zinc using the publicly available codebase provided by the paper’s authors and then generate 20K molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' We calculate metrics based on the molecules provided by Fragment-base-DGM[27] on Zinc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' We generate 20K molecules using the pre-trained model provided by hgraph2graph[13] on ChEMBL and calculate metrics for evaluation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' We do not train the rest of the baseline models by ourselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' For QM9 and ChEMBL, we take some results of baseline models from Bagal et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' [9], Cao and Kipf [60], O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Mahmood et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' [11] and GuacaMol[37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Finally, we follow the open-source implementation of the GuacaMol[37] benchmark baselines to calculate metrics for all baseline and new designed models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 5 Conclusion and Outlook One of the challenging problems in designing drug molecules is the vast chemical space to be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Fragment-based approaches identify a subset of chemical moieties responsible for key molecular recognitions early on and this allows scientists to devote their time in a much reduced and relevant chemical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In general, if atoms are analogized to characters in natural language processing, fragments could be analogized to words or phrases as functional "units", then molecule could be a fragment based discrete structured data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' To train a general-purpose string generative mode to learn the knowledge from discrete structured data, we need to prepare large amount of valid combinations of the structures, which is time consuming and will also face the challenge of insufficient effective data 26 practical in domains like target-oriented drug discovery.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Our proposed t-SMILES uses SMILES instead of fragment dictionaries id, re-encodes fragment-based molecular tree through a full binary tree and generates molecular sequence with multiscale structural information, thus providing a new idea for fragment-based molecular design.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In this way, powerful and rapidly developing sequence- based solutions can be applied to fragment-based molecular tasks in the same way as classical SMILES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' In addition, compared with classical SMILES which is relatively difficult to be augmented, by using different fragmentation algorithms, the training dataset is easier and more efficiently to be expanded on t-SMILES to explore different chemical spaces without having to change anything of the architecture of generation model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Scalability Our proposed t-SMILES solution supports any effective substructures types and patterns as long as they could be used to obtain chemically valid molecular fragments and ultimately construct valid acyclic molecular trees.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' So that, with the invaluable accumulation of drug fragments by experienced chemists, it’s possible that the t-SMILES algorithm combines this experience with a powerful sequence based deep neural network model to help chemists better explore chemical space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' If in some special cases, only one specific fragment space needs to be explored, and expansion is not required, then in t-SMILES algorithm, tree nodes could be easily encoded by dictionary id instead of SMILES.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Or we could replace the newly generated fragments with the fragments from the training data set according to specified rules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The encoding logic and algorithm flow of t-SMILES remain unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' To be improved Our experiments show that how to segment, assemble molecular fragments and how to optimize molecular are also key steps controlling the quality of generated molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Therefore, this challenging topic could serve as a starting point for further research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 27 With the rapid development of NLP technology, Transform-based models have been proved to enable text generation with human-like capabilities if trained on enough data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' SMILES-based models have proved that the sequence-based NLP models are powerful tools to generate molecules.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Then, t-SMILES makes it easily to use sequence-based NLP models to generate molecules fragment-by-fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Therefore, the investigation of t-SMILES based generation models or other fragment-based tasks would be served as interesting topic for further study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='List of abbreviations ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='SOTA: State-of-the-art ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='FBDD: Fragment-based drug discovery ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='AMT: Acyclic Molecular Tree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='FBT: Full Binary Tree ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='BFS: Breadth First Search ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='DFS: Depth First Search ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='SMILES: Simplified Molecular Input Line Entry Specification ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='plogP: Penalized logP ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='SAS: Synthetic Accessibility score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='QED: Quantitative Estimate of Drug-likeness ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='NPS: Natural Product-likeness score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='FCD: ChemNet distance score ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='KLD: KL divergence ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='Conflicts of Interest ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='The authors declare that they have no competing interests.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 28 Author Contributions Ruqin Yu and Juanni Wu designed the study and manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Juanni Wu conceived the project, constructed the algorithms and Python script, performed the experiments, informatics analyses, and wrote the draft manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Tong Wang and Yue Chen participated in the discussion and experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Lijuan Tang and Hailong Wu participated in the discussion and funding acquisition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' All authors contributed to manuscript editing, revising and have approved the final version of the manuscript.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Acknowledgements This research was funded by the National Natural Science Foundation of China including No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 21874040 and No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' 22174036.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Data availability The datasets used in this study are publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' They are referenced in the Datasets and Evaluation Metrics part of the Experiments section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' The processed data used in this study can be found at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='com/juanniwu/t-SMILES/ Code availability Code, pretrained t-SMILES models, training and generation scripts for this work and lists of generated molecules can be found at: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='com/juanniwu/t-SMILES/ The code of baseline models used to in this work are publicly available.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' We are gratefully acknowledging all authors of these researches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' [1] MolGPT[9]: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='com/devalab/molgpt [2] MGM[11]：https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='com/nyu-dl/dl4chem-mgm 29 [3] JTVAE[12]：https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='com/wengon-jin/icml18-jtnn [4] hgraph2graph[13]: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='com/wengong-jin/hgraph2graph [5] FragDGM[27]: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='com/marcopodda/fragment-based-dgm [6] Guacamol[37]：https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='com/BenevolentAI/guacamol_baselines [7] GPT2: https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content='com/samwisegamjeee/pytorch-transformers Reference 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Shuker SB, Hajduk P, Meadows R, Fesik S (1996) Discovering High-Affinity Ligands for Proteins: SAR by NMR.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/ddAzT4oBgHgl3EQf3f4p/content/2301.01829v1.pdf'} +page_content=' Science (80- ) 274:1531–1534.' metadata={'source': 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Liu2, 1, 3, † +1Beijing Academy of Quantum Information Sciences, Beijing 100193, China +2State Key Laboratory of Low Dimensional Quantum Physics, +Department of Physics, Tsinghua University, Beijing, 100084, China +3Frontier Science Center for Quantum Information, Beijing 100184, China +(Dated: January 6, 2023) +Noise remains the major obstacle to scalable quantum computation. Quantum benchmarking +methods provide key information on noise properties for quantum processor calibration, quantum +error mitigation, and quantum error correction. However, current benchmarking methods, such as +randomized benchmarking or its variants, can only evaluate the performance of some particular +subsets of quantum gates. +Moreover, due to the randomization inherent in these protocols, the +figure of merit they actually measure is not the fidelity of individual target gate but the average +of the fidelities of some random circuit cycles incorporating the target. To overcome these limits, +we propose the channel spectrum benchmarking (CSB), a method to infer the noise properties of +the target quantum process, such as process fidelity, from the eigenvalues of its noisy quantum +channel. The noisy eigenvalues can be estimated by the circuits of control-free phase estimation +in a state-preparation and measurement error resilient manner. Our method can benchmark uni- +versal quantum processes and is scalable to many-qubit quantum processes. We demonstrate the +performance of our method using simulated experiments, including the single-qubit Pauli rotations, +2-qubit fermionic simulation gates, a 3-qubit cycle implementing the Toffoli gate, and a 10-qubit +cycle implementing the Ising Hamiltonian evolution operator. Our method will pave an important +way for the development of cleaner and large-scale quantum devices. +I. +INTRODUCTION +The performance of today’s quantum computers is +severely affected by noise and the limited number of +qubits [1]. Quantum error correction and fault-tolerant +schemes may someday unlock the full potential of quan- +tum computation [2–7], but more precise gate opera- +tions must be developed beforehand. +It is crucial and +necessary to obtain information on the gate noise char- +acteristics and their performance benchmarks in order +to calibrate and optimize these gate operations [8–10]. +Nonetheless, there is a trade-off between noise informa- +tion obtained and the resource overhead for their testing +experiments [11]. Process tomography [12, 13] is a typical +technique for reconstructing the matrix representation of +a quantum process, with which the full information of +noise is at hand. However, process tomography has expo- +nentially increasing experimental costs and suffers from +state-preparation and measurement (SPAM) errors. Al- +though its variant, the gate-set tomography [14–18], can +handle SPAM errors, the experimental costs cannot be +reduced. +In reality, for probing noise strength or noise types +of a gate, the full reconstruction of the noisy process is +not necessary [19, 20]. For instance, the average gate fi- +delity, that measures the average performance of the im- +plemented noisy gates, can be efficiently obtained by ran- +domized benchmarking (RB) [21–26]. The RB protocol +is insensitive to SPAM errors and its variants [8, 27–29] +∗ guyw@baqis.ac.cn +† dongeliu@mail.tsinghua.edu.cn +can be applied to benchmark devices with larger system +size. It is important to note that protocols like random- +ized benchmarking do not directly measure the fidelity of +individual quantum gates, but rather the average fidelity +of some random circuit fragments [30–32]. To determine +the fidelity of a specific gate, additional techniques such +as interleaved RB [33] or modifying the sampling distri- +bution of random circuits [27, 29] must be used, which +can induce more experimental cost and is prone to a large +systematic uncertainty [34]. Additionally, to simplify the +functional form of measured signals in RB methods, it +is often necessary to use group twirling, which limits the +types of gates that can be benchmarked. +As a conse- +quence, the RB protocols based on random Clifford cir- +cuits can only be applied to benchmark the Clifford gates; +however, the important non-Clifford gates have to rely +on more complicated random circuit sets in which their +native gates belong to other groups instead of Clifford +group, e.g. dihedral groups [35, 36]. +In order to overcome the two limitations, we intro- +duce channel spectrum benchmarking (CSB), a scalable +protocol to estimate the individual noise properties of +a universal quantum process from the noisy eigenvalues +of its corresponding quantum channel. We estimate the +noisy eigenvalues by control-free phase estimation cir- +cuits [40–44] that is robust to SPAM errors. With a re- +lationship between ideal and noisy eigenvalues, which is +derived from the first order perturbation theory [45], we +can infer the diagonal entries of the matrix of pure noise +process under a basis composed of the eigen-operators +of the ideal gate. From these diagonal entries, we can +estimate some noise properties, for examples, process in- +fidelity, stochastic infidelity (a quantity similar to uni- +arXiv:2301.02056v1 [quant-ph] 5 Jan 2023 + +2 +Gates +Fidelity +Conditions for Scalablity +CSB +universal +• General case: target +• Strong unitary error and RC: +target + twirling gates +• eigen-decomposition of target gate is possible +• initial state preparation is efficient +Clifford RB [22, 23] Clifford +ave. among Clifford gates +not scalable due to compilation issue [27] +Mirror RB [29] +Clifford +ave. among rand. cycles +only applicable to Clifford gates +CB [28] +U m = I +target + twirling gates +target gate is Clifford +XEB [8] +universal +ave. among rand. cycles +circuits can be classically simulated +TABLE I. Comparison with other leading benchmarking protocols. We compare our CSB protocol with other benchmarking +protocols under three aspects: (1) what gates they can benchmark; (2) what type of fidelity they actually measure; (3) under +what conditions they can be scalable to many-qubit systems. Usually, our CSB measures the fidelity of the target gate. But +for strong unitary error, we need to perform randomized compiling (RC) [37, 38] with twirling gates to convert unitary error +to stochastic error in order to obtain a better performance. When benchmarking a circuit fragment, the twirling gates can be +merged into target gate, our method still measures the individual fidelity of the target as shown in Sec. IV C and IV D. But, +when benchmarking native gates, the twirling gates can not be merged (see Appendix. C). In this case, our method measures the +average fidelity of the compositions of the target gate and twirling gates. Our CSB is scalable as long as eigen-decomposition of +target is possible and the number of single and two-qubit gates in the circuits preparing initial states scales at most polynomial +with the number of qubits. Clifford RB and Mirror RB use random Clifford circuits to simplify noise and thus only apply to +Clifford gates. The fidelity they actually measure is the average of fidelities among random Clifford cycles. Mirror RB can +be scalable but Clifford RB cannot. For cycle benchmarking (CB), the gate or cycle U that can be benchmarked must satisfy +U m = I where m is an integer. CB uses Pauli twirling to simplify noise and thus measures the fidelity of composition of target +and twirling gates. It needs to compute the output Pauli operator of ideal circuits, which is possible only when the target +gate is Clifford for large systems. XEB uses random universal circuits to simplify noise, so it measures the average of fidelities +among some random circuit cycles generated with a same sampling distribution. It requires the classical simulation of circuits +to obtain the ideal probabilities of sampled bit strings, which limits its scalability. Additionally, our CSB and XEB can directly +measure how close the noise is to unitary error, while RB methods need extra procedures to measure this information [39]. +tarity [39]) and over-rotation angle. We demonstrate the +performance of our method with some simulated exper- +iments, 1-qubit Pauli rotaional gates, 2-qubit fermionic- +simulation (Fsim) gates, 3-qubit circuit fragment imple- +menting Toffoli gate, and 10-qubit circuit fragment im- +plementing Ising evolution operator. In all experiments, +our CSB method can accurately estimate the noise prop- +erties. To get a more clear picture of the performance +of our CSB, in Table I, we compare our CSB protocol +with other leading benchmarking protocols under three +aspects: (1) what gates they can benchmark; (2) what +type of fidelity they actually measure; (3) under what +conditions they can be scalable to many-qubit systems. +II. +QUANTUM CHANNEL, FIDELITY, AND +CHANNEL SPECTRUM +In this section, we provide some preliminaries about +quantum channel, the fidelity of implemented noisy gates, +and the relationship between the fidelity of a gate and the +channel spectrum of its noisy implementation. +Consider a quantum gate U acting on a d-dimensional +space with eigenvalues eiλa and eigenstates |φa⟩ such that +U|φa⟩ = eiλa|φa⟩. Because of noise, the actual imple- +mentation of the gate should be denoted as a quantum +channel �U = EU, or say completely-positive and trace- +preserving (CPTP) map [12], where U is the correspond- +ing quantum channel of the ideal gate U and E is a pure +noise process. +Quantum channels are usually denoted +by a set of Kraus operators, for example, U = UρU † +and E(ρ) = � +k EkρE† +k where ρ is an arbitrary operator. +Quantum channels can also be represented by a matrix +on the basis of d2 dimensional operator space, for ex- +ample, Pauli operators. We will use the two representa- +tions interchangeably and the same symbols for both the +abstract quantum channels and their matrix representa- +tions. +One can use some fidelity measures to assess the per- +formance of the implemented noisy gate �U, such as the +process fidelity (or referred to as entanglement fidelity) +which is defined as +F(U, �U) = tr +� +I ⊗ U(|α⟩⟨α|) I ⊗ �U(|α⟩⟨α|) +� +(1) +where |α⟩ = +1 +√ +d +�d +i=1 |i⟩⊗|i⟩ is the maximally entangled +state. The process fidelity is closely related to another +ubiquitous measure, the average gate fidelity [46] +Fave(U, �U) = +ˆ +dψ tr +� +U(|ψ⟩⟨ψ|) �U(|ψ⟩⟨ψ|) +� += dF + 1 +d + 1 . +(2) +It has been proven that the process fidelity only depends +on the trace of the pure noise E [46], that is +F(U, �U) = +tr +� +U† �U +� +d2 += tr{E} +d2 +. +(3) +Current benchmarking methods, for example, random- +ized benchmarking and its variants, measure the infor- +mation of tr{E} on a basis composed of Pauli operators. + +3 +In these protocols, Clifford twirling or Pauli twirling are +used to simplify the noise matrix E, that is, only diag- +onal entries of E on the Pauli basis are kept, such that +the relevant figure of merit can be extracted easily from +measured signals. The twirling operations need to be per- +formed by running some random circuits. This causes RB +type of methods only apply to some subsets of quantum +gates and only measure the average fidelity of a set of +gates. +Distracting from the Pauli operator basis, one can note +that the ideal channel U also induces a natural opera- +tor basis composed of its eigen-operators |φa⟩⟨φb| (corre- +sponding eigenvalues are ei(λa−λb)). If we can measure +the diagonal entries of noise E in this basis, we can also +estimate the gate fidelity. This can be achievable by a +relationship between the eigenvalues of noisy gate �U and +those of ideal gate U [45], that is +gabeiλab ≈ ei(λa−λb)tr +� +(|φa⟩⟨φb|)†E(|φa⟩⟨φb|) +� +(4) +where gab and λab is the amplitude and phase of an eigen- +value of �U with eigen-operator Mab, that is �U(Mab) = +gabeiλabMab. +For the spectrum of quantum channels, +there are some useful properties [47]: (1) the eigenval- +ues lie in the unit disc of complex plain, i.e., 0 ≤ gab ≤ 1 +(2) the eigenvalues and eigen-operators always come in +conjugate pairs, i.e., for every eigenvalue gabeiλab we have +�U(M † +ab) = gabe−iλabM † +ab. +The relationship Eq. (4) is derived from the first order +perturbation theory [45] (also see Appendix A). Thus a +diagonal entry of E in the basis composed of |φa⟩⟨φb| can +be obtained +Eab,ab ≈ gabeiλabe−i(λa−λb) . +(5) +As long as we can measure the noisy eigenvalues gabeiλab +of �U and identify their corresponding ideal eigenvalues +ei(λa−λb), we obtain the diagonal entries of Eab,ab by +Eq. (5). +If we can uniformly at random sample some +noisy eigenvalues gabeiλab or equivalently Eab,ab, then we +can use the average of these samples to obtain an esti- +mate of process fidelity F = tr{E}/d2. Because all the +diagonal entries have amplitude smaller than 1, we can +infer the number of samples needed from the Hoeffding’s +inequality [48], that is, let X1, · · · , XK be independent +bounded random variables with ai ≤ Xi ≤ bi for all +i ∈ [K] and denote their average X = 1 +K +� +i Xi, then for +any ϵ > 0 it holds that +P +������X − 1 +K +� +i +E(Xi) +����� ≥ ϵ +� +≤ 2 exp +� +− +−2K2ϵ2 +� +i(bi − ai)2 +� +. +(6) +This inequality bounds the probability that the empirical +average X deviates from the average of expectation val- +ues of these random variables with a distance ϵ. Assume +we have K samples of diagonal entries Eab,ab sampled +from a uniform distribution, so the expectation value of +each sampled diagonal entry is E(Eab,ab) = tr{E} +d2 += F. +We take the average value of these samples as our esti- +mate of the process fidelity, that is ˆF = +1 +K +� +ab Eab,ab. +Thus, the needed number of diagonal entries Eab,ab to +estimate the process fidelity within an error ϵ with the +probability 1 − δ, or say P(| ˆF − F| ≤ ϵ) = 1 − δ, is +K = log(2/δ) +2ϵ2 +, +(7) +which is independent of the system dimension. +Besides the process fidelity, the noisy eigenvalues can +also be used to infer the noise strength of stochastic noise +only. +Since the amplitudes of eigenvalues are only af- +fected by stochastic noise and not changed under unitary +noise, we can use those amplitudes to define a quantity +referred as stochastic fidelity +Fsto = +� +1 +d2 +� +ab +g2 +ab . +(8) +to assess the impact of stochastic noise only. As shown +in Ref. [49], the stochastic fidelity is related to the uni- +tarity defined in Ref. [39]. Both quantities measure how +close the noise E is to a unitary channel and are use- +ful for error-budget. We emphasize that, compared to +the stochastic errors, the unitary errors may cause more +subtle and complicated problems in quantum error cor- +rection and fault-tolerant quantum computation [50–54]. +As a result, differentiating between stochastic and uni- +tary errors can assist us in recognizing their respective +impacts, and in addition, can help to calibrate and tailor +the error types. +We can also estimate the actual values of some uni- +tary parameters of a native gate, from the phases λab +of noisy eigenvalues. This is achieved by identifying the +relationship between these unitary parameters and some +eigenvalues of the gate, which is similar as the robust +phase estimation [40] and Floquet calibration [43]. +III. +CHANNEL SPECTRUM BENCHMARKING +In this section, we present a practical procedure, which +we refer as Channel Spectrum Benchmarking (CSB), to +measure the individual fidelity of a universal process U +no matter it’s a native gate or a circuit fragment. +The estimate of fidelity of the gate U requires a uni- +form sample of diagonal entries of E, which is identical to +a uniform sample of noisy eigenvalues gabeiλab. The noisy +eigenvalues can be estimated by the circuits of control- +free phase estimation depicted in Fig. 1. In these circuits, +we first prepare a state ρ, then repeatedly apply the tar- +get gate U for L times, and finally measure the expecta- +tion value of an operator O. We denote the noisy version +of ρ and O as �ρ and �O. The noisy eigen-operators Mab +of �U can be used as a basis (not necessarily orthonormal) +to expand the initial state �ρ, that is +�ρ = +� +ab +tr +� +G† +ab�ρ +� +Mab +(9) + +4 +|0⟩ +𝑈𝑠 +𝑈 +𝐿 +prepare 𝑐𝑎 𝜙𝑎 + 𝑐𝑏|𝜙𝑏⟩ +measurement +𝑈 +𝑈𝑠 +† +Estimate noisy eigenvalues with +matrix pencil method and compute +diagonal entries by Eq. (5) . +1. Uniformly at random sample 𝐾 +pairs of eigenstates 𝜙𝑎 , 𝜙𝑏 }. +2. For each pair of eigenstates, +do step 3. +4. Compute process fidelity and +stochastic fidelity by Eq. (13) and +Eq. (14) . +|0⟩ +𝑈 +𝐿 +prepare 𝑐𝑎 𝜙𝑎 + 𝑐𝑏|𝜙𝑏⟩ +measurement +|0⟩ +𝑈 +𝐿 +prepare 𝑐𝑎 𝜙𝑎 + 𝑐𝑏|𝜙𝑏⟩ +measurement +3. For each length 𝐿 in [0, 𝐿max], +run benchmarking circuits. +FIG. 1. The procedures of channel spectrum benchmarking. The benchmarking circuits are composed of three parts: the first +part Us prepares the initial state |ψ⟩ = ca|φa⟩ + cb|φb⟩, which is a superposition of two eigenstates of target gate U; then the +target gate U is repeated L times, where L is an integer in [0, Lmax]; finally, the operator O = |ψ⟩⟨ψ| is measured. +where Gab is the corresponding left eigen-operator of Mab +and they satisfy tr +� +G† +abMa′b′ +� += δab,a′b′. Then we can +show that the expectation value of O at length L under +noise is +� +�O +� +L = tr +� +�O �UL(�ρ) +� += +� +ab +tr +� +�OMab +� +tr +� +G† +ab�ρ +� +(gabeiλab)L +(10) +This is a damping oscillating function. From the time +series data +� +�O +� +L at different depth L, we can extract the +noisy eigenvalues via signal processing methods, such as +matrix pencil method [55–57]. +By selecting an appropriate initial state ρ and measure- +ment operator O, we can control the number of eigenval- +ues presented in the resulting signals. The presence of +too many different eigenvalues in the signals can pose +some difficulties. These include: (1) it may be difficult +to extract the eigenvalues from the measured signals, as +this may require more data or equivalently a larger depth +L (which is limited by the damping rate gab); (2) it can +be harder to identify the corresponding ideal eigenvalue +for a given noisy counterpart; (3) it can be difficult to +maintain a uniform sample of the diagonal entries of E. +To address these issues, we prepare the initial state and +measurement operator as follows: +|ψ⟩ = ca|φa⟩ + cb|φb⟩ +ρ = O = |ψ⟩⟨ψ| +(11) +which are superposition of two eigenvectors only. +For +this type of initial state and measurement operator, +only the corresponding noisy eigenvalues of these ideal +eigen-operators +{|φa⟩⟨φb|, |φb⟩⟨φa|, |φa⟩⟨φa|, |φb⟩⟨φb|} +(presented in the selected initial state) possess a major +portion in the measured signal. +Thus, as illustrated in Fig. 1, we propose the proce- +dures of channel spectrum benchmarking below. +1. Uniformly at random sample K pairs of eigenstates +{|φa⟩, |φb⟩} of target unitary operator U. +2. For each pair of eigenstates, do step 3, i.e., running +phase estimation circuits. +3. In phase estimation circuits, one first prepares the +initial state |ψ⟩ = ca|φa⟩ + cb|φb⟩, then repeatedly +apply the target gate U for L times where L takes +successive integers in [0, Lmax] , finally measure the +probability ⟨O⟩L of obtaining O = |ψ⟩⟨ψ|. +3a. Estimate the noisy eigenvalues gabeiλab (am- +plitudes and phases) from the time series data +� +�O +� +L by matrix pencil method. + +5 +3b. Identify the ideal counterparts of the mea- +sured noisy eigenvalues. +3c. Compute the diagonal entries of E by Eq. (5). +4. Compute the process fidelity by Eq. (13) and +stochastic fidelity by Eq. (14). +Step 1 ensures the estimated diagonal entries are uni- +form samples. We require the amplitude of two coeffi- +cients ca, cb are comparable and the initial state |ψ⟩ can +be efficiently prepared. +In the simulated experiments, +we always choose ca = cb = +1 +√ +2. The number of initial +states K is independent of system dimension d and only +depends on desired precision referring to Eq. (7), which +is guaranteed by Hoeffding’s inequality. So our method +has the potential scalable to many-qubit systems. +In the phase estimation circuits of step 3, we choose +the length L from [0, Lmax]. The maximum length Lmax +and the number of initial states K determine the total +number of benchmarking circuits Nc = K(Lmax + 1). +When combined with the number of shots Ns needed to +run each circuit to collect statistic, the total experimen- +tal cost is NcNs = K(Lmax + 1)Ns. The choice of Lmax +and Ns also depend only on desired precision and not on +the system dimension. Previous work has shown that the +uncertainty of estimated eigenvalues is inversely propor- +tional to the length L, the so-called Heisenberg scaling +[40, 43]. Therefore, if higher precision is desired, it is gen- +erally better to increase Lmax rather than the number of +shots Ns per circuit, before the signals are completely +degraded. +In step 3a, the noisy eigenvalues are estimated using +the matrix pencil method [55–57]. This method is well- +suited for this task because it involves a singular value +decomposition of the data Hankel matrix, which allows +us to keep only the components with non-trivial singu- +lar values, i.e., those caused by noisy eigenvalues of ideal +eigen-operators shown in initial state. This process can +reduce sampling error and eliminate unwanted eigenval- +ues due to SPAM errors or the effect of degenerate eigen- +values (for both the phase and the amplitude) of ideal +channel. In our simulated experiments, when using an +initial state with unequal phases λa, λb, the number of +obtained noisy eigenvalues is at most four. +In step 3b, our goal is to match the obtained noisy +eigenvalues from matrix pencil method to their corre- +sponding ideal counterparts such that we can compute +the diagonal entries of E by Eq. (5). For a initial state +whose two eigenstates |φa⟩, |φb⟩ have equal eigenvalues, +this process is not needed because all ideal channel eigen- +values are 1. However, for a initial state with unequal +two eigenvalues, there are three ideal channel eigenvalues +{ei(λa−λb), e−i(λa−λb), 1}. To match the obtained noisy +eigenvalues to the three ideal ones, we calculate the dis- +tance between the phases of the estimated noisy eigen- +values and the ideal phase λa − λb of the eigenvalue of +the particular eigen-operator |φa⟩⟨φb|. The noisy eigen- +value with the smallest distance is chosen as the noisy +counterpart of the ideal eigenvalue ei(λa−λb). Similarly, +the noisy counterpart of e−i(λa−λb) is also determined. +The remaining noisy eigenvalues are considered as the +counterparts of the ideal eigenvalue 1. +This criterion +assumes that the magnitude of the actual phase error +δλ = λab − (λa − λb) is small, more precisely we require +|δλ| ≪ |λa − λb|. +(12) +After calculating the diagonal entries using Eq. (5), +we divide them into two categories based on the ideal +eigenvalue of the associated basis |φa⟩⟨φb|: one is the +trivial operator subspace with λa = λb (or say the opera- +tor subspace spanned by the eigen-operators with eigen- +value 1), the other is the non-trivial operator subspace +with λa ̸= λb. +Because the dimension of trivial sub- +space dts is usually very different from the dimension of +non-trivial subspace dns, the probability of sampling an +entry in the two subspace are very different. For exam- +ple, for a many-qubit gate U with non-degenerate oper- +ator spectrum, the trivial subspace is spanned by all the +eigen-operators with the form |φa⟩⟨φa|, whose dimension +dts = d is much smaller than dns = d2 − d. If there are +some degeneracy in the spectrum of the operator U, that +is λa = λb for two different eigenstates |φa⟩, |φb⟩, the +trivial subspace can include the eigen-operators of the +form |φa⟩⟨φb|. In step 1, we assign the same probability +for the two subspaces. Therefore, we need to separately +compute the average of the diagonal entries in the triv- +ial and non-trivial subspaces. Finally, the estimator of +the process fidelity is obtained by combining these two +averages, that is +ˆF = dts Eab,ab|λa=λb + dns Eab,ab|λa̸=λb +d2 +(13) +where Eab,ab is the average value of sampled entries. Sim- +ilarly, the estimator for stochastic fidelity is +ˆFsto = +� +dts g2 +ab,ab|λa=λb + dns g2 +ab,ab|λa̸=λb +d2 +. +(14) +IV. +SIMULATED EXPERIMENTS +In this section, we perform simulated experiments to +show the performance of our CSB protocol, including +single-qubit Pauli rotation gates, two-qubit Fermionic +simulation (Fsim) gates, three-qubit Toffoli gate, and +an Ising Hamiltonian evolution operator with 10 qubits. +Throughout this work, each benchmarking circuit is re- +peated Ns = 104 times to collect enough statistic. +A. +Single-qubit Pauli rotation gates +Here we measure the fidelity of single-qubit rotation +gates, that is +Rσ(θ) = e−i θ +2 σ +(15) + +6 +10 +3 +10 +2 +10 +1 +Probability of stochastic error p +10 +3 +10 +2 +Infidelity +(a) +Act. proc. infidelity +Act. stoch. infidelity +Est. proc. infidelity +Est. stoch. infidelity +0.02 +0.01 +0.00 +0.01 +0.02 +Angle error +Act. angle error +Est. angle error +10 +3 +10 +2 +10 +1 +Angle of unitary error +10 +3 +6 × 10 +4 +2 × 10 +3 +3 × 10 +3 +Infidelity +(b) +10 +3 +10 +2 +10 +1 +Angle error +FIG. 2. +Benchmarking of T gate. +In (a), we fix the uni- +tary error (δθ = −0.01) and vary the probability of stochastic +error. In (b), we fix the stochastic error (δp = 0.001) and +vary the angle of unitary error. The actual process infidelity +and stochastic infidelity is obtained by first computing the +channel of noisy gate and then using Eq. (3) and Eq. (8). In +both cases, we accurately estimate process infidelity, stochas- +tic infidelity and the angle of unitary error. The accuracy of +estimation can be further improved by increasing the circuit +length or shots for each circuit. +where θ is the rotational angle and σ is a Pauli matrix +describing the direction of the rotational axis. This type +of unitary operator has two eigenvalues e−i θ +2 and ei θ +2 . +The dimension of the trivial eigen-operator subspace is +2, which is the same as the dimension of the non-trivial +eigen-operator subspace. Our choice of initial state only +generates one operator 1 +2(|φa⟩⟨φa| + |φb⟩⟨φb|) in the triv- +ial subspace, which means that we may only obtain one +noisy eigenvalue in this subspace, potentially leading to +an inaccurate estimation of the process fidelity. To ad- +dress this issue, we also prepare another initial state to +run phase estimation circuits, that is one of the eigen- +states of Rσ(θ) in addition to the superposition state. +This results in a value of K = 2. At the same circuit +length, we sum the measured probabilities of the two +types of circuits with the two initial states, allowing us +to extract all the noisy eigenvalues simultaneously. +Fig. 2 shows the results of benchmarking RZ( π +4 ) (also +known as T gate). In this simulation, the noise model +consists of a combination of stochastic errors (includ- +ing T1 and T2 errors with equal probabilities δp) and +over/under-rotation errors with angle δθ. In Fig. 2(a), we +fix the unitary error (δθ = −0.01) and vary the probabil- +ity of stochastic error. In Fig. 2(b), we fix the stochastic +error (δp = 0.001) and vary the angle of unitary error. In +both cases, we are able to accurately estimate the pro- +cess fidelity of the gate. +As a byproduct, we can also +estimate the angle of the unitary error by comparing the +phases of some noisy eigenvalues to their corresponding +ideal values. This is a more sensitive probe of unitary +errors than infidelity, as shown in Fig. 2(b), where the +process infidelity remains almost unchanged when δθ is +varied from 10−3 to 10−2. +In this simulation, we set Lmax = 50, except when +stochastic probability δp = 10−3, where Lmax = 100. It +is worth noting that the accuracy of the estimation can be +further improved by increasing the length of the bench- +marking circuits. However, increasing Lmax directly also +increases the number of circuits used, which leads to +higher costs. Instead, we can repeat the target gate U a +certain number of times (Nrep times) to create a new tar- +get gate, U ′ = U Nrep. Correspondingly, the noisy eigen- +value we estimate becomes (gabeiλab)Nrep. But remember +we need to determine the ideal eigenvalue from phase +difference, thus as a result of Eq. (12), we require +Nrep|δλ| ≪ |λa − λb|. +(16) +B. +Two-qubit Fsim gates +Here, +we +benchmark +the +two-qubit +fermionic- +simulation (Fsim) gates [8], i.e., +Fsim(θ, φ) = +� +�� +1 +0 +0 +0 +0 +cos θ +−i sin θ +0 +0 −i sin θ +cos θ +0 +0 +0 +0 +eiφ +� +�� +(17) +where θ is the iswap angle and φ is the control phase +angle. We omit some phase parameters that can be freely +adjusted by Z rotations. +For the preparation of initial states, we consider all +pairs of eigenstates (K = 6). The choice of Lmax is 50 +or 100 (for δp = 10−3). +In this simulation, the noise +model includes T1, T2 noise with equal probabilities δp +for all single-qubit gates. For two-qubit gates, each qubit +experiences the same errors as single-qubit gates, as well +as an over-rotation unitary error with angle errors δθ and +δφ. +We benchmark a specific Fsim gates with θ = π +4 , φ = π +2 , +as shown in Fig. 3. In Fig. 3(a), we fix the unitary error +with δθ = −0.01, δφ = −0.02 and vary the probability of +stochastic error δp. We accurately estimate all infidelities +in this case. However, the estimation of the angle of the +unitary error becomes less accurate when the stochastic +error is too strong, as the signal decays too quickly to +accumulate enough information to estimate the angle. +In Fig. 3(b), we fix the probability of stochastic error +with δp = 0.001 and vary the angles of unitary error +with δθ = 0.5δφ = 10−3 ∼ 10−1. Again, we accurately +estimate all infidelities and angles of the unitary error. + +7 +10 +3 +10 +2 +10 +1 +Probability of stochastic error p +10 +3 +10 +2 +10 +1 +Infidelity +(a) +Act. proc. infidelity +Act. stoch. infidelity +Est. proc. infidelity +Est. stoch. infidelity +0.04 +0.03 +0.02 +0.01 +0.00 +Angle error +Act. error +Act. error +Est. error +Est. error +10 +3 +10 +2 +10 +1 +Angle of unitary error +10 +3 +10 +2 +Infidelity +(b) +10 +3 +10 +2 +10 +1 +Angle error +FIG. 3. Benchmarking of a Fsim gate with θ = +π +4 , φ = +π +2 . +In (a), we fix the unitary error with δθ = −0.01, δφ = −0.02 +and vary the probability of stochastic error δp. In Fig. 3(b), +we fix the probability of stochastic error with δp = 0.001 and +vary the angles of unitary error with δθ = 0.5δφ = 10−3 ∼ +10−1. +We always accurately estimate the process infidelity +and the stochastic infidelity of the gate. But, the accuracy +of estimating the angles of the unitary error is compromised +when there is a high level of stochastic noise, as the signal +degrades quickly and there is not enough data to accurately +estimate the angles. +C. +Three-qubit Toffoli gate +In this study, we evaluate the performance of the three- +qubit Toffoli gate, which is not a native gate but rather +a circuit fragment composed of 1-qubit and 2-qubit gates +as shown in Fig. 4(c). We randomly select K = 10 pairs +of eigenstates as the initial state and set Lmax = 50. +In the simulated noise model, all single-qubit gates are +subject to T1, T2 noise with equal probability δp. For the +two-qubit gates, each qubit experiences the same type of +stochastic error as the single-qubit gates, followed by a +unitary error of the Fsim type with error angles δθ = δφ. +The Toffoli operator has a highly degenerate spectrum, +which creates two challenges for our method. First, when +sampling noisy eigen-operators, we need them to be uni- +formly distributed, but for degenerate eigenvalues, the +noisy eigen-operators are superpositions of ideal ones in +the degenerate subspace, which are determined by the de- +tails of the noise, see Appendix A. This makes it difficult +to generate a uniform sample of noisy eigen-operators. +Second, the degenerate eigenvalue may be split by noise +into many eigenvalues in the signal, making it harder to +extract the noisy eigenvalues and each eigenvalue may +only occupy a small portion of the signal, making them +0.002 +0.004 +0.006 +0.008 +0.010 +Probability of stochastic error p +0.00 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +Infidelity +Act. proc. infidelity +Act. stoch. infidelity +Est. proc. infidelity +Est. stoch. infidelity +Est. proc. infidelity (varied circ) +Est. stoch. infidelity (varied circ) +0.02 +0.04 +0.06 +0.08 +0.10 +Angle of unitary error +0.02 +0.03 +0.04 +0.05 +0.06 +0.07 +0.08 +0.09 +0.10 +Infidelity +Est. proc. infidelity (varied circ; RC) +Est. stoch. infidelity (varied circ; RC) +(a) +(b) +(c) +q0 +q1 +q2 +0, 0, 0 +U3 +0, 0, 0 +U3 +/2, 0, +U3 +0, 0, 0 +U3 +0, 0, 0 +U3 +0, 0, +/4 +U3 +0, 0, 0 +U3 +0, 0, 0 +U3 +0, 0, /4 +U3 +0, 0, 0 +U3 +0, 0, 0 +U3 +0, 0, +/4 +U3 +0, 0, 0 +U3 +0, 0, +/4 +U3 +0, 0, /4 +U3 +0, 0, 0 +U3 +0, 0, +/4 +U3 +/2, 0, +U3 +0, 0, /4 +U3 +0, 0, /2 +U3 +0, 0, 0 +U3 +FIG. 4. Bechmarking of Toffoli circuit fragment. We fix the +unitary error (δθ = 0.01) and vary stochastic error in (a), and +fix stochastic error (δp = 0.001) and vary unitary error in +(b). The circuit implementing Toffoli gate is presented in (c). +Due to the highly degenerate spectrum of the Toffoli gate, the +estimate of the infidelity is unreliable. However, the degener- +acy can be removed by changing the last layer of single-qubit +gates. With the varied circuit, we accurately estimate the in- +fidelity of the Toffoli circuit under weak unitary error in (a). +For strong unitary error, we perform randomized compiling to +the benchmarking circuits, converting the unitary error into +stochastic error. As a result, the varied circuit also accurately +estimates the process infidelity of Toffoli circuit under strong +unitary error, as shown in (b). +more susceptible to errors. The impact of the highly de- +generate spectrum on the estimate of gate noise is demon- +strated by the simulated results in Fig. 4(a),(b). +Usually, some of degeneracy can be removed by ap- +pending a layer of single-qubit gates to the target gate +or circuit fragment. For the Toffoli circuit, we append +RZ( π +2 )⊗RZ( 2π +3 )⊗RX( 4π +5 ) to the Toffoli circuit and com- +bine this layer with the last layer of the Toffoli circuit. +The choice of appended layer should keep the state prepa- +ration of the new target gate efficient. Here our choice +does not change the eigenstates. For the angle parame- +ters in the appended gates, one can design an optimiza- +tion algorithm to choose the parameters that maximize +the distance between eigenvalues. The appended layer of +gates results in a varied circuit with a similar structure to + +8 +0.002 +0.004 +0.006 +0.008 +0.010 +Probability of stochastic error p +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +0.35 +0.40 +0.45 +Infidelity +Act. proc. infidelity +Est. proc. infidelity +Est. stoch. infidelity +0.02 +0.04 +0.06 +0.08 +0.10 +Angle of unitary error +0.06 +0.08 +0.10 +0.12 +0.14 +0.16 +0.18 +0.20 +Infidelity +Act. proc. infidelity +Est. proc. infidelity +Est. stoch. infidelity +Est. proc. infidelity (RC) +Est. stoch. infidelity (RC) +(a) +(b) +(c) q0 +q1 +q2 +q3 +q4 +q5 +q6 +q7 +q8 +q9 +0, 0, +0.628 +U3 +0, 0, 0.929 +U3 +0, 0, +1.17 +U3 +0, 0, 0.626 +U3 +0, 0, 1.79 +U3 +0, 0, +0.745 +U3 +0, 0, 0.487 +U3 +0, 0, 0.742 +U3 +0, 0, +0.758 +U3 +0, 0, +1.68 +U3 +0, 0, 0 +U3 +0, 0, 1.4 +U3 +0, 0, 0 +U3 +0, 0, 0.198 +U3 +0, 0, 0 +U3 +0, 0, 1.25 +U3 +0, 0, 0 +U3 +0, 0, +1.05 +U3 +0, 0, 0 +U3 +0, 0, +0.0943 +U3 +0, 0, 0 +U3 +0, 0, 1.01 +U3 +0, 0, 0 +U3 +0, 0, 1.03 +U3 +0, 0, 0 +U3 +0, 0, +1.93 +U3 +0, 0, 0 +U3 +0, 0, 0.222 +U3 +0, 0, 0 +U3 +0, 0, +0.393 +U3 +FIG. 5. Benchmarking of a 10-qubit Ising evolution operator. We fix unitary error (δθ = 0.01) and vary stochastic error in +(a), and fix stochastic error (δp = 0.001) and vary unitary error in (b). The circuit implementing Ising evolution operator is +presented in (c). The actual fidelity is not computed from the channel of the circuit, but rather inferred from the product of +the fidelity of all single-qubit and two-qubit gates. However, the actual stochastic infidelity can not be reliably inferred by this +procedure. We accurately estimate infidelity of the Ising evolution operator under weak unitary error (a) and strong unitary +error with RC (b). +the original Toffoli circuit (only the last layer is changed) +and they should possess similar noise properties. In the +case of strong stochastic error and weak unitary error +(δθ = 0.01) in Fig. 4(a), the benchmarking of the varied +circuit provides a very accurate estimate of the process +infidelity and the stochastic infidelity of the original Tof- +foli circuit. +However, there is a significant difference between the +estimated and actual process infidelity when the unitary +error is very strong, as shown in Fig. 4(b) (with fixed +stochastic error δp = 0.001). +In the Appendix B, we +show that our method may under-estimate the process +infidelity in the presence of certain strong unitary errors. +One way to address this issue is to introduce random +gates into the benchmarking circuits to convert the uni- +tary errors to stochastic errors [37, 38, 58]. +The Ap- +pendix C describes a procedure for transforming noise in +the native gates to stochastic errors using random gates +from the symmetry group of the target U. For bench- +marking circuit fragments, we use a technique called ran- +domized compiling [37, 38] to achieve this. Randomized +compiling (RC) is a method that transforms the noise +in the circuit into stochastic Pauli errors while maintain- +ing the circuit structure and depth. After RC, the noise +type of a circuit cycle is changed, but the fidelity of the +cycle and the circuit structure remains unchanged. As +long as there is no repeated structure in U where unitary +error can coherently build up and increase the infidelity +quadratically with the circuit depth [59] (this is a case +where RC should be introduced to suppress the unitary +noise), we expect the fidelity of the circuit U to remain +unchanged after RC. For each original circuit, we gener- +ate Nr = 10 random circuits by RC and each random +circuit is run 103 times to keep the cost unchanged. As +shown in Fig. 4(b), after RC the varied circuit can ac- +curately estimate the process infidelity of Toffoli circuit +under unitary noise. +D. +Ten-qubit Ising evolution operator +Our method is practically scalable if the following two +requirements are met: +1. The eigenvalues and eigenvectors of target unitary +operator U can be efficiently computed. +2. The initial state can be efficiently prepared, i.e., the +number of 1-qubit and 2-qubit gates needed for the +preparation should at most scale polynomial with +the number of qubits. +In general, these two requirements are not always sat- +isfied. However, for certain types of unitary operators, +such as the evolution operator of an Ising Hamiltonian, +these requirements can be met. For an Ising Hamiltonian, +the eigenvectors are known and are simply the computa- +tional basis states. Given an eigenstate, the eigenvalue +can be efficiently computed. +The initial state of a superposition of two computa- +tional basis states |x⟩ = |x0, · · · , xi, · · · , xN−1⟩, |y⟩ = +|y0, · · · , yi, · · · , yN−1⟩ can be prepared as follows: first, +for the qubit i, if xi = yi, the state can be prepared +by an X gate if xi = yi = 1; then, for the state of re- +maining qubits with xi ̸= yi, if we only have one such +qubit, a Hadamard gate H can be applied; if there is +more than one qubit with xi ̸= yi, one can first prepare +a GHZ state on these qubits and then apply some X +gates to obtain the target state. Therefore, the prepa- +ration of such states cost at most N 1-qubit and N 2- +qubit gates. Additionally, for the evolution operator of +the Hamiltonian that can be obtained by performing lo- +cal unitary transformation on an Ising Hamiltonian, i.e., +H = � +i UiHIsing +� +i U † +i , the initial states can also be +obtained in the similar way with additional two layers +of single-qubit gates � Ui, � U † +i . +Thus, this type of +evolution operators is a good example for benchmarking +many-qubit quantum systems. + +9 +Here we benchmark the evolution operator of a 1- +dimensional Ising ring H = �10 +i=1 hiZi + Ji,i+1ZiZi+1, +where hi, Ji,i+1 are randomly chosen. +The circuit is +shown as in Fig. 5(c). We sample K = 10 pairs of eigen- +states and set Lmax = 50. The noise model is the same as +the case in Sec. IV C. The actual infidelity is inferred from +the infidelity of single-qubit and two-qubit gates, because +our computer is not powerful enough to compute the +quantum channel of a 10-qubit circuit. Our method ac- +curately estimates process infidelity under both weak and +strong unitary error (with RC), as shown in Fig. 5(a),(b). +V. +CONCLUSION AND OUTLOOK +In this work, we introduced a procedure called chan- +nel spectrum benchmarking, which infers the noise prop- +erties of a quantum gate from the eigenvalues of noisy +channel representing the gate. In the protocol, we first +choose the initial state using a superposition of randomly +sampled pair of eigenstates of the target gate, and then, +we use control-free phase estimation circuits to estimate +the noisy eigenvalues in a SPAM error-resistant manner. +This choice of initial state simplifies the data processing +because the measured signal only contains a few eigen- +values, which can be extracted using signal processing +methods such as the matrix pencil method. By compar- +ing the noisy eigenvalues to their ideal counterparts, we +can estimate noise properties such as the process infi- +delity, stochastic infidelity, and some over-rotation angle +errors. Our method can be applied to any quantum gate, +but performs better on gates with non-degenerate oper- +ator spectrum. For gates with highly degenerate spec- +trum, we can append a layer of single-qubit gates to re- +move the degeneracy while maintaining a similar circuit +structure. Some types of unitary error can also affect the +performance, which can be addressed using randomiza- +tion techniques like randomized compiling. Our method +is scalable to many-qubit systems as long as the eigen- +decomposition can be computed and the initial state can +be efficiently prepared, such as the evolution operator of +an Ising-type Hamiltonian. +The requirements for the scalability of our method +could be relaxed. In principle, we do not need to obtain +the complete set of the eigenmodes for the target gate +operator, a few samples of eigenvalues and eigenstates +are sufficient. For initial state preparation, there are ex- +isting methods for preparing arbitrary states [60–63], but +it would be interesting to develop a more efficient algo- +rithm for preparing the particular type of initial states +in our method. A variational algorithm [64] may be able +to efficiently prepare these states for most target gates, +because we have the freedom to choose the coefficients of +the superposition states and do not need perfect prepa- +ration. Our method can be scaled up in a way similar to +simultaneous randomized benchmarking [65, 66], where +some few-qubit gates are simultaneously benchmarked on +different subsets of a many-qubit system such that the ef- +fect of crosstalk [67] can be detected. +One immediate use of benchmarking is to calibrate +quantum gates using the measured figures of merit as a +cost function [8–10]. Our method can provide more spe- +cific information (process infidelity, stochastic infidelity +and over-rotation angle of the target gate) about the cal- +ibrating gate, so it is expected to perform better on this +task than other benchmarking methods. A detailed com- +parison of different benchmarks for calibration will be a +topic for future research. Additionally, our method can +be used to calibrate universal gates, including not only 1 +or 2-qubit native gates, but also many-qubit native gates +such as MS gates [68, 69] used in ion trap systems. It +may also be interesting to use our method to calibrate +certain circuit fragments that are commonly used in al- +gorithms, such as the trotterized Hamiltonian evolution +operator in quantum simulation and the Grover iteration +operator in Grover’s search algorithm. +ACKNOWLEDGMENTS +The work is supported by the National Natural Sci- +ence Foundation of China (Grant No. +12147123 and +11974198) and Beijing Natural Science Foundation (No. +Z220002). +Source code for the simulated experiments +is available at this site https://github.com/yanwu-gu/ +channel-spectrum-benchmarking. +Appendix A: The relationship between noisy and +ideal eigenvalues of quantum channels +In this section, we derive the relationship between the +noisy channel eigenvalues of a gate and its correspond- +ing ideal counterparts with the first order perturbation +theory. +Consider a gate U acting on a d-dimensional space with +eigen-decomposition U|φa⟩ = eiλa|φa⟩. As the actual im- +plementation of a gate U is inevitably associated with +some noise, it is more convenient to use quantum chan- +nels rather than quantum operators. Quantum channels +are completely-positive trace-preserving (CPTP) maps, +which transform one operator to another. The action of +a quantum channel E on an arbitrary operator O can +be characterized by a set of Kraus operators Ek, i.e., +E(O) = � +k EkOE† +k. We denote the corresponding uni- +tary channel of the unitary operator U as U, whose action +on an operator O is U(O) = UOU †. Thus, the unitary +channel U has the eigen-decomposition +U(|φa⟩⟨φb|) = U|φa⟩⟨φb|U † = ei(λa−λb)|φa⟩⟨φb|. +(A1) +Quantum channels are linear maps that can be repre- +sented as matrices under a set of the basis operators +of the operator space, such as eigen-operators |φa⟩⟨φb|. +Meanwhile, operators are represented as vectors. The as- +sociated inner product between two operators A and B is + +10 +the Hilbert-Schmidt inner product tr +� +A†B +� +. Therefore, +in this representation U is a unitary matrix. +Let us append a noise channel E to U, with the noisy +version of U denoted as �U = EU. We investigate the re- +lationship between the eigenvalues of �U and those of U. +If the noise is relatively weak, the problem is an eigen- +value perturbation of unitary matrix. Given the close re- +lationship between unitary and Hermitian matrices, one +can use Hermitian matrix perturbation theory to get the +correction of eigenvalues and eigenstates, assuming a di- +agonalizable noisy gate �U. In most cases, the assumption +should be met in actual devices, since diagonalizable ma- +trices are dense in the space of all matrices, meaning that +any non-diagonalizable matrix can be deformed into a di- +agonalizable one by a small perturbation. In the follow- +ing, we apply Hermitian perturbation theory to obtain +the first order correction of the eigenvalues, and obtain +the relationship between the noisy eigenvalues and ideal +ones. +Define the eigenvalues and eigen-operators of �U as +gabeiλab and Mab, that is +�U(Mab) = gabeiλabMab . +(A2) +The perturbation matrix is +∆ = �U − U = (E − I) U . +(A3) +We assume the perturbation is small in terms of some +norm, such as the diamond norm ∥∆∥⋄ = δ [46]. Then, +for a non-degenerate eigenvalue ei(λa−λb) with eigen- +operator |φa⟩⟨φb|, the first order correction is +ϵab +1 = tr +� +(|φa⟩⟨φb|)†∆(|φa⟩⟨φb|) +� += tr +� +(|φa⟩⟨φb|)†(E − I) U(|φa⟩⟨φb|) +� += ei(λa−λb) � +tr +� +(|φa⟩⟨φb|)†E(|φa⟩⟨φb|) +� +− 1 +� +.(A4) +Thus, the noisy eigenvalue is approximated as +gabeiλab ≈ ei(λa−λb) + ϵab +1 += ei(λa−λb)tr +� +(|φa⟩⟨φb|)†E(|φa⟩⟨φb|) +� +. (A5) +In this case, the noisy eigen-operator Mab is approxi- +mated by +Mab ≈ |φa⟩⟨φb| + O(δ) +(A6) +where O(δ) are some correction terms with the first order +of δ. +For degenerate eigenvalues eiλn with eigen-operators +|φa⟩⟨φb| satisfying λa − λb = λn, these eigen-operators +span a subspace. The ab, a′b′-entry of the perturbation +matrix projected in this subspace is +∆(n) +ab,a′b′ = tr +� +(|φa⟩⟨φb|)†(E − I) U(|φa′⟩⟨φb′|) +� += eiλn � +tr +� +(|φa⟩⟨φb|)†E(|φa′⟩⟨φb′|) +� +− δaa′δbb′� += eiλn(E(n) +ab,a′b′ − I(n) +ab,a′b′) +(A7) +where E(n) +ab,a′b′ and I(n) +ab,a′b′ are the entries of pure noise +map E and Identity map I projected in this degenerate +subspace. In the degenerate case, the first order correc- +tions to the eigenvalue eiλn of U are the eigenvalues of +the perturbation matrix ∆(n). It’s easy to find that the +matrix ∆(n) and the matrix E(n) have the same eigen- +operators M 0 +pq, which are the superposition of eigen- +operators |φa⟩⟨φb| in this degenerate subspace. They are +also the corresponding unperturbed eigen-operators of +noisy eigen-operator Mpq of �U, that is Mpq ≈ M 0 +pq+O(δ). +In this case, the eigenvalue gpqeiλpq of �U is +gpqeiλpq ≈ eiλn + tr +� +G0† +pq∆(n)(M 0 +pq) +� += eiλn + eiλn � +tr +� +G0† +pqE(n)(M 0 +pq) +� +− 1 +� += eiλntr +� +G0† +pqE(n)(M 0 +pq) +� +(A8) +where G0 +pq is the corresponding left eigen-operator of M 0 +pq +and they satisfy tr +� +G0† +pqM 0 +p′q′ +� += δpq,p′q′. Therefore, the +Eq. (A8) has the same form as Eq. (A5) but with a basis +of a different form. +Appendix B: Perturbation of channel eigenvalues +under pure unitary error +Here we consider the noisy eigenvalues of a quantum +gate under a pure unitary error +V = e−iHeδ +(B1) +where He is the Hamiltonian of error and δ characterize +the error strength. Assume the target gate U = e−iHθ. +Thus the operator of noisy gate is �U = V U. In this case, +the process fidelity is +F = |tr{V }|2 +d2 +≈ |tr +� +I − iHeδ − 1 +2H2 +e )δ2� +|2 +d2 += d2 − dtr +� +H2 +e +� +δ2 +d2 += 1 − tr +� +H2 +e +� +d +δ2 +(B2) +where we use the property tr{He} = 0 and keep the term +up to O(δ2). This is a well-known result that unitary +error with some matrix norm δ has process infidelity of +order O(δ2) [46]. +We then analyze how eigenvalues of a quantum chan- +nel U change under such unitary error with perturbation +theory. Now the effect of the noisy gate �U on an oper- +ator O is �U(O) = V UOU †V †. Represent all quantum +channels as matrices, the perturbation matrix is +∆ = �U − U . +(B3) + +11 +For a non-degenerate eigen-operator |φa⟩⟨φb| with eigen- +value ei(λa−λb), the first order correction is +ϵab +1 = tr +� +(|φa⟩⟨φb|)†∆(|φa⟩⟨φb|) +� += ei(λa−λb) � +tr +� +(|φa⟩⟨φb|)†V |φa⟩⟨φb|V †� +− 1 +� +. +(B4) +To further expand this equation, we will use the Baker- +Hausdorff (BH) lemma +eXY e−X = eadX(Y ) = +∞ +� +n=0 +adn +X(Y ) +n! +(B5) +where ad is a map on operators with the effect adX(Y ) = +[X, Y ]. Then the first order correction is +ϵab +1 = ei(λa−λb) � +tr +� +(|φa⟩⟨φb|)†e−iδadHe (|φa⟩⟨φb|) +� +− 1 +� +≈ ei(λa−λb) +� +tr +� +(|φa⟩⟨φb|)† +� +I − iδadHe − 1 +2δ2ad2 +He +� +(|φa⟩⟨φb|) +� +− 1 +� += ei(λa−λb) +� +����−iδ (⟨φa|He|φa⟩ − ⟨φb|He|φb⟩) +� +�� +� +ϵab +1,1 +−1 +2δ2tr +� +(|φa⟩⟨φb|)†ad2 +He(|φa⟩⟨φb|) +� +� +�� +� +ϵab +1,2 +� +���� . +(B6) +If we consider only the first-order correction, the noisy +eigenvalue is +gabeiλab ≈ ei(λa−λb) + ϵab +1 = ei(λa−λb)(1 + ϵab +1,1 + ϵab +1,2). +(B7) +From Eq. (3) and Eq. (5), we get the estimate of process +fidelity +ˆF = 1 + 1 +d2 +� +ab +ϵab +1,1 + 1 +d2 +� +ab +ϵab +1,2 +(B8) +where the term with order O(δ) is +1 +d2 +� +ab +ϵab +1,1 += 1 +d2 +� +ab +−iδ (⟨φa|He|φa⟩ − ⟨φb|He|φb⟩) += 0 +(B9) +and the term with order O(δ2) is +1 +d2 +� +ab +ϵab +1,2 += − δ2 +2d2 +� +ab +tr +� +(|φa⟩⟨φb|)†[He, [He, |φa⟩⟨φb|]] +� += − δ2 +2d2 +� +ab +⟨φa|H2 +e |φa⟩ − 2⟨φa|He|φa⟩⟨φb|He|φb⟩ + ⟨φb|H2 +e |φb⟩ += − δ2 +2d2 (2dtr +� +H2 +e +� +− 2tr{He}2) += −1 +dtr +� +H2 +e +� +δ2 . +(B10) +This coincides with the expression in Eq. (B2). +However, due to the first order correction only contributing a term with order O(δ2), we must also take into account +the second order correction to the eigenvalues. The second order correction is + +12 +ϵab +2 = +� +mn̸=ab +|tr +� +(|φm⟩⟨φn|)†∆(|φa⟩⟨φb|) +� +|2 +ei(λa−λb) − ei(λm−λn) += +� +mn̸=ab +|tr +� +(|φm⟩⟨φn|)†V |φa⟩⟨φb|V †� +|2 +ei(λa−λb) − ei(λm−λn) += +� +mn̸=ab +|⟨φm|V |φa⟩|2|⟨φn|V |φb⟩|2 +ei(λa−λb) − ei(λm−λn) +≈ +� +mn̸=ab +� +δam + (⟨φm|He|φa⟩⟨φa|He|φm⟩ − ⟨φm|H2 +e |φa⟩δam)δ2� � +δbn + (⟨φn|He|φb⟩⟨φb|He|φn⟩ − ⟨φn|H2 +e |φb⟩δbn)δ2� +ei(λa−λb) − ei(λm−λn) += +� +m̸=a +⟨φm|He|φa⟩⟨φa|He|φm⟩δ2 +ei(λa−λb) − ei(λm−λb) ++ +� +n̸=b +⟨φn|He|φb⟩⟨φb|He|φn⟩δ2 +ei(λa−λb) − ei(λa−λn) +. +(B11) +If the error Hamiltonian He is diagonal under the basis of +eigenvectors of U, the second order correction is ϵab +2 = 0 +up to the second order O(δ2). There is no problem for +our method. +However, except the special case, there is some dis- +crepancy between the process fidelity estimated using our +method and the actual value, due to the presence of the +term ϵab +2 in the noisy eigenvalue gabeiλab. +Here, we can directly compute the noisy eigenvalues of +the channel �U from the eigenvalues of the operator �U and +give the analytical form of estimated process fidelity by +our method. We first compute the Hamiltonian of �U by +Baker-Campbell-Hausdorff formula +H′ = log (V U) = log (e−iHeδe−iHθ) +≈ −iHθ − iHeδ − 1 +2ad−iHθ(−iHeδ) + 1 +12ad2 +−iHθ(−iHeδ) − +1 +720ad4 +−iHθ(−iHeδ) + · · · +(B12) +where we only keep the terms up to the order O(δ) and +omit some terms with the ad map. +We can compute +the eigenvalues of H′ comparing to those of the −iHθ +with the first order perturbation theory . The first order +correction to the eigenvalue iλa with eigenstate |φa⟩ is +ϵa +1 = ⟨φa|H′ − (−iHθ)|φa⟩ += −iδ⟨φa|He|φa⟩ +(B13) +where these terms with ad map are all zeros because +⟨φa|ad−iHθ(O)|φa⟩ += −iθ⟨φa|(HO − OH)|φa⟩ += iλa(⟨φa|O|φa⟩ − ⟨φa|O|φa⟩) = 0 +(B14) +where O is an any operator. Then the noisy eigenvalue +of |φa⟩⟨φb| is +gabeiλab ≈ eiλa+ϵa +1−iλb−ϵb +1 . +(B15) +Thus our estimator for process fidelity is +ˆF = 1 +d2 +� +ab +eϵa +1−ϵb +1 += 1 − +� +a⟨φa|He|φa⟩2 +d +δ2 . +(B16) +Because the term � +a⟨φa|He|φa⟩2 is always smaller than +the term tr +� +H2 +e +� +except when He is a diagonal matrix un- +der the basis |φa⟩, our method under-estimates process +infidelity under unitary error in general. This problem +can be fixed by introducing some randomization proce- +dure into the benchmarking circuits to convert unitary +errors to stochastic errors [37, 38, 58]. +Appendix C: Randomized compiling with the +symmetric group of the target gate +For a circuit composed of single-qubit and two-qubit +gates, randomized compiling (RC) is a standard proce- +dure to tailor the noise into stochastic Pauli noise with +Pauli twirling. Here, we consider another case that the +circuit is repetitions of a native gate U, that is U L. In +the spirit of RC, if considering U as hard gate, we need +a twirling group T, whose element Ti should be trans- +formed to another Tj under the conjugate operation of +U, that is UTiU † = Tj. +A simple example of this type of groups is a symmetric +group of U +T = {T : U †TU = T} . +(C1) + +13 +0.02 +0.04 +0.06 +0.08 +0.10 +Angle of unitary error +0.0005 +0.0010 +0.0015 +0.0020 +0.0025 +0.0030 +0.0035 +Infidelity +Act. proc. infidelity +Est. proc. infidelity (RC) +Est. stoch. infidelity (RC) +FIG. 6. Benchmarking of T gate with randomized compiling. +In simulation, stochastic error is fixed (δp = 0.001) and uni- +tary is RX(δθ) with varied error angle δθ. The twirling group +is T = {I, Z}. For each original circuit, we generate Nr = 10 +random circuits and each random circuit is run for Ns = 103 +times. +For the sequence of U L, random gates from the twirling +group T are introduced before each application of U, but +the effect of these random gates should be cancelled be- +fore the next U is applied. Finally, we get a new random +sequence +T † +LU(TLT † +L−1) · · · U(TiT † +i−1) · · · U(T2T † +1 )UT1 +(C2) +where the gates in parentheses should be implemented as +one gate. In actual implementation, all the gates should +be associated with a noise, and the gate sequence is de- +noted as composition of quantum channels +T † +LET EUTLT † +L−1ET · · · EUT2T † +1 ET EUT1ET += T † +LET ETLUT † +L−1ET · · · ET2UT † +1 ET ET1UET +(C3) +where we use the property that the gates in group T +commute with U. We assume the noise of each twirling +gate is the same quantum channel ET for simplicity, but +this assumption can be relaxed [37]. +After averaging many such random sequences we get +� +1 +NT +� +TL +T † +LET ETL +� +U · · · +� +1 +NT +� +T1 +T † +1 ET ET1 +� +UET +(C4) +where NT is the number of gates in the twirling group T. +The effect of this randomization procedure is to trans- +form the noise to stochastic noise, that is applying a +random quantum channel T †ET ET with probability +1 +NT . +One can use group representation theory to get a sim- +pler form of the noise. But in our case, this subtlety is +not necessary. This procedure is similar to the Ref. [58]. +However, we do not require the twirling group is abelian +and do not need the assumption that there is no equal ir- +reducible representation for symmetric group. Thus our +method has high flexibility to choose twirling group. +We use a simulated experiment to show the perfor- +mance of this procedure. We benchmark T gate under +a unitary error RX(δθ) with varied error angle δθ and +fixed stochastic error δp = 0.001. We choose T = {I, Z} +as twirling group. For each original circuit, we generate +Nr = 10 random circuits and each random circuit is run +for Ns = 103 times. +The theory in Appendix B shows that the process in- +fidelity estimated by our method is of the order O(δθ4) +without the use of randomized compiling. However, by +introducing randomized compiling, our method can ac- +curately estimate the process infidelity, as demonstrated +in Fig. 6. It is important to note that the process in- +fidelity measured using randomized compiling on native +gate includes the noise from both the target gate and the +twirling gates, since the twirling gates are not merged +into the original circuit in the same way as when using +randomized compiling on circuit fragments. For simplic- +ity, we did not add noise to the twirling gates in this +case. 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A 62, 022311 (2000). + diff --git a/dtA0T4oBgHgl3EQfG__M/content/tmp_files/load_file.txt b/dtA0T4oBgHgl3EQfG__M/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..2bf1eeeeffb5a64315eb2d8533c4b37474cd97e5 --- /dev/null +++ b/dtA0T4oBgHgl3EQfG__M/content/tmp_files/load_file.txt @@ -0,0 +1,1011 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf,len=1010 +page_content='Benchmarking universal quantum gates via channel spectrum Yanwu Gu,1, 2, ∗ Wei-Feng Zhuang,1 Xudan Chai,1, 2 and Dong E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Liu2, 1, 3, † 1Beijing Academy of Quantum Information Sciences, Beijing 100193, China 2State Key Laboratory of Low Dimensional Quantum Physics, Department of Physics, Tsinghua University, Beijing, 100084, China 3Frontier Science Center for Quantum Information, Beijing 100184, China (Dated: January 6, 2023) Noise remains the major obstacle to scalable quantum computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Quantum benchmarking methods provide key information on noise properties for quantum processor calibration, quantum error mitigation, and quantum error correction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' However, current benchmarking methods, such as randomized benchmarking or its variants, can only evaluate the performance of some particular subsets of quantum gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Moreover, due to the randomization inherent in these protocols, the figure of merit they actually measure is not the fidelity of individual target gate but the average of the fidelities of some random circuit cycles incorporating the target.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' To overcome these limits, we propose the channel spectrum benchmarking (CSB), a method to infer the noise properties of the target quantum process, such as process fidelity, from the eigenvalues of its noisy quantum channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The noisy eigenvalues can be estimated by the circuits of control-free phase estimation in a state-preparation and measurement error resilient manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Our method can benchmark uni- versal quantum processes and is scalable to many-qubit quantum processes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We demonstrate the performance of our method using simulated experiments, including the single-qubit Pauli rotations, 2-qubit fermionic simulation gates, a 3-qubit cycle implementing the Toffoli gate, and a 10-qubit cycle implementing the Ising Hamiltonian evolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Our method will pave an important way for the development of cleaner and large-scale quantum devices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' INTRODUCTION The performance of today’s quantum computers is severely affected by noise and the limited number of qubits [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Quantum error correction and fault-tolerant schemes may someday unlock the full potential of quan- tum computation [2–7], but more precise gate opera- tions must be developed beforehand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' It is crucial and necessary to obtain information on the gate noise char- acteristics and their performance benchmarks in order to calibrate and optimize these gate operations [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Nonetheless, there is a trade-off between noise informa- tion obtained and the resource overhead for their testing experiments [11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Process tomography [12, 13] is a typical technique for reconstructing the matrix representation of a quantum process, with which the full information of noise is at hand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' However, process tomography has expo- nentially increasing experimental costs and suffers from state-preparation and measurement (SPAM) errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Al- though its variant, the gate-set tomography [14–18], can handle SPAM errors, the experimental costs cannot be reduced.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In reality, for probing noise strength or noise types of a gate, the full reconstruction of the noisy process is not necessary [19, 20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For instance, the average gate fi- delity, that measures the average performance of the im- plemented noisy gates, can be efficiently obtained by ran- domized benchmarking (RB) [21–26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The RB protocol is insensitive to SPAM errors and its variants [8, 27–29] ∗ guyw@baqis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='cn † dongeliu@mail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='tsinghua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='edu.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='cn can be applied to benchmark devices with larger system size.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' It is important to note that protocols like random- ized benchmarking do not directly measure the fidelity of individual quantum gates, but rather the average fidelity of some random circuit fragments [30–32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' To determine the fidelity of a specific gate, additional techniques such as interleaved RB [33] or modifying the sampling distri- bution of random circuits [27, 29] must be used, which can induce more experimental cost and is prone to a large systematic uncertainty [34].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Additionally, to simplify the functional form of measured signals in RB methods, it is often necessary to use group twirling, which limits the types of gates that can be benchmarked.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' As a conse- quence, the RB protocols based on random Clifford cir- cuits can only be applied to benchmark the Clifford gates;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' however, the important non-Clifford gates have to rely on more complicated random circuit sets in which their native gates belong to other groups instead of Clifford group, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' dihedral groups [35, 36].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In order to overcome the two limitations, we intro- duce channel spectrum benchmarking (CSB), a scalable protocol to estimate the individual noise properties of a universal quantum process from the noisy eigenvalues of its corresponding quantum channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We estimate the noisy eigenvalues by control-free phase estimation cir- cuits [40–44] that is robust to SPAM errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' With a re- lationship between ideal and noisy eigenvalues, which is derived from the first order perturbation theory [45], we can infer the diagonal entries of the matrix of pure noise process under a basis composed of the eigen-operators of the ideal gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' From these diagonal entries, we can estimate some noise properties, for examples, process in- fidelity, stochastic infidelity (a quantity similar to uni- arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='02056v1 [quant-ph] 5 Jan 2023 2 Gates Fidelity Conditions for Scalablity CSB universal General case: target Strong unitary error and RC: target + twirling gates eigen-decomposition of target gate is possible initial state preparation is efficient Clifford RB [22, 23] Clifford ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' among Clifford gates not scalable due to compilation issue [27] Mirror RB [29] Clifford ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' among rand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' cycles only applicable to Clifford gates CB [28] U m = I target + twirling gates target gate is Clifford XEB [8] universal ave.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' among rand.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' cycles circuits can be classically simulated TABLE I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Comparison with other leading benchmarking protocols.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We compare our CSB protocol with other benchmarking protocols under three aspects: (1) what gates they can benchmark;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (2) what type of fidelity they actually measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (3) under what conditions they can be scalable to many-qubit systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Usually, our CSB measures the fidelity of the target gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' But for strong unitary error, we need to perform randomized compiling (RC) [37, 38] with twirling gates to convert unitary error to stochastic error in order to obtain a better performance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' When benchmarking a circuit fragment, the twirling gates can be merged into target gate, our method still measures the individual fidelity of the target as shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' IV C and IV D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' But, when benchmarking native gates, the twirling gates can not be merged (see Appendix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In this case, our method measures the average fidelity of the compositions of the target gate and twirling gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Our CSB is scalable as long as eigen-decomposition of target is possible and the number of single and two-qubit gates in the circuits preparing initial states scales at most polynomial with the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Clifford RB and Mirror RB use random Clifford circuits to simplify noise and thus only apply to Clifford gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The fidelity they actually measure is the average of fidelities among random Clifford cycles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Mirror RB can be scalable but Clifford RB cannot.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For cycle benchmarking (CB), the gate or cycle U that can be benchmarked must satisfy U m = I where m is an integer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' CB uses Pauli twirling to simplify noise and thus measures the fidelity of composition of target and twirling gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' It needs to compute the output Pauli operator of ideal circuits, which is possible only when the target gate is Clifford for large systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' XEB uses random universal circuits to simplify noise, so it measures the average of fidelities among some random circuit cycles generated with a same sampling distribution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' It requires the classical simulation of circuits to obtain the ideal probabilities of sampled bit strings, which limits its scalability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Additionally, our CSB and XEB can directly measure how close the noise is to unitary error, while RB methods need extra procedures to measure this information [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' tarity [39]) and over-rotation angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We demonstrate the performance of our method with some simulated exper- iments, 1-qubit Pauli rotaional gates, 2-qubit fermionic- simulation (Fsim) gates, 3-qubit circuit fragment imple- menting Toffoli gate, and 10-qubit circuit fragment im- plementing Ising evolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In all experiments, our CSB method can accurately estimate the noise prop- erties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' To get a more clear picture of the performance of our CSB, in Table I, we compare our CSB protocol with other leading benchmarking protocols under three aspects: (1) what gates they can benchmark;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (2) what type of fidelity they actually measure;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (3) under what conditions they can be scalable to many-qubit systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' II.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' QUANTUM CHANNEL, FIDELITY, AND CHANNEL SPECTRUM In this section, we provide some preliminaries about quantum channel, the fidelity of implemented noisy gates, and the relationship between the fidelity of a gate and the channel spectrum of its noisy implementation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Consider a quantum gate U acting on a d-dimensional space with eigenvalues eiλa and eigenstates |φa⟩ such that U|φa⟩ = eiλa|φa⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Because of noise, the actual imple- mentation of the gate should be denoted as a quantum channel �U = EU, or say completely-positive and trace- preserving (CPTP) map [12], where U is the correspond- ing quantum channel of the ideal gate U and E is a pure noise process.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Quantum channels are usually denoted by a set of Kraus operators, for example, U = UρU † and E(ρ) = � k EkρE† k where ρ is an arbitrary operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Quantum channels can also be represented by a matrix on the basis of d2 dimensional operator space, for ex- ample, Pauli operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We will use the two representa- tions interchangeably and the same symbols for both the abstract quantum channels and their matrix representa- tions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' One can use some fidelity measures to assess the per- formance of the implemented noisy gate �U, such as the process fidelity (or referred to as entanglement fidelity) which is defined as F(U, �U) = tr � I ⊗ U(|α⟩⟨α|) I ⊗ �U(|α⟩⟨α|) � (1) where |α⟩ = 1 √ d �d i=1 |i⟩⊗|i⟩ is the maximally entangled state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The process fidelity is closely related to another ubiquitous measure, the average gate fidelity [46] Fave(U, �U) = ˆ dψ tr � U(|ψ⟩⟨ψ|) �U(|ψ⟩⟨ψ|) � = dF + 1 d + 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (2) It has been proven that the process fidelity only depends on the trace of the pure noise E [46], that is F(U, �U) = tr � U† �U � d2 = tr{E} d2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (3) Current benchmarking methods, for example, random- ized benchmarking and its variants, measure the infor- mation of tr{E} on a basis composed of Pauli operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 3 In these protocols, Clifford twirling or Pauli twirling are used to simplify the noise matrix E, that is, only diag- onal entries of E on the Pauli basis are kept, such that the relevant figure of merit can be extracted easily from measured signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The twirling operations need to be per- formed by running some random circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' This causes RB type of methods only apply to some subsets of quantum gates and only measure the average fidelity of a set of gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Distracting from the Pauli operator basis, one can note that the ideal channel U also induces a natural opera- tor basis composed of its eigen-operators |φa⟩⟨φb| (corre- sponding eigenvalues are ei(λa−λb)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' If we can measure the diagonal entries of noise E in this basis, we can also estimate the gate fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' This can be achievable by a relationship between the eigenvalues of noisy gate �U and those of ideal gate U [45], that is gabeiλab ≈ ei(λa−λb)tr � (|φa⟩⟨φb|)†E(|φa⟩⟨φb|) � (4) where gab and λab is the amplitude and phase of an eigen- value of �U with eigen-operator Mab, that is �U(Mab) = gabeiλabMab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For the spectrum of quantum channels, there are some useful properties [47]: (1) the eigenval- ues lie in the unit disc of complex plain, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=', 0 ≤ gab ≤ 1 (2) the eigenvalues and eigen-operators always come in conjugate pairs, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=', for every eigenvalue gabeiλab we have �U(M † ab) = gabe−iλabM † ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The relationship Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (4) is derived from the first order perturbation theory [45] (also see Appendix A).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Thus a diagonal entry of E in the basis composed of |φa⟩⟨φb| can be obtained Eab,ab ≈ gabeiλabe−i(λa−λb) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (5) As long as we can measure the noisy eigenvalues gabeiλab of �U and identify their corresponding ideal eigenvalues ei(λa−λb), we obtain the diagonal entries of Eab,ab by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' If we can uniformly at random sample some noisy eigenvalues gabeiλab or equivalently Eab,ab, then we can use the average of these samples to obtain an esti- mate of process fidelity F = tr{E}/d2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Because all the diagonal entries have amplitude smaller than 1, we can infer the number of samples needed from the Hoeffding’s inequality [48], that is, let X1, · · · , XK be independent bounded random variables with ai ≤ Xi ≤ bi for all i ∈ [K] and denote their average X = 1 K � i Xi, then for any ϵ > 0 it holds that P ������X − 1 K � i E(Xi) ����� ≥ ϵ � ≤ 2 exp � − −2K2ϵ2 � i(bi − ai)2 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (6) This inequality bounds the probability that the empirical average X deviates from the average of expectation val- ues of these random variables with a distance ϵ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Assume we have K samples of diagonal entries Eab,ab sampled from a uniform distribution, so the expectation value of each sampled diagonal entry is E(Eab,ab) = tr{E} d2 = F.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We take the average value of these samples as our esti- mate of the process fidelity, that is ˆF = 1 K � ab Eab,ab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Thus, the needed number of diagonal entries Eab,ab to estimate the process fidelity within an error ϵ with the probability 1 − δ, or say P(| ˆF − F| ≤ ϵ) = 1 − δ, is K = log(2/δ) 2ϵ2 , (7) which is independent of the system dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Besides the process fidelity, the noisy eigenvalues can also be used to infer the noise strength of stochastic noise only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Since the amplitudes of eigenvalues are only af- fected by stochastic noise and not changed under unitary noise, we can use those amplitudes to define a quantity referred as stochastic fidelity Fsto = � 1 d2 � ab g2 ab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (8) to assess the impact of stochastic noise only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' As shown in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' [49], the stochastic fidelity is related to the uni- tarity defined in Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' [39].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Both quantities measure how close the noise E is to a unitary channel and are use- ful for error-budget.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We emphasize that, compared to the stochastic errors, the unitary errors may cause more subtle and complicated problems in quantum error cor- rection and fault-tolerant quantum computation [50–54].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' As a result, differentiating between stochastic and uni- tary errors can assist us in recognizing their respective impacts, and in addition, can help to calibrate and tailor the error types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We can also estimate the actual values of some uni- tary parameters of a native gate, from the phases λab of noisy eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' This is achieved by identifying the relationship between these unitary parameters and some eigenvalues of the gate, which is similar as the robust phase estimation [40] and Floquet calibration [43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' III.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' CHANNEL SPECTRUM BENCHMARKING In this section, we present a practical procedure, which we refer as Channel Spectrum Benchmarking (CSB), to measure the individual fidelity of a universal process U no matter it’s a native gate or a circuit fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The estimate of fidelity of the gate U requires a uni- form sample of diagonal entries of E, which is identical to a uniform sample of noisy eigenvalues gabeiλab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The noisy eigenvalues can be estimated by the circuits of control- free phase estimation depicted in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In these circuits, we first prepare a state ρ, then repeatedly apply the tar- get gate U for L times, and finally measure the expecta- tion value of an operator O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We denote the noisy version of ρ and O as �ρ and �O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The noisy eigen-operators Mab of �U can be used as a basis (not necessarily orthonormal) to expand the initial state �ρ, that is �ρ = � ab tr � G† ab�ρ � Mab (9) 4 |0⟩ 𝑈𝑠 𝑈 𝐿 prepare 𝑐𝑎 𝜙𝑎 + 𝑐𝑏|𝜙𝑏⟩ measurement 𝑈 𝑈𝑠 † Estimate noisy eigenvalues with matrix pencil method and compute diagonal entries by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (5) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Uniformly at random sample 𝐾 pairs of eigenstates 𝜙𝑎 , 𝜙𝑏 }.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For each pair of eigenstates, do step 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Compute process fidelity and stochastic fidelity by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (13) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (14) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' |0⟩ 𝑈 𝐿 prepare 𝑐𝑎 𝜙𝑎 + 𝑐𝑏|𝜙𝑏⟩ measurement |0⟩ 𝑈 𝐿 prepare 𝑐𝑎 𝜙𝑎 + 𝑐𝑏|𝜙𝑏⟩ measurement 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For each length 𝐿 in [0, 𝐿max], run benchmarking circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The procedures of channel spectrum benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The benchmarking circuits are composed of three parts: the first part Us prepares the initial state |ψ⟩ = ca|φa⟩ + cb|φb⟩, which is a superposition of two eigenstates of target gate U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' then the target gate U is repeated L times, where L is an integer in [0, Lmax];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' finally, the operator O = |ψ⟩⟨ψ| is measured.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' where Gab is the corresponding left eigen-operator of Mab and they satisfy tr � G† abMa′b′ � = δab,a′b′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Then we can show that the expectation value of O at length L under noise is � �O � L = tr � �O �UL(�ρ) � = � ab tr � �OMab � tr � G† ab�ρ � (gabeiλab)L (10) This is a damping oscillating function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' From the time series data � �O � L at different depth L, we can extract the noisy eigenvalues via signal processing methods, such as matrix pencil method [55–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' By selecting an appropriate initial state ρ and measure- ment operator O, we can control the number of eigenval- ues presented in the resulting signals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The presence of too many different eigenvalues in the signals can pose some difficulties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' These include: (1) it may be difficult to extract the eigenvalues from the measured signals, as this may require more data or equivalently a larger depth L (which is limited by the damping rate gab);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (2) it can be harder to identify the corresponding ideal eigenvalue for a given noisy counterpart;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (3) it can be difficult to maintain a uniform sample of the diagonal entries of E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' To address these issues, we prepare the initial state and measurement operator as follows: |ψ⟩ = ca|φa⟩ + cb|φb⟩ ρ = O = |ψ⟩⟨ψ| (11) which are superposition of two eigenvectors only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For this type of initial state and measurement operator, only the corresponding noisy eigenvalues of these ideal eigen-operators {|φa⟩⟨φb|, |φb⟩⟨φa|, |φa⟩⟨φa|, |φb⟩⟨φb|} (presented in the selected initial state) possess a major portion in the measured signal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Thus, as illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 1, we propose the proce- dures of channel spectrum benchmarking below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Uniformly at random sample K pairs of eigenstates {|φa⟩, |φb⟩} of target unitary operator U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For each pair of eigenstates, do step 3, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=', running phase estimation circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In phase estimation circuits, one first prepares the initial state |ψ⟩ = ca|φa⟩ + cb|φb⟩, then repeatedly apply the target gate U for L times where L takes successive integers in [0, Lmax] , finally measure the probability ⟨O⟩L of obtaining O = |ψ⟩⟨ψ|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Estimate the noisy eigenvalues gabeiλab (am- plitudes and phases) from the time series data � �O � L by matrix pencil method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 5 3b.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Identify the ideal counterparts of the mea- sured noisy eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 3c.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Compute the diagonal entries of E by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Compute the process fidelity by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (13) and stochastic fidelity by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (14).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Step 1 ensures the estimated diagonal entries are uni- form samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We require the amplitude of two coeffi- cients ca, cb are comparable and the initial state |ψ⟩ can be efficiently prepared.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In the simulated experiments, we always choose ca = cb = 1 √ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The number of initial states K is independent of system dimension d and only depends on desired precision referring to Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (7), which is guaranteed by Hoeffding’s inequality.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' So our method has the potential scalable to many-qubit systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In the phase estimation circuits of step 3, we choose the length L from [0, Lmax].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The maximum length Lmax and the number of initial states K determine the total number of benchmarking circuits Nc = K(Lmax + 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' When combined with the number of shots Ns needed to run each circuit to collect statistic, the total experimen- tal cost is NcNs = K(Lmax + 1)Ns.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The choice of Lmax and Ns also depend only on desired precision and not on the system dimension.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Previous work has shown that the uncertainty of estimated eigenvalues is inversely propor- tional to the length L, the so-called Heisenberg scaling [40, 43].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Therefore, if higher precision is desired, it is gen- erally better to increase Lmax rather than the number of shots Ns per circuit, before the signals are completely degraded.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In step 3a, the noisy eigenvalues are estimated using the matrix pencil method [55–57].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' This method is well- suited for this task because it involves a singular value decomposition of the data Hankel matrix, which allows us to keep only the components with non-trivial singu- lar values, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=', those caused by noisy eigenvalues of ideal eigen-operators shown in initial state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' This process can reduce sampling error and eliminate unwanted eigenval- ues due to SPAM errors or the effect of degenerate eigen- values (for both the phase and the amplitude) of ideal channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In our simulated experiments, when using an initial state with unequal phases λa, λb, the number of obtained noisy eigenvalues is at most four.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In step 3b, our goal is to match the obtained noisy eigenvalues from matrix pencil method to their corre- sponding ideal counterparts such that we can compute the diagonal entries of E by Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (5).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For a initial state whose two eigenstates |φa⟩, |φb⟩ have equal eigenvalues, this process is not needed because all ideal channel eigen- values are 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' However, for a initial state with unequal two eigenvalues, there are three ideal channel eigenvalues {ei(λa−λb), e−i(λa−λb), 1}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' To match the obtained noisy eigenvalues to the three ideal ones, we calculate the dis- tance between the phases of the estimated noisy eigen- values and the ideal phase λa − λb of the eigenvalue of the particular eigen-operator |φa⟩⟨φb|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The noisy eigen- value with the smallest distance is chosen as the noisy counterpart of the ideal eigenvalue ei(λa−λb).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Similarly, the noisy counterpart of e−i(λa−λb) is also determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The remaining noisy eigenvalues are considered as the counterparts of the ideal eigenvalue 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' This criterion assumes that the magnitude of the actual phase error δλ = λab − (λa − λb) is small, more precisely we require |δλ| ≪ |λa − λb|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (12) After calculating the diagonal entries using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (5), we divide them into two categories based on the ideal eigenvalue of the associated basis |φa⟩⟨φb|: one is the trivial operator subspace with λa = λb (or say the opera- tor subspace spanned by the eigen-operators with eigen- value 1), the other is the non-trivial operator subspace with λa ̸= λb.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Because the dimension of trivial sub- space dts is usually very different from the dimension of non-trivial subspace dns, the probability of sampling an entry in the two subspace are very different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For exam- ple, for a many-qubit gate U with non-degenerate oper- ator spectrum, the trivial subspace is spanned by all the eigen-operators with the form |φa⟩⟨φa|, whose dimension dts = d is much smaller than dns = d2 − d.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' If there are some degeneracy in the spectrum of the operator U, that is λa = λb for two different eigenstates |φa⟩, |φb⟩, the trivial subspace can include the eigen-operators of the form |φa⟩⟨φb|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In step 1, we assign the same probability for the two subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Therefore, we need to separately compute the average of the diagonal entries in the triv- ial and non-trivial subspaces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Finally, the estimator of the process fidelity is obtained by combining these two averages, that is ˆF = dts Eab,ab|λa=λb + dns Eab,ab|λa̸=λb d2 (13) where Eab,ab is the average value of sampled entries.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Sim- ilarly, the estimator for stochastic fidelity is ˆFsto = � dts g2 ab,ab|λa=λb + dns g2 ab,ab|λa̸=λb d2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (14) IV.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' SIMULATED EXPERIMENTS In this section, we perform simulated experiments to show the performance of our CSB protocol, including single-qubit Pauli rotation gates, two-qubit Fermionic simulation (Fsim) gates, three-qubit Toffoli gate, and an Ising Hamiltonian evolution operator with 10 qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Throughout this work, each benchmarking circuit is re- peated Ns = 104 times to collect enough statistic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Single-qubit Pauli rotation gates Here we measure the fidelity of single-qubit rotation gates, that is Rσ(θ) = e−i θ 2 σ (15) 6 10 3 10 2 10 1 Probability of stochastic error p 10 3 10 2 Infidelity (a) Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='02 Angle error Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' angle error Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' angle error 10 3 10 2 10 1 Angle of unitary error 10 3 6 × 10 4 2 × 10 3 3 × 10 3 Infidelity (b) 10 3 10 2 10 1 Angle error FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Benchmarking of T gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In (a), we fix the uni- tary error (δθ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='01) and vary the probability of stochastic error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In (b), we fix the stochastic error (δp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='001) and vary the angle of unitary error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The actual process infidelity and stochastic infidelity is obtained by first computing the channel of noisy gate and then using Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (3) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In both cases, we accurately estimate process infidelity, stochas- tic infidelity and the angle of unitary error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The accuracy of estimation can be further improved by increasing the circuit length or shots for each circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' where θ is the rotational angle and σ is a Pauli matrix describing the direction of the rotational axis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' This type of unitary operator has two eigenvalues e−i θ 2 and ei θ 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The dimension of the trivial eigen-operator subspace is 2, which is the same as the dimension of the non-trivial eigen-operator subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Our choice of initial state only generates one operator 1 2(|φa⟩⟨φa| + |φb⟩⟨φb|) in the triv- ial subspace, which means that we may only obtain one noisy eigenvalue in this subspace, potentially leading to an inaccurate estimation of the process fidelity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' To ad- dress this issue, we also prepare another initial state to run phase estimation circuits, that is one of the eigen- states of Rσ(θ) in addition to the superposition state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' This results in a value of K = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' At the same circuit length, we sum the measured probabilities of the two types of circuits with the two initial states, allowing us to extract all the noisy eigenvalues simultaneously.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 2 shows the results of benchmarking RZ( π 4 ) (also known as T gate).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In this simulation, the noise model consists of a combination of stochastic errors (includ- ing T1 and T2 errors with equal probabilities δp) and over/under-rotation errors with angle δθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 2(a), we fix the unitary error (δθ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='01) and vary the probabil- ity of stochastic error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 2(b), we fix the stochastic error (δp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='001) and vary the angle of unitary error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In both cases, we are able to accurately estimate the pro- cess fidelity of the gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' As a byproduct, we can also estimate the angle of the unitary error by comparing the phases of some noisy eigenvalues to their corresponding ideal values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' This is a more sensitive probe of unitary errors than infidelity, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 2(b), where the process infidelity remains almost unchanged when δθ is varied from 10−3 to 10−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In this simulation, we set Lmax = 50, except when stochastic probability δp = 10−3, where Lmax = 100.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' It is worth noting that the accuracy of the estimation can be further improved by increasing the length of the bench- marking circuits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' However, increasing Lmax directly also increases the number of circuits used, which leads to higher costs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Instead, we can repeat the target gate U a certain number of times (Nrep times) to create a new tar- get gate, U ′ = U Nrep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Correspondingly, the noisy eigen- value we estimate becomes (gabeiλab)Nrep.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' But remember we need to determine the ideal eigenvalue from phase difference, thus as a result of Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (12), we require Nrep|δλ| ≪ |λa − λb|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (16) B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Two-qubit Fsim gates Here, we benchmark the two-qubit fermionic- simulation (Fsim) gates [8], i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=', Fsim(θ, φ) = � �� 1 0 0 0 0 cos θ −i sin θ 0 0 −i sin θ cos θ 0 0 0 0 eiφ � �� (17) where θ is the iswap angle and φ is the control phase angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We omit some phase parameters that can be freely adjusted by Z rotations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For the preparation of initial states, we consider all pairs of eigenstates (K = 6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The choice of Lmax is 50 or 100 (for δp = 10−3).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In this simulation, the noise model includes T1, T2 noise with equal probabilities δp for all single-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For two-qubit gates, each qubit experiences the same errors as single-qubit gates, as well as an over-rotation unitary error with angle errors δθ and δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We benchmark a specific Fsim gates with θ = π 4 , φ = π 2 , as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 3(a), we fix the unitary error with δθ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='01, δφ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='02 and vary the probability of stochastic error δp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We accurately estimate all infidelities in this case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' However, the estimation of the angle of the unitary error becomes less accurate when the stochastic error is too strong, as the signal decays too quickly to accumulate enough information to estimate the angle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 3(b), we fix the probability of stochastic error with δp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='001 and vary the angles of unitary error with δθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='5δφ = 10−3 ∼ 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Again, we accurately estimate all infidelities and angles of the unitary error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 7 10 3 10 2 10 1 Probability of stochastic error p 10 3 10 2 10 1 Infidelity (a) Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='00 Angle error Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' error Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' error Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' error Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' error 10 3 10 2 10 1 Angle of unitary error 10 3 10 2 Infidelity (b) 10 3 10 2 10 1 Angle error FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Benchmarking of a Fsim gate with θ = π 4 , φ = π 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In (a), we fix the unitary error with δθ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='01, δφ = −0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='02 and vary the probability of stochastic error δp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 3(b), we fix the probability of stochastic error with δp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='001 and vary the angles of unitary error with δθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='5δφ = 10−3 ∼ 10−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We always accurately estimate the process infidelity and the stochastic infidelity of the gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' But, the accuracy of estimating the angles of the unitary error is compromised when there is a high level of stochastic noise, as the signal degrades quickly and there is not enough data to accurately estimate the angles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Three-qubit Toffoli gate In this study, we evaluate the performance of the three- qubit Toffoli gate, which is not a native gate but rather a circuit fragment composed of 1-qubit and 2-qubit gates as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 4(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We randomly select K = 10 pairs of eigenstates as the initial state and set Lmax = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In the simulated noise model, all single-qubit gates are subject to T1, T2 noise with equal probability δp.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For the two-qubit gates, each qubit experiences the same type of stochastic error as the single-qubit gates, followed by a unitary error of the Fsim type with error angles δθ = δφ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The Toffoli operator has a highly degenerate spectrum, which creates two challenges for our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' First, when sampling noisy eigen-operators, we need them to be uni- formly distributed, but for degenerate eigenvalues, the noisy eigen-operators are superpositions of ideal ones in the degenerate subspace, which are determined by the de- tails of the noise, see Appendix A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' This makes it difficult to generate a uniform sample of noisy eigen-operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Second, the degenerate eigenvalue may be split by noise into many eigenvalues in the signal, making it harder to extract the noisy eigenvalues and each eigenvalue may only occupy a small portion of the signal, making them 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='010 Probability of stochastic error p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='00 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='40 Infidelity Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity (varied circ) Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity (varied circ) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='10 Angle of unitary error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='07 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='09 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='10 Infidelity Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity (varied circ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' RC) Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity (varied circ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' RC) (a) (b) (c) q0 q1 q2 0, 0, 0 U3 0, 0, 0 U3 /2, 0, U3 0, 0, 0 U3 0, 0, 0 U3 0, 0, /4 U3 0, 0, 0 U3 0, 0, 0 U3 0, 0, /4 U3 0, 0, 0 U3 0, 0, 0 U3 0, 0, /4 U3 0, 0, 0 U3 0, 0, /4 U3 0, 0, /4 U3 0, 0, 0 U3 0, 0, /4 U3 /2, 0, U3 0, 0, /4 U3 0, 0, /2 U3 0, 0, 0 U3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Bechmarking of Toffoli circuit fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We fix the unitary error (δθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='01) and vary stochastic error in (a), and fix stochastic error (δp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='001) and vary unitary error in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The circuit implementing Toffoli gate is presented in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Due to the highly degenerate spectrum of the Toffoli gate, the estimate of the infidelity is unreliable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' However, the degener- acy can be removed by changing the last layer of single-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' With the varied circuit, we accurately estimate the in- fidelity of the Toffoli circuit under weak unitary error in (a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For strong unitary error, we perform randomized compiling to the benchmarking circuits, converting the unitary error into stochastic error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' As a result, the varied circuit also accurately estimates the process infidelity of Toffoli circuit under strong unitary error, as shown in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' more susceptible to errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The impact of the highly de- generate spectrum on the estimate of gate noise is demon- strated by the simulated results in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 4(a),(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Usually, some of degeneracy can be removed by ap- pending a layer of single-qubit gates to the target gate or circuit fragment.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For the Toffoli circuit, we append RZ( π 2 )⊗RZ( 2π 3 )⊗RX( 4π 5 ) to the Toffoli circuit and com- bine this layer with the last layer of the Toffoli circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The choice of appended layer should keep the state prepa- ration of the new target gate efficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Here our choice does not change the eigenstates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For the angle parame- ters in the appended gates, one can design an optimiza- tion algorithm to choose the parameters that maximize the distance between eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The appended layer of gates results in a varied circuit with a similar structure to 8 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='002 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='004 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='006 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='008 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='010 Probability of stochastic error p 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='15 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='20 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='25 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='30 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='35 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='45 Infidelity Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='10 Angle of unitary error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='10 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='12 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='14 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='16 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='18 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='20 Infidelity Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity (RC) Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity (RC) (a) (b) (c) q0 q1 q2 q3 q4 q5 q6 q7 q8 q9 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='628 U3 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='929 U3 0, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='17 U3 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='626 U3 0, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='79 U3 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='745 U3 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='487 U3 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='742 U3 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='758 U3 0, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='68 U3 0, 0, 0 U3 0, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='4 U3 0, 0, 0 U3 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='198 U3 0, 0, 0 U3 0, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='25 U3 0, 0, 0 U3 0, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='05 U3 0, 0, 0 U3 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='0943 U3 0, 0, 0 U3 0, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='01 U3 0, 0, 0 U3 0, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='03 U3 0, 0, 0 U3 0, 0, 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='93 U3 0, 0, 0 U3 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='222 U3 0, 0, 0 U3 0, 0, 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='393 U3 FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Benchmarking of a 10-qubit Ising evolution operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We fix unitary error (δθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='01) and vary stochastic error in (a), and fix stochastic error (δp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='001) and vary unitary error in (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The circuit implementing Ising evolution operator is presented in (c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The actual fidelity is not computed from the channel of the circuit, but rather inferred from the product of the fidelity of all single-qubit and two-qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' However, the actual stochastic infidelity can not be reliably inferred by this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We accurately estimate infidelity of the Ising evolution operator under weak unitary error (a) and strong unitary error with RC (b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' the original Toffoli circuit (only the last layer is changed) and they should possess similar noise properties.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In the case of strong stochastic error and weak unitary error (δθ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='01) in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 4(a), the benchmarking of the varied circuit provides a very accurate estimate of the process infidelity and the stochastic infidelity of the original Tof- foli circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' However, there is a significant difference between the estimated and actual process infidelity when the unitary error is very strong, as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 4(b) (with fixed stochastic error δp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='001).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In the Appendix B, we show that our method may under-estimate the process infidelity in the presence of certain strong unitary errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' One way to address this issue is to introduce random gates into the benchmarking circuits to convert the uni- tary errors to stochastic errors [37, 38, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The Ap- pendix C describes a procedure for transforming noise in the native gates to stochastic errors using random gates from the symmetry group of the target U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For bench- marking circuit fragments, we use a technique called ran- domized compiling [37, 38] to achieve this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Randomized compiling (RC) is a method that transforms the noise in the circuit into stochastic Pauli errors while maintain- ing the circuit structure and depth.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' After RC, the noise type of a circuit cycle is changed, but the fidelity of the cycle and the circuit structure remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' As long as there is no repeated structure in U where unitary error can coherently build up and increase the infidelity quadratically with the circuit depth [59] (this is a case where RC should be introduced to suppress the unitary noise), we expect the fidelity of the circuit U to remain unchanged after RC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For each original circuit, we gener- ate Nr = 10 random circuits by RC and each random circuit is run 103 times to keep the cost unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 4(b), after RC the varied circuit can ac- curately estimate the process infidelity of Toffoli circuit under unitary noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Ten-qubit Ising evolution operator Our method is practically scalable if the following two requirements are met: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The eigenvalues and eigenvectors of target unitary operator U can be efficiently computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The initial state can be efficiently prepared, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=', the number of 1-qubit and 2-qubit gates needed for the preparation should at most scale polynomial with the number of qubits.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In general, these two requirements are not always sat- isfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' However, for certain types of unitary operators, such as the evolution operator of an Ising Hamiltonian, these requirements can be met.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For an Ising Hamiltonian, the eigenvectors are known and are simply the computa- tional basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Given an eigenstate, the eigenvalue can be efficiently computed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The initial state of a superposition of two computa- tional basis states |x⟩ = |x0, · · · , xi, · · · , xN−1⟩, |y⟩ = |y0, · · · , yi, · · · , yN−1⟩ can be prepared as follows: first, for the qubit i, if xi = yi, the state can be prepared by an X gate if xi = yi = 1;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' then, for the state of re- maining qubits with xi ̸= yi, if we only have one such qubit, a Hadamard gate H can be applied;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' if there is more than one qubit with xi ̸= yi, one can first prepare a GHZ state on these qubits and then apply some X gates to obtain the target state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Therefore, the prepa- ration of such states cost at most N 1-qubit and N 2- qubit gates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Additionally, for the evolution operator of the Hamiltonian that can be obtained by performing lo- cal unitary transformation on an Ising Hamiltonian, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=', H = � i UiHIsing � i U † i , the initial states can also be obtained in the similar way with additional two layers of single-qubit gates � Ui, � U † i .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Thus, this type of evolution operators is a good example for benchmarking many-qubit quantum systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 9 Here we benchmark the evolution operator of a 1- dimensional Ising ring H = �10 i=1 hiZi + Ji,i+1ZiZi+1, where hi, Ji,i+1 are randomly chosen.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The circuit is shown as in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 5(c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We sample K = 10 pairs of eigen- states and set Lmax = 50.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The noise model is the same as the case in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' IV C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The actual infidelity is inferred from the infidelity of single-qubit and two-qubit gates, because our computer is not powerful enough to compute the quantum channel of a 10-qubit circuit.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Our method ac- curately estimates process infidelity under both weak and strong unitary error (with RC), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 5(a),(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' V.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' CONCLUSION AND OUTLOOK In this work, we introduced a procedure called chan- nel spectrum benchmarking, which infers the noise prop- erties of a quantum gate from the eigenvalues of noisy channel representing the gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In the protocol, we first choose the initial state using a superposition of randomly sampled pair of eigenstates of the target gate, and then, we use control-free phase estimation circuits to estimate the noisy eigenvalues in a SPAM error-resistant manner.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' This choice of initial state simplifies the data processing because the measured signal only contains a few eigen- values, which can be extracted using signal processing methods such as the matrix pencil method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' By compar- ing the noisy eigenvalues to their ideal counterparts, we can estimate noise properties such as the process infi- delity, stochastic infidelity, and some over-rotation angle errors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Our method can be applied to any quantum gate, but performs better on gates with non-degenerate oper- ator spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For gates with highly degenerate spec- trum, we can append a layer of single-qubit gates to re- move the degeneracy while maintaining a similar circuit structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Some types of unitary error can also affect the performance, which can be addressed using randomiza- tion techniques like randomized compiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Our method is scalable to many-qubit systems as long as the eigen- decomposition can be computed and the initial state can be efficiently prepared, such as the evolution operator of an Ising-type Hamiltonian.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The requirements for the scalability of our method could be relaxed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In principle, we do not need to obtain the complete set of the eigenmodes for the target gate operator, a few samples of eigenvalues and eigenstates are sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For initial state preparation, there are ex- isting methods for preparing arbitrary states [60–63], but it would be interesting to develop a more efficient algo- rithm for preparing the particular type of initial states in our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' A variational algorithm [64] may be able to efficiently prepare these states for most target gates, because we have the freedom to choose the coefficients of the superposition states and do not need perfect prepa- ration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Our method can be scaled up in a way similar to simultaneous randomized benchmarking [65, 66], where some few-qubit gates are simultaneously benchmarked on different subsets of a many-qubit system such that the ef- fect of crosstalk [67] can be detected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' One immediate use of benchmarking is to calibrate quantum gates using the measured figures of merit as a cost function [8–10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Our method can provide more spe- cific information (process infidelity, stochastic infidelity and over-rotation angle of the target gate) about the cal- ibrating gate, so it is expected to perform better on this task than other benchmarking methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' A detailed com- parison of different benchmarks for calibration will be a topic for future research.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Additionally, our method can be used to calibrate universal gates, including not only 1 or 2-qubit native gates, but also many-qubit native gates such as MS gates [68, 69] used in ion trap systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' It may also be interesting to use our method to calibrate certain circuit fragments that are commonly used in al- gorithms, such as the trotterized Hamiltonian evolution operator in quantum simulation and the Grover iteration operator in Grover’s search algorithm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' ACKNOWLEDGMENTS The work is supported by the National Natural Sci- ence Foundation of China (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 12147123 and 11974198) and Beijing Natural Science Foundation (No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Z220002).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Source code for the simulated experiments is available at this site https://github.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='com/yanwu-gu/ channel-spectrum-benchmarking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Appendix A: The relationship between noisy and ideal eigenvalues of quantum channels In this section, we derive the relationship between the noisy channel eigenvalues of a gate and its correspond- ing ideal counterparts with the first order perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Consider a gate U acting on a d-dimensional space with eigen-decomposition U|φa⟩ = eiλa|φa⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' As the actual im- plementation of a gate U is inevitably associated with some noise, it is more convenient to use quantum chan- nels rather than quantum operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Quantum channels are completely-positive trace-preserving (CPTP) maps, which transform one operator to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The action of a quantum channel E on an arbitrary operator O can be characterized by a set of Kraus operators Ek, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=', E(O) = � k EkOE† k.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We denote the corresponding uni- tary channel of the unitary operator U as U, whose action on an operator O is U(O) = UOU †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Thus, the unitary channel U has the eigen-decomposition U(|φa⟩⟨φb|) = U|φa⟩⟨φb|U † = ei(λa−λb)|φa⟩⟨φb|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (A1) Quantum channels are linear maps that can be repre- sented as matrices under a set of the basis operators of the operator space, such as eigen-operators |φa⟩⟨φb|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Meanwhile, operators are represented as vectors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The as- sociated inner product between two operators A and B is 10 the Hilbert-Schmidt inner product tr � A†B � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Therefore, in this representation U is a unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Let us append a noise channel E to U, with the noisy version of U denoted as �U = EU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We investigate the re- lationship between the eigenvalues of �U and those of U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' If the noise is relatively weak, the problem is an eigen- value perturbation of unitary matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Given the close re- lationship between unitary and Hermitian matrices, one can use Hermitian matrix perturbation theory to get the correction of eigenvalues and eigenstates, assuming a di- agonalizable noisy gate �U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In most cases, the assumption should be met in actual devices, since diagonalizable ma- trices are dense in the space of all matrices, meaning that any non-diagonalizable matrix can be deformed into a di- agonalizable one by a small perturbation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In the follow- ing, we apply Hermitian perturbation theory to obtain the first order correction of the eigenvalues, and obtain the relationship between the noisy eigenvalues and ideal ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Define the eigenvalues and eigen-operators of �U as gabeiλab and Mab, that is �U(Mab) = gabeiλabMab .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (A2) The perturbation matrix is ∆ = �U − U = (E − I) U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (A3) We assume the perturbation is small in terms of some norm, such as the diamond norm ∥∆∥⋄ = δ [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Then, for a non-degenerate eigenvalue ei(λa−λb) with eigen- operator |φa⟩⟨φb|, the first order correction is ϵab 1 = tr � (|φa⟩⟨φb|)†∆(|φa⟩⟨φb|) � = tr � (|φa⟩⟨φb|)†(E − I) U(|φa⟩⟨φb|) � = ei(λa−λb) � tr � (|φa⟩⟨φb|)†E(|φa⟩⟨φb|) � − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (A4) Thus, the noisy eigenvalue is approximated as gabeiλab ≈ ei(λa−λb) + ϵab 1 = ei(λa−λb)tr � (|φa⟩⟨φb|)†E(|φa⟩⟨φb|) � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (A5) In this case, the noisy eigen-operator Mab is approxi- mated by Mab ≈ |φa⟩⟨φb| + O(δ) (A6) where O(δ) are some correction terms with the first order of δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For degenerate eigenvalues eiλn with eigen-operators |φa⟩⟨φb| satisfying λa − λb = λn, these eigen-operators span a subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The ab, a′b′-entry of the perturbation matrix projected in this subspace is ∆(n) ab,a′b′ = tr � (|φa⟩⟨φb|)†(E − I) U(|φa′⟩⟨φb′|) � = eiλn � tr � (|φa⟩⟨φb|)†E(|φa′⟩⟨φb′|) � − δaa′δbb′� = eiλn(E(n) ab,a′b′ − I(n) ab,a′b′) (A7) where E(n) ab,a′b′ and I(n) ab,a′b′ are the entries of pure noise map E and Identity map I projected in this degenerate subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In the degenerate case, the first order correc- tions to the eigenvalue eiλn of U are the eigenvalues of the perturbation matrix ∆(n).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' It’s easy to find that the matrix ∆(n) and the matrix E(n) have the same eigen- operators M 0 pq, which are the superposition of eigen- operators |φa⟩⟨φb| in this degenerate subspace.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' They are also the corresponding unperturbed eigen-operators of noisy eigen-operator Mpq of �U, that is Mpq ≈ M 0 pq+O(δ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In this case, the eigenvalue gpqeiλpq of �U is gpqeiλpq ≈ eiλn + tr � G0† pq∆(n)(M 0 pq) � = eiλn + eiλn � tr � G0† pqE(n)(M 0 pq) � − 1 � = eiλntr � G0† pqE(n)(M 0 pq) � (A8) where G0 pq is the corresponding left eigen-operator of M 0 pq and they satisfy tr � G0† pqM 0 p′q′ � = δpq,p′q′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Therefore, the Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (A8) has the same form as Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (A5) but with a basis of a different form.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Appendix B: Perturbation of channel eigenvalues under pure unitary error Here we consider the noisy eigenvalues of a quantum gate under a pure unitary error V = e−iHeδ (B1) where He is the Hamiltonian of error and δ characterize the error strength.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Assume the target gate U = e−iHθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Thus the operator of noisy gate is �U = V U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In this case, the process fidelity is F = |tr{V }|2 d2 ≈ |tr � I − iHeδ − 1 2H2 e )δ2� |2 d2 = d2 − dtr � H2 e � δ2 d2 = 1 − tr � H2 e � d δ2 (B2) where we use the property tr{He} = 0 and keep the term up to O(δ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' This is a well-known result that unitary error with some matrix norm δ has process infidelity of order O(δ2) [46].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We then analyze how eigenvalues of a quantum chan- nel U change under such unitary error with perturbation theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Now the effect of the noisy gate �U on an oper- ator O is �U(O) = V UOU †V †.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Represent all quantum channels as matrices, the perturbation matrix is ∆ = �U − U .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (B3) 11 For a non-degenerate eigen-operator |φa⟩⟨φb| with eigen- value ei(λa−λb), the first order correction is ϵab 1 = tr � (|φa⟩⟨φb|)†∆(|φa⟩⟨φb|) � = ei(λa−λb) � tr � (|φa⟩⟨φb|)†V |φa⟩⟨φb|V †� − 1 � .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (B4) To further expand this equation, we will use the Baker- Hausdorff (BH) lemma eXY e−X = eadX(Y ) = ∞ � n=0 adn X(Y ) n!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (B5) where ad is a map on operators with the effect adX(Y ) = [X, Y ].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Then the first order correction is ϵab 1 = ei(λa−λb) � tr � (|φa⟩⟨φb|)†e−iδadHe (|φa⟩⟨φb|) � − 1 � ≈ ei(λa−λb) � tr � (|φa⟩⟨φb|)† � I − iδadHe − 1 2δ2ad2 He � (|φa⟩⟨φb|) � − 1 � = ei(λa−λb) � ����−iδ (⟨φa|He|φa⟩ − ⟨φb|He|φb⟩) � �� � ϵab 1,1 −1 2δ2tr � (|φa⟩⟨φb|)†ad2 He(|φa⟩⟨φb|) � � �� � ϵab 1,2 � ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (B6) If we consider only the first-order correction, the noisy eigenvalue is gabeiλab ≈ ei(λa−λb) + ϵab 1 = ei(λa−λb)(1 + ϵab 1,1 + ϵab 1,2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (B7) From Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (3) and Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (5), we get the estimate of process fidelity ˆF = 1 + 1 d2 � ab ϵab 1,1 + 1 d2 � ab ϵab 1,2 (B8) where the term with order O(δ) is 1 d2 � ab ϵab 1,1 = 1 d2 � ab −iδ (⟨φa|He|φa⟩ − ⟨φb|He|φb⟩) = 0 (B9) and the term with order O(δ2) is 1 d2 � ab ϵab 1,2 = − δ2 2d2 � ab tr � (|φa⟩⟨φb|)†[He, [He, |φa⟩⟨φb|]] � = − δ2 2d2 � ab ⟨φa|H2 e |φa⟩ − 2⟨φa|He|φa⟩⟨φb|He|φb⟩ + ⟨φb|H2 e |φb⟩ = − δ2 2d2 (2dtr � H2 e � − 2tr{He}2) = −1 dtr � H2 e � δ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (B10) This coincides with the expression in Eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (B2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' However, due to the first order correction only contributing a term with order O(δ2), we must also take into account the second order correction to the eigenvalues.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The second order correction is ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='12 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='ϵab ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='2 = ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='mn̸=ab ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='|tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='(|φm⟩⟨φn|)†∆(|φa⟩⟨φb|) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='ei(λa−λb) − ei(λm−λn) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='mn̸=ab ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='|tr ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='(|φm⟩⟨φn|)†V |φa⟩⟨φb|V †� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='ei(λa−λb) − ei(λm−λn) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='mn̸=ab ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='|⟨φm|V |φa⟩|2|⟨φn|V |φb⟩|2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='ei(λa−λb) − ei(λm−λn) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='≈ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='mn̸=ab ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='δam + (⟨φm|He|φa⟩⟨φa|He|φm⟩ − ⟨φm|H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='e |φa⟩δam)δ2� � ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='δbn + (⟨φn|He|φb⟩⟨φb|He|φn⟩ − ⟨φn|H2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='e |φb⟩δbn)δ2� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='ei(λa−λb) − ei(λm−λn) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='= ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='m̸=a ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='⟨φm|He|φa⟩⟨φa|He|φm⟩δ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='ei(λa−λb) − ei(λm−λb) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='� ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='n̸=b ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='⟨φn|He|φb⟩⟨φb|He|φn⟩δ2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='ei(λa−λb) − ei(λa−λn) ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (B11) If the error Hamiltonian He is diagonal under the basis of eigenvectors of U, the second order correction is ϵab 2 = 0 up to the second order O(δ2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' There is no problem for our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' However, except the special case, there is some dis- crepancy between the process fidelity estimated using our method and the actual value, due to the presence of the term ϵab 2 in the noisy eigenvalue gabeiλab.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Here, we can directly compute the noisy eigenvalues of the channel �U from the eigenvalues of the operator �U and give the analytical form of estimated process fidelity by our method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We first compute the Hamiltonian of �U by Baker-Campbell-Hausdorff formula H′ = log (V U) = log (e−iHeδe−iHθ) ≈ −iHθ − iHeδ − 1 2ad−iHθ(−iHeδ) + 1 12ad2 −iHθ(−iHeδ) − 1 720ad4 −iHθ(−iHeδ) + · · · (B12) where we only keep the terms up to the order O(δ) and omit some terms with the ad map.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We can compute the eigenvalues of H′ comparing to those of the −iHθ with the first order perturbation theory .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The first order correction to the eigenvalue iλa with eigenstate |φa⟩ is ϵa 1 = ⟨φa|H′ − (−iHθ)|φa⟩ = −iδ⟨φa|He|φa⟩ (B13) where these terms with ad map are all zeros because ⟨φa|ad−iHθ(O)|φa⟩ = −iθ⟨φa|(HO − OH)|φa⟩ = iλa(⟨φa|O|φa⟩ − ⟨φa|O|φa⟩) = 0 (B14) where O is an any operator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Then the noisy eigenvalue of |φa⟩⟨φb| is gabeiλab ≈ eiλa+ϵa 1−iλb−ϵb 1 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (B15) Thus our estimator for process fidelity is ˆF = 1 d2 � ab eϵa 1−ϵb 1 = 1 − � a⟨φa|He|φa⟩2 d δ2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (B16) Because the term � a⟨φa|He|φa⟩2 is always smaller than the term tr � H2 e � except when He is a diagonal matrix un- der the basis |φa⟩, our method under-estimates process infidelity under unitary error in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' This problem can be fixed by introducing some randomization proce- dure into the benchmarking circuits to convert unitary errors to stochastic errors [37, 38, 58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Appendix C: Randomized compiling with the symmetric group of the target gate For a circuit composed of single-qubit and two-qubit gates, randomized compiling (RC) is a standard proce- dure to tailor the noise into stochastic Pauli noise with Pauli twirling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Here, we consider another case that the circuit is repetitions of a native gate U, that is U L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In the spirit of RC, if considering U as hard gate, we need a twirling group T, whose element Ti should be trans- formed to another Tj under the conjugate operation of U, that is UTiU † = Tj.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' A simple example of this type of groups is a symmetric group of U T = {T : U †TU = T} .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' (C1) 13 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='08 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='10 Angle of unitary error 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='0005 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='0010 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='0015 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='0020 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='0025 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='0030 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='0035 Infidelity Act.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity (RC) Est.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' stoch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' infidelity (RC) FIG.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Benchmarking of T gate with randomized compiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In simulation, stochastic error is fixed (δp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='001) and uni- tary is RX(δθ) with varied error angle δθ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The twirling group is T = {I, Z}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For each original circuit, we generate Nr = 10 random circuits and each random circuit is run for Ns = 103 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For the sequence of U L, random gates from the twirling group T are introduced before each application of U, but the effect of these random gates should be cancelled be- fore the next U is applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Finally, we get a new random sequence T † LU(TLT † L−1) · · · U(TiT † i−1) · · · U(T2T † 1 )UT1 (C2) where the gates in parentheses should be implemented as one gate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' In actual implementation, all the gates should be associated with a noise, and the gate sequence is de- noted as composition of quantum channels T † LET EUTLT † L−1ET · · · EUT2T † 1 ET EUT1ET = T † LET ETLUT † L−1ET · · · ET2UT † 1 ET ET1UET (C3) where we use the property that the gates in group T commute with U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We assume the noise of each twirling gate is the same quantum channel ET for simplicity, but this assumption can be relaxed [37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' After averaging many such random sequences we get � 1 NT � TL T † LET ETL � U · · · � 1 NT � T1 T † 1 ET ET1 � UET (C4) where NT is the number of gates in the twirling group T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The effect of this randomization procedure is to trans- form the noise to stochastic noise, that is applying a random quantum channel T †ET ET with probability 1 NT .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' One can use group representation theory to get a sim- pler form of the noise.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' But in our case, this subtlety is not necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' This procedure is similar to the Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' [58].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' However, we do not require the twirling group is abelian and do not need the assumption that there is no equal ir- reducible representation for symmetric group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' Thus our method has high flexibility to choose twirling group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We use a simulated experiment to show the perfor- mance of this procedure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We benchmark T gate under a unitary error RX(δθ) with varied error angle δθ and fixed stochastic error δp = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content='001.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' We choose T = {I, Z} as twirling group.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For each original circuit, we generate Nr = 10 random circuits and each random circuit is run for Ns = 103 times.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' The theory in Appendix B shows that the process in- fidelity estimated by our method is of the order O(δθ4) without the use of randomized compiling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' However, by introducing randomized compiling, our method can ac- curately estimate the process infidelity, as demonstrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' It is important to note that the process in- fidelity measured using randomized compiling on native gate includes the noise from both the target gate and the twirling gates, since the twirling gates are not merged into the original circuit in the same way as when using randomized compiling on circuit fragments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} +page_content=' For simplic- ity, we did not add noise to the twirling gates in this case.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtA0T4oBgHgl3EQfG__M/content/2301.02056v1.pdf'} diff --git a/dtFAT4oBgHgl3EQf6x4A/content/tmp_files/2301.08740v1.pdf.txt b/dtFAT4oBgHgl3EQf6x4A/content/tmp_files/2301.08740v1.pdf.txt new file mode 100644 index 0000000000000000000000000000000000000000..d598728b4b66a9d55725be8483cab4419bf36959 --- /dev/null +++ b/dtFAT4oBgHgl3EQf6x4A/content/tmp_files/2301.08740v1.pdf.txt @@ -0,0 +1,2316 @@ +Scaling approaches to steady wall-induced turbulence +Paul C. Fife +August 12, 2006 +Abstract +The problem of discerning key features of steady turbulent flow adjacent to a wall has drawn +the attention of some of the most noted fluid dynamicists of all time. Standard examples of +such features are found in the mean velocity profiles of turbulent flow in channels, pipes or +boundary layers. The aim of this review article is to expound the essence of some elementary +theoretical efforts in this regard. Possibly the best known of them, and certainly the simplest, +is the argument (obtained independently) by Izakson (1937) and Millikan (1939). They showed +that if an inner scaling and an outer scaling for the profile are valid near the wall and near the +center of the flow (or the edge of the boundary layer), respectively, and if there is an overlap +region where both scalings are valid, then the profile must be logarithmic in that common region. +That theoretical justification has been used and expanded upon by innumerable authors for over +60 years, and at the present time is still rightly enjoying popularity. +Although background discussions of several related topics are included in the present article, +for example the classical ideas of Prandtl and von Karman, the main foci will be on (i) a +careful examination of the Izakson-Millikan argument, together with a presentation of a better +mathematical justification for its conclusion; and (ii) a detailed clarification of a newer approach +to gaining theoretical understanding of the mean velocity and Reynolds stress profiles based on +the “search for scaling patches”. +The two approaches share common goals, they are both heavily involved with scaling con- +cepts, and many results are similar, but the logical trains of thought are entirely different. The +first, as mentioned, dates back to the 30’s and the second was introduced in a series of recent +papers by Fife et al, Wei et al, and Klewicki et al. +Our emphasis will be on the question of how and how well these arguments supply insight +into the structure of the mean flow profiles. Although empirical results may initiate the search +for explanations, they will be viewed simply as means to that end. +Contents +1 +Introduction +2 +2 +The Navier-Stokes equations and Reynolds averaging +3 +3 +Early scaling concepts of Prandtl and von Karman +5 +4 +The notion of scaling +7 +4.1 +Scaling patches . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +8 +4.2 +A classical example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +9 +4.3 +Inner, outer, and overlapping regions . . . . . . . . . . . . . . . . . . . . . . . . . . . +11 +4.4 +A uniform approximation +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +1 + +4.5 +A generalization +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +13 +4.6 +The Izakson-Millikan observation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +14 +4.7 +A more realistic version +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +15 +5 +The mean structure of turbulent Couette flow in a channel +16 +5.1 +The differential equations +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +17 +5.2 +The friction velocity and boundary conditions . . . . . . . . . . . . . . . . . . . . . . +18 +5.3 +Dimensionless variables in the core region . . . . . . . . . . . . . . . . . . . . . . . . +19 +5.4 +The wall layer, rescaling, and law of the wall +. . . . . . . . . . . . . . . . . . . . . . +20 +5.5 +Velocity in the core . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +21 +5.6 +Application of the Izakson-Millikan reasoning . . . . . . . . . . . . . . . . . . . . . . +22 +5.7 +Observations on the foregoing procedure . . . . . . . . . . . . . . . . . . . . . . . . . +23 +5.8 +The uniformizing effect of turbulence and some possible implications. . . . . . . . . . +24 +6 +The search for scaling patches +24 +6.1 +Adjusted Reynolds stresses, balance exchange phenomena, and the identification of +patches +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +25 +6.2 +The locations of the scaling patches +. . . . . . . . . . . . . . . . . . . . . . . . . . . +28 +6.3 +More on the locations of the patches . . . . . . . . . . . . . . . . . . . . . . . . . . . +29 +6.4 +The case A = constant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +30 +6.4.1 +Characteristic length as it depends on location +. . . . . . . . . . . . . . . . . +30 +6.4.2 +The Reynolds stress +. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +31 +6.4.3 +The mean velocity profile . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +31 +6.5 +Relaxing the assumption that A is constant . . . . . . . . . . . . . . . . . . . . . . . +32 +6.6 +Approximate constancy of A in interior zones . . . . . . . . . . . . . . . . . . . . . . +33 +6.7 +Evidence for logarithmic growth, i.e. σ = 0 +. . . . . . . . . . . . . . . . . . . . . . . +33 +6.8 +The inner scaling patch at the wall and the outer scaling patch at the midline . . . . +34 +6.9 +Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +34 +7 +Turbulent Poiseuille flow in a channel +35 +7.1 +Differences from Couette flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +35 +7.2 +Hierarchy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +36 +7.3 +Behavior near the wall . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . +37 +8 +Other applications and conclusion +39 +1 +Introduction +Every theoretical investigation of highly turbulent fluid dynamics is necessarily incomplete, because +accepted accurate models such as the Navier-Stokes equations lie beyond the scope of full solution +by existing methods, whether those methods be numerical or analytical. +Faced with this failing, researchers have often turned to seeking theoretical information through +partial analyses, incomplete models, or reasoning which is not fully based on rigorous deduction. +The search for such approaches is bound to yield, and historically has yielded, some fruitful avenues +leading to insightful, though perhaps tentative, conclusions. +The goals of this paper are to examine, and elaborate on, some of the major elementary attempts +in this vein to gain some understanding of the mechanisms behind steady (in the mean) turbulent +2 + +incompressible flow bounded by a wall, as well as to expound some recent developments. +The +prototypical examples will be rapid flow in a channel or pipe forced by an imposed pressure gradient +or the differential motion of the walls, and turbulent boundary layers of various types. There has +been an enormous amount of work along these lines, and a complete review of it will not be given. +Rather we look only at some efforts to gain insight into these mechanisms as they affect simple +mean flow quantities, through the use of scaling concepts and averaged Navier-Stokes equations. +These efforts have been diverse, and when compared one with another have sometimes relied on +assumptions which appear to be incompatible. Nevertheless despite any seeming incongruities, the +various approaches should not necessarily be thought of as competing among themselves. Rather by +contributing their individual insights, they collectively add to our understanding. No one approach +can ever rationally be viewed as the last word on the subject. +So to summarize, the goal in these notes is to understand what phenomena relating to average +events occur in wall-bounded turbulent flow and, more importantly, why they occur. This means +that our approach will focus on supplying theoretical explanations for observed features of the +flow, as well as providing predictions of features which can then be correlated with empirical data. +When possible, we will not be content with pointing to experimental evidence for properties of the +flow, but will strive to develop reasons why those properties should hold. These reasons will not +generally be rigorous, and at times, of necessity, they will be partly based on empirical findings. +But to advance the goal of basic understanding, they will, as far as possible, be grounded in +theory, i.e. in accepted mathematical models. Mostly, these models will be built from the averaged +Navier-Stokes equations. +2 +The Navier-Stokes equations and Reynolds averaging +Our mathematical models use the symbols u = (u1, u2, u3), p, t and x = (x1, x2, x3) to denote the +velocity, pressure, time and space variables, with µ and ρ the material constants of viscosity and +fluid density. +The standard incompressible Navier-Stokes equations when there is no body force can be written +as follows: +∂u +∂t + (u·∇)u = ν∇2u − 1 +ρ∇p, +(1) +where ν = µ/ρ is the kinematic viscosity; and +div u = 0. +(2) +The equations (1) (a vector equation which has three components) and (2) are four equations in +all, for four unknown functions: the three components of u plus the pressure p. +These equations are almost universally recognized as an accurate representation, on the macro- +scopic level, of flows of many standard kinds of incompressible fluids. Generalizations exist in many +forms. The second (nonlinear) term on the left of (1) is called the “inertia term”, and the first +term on the right is the “viscosity term”. Broad features of the solutions are typically governed by +the order of magnitude of the Reynolds number, a ratio of the effects of these terms. +Here we start with a very common framework for studying turbulent motions called Reynolds +averaging [1]. It is a short cut, and as such does not supply all the desired information about any +given flow scenario. A wealth of details about fluid motions is erased. The modeler is left with the +task of partially filling the resulting deficiency with assumptions about features of the fluid motion +in its disorganized state. These attempts are bound to be hit or miss to some extent, but as we +3 + +will see, there may be tests which one can apply to gauge the validity of such assumptions. For +example, two or more avenues to obtaining models may yield similar results, which would tend +to corroborate both. A very common remedy for the underdetermined nature of the Reynolds +averaged equations is to posit closure relations. That approach is especially useful in numerical +simulations. But the magnitude of these endeavors precludes their inclusion here. +The goal, then, is to use the Navier-Stokes equations in their averaged form, together with scal- +ing considerations, to gain insight into some basic properties of wall-induced turbulence. Specifi- +cally, we focus on features of the mean velocity and Reynolds stress profiles, as functions of distance +from the wall. The Reynolds stress is one of the measures of turbulence activity. There have in fact +been several distinct approaches to the task of using the averaged equations to provide insight into +wall-induced turbulence. That fact, together with the apparent simplicity of the formulation within +that framework, makes the latter an ideal arena in which to explore the interaction of mathematical +and intuitive tools to tackle a very complex problem. +We now review the well-known ideas behind the process of Reynolds averaging [1]. One supposes +that at each spatial location x and time t, the velocity and the other flow quantities have well defined +average values. This could be an imagined “ensemble” average over many similar observations, or +over some small (but not too small) region in space-time containing the point in question. In the +case of “steady” fully developed turbulence, it could be the time average. The mean (average) +values may then depend on space and time, but on scales perhaps larger than those over which the +average is taken. +Thus, we can write the velocity vector as its mean plus its fluctuation about the mean: +u(x, t) = U(x, t) + u′(x, t), +(3) +with a similar decomposition for vorticity, pressure, etc. Averages will be taken of not only primary +flow variables, but also of products of them. The averaging operation is denoted by angle brackets— +for example ⟨u⟩ = U; ⟨∂iu⟩ = ∂iU for any derivative ∂i = +∂ +∂xi or +∂ +∂t; ⟨u′⟩ = 0; ⟨Uiu′ +j⟩ = 0. +We substitute (3) into the Navier-Stokes equations (1), (2) and then take the average of the +resulting equations to obtain +∂tU − ν∇2U + (U·∇)U + ⟨u′·∇u′⟩ + 1 +ρ∇P = 0, +(4) +∇·U = 0. +(5) +Note also that +∇·u′ = 0. +(6) +The i-th component of the 4th term in (4) is +⟨u′ +j∂ju′ +i⟩ = ∂j⟨u′ +iu′ +j⟩ +(summing over the repeated index), the equality by virtue of (6). This shows that the 4th term, +being a divergence, acts as a pseudo-stress gradient. Accordingly, we define the “Reynolds stress +tensor” τ by +τij = −⟨u′ +iu′ +j⟩, +(7) +which is symmetric, as any stress tensor should be. Therefore (4) becomes +∂tU − ν∇2U + (U·∇)U + ∇ +�1 +ρP − τ +� += 0, +(8) +4 + +where the last term can also be written ∇·T , where +Tij = 1 +ρPδij − τij. +An intuitive feel for the role of the tensor τ in the transport of momentum can be gained by +looking closely at the case i = 1, j = 2. We relabel u′ +i = u′, u′ +2 = v′, x1 = x, x2 = y, and think +of the coordinates x1 and x2 as being horizontal and vertical, respectively. Then the average ⟨u′v′⟩ +appearing in (7) involves the horizontal and vertical components of the fluctuating part of the +velocity. Imagine a particle whose instantaneous velocity has both a positive x-component u′ and +a positive y component v′. The former says that the particle is bearing some x-momentum, and +the latter says that what it bears is being transported in the vertical (y) direction. The magnitude +of this vertical transport of momentum is proportional to the product of the two components, u′v′. +If one or both of these two components changes sign, this proportionality property is still seen to +hold. Therefore ⟨u′v′⟩ is proportional to the mean vertical transport of x-momentum (which could +be positive or negative) by means of turbulent fluctuations. This vertical transport represents the +mean force per unit area exerted by the fluid above the point under consideration on the fluid +below that point. Newton’s first law characterizes a force as a rate of change of momentum; in this +case, that rate of change is given by vertical transport of horizontal momentum-bearing particles. +Finally, the y-derivative +∂ +∂y⟨u′v′⟩, being a scaled difference of forces exerted at two nearby points, +is the net x-component of the mean force per unit mass exerted on particles at that location by +the turbulent fluctuations. +Generalizing these notions leads to an intuitive recognition of the last term in (8), −∇τ, as a +force produced by turbulent fluctuations. +If we knew the tensor τ, then we could use that knowledge in (8) so that (8) and (5) would +constitute a closed system for the determination of U and P. But nature is not so kind, and τ +is not known. Many attempts have been made in the past to remedy this basic defect, by writing +down other equations for the determination of τ. These models propose a “closure” mechanism to +produce a closed system for the mean values of various flow quantities. In all cases, these other +equations are at least partly ad hoc, and in most cases partly empirical. +3 +Early scaling concepts of Prandtl and von Karman +There have been many turbulence theories utilizing the ideas of Reynolds averaging; in fact the +ones to be discussed in this paper are important examples. Prandtl and von Karman were, at an +early stage, responsible for well known models in this category. Their concepts also included types +of characteristic lengths, which are allied to the scaling theme to be developed below. Therefore +we begin with brief discussions of some of their ideas. +In 1925, Prandtl [2] proposed a conceptual framework designed to provide insight into the +mechanisms producing turbulence in shear flow, and also to provide relationships among key fluid +dynamical quantities. +One aim of his was to relate the flux of momentum caused by turbulent fluctuations to the +gradient of mean momentum. That there is such a relationship was simply an assumption springing +from an analogy with conventional diffusion of quantities like heat in a stationary medium. In that +setting, Fick’s law asserts that the flux is proportional to the gradient of the concentration of +the substance (like heat) that is being transported. In the present situation, since ⟨u′v′⟩ is (see +previous section) a measure of the mean turbulent transport, in the vertical direction, of horizontal +5 + +momentum, the analog of the heat equation for the mean x-momentum U would be +⟨u′v′⟩ = −ǫdU +dy . +(9) +for some pseudo-diffusion coefficient ǫ. That coefficient is generally called the coefficient of eddy +diffusion, because one could argue that the transfer of momentum is caused by many eddies in the +fluid. +To say more about ǫ, Prandtl introduced the concept of mixing length ℓ, which more or less +represents the distance a fluid particle typically travels before transferring its momentum to another +particle. Various versions of Prandl’s theory have been involved with characterizing ℓ as it depends +on local or global properties of the flow. +Since we will be dealing with vague definitions, the symbol “∼” will be used to indicate a +relation which is not precisely defined, but rather should be considered as part of a modeling +concept. Moreover if numerical values are given to the two sides of the relation, they will be equal +only in order of magnitude. +Prandtl’s ideas related the left side of (9) directly to other local quantities as follows. +Suppose that the gradient dU +dy is responsible, in a linear manner, for the local characteristic mag- +nitude of the velocity fluctuations: |u′| ∼ k1 +���dU +dy +��� ; |v′| ∼ k2 +��� dU +dy +���. The proportionality coefficients ki +in these relations have to have the dimensions of length, and the most natural local characteristic +length is the mixing length ℓ. So we set ki ∼ ℓ to obtain +|u′| ∼ ℓ +���� +dU +dy +���� ; |v′| ∼ ℓ +���� +dU +dy +���� . +(10) +Then the average |⟨u′v′⟩| will be related to the quantity on the right of (10) squared: +|⟨u′v′⟩| ∼ ℓ2 +���� +dU +dy +���� +2 +. +(11) +Now apply this to (9), at the same time ensuring that signs are chosen so that ǫ > 0. We find +that +ǫ ∼ ℓ2 +���� +dU +dy +���� , +(12) +so that +⟨u′v′⟩ ∼ −ℓ2 +���� +dU +dy +���� +dU +dy . +(13) +It seems intuitive that the local distance ℓ should grow shorter as one draws near the wall, +because the wall exerts a constraint on the motions; and Prandtl sometimes proposed a linear +relation +ℓ ∼ y. +(14) +This is probably not a good approximation very near the wall, because it would predict that ⟨u′v′⟩ +grows like y2 as one moves away from the wall, whereas there is abundant evidence that the growth +rate is like y3. +This, then, is how Prandtl and others proposed handling a truly complex phenomenon (forces +caused by the turbulent transport of momentum) by relating it to a much simpler quantity (the +gradient of mean velocity). This proposal had very limited success in explaining the processes of +turbulence, although there was often good correlation with experimental data. +6 + +Von Karman [3] proposed his similarity hypothesis in 1930; a prominent ingredient was the +assumption that ǫ and ℓ should be characterizable in terms of local properties of the fluid. For +example, (14) would not qualify because the distance y from the wall is not a local property. One +might argue that the similarity hypothesis may approximate conditions far enough away from the +walls and centerline. +This hypothesis proceeds from rescaling considerations, the idea being that if lengths for the +velocity fluctuations in a neighborhood of a point in the flow are rescaled with a characteristic +length ℓ (which we can identify as essentially the mixing length above), then key hydrodynamic +quantities in that neighborhood are, except for an appropriate rescaling factor, functions only of +the rescaled lengths; not of the position in the flow. +One then searches for a local quantity with the dimensions of length that one could use to +characterize ℓ. The possibly simplest choice would be the ratio +ℓ ∼ −dU +dy +� +d2U +dy2 . +(15) +(The minus sign comes about from noting that the ratio itself is always negative.) +Note that if this and (14) are both true, one could solve for U(y) to get +U(y) ∼ C1| ln y| + C2, +(16) +an example of the renowned logarithmic profile. +Other theories, as we shall see, are in partial agreement with this result and suggest other +information as well. In particular, the theories brought out here in Secs. 5 and 6 involve ideas +reminiscent of those of Prandtl and von Karman, but the differences outweigh the similarities. +The main approaches to wall-induced turbulence that we shall examine are heavily involved +with various scaling concepts, and so we make a digression here to discuss some ideas basic to that +subject. +4 +The notion of scaling +Much of this section will consist of a review of scaling concepts, mostly well known, relating to the +remainder of the paper. It will be useful to give a formal definition of scaling patch (less well known) +in Sec. 4.1 and to formulate many of our results in terms of those patches. For example, it will be +important to know that the overlapping regions of the classical example sketched in Secs. 4.2 and +4.3 are not scaling patches. The classical Izakson-Millikan observation covered in Sec. 4.6 is best +known in the setting of turbulent flows, but is given here in a more general mathematical context +and in the form of an approximative statement (Sec. 4.7) which is mathematically rigorous. The +implications for wall-induced turbulent flow are discussed later in Sec. 5.6 and, more importantly, +in Sec. 5.7. +Models involving small or large parameters are commonplace in the natural sciences; in more +cases than not there are processes making up the action which operate on more than one, often +many, different space and time scales. The phenomenon being studied can then most clearly and +naturally be represented, in certain subdomains, in terms of functions of “rescaled variables”, or +of a combination of rescaled variables. Here rescaling means that new dependent and independent +variables are defined, in differential form, as linear transformations of the original ones, the coef- +ficients in the transformation generally being functions of the original small or large parameters. +7 + +Multiscaling refers to the event that more than one scaling are appropriately used, either in different +subdomains or simultaneously in the same subdomain. +Our focus will be on differential equations containing parameters which are supposed to be +small or large. +The ultimate goal here will be to gain some understanding of fluid motions by applying scaling +concepts to the averaged Navier-Stokes equations. The recognition that the dynamics of turbulent +fluids operate on a great many space and time scales has been a cornerstone of well known investi- +gations into those processes, including the construction of mathematical models. Our goal is more +limited, yet still daunting: to surmise information about important mean flow quantities on the +basis of the averaged, rather than the original, Navier-Stokes equations. Rather than positing, at +each specific location in the flow domain, an array of length scales, as would be the case for the +microscopic turbulent motions, the mean flow profiles will themselves have unique characteristic +lengths associated with each such location. +Our principal technique will be, first, to attempt to ascertain the local scaling behavior of those +mean quantities by means of the averaged equations together with judicious assumptions, and then +to derive further information about the mean profiles themselves. +In most of the following, there will be one independent small parameter called ǫ, although +another parameter β, which will also be small, will appear in some sections. The coefficients in +the scaling transformations will depend on ǫ, and possibly on β, and their orders of magnitude will +be of primary importance. The notation O(1) will refer to a quantity or function which generally +depends on ǫ, but is bounded above and below by positive constants independent of ǫ and β as those +parameters approach 0. When the lower bound is not assumed, we will generally write ≤ O(1). +The meaning of other order relations hopefully will be clear from the context. +4.1 +Scaling patches +We shall be dealing with functions of a single independent variable, and so the following discussion +will apply to that case only. Moreover, the independent variable will represent a space coordinate. +Finally, all statements about magnitudes in the following are to be understood in the order of +magnitude sense relative to the small or large parameter under consideration. For example if ǫ is +a small parameter, then ǫ has smaller order of magnitude than ǫ1/2 as ǫ → 0. To repeat what was +said above, we shall use the convention that a quantity is O(1) with respect to ǫ if it is bounded, +and also bounded away from zero, by positive constants independent of ǫ as ǫ → 0. +Consider an interval in the domain of the independent space variable, and a single rescaling (of +dependent and independent variables) in that subdomain. The interval may depend on the scaling, +hence on ǫ, and we assume the interval has size O(1) when measured in the rescaled distance +variable. +A “characteristic length” for the interval can be defined in terms of the scaling of the original +space variable (call it x), which produces a new space variable (call it ˆx). Using differentials, we +have, say, dx = α dˆx, with scaling coefficient α. Then a characteristic length for that subdomain +will be α. As ˆx traverses the O(1) length of the interval, x changes by an amount O(α). The +stipulation that the size of the subdomain (called a “patch” below) be of size O(1) in the rescaled +variable is designed thereby so that α is a proper definition of characteristic length. Importantly, +a patch cannot be artificially enlarged by adjoining a section in which the characteristic length is +larger. +The rescaling (including that of the dependent as well as of the independent variables) can +be thought of as being “natural” for a given problem if, when the solution is expressed in terms +8 + +of the rescaled variables, the scaled dependent variables are seen to undergo, in that subdomain, +variations which are not too large and not too small. This rate of variation could be gauged by +the magnitude of the rescaled derivatives. The requirement “not too large” then would be taken +to mean that all derivatives, of orders 1 up to some appropriate order, are bounded in magnitude +independently of the parameters in the problem. It usually happens that some of these derivatives +are necessarily zero or very small in places, so the corresponding (opposite) criterion cannot be +imposed to gauge the satisfaction of the requirement “not too small”. Instead, a proper criterion +would be that the characteristic length α associated with the scaling under consideration cannot be +decreased without the above criterion “not too large” being violated. In this, “decreased” means +in the order of magnitude sense: if α is replaced by a different function of ǫ with smaller order of +magnitude, so that in the newly scaled variables the derivatives are correspondingly larger, then +the magnitudes of some of these derivatives must not be bounded independently of ǫ. +Such an interval, together with its natural scaling, will in the following be called a scaling patch. +This appears to be a reasonable meaning for the concept of natural scaling in patches, but it +ignores, so far, the important question, how does one find the scaling patches? That is, how does +one determine the proper scaling in the proper locations? The search for scaling patches in wall- +bounded turbulent flow will be the primary activity in Section 6. But it is not an easy question in +general, and there are few easy mathematically rigorous criteria which can be applied to determine +them. Nevertheless, there are nonrigorous arguments, most easily introduced through examples. +Here is a straightforward one. +4.2 +A classical example +The following is an elementary textbook model example of a problem in which scaling consider- +ations, and in particular scaling patches, are very pertinent to understanding salient features of +the solution. The solution, it turns out, can be written down explicitly, but we choose a different +approach in order to better bring forth the ideas. Although it is well known, this example is chosen +because we want to put forward a slightly nonstandard point of view, and it bears some similarity +with the much more difficult averaged wall-induced turbulence problems which will be discussed +later. Both scenarios have traditional inner and outer scaling regions. On the other hand there are +also striking differences. +Let ǫ be a small positive parameter, i.e. 0 < ǫ ≪ 1. We wish to solve the following boundary +value problem for u(η): +ǫ2 d2u +dη2 − u + g(η) = 0 +for 0 < η < 1, +(17) +u(0) = α, +u(1) = β, +(18) +where α and β are fixed numbers and g is any given smooth function. To make things simple, we +suppose that g is independent of ǫ, and that it is not a constant (i.e. it has nontrivial variation). +The solution of course will depend on ǫ as well as on η, and the desire is to find that dependence for +all small enough values of ǫ, say for 0 < ǫ ≤ ǫ0, where ǫ0 is a fixed small number. This dependence +of the solution on both η and ǫ is sometimes expressed by writing u(η; ǫ). +As a first guess for an approximate solution, we try discarding the first term in (17), which has +a small factor ǫ2. We obtain the “reduced” problem +u = g(η). +(19) +9 + +The function u = g(η) approximately satisfies (17) in a formal sense, but is generally far from +satisfying the two boundary conditions (18) (unless by unlikely accident g(0) = α or g(1) = β). So +we must either scrap that solution altogether, or doctor it up. The latter is possible. +This is where the search for scaling patches comes in. Two things are clear: the locations of +the trouble are at the two boundaries η = 0 and η = 1. And secondly, somehow the discarded +term ǫ2 d2u +dη2 at those two locations must be important after all. We try constructing an “internal” +patch by excluding neighborhoods of the two troublesome boundary points as follows. Take any +number δ > 0 independent of ǫ, and use, for the patch, the original scaling, leading to (19), in the +subdomain defined by {η : δ < η < 1−δ}. This interval, together with the original scaling in which +the variables η and u remain unchanged, will be the first scaling patch. To lowest approximation in +the parameter ǫ, the differential equation (17) becomes (19). An important part of our argument +later will be centered around the issue of how far we can enlarge this interval by letting δ depend +on ǫ. +Now let us also try to construct a scaling patch which encompasses the left boundary η = 0. +The appearance of the differential equation can be changed by rescaling in such a way as to render +the derivative term overtly important. The most natural way is by passing to a new independent +variable y by +y = η +ǫ , +U = u. +(20) +Then we consider U to be a function of y in some subdomain consisting of values of η near 0. +We shall be conservative at first and take that subdomain as {0 ≤ y < y0}, where y0 is independent +of ǫ. Later, we see about extending the interval by letting y0 depend on ǫ +The rescaled differential equation is +d2U +dy2 − U(y) + g(ǫy) = 0. +(21) +Neglecting ǫ in (21) leads to +d2U +dy2 − U(y) + g(0) = 0. +(22) +The boundary condition at η = 0 (the same as y = 0) now becomes +U(0) = α. +(23) +It is proposed, then, that the scaling given by (20) in the interval {0 ≤ y < y0} be our second +scaling patch, and that the solution in that patch be approximately a solution of (22) and (23). +Moreover it is reasonable to impose the condition that U be bounded when y grows large (like +O(1/ǫ)). This provides a unique solution and the formal approximation +U(y) ≈ g(0) +�1 − e−y� + αe−y. +(24) +This, of course, is only an approximation which is put forward for further verification. For +one thing, notice that limy→∞ U(y) = g(0) and that the original approximation u = g(η) is also +close to the value g(0) for η ≪ 1. Thus in a sense the approximation at the left boundary meshes +smoothly with the supposed approximation in the interior of the interval. Let us dwell on the idea +of smooth meshing a little more carefully. Let ω be any positive number, arbitrarily small. We will +show that there are large values of y and small values of ǫ for which the boundary approximation +U(y) and the interior approximation g(η) = g(ǫy) are both closer than ω to g(0), hence arbitrarily +10 + +close to each other. Let y1 be a large number, depending on ω, such that |U(y) − g(0)| < 1 +2ω for +all y > y1. Such a number exists, as you can see from (24). Next, let η1 be a small number, again +depending on ω, such that |g(η) − g(0)| < 1 +2ω for all η < η1. Such a number η1 exists because of +the continuity of the function g. Since η = ǫy, both of these statements will be valid if ǫ is chosen +such that ǫ < η1 +y1 , and y is in the interval y1 < y < η1/ǫ. Since both are valid for these values of y, +necessarily |U(y)−g(0)| < ω. The conclusion is that if ǫ is small enough, depending on ω, there are +places where the inner and outer approximations are closer together than ω, which can be chosen +as small as desired. This is what we mean by the two approximations meshing smoothly together. +In summary, we have found (a) what appears to be a reasonable approximation for the solution +u(η; ǫ) in regions of the interval [0, 1] not too close to either boundary, and also (b) a reasonable +approximation in regions close to the left boundary. +The same procedure yields a third scaling patch near the right hand boundary. The solution in +it undergoes a similar exponential (in terms of a rescaled variable) transition from the value β to +g(1) as we move left from that point. The rescaling in that case is +z = 1 − η +ǫ +, +V (z) = u(η) = u(1 − ǫz), +and the right inner solution in that patch is +V (z) ≈ g(1) +�1 − e−z� + βe−z. +(25) +In the interior, the solution is approximately a function of only η with no ǫ-dependence, so +no new scaling is necessary. In a region which excludes small neighborhoods of each boundary +(we will discuss how small later), the original variables are the natural ones in our adopted sense. +That region, together with the original unscaled variables, constitutes one scaling patch. The small +interval y < y0, i.e. η < ǫy0), together with the rescaling (20), constitutes a second scaling patch, +in which the solution undergoes, in an exponential manner, a transition from the imposed value α +to the value g(0) associated with the first scaling patch, but extended to the forbidden boundary +point. At this point the two patches do not touch each other but, as we shall see, the ranges of +validity of the corresponding approximations can be enlarged so that they overlap. +Thus through proper “scaling”, the original problem may be clarified and simplified in certain +subranges of the range of independent variables. +The outlined procedure for finding scaling patches in the example problem is only heuristic, +but it can be made rigorous in this and other cases. +Moreover, we have only found the most +basic (lowest order) approximation. Successive higher order approximations, with errors of orders +O(ǫ), O(ǫ2), etc., can be found and proved to be correct in this and in a large class of similar +problems, such as elliptic boundary value problems when there are several independent variables. +4.3 +Inner, outer, and overlapping regions +We have constructed independent approximate representations of the solution in three subdomains: +(19) for η not too close to either boundary point, (24) near the left boundary, and (25) near the +right one. +Traditionally, these three approximate solutions are called the outer, left inner, and right inner +solutions. We denote the outer solution by uo(η) and the left inner by U(y). In the following, we +shall usually disregard the right inner approximation because its analysis follows that of the left +inner solution. +11 + +We now ask about the possibility of extending those subdomains, while retaining the validity +of the approximate representations. We do this by introducing a family of possible intermediate +scalings parameterized by an exponent γ in the range 0 < γ < 1. The independent variable η is +rescaled to obtain a new variable s by +η = ǫγs, +y = ǫγ−1s. +(26) +We are excluding the choice γ = 0, which would correspond to the original scaling, and γ = 1, +corresponding to (20). Set Uγ(s) = u(η) = u(ǫγs). +The rescaled differential equation (17) becomes +ǫ2−2γ d2Uγ +ds2 − Uγ + g(ǫγs) = 0, +(27) +and the boundary condition is U(0) = α. We ask, can the region defined by {0 < s0 < s < s1}, +where s0, s1 are independent of ǫ, be the location of a new scaling patch? (The answer will turn +out to be no in the strict sense, but we can try.) If so, the corresponding solution should, to lowest +order as ǫ→0, hence ǫγ→0, satisfy (27), which to lowest order comes out to be +Uγ(s) = g(0) +(28) +(since 0 < γ < 1). That is a very simple approximate solution indeed. +Keeping that in mind, we now examine the left inner and the outer solutions (24) and (19) in the +proposed patch. For the left inner, we obtain the expression U +�ǫγ−1s +� ≈ [g(0) (1 − e−y) + αe−yg(0)]y=eγ−1s. +To find the lowest order version of this expression, just let ǫ→0. We get, again to lowest order, the +same result as (28): +U +� +ǫγ−1s +� +≈ g(0). +(29) +In the case of the outer solution, a similar procedure yields +uo (ǫγs) ≈ g(0). +(30) +This means that in any of the proposed intermediate scaling patches (as it will turn out, they +are not true patches), the inner and outer solutions are both approximate solutions of the rescaled +equation (27), so that in this sense each of them is a valid approximation to the true solution, for +small enough ǫ . +This verifies that the original subdomains of the inner and outer scalings can be extended to +include the intermediate regions, the corresponding approximations continuing to be valid. In other +words, there is a region of overlap, in which both the inner and outer solutions are valid. In fact, +there are many overlap regions, depending on the choices of γ, s0, and s1. These are all approximate +solutions, and the main effective differences among them lies in the accuracy of the approximation. +Bear in mind that ǫγ is approximated by 0 better when γ is larger. +Finally, note that the approximate solutions in any of these overlap regions are not very interest- +ing: they are all constant to lowest order, equal to g(0)! This implies that the regions {s0 < s < s1}, +together with the scaling (26), do not form legitimate scaling patches! The reason is that the solu- +tion does not satisfy one of the stated criteria in section 4.1, namely that the length scale α, which +by (26) is equal to ǫγ, can certainly be reduced without the scaled dependent variable, which we +have seen is constant, having unbounded derivatives. +The same procedure yields overlapping regions near the right boundary point, in which the +approximate solutions are all another constant, g(1). +12 + +4.4 +A uniform approximation +The separate approximations in the three specified regions, namely uo(η) = g(η) (19), U(y) (24) and +V (z) (25), can be combined into a single expression, providing a uniform approximation throughout +the interval [0, 1]. For this, we define +F(y) = U(y) − g(0), +H(z) = V (z) − g(1). +(31) +The uniform approximation is then +Uunif(η; ǫ) = uo(η) + F(y) + H(z) = uo(η) + F(η/ǫ) + H((1 − η)/ǫ). +(32) +Its validity can be verified directly in each of the regions considered above. This form consists of a +sum of terms, each a function of one of the scaled variables. +4.5 +A generalization +It is natural to ask whether the outlined features in the overlap region of that classical example +apply also to a wider class of functions, and indeed they do. Rather than starting with a differential +equation, we take a general class of functions expressed in the form of a sum of individual functions +of the three scaled variables η, y, z separately, as in (32). So we consider any function in the +form (32), irrespective of whether it is a solution or approximate solution of some problem. Such +a function may still have, depending on uo, F and H, an outer approximation and two inner +approximations. +It may also have a region of overlap between the outer and one of the inner +solutions. +No generality is lost by taking H = 0, and we do that. Let ǫ ≪ 1, y = η +ǫ, and uo(η) and F(y) +be any functions such that uo(η) has a limit as η→0 and F has a limit as y→∞: +lim +η→0 uo(η) = uo(0); +lim +y→∞ F(y) = G. +(33) +We call η the outer variable and y the inner variable. Finally, let +u(η; ǫ) = uo(η) + F(y) = uo(η) + F(η/ǫ). +(34) +This will be the central prototypical two-scaled function which will be approximated differently in +an inner and an outer region. +In a region {η > δ} for any fixed δ > 0, η/ǫ→∞ uniformly as ǫ→0, so that F can be approxi- +mated by G: +u(η, ǫ) ≈ uo(η) + G ≡ uout(η), +(35) +which we call the outer representation of u. Similarly near η = 0, we replace η by 0 in the first +term on the right of (34) to obtain the inner representation +uin(y) ≡ uo(0) + F(y). +(36) +Now consider a family of intermediate regions parameterized by γ ∈ (0, 1) in which s, defined +by (26), is confined to the interval s0 ≤ s ≤ s1, the si being specified constants. In such a region, +we find the approximations +uin ≈ uo(0) + F(ǫγ−1s) ≈ uo(0) + G, +(37) +13 + +uout ≈ uo(ǫγs) + G ≈ uo(0) + G, +(38) +u(η, ǫ) ≈ uo(ǫγs) + F(ǫγ−1s) ≈ uo(0) + G. +(39) +The conclusions are, first, that the inner and outer representations are both valid in the interme- +diate region, which is therefore an overlap zone. Secondly, in the overlap region, the approximation +is equal to the constant uo(0) + G. +These conclusions are the same, in our generic class of examples, as those for the classical +example in section 4.2. +4.6 +The Izakson-Millikan observation +In [4] and [5], Izakson and Millikan independently considered functions of the form (34) without +imposing limit conditions such as (33). Their context was the averaged equations of wall-induced +turbulence, but their reasoning is valid in the present more general scenario. They showed, more +or less, that an assumption of the existence of an overlap region is sufficient to imply that the +approximations in that region are either constant or logarithmic. Beginning in Sec. 6, a different +approach, but with a similar objective, will be discussed in detail. +Its method, limited to the +turbulence problem, will be based on a search for scaling patches +In the Izakson-Millikan argument, then, no special conditions are required at either boundary +of the interval I (below) on which our functions are defined. +Moreover, instead of providing +functions of inner and outer variables and asking whether there is an overlap region of validity +of the corresponding inner and outer approximations, we now do just the opposite. We assume +that there is such a region of common validity, and ask what more, if anything, that implies about +those approximations. The answer is surprising. +Assume, for some unknown functions uo(η), U(y) (where y = η/ǫ as always), G(ǫ) and interval +I, that +uo(η) + G(ǫ) = U(y) +(40) +for all values of η ∈ I and all values of ǫ in some interval K. Since η and ǫ can vary independently +of each other in their respective domains, η and y also vary independently of each other, for η ∈ I +and y ∈ J, J here being the set of all values of y = η/ǫ for η ∈ I and ǫ in its previously allocated +interval of variation K. The assumed identity (40) asserts that there is a region of overlap where +the outer function uo(η) (plus a correction G(ǫ)) equals the inner function U(y). All these functions +are at present unknown. It turns out that the correction G is necessary to include, because without +it, (40) is too stringent a condition to be fulfilled in general. +First, in (40) make ǫ a fixed number in K, and of course y = η/ǫ for all η ∈ I. Differentiating +(40) with respect to η and multiplying by η, we obtain +ηu′ +o(η) = η +ǫ U′(y) = yU′(y). +(41) +This is valid for every chosen value of ǫ ∈ K, so is true for all y ∈ J as well as η ∈ I. But since, as +remarked before, η and y vary independently of each other, each side of (41) must be a constant. +For example η can be allowed to vary throughout I while y is held constant, which of course implies +that the right side of (41) is held constant. That means the left side is also constant. Therefore for +some constant A, +ηu′ +o(η) = A; +yU′(y) = A. +(42) +Integrating these two equations, we find, for some other constants B and C, +uo(η) = A ln η + B, +U(y) = A ln y + C = A ln η + C − A ln ǫ, +(43) +14 + +the last equation a result of the identity y = η/ǫ. Since uo is assumed to be a function only of η, +necessarily B is a constant independent of ǫ, and the same holds true of C for a similar reason. +The function G can now be determined if we substitute the expressions (43) back into (40). Doing +so while noting that ln y = ln η − ln ǫ yields +G(ǫ) = −A ln ǫ + C − B. +(44) +In short, the three functions in question must be of the form (43), (44), where A, B, C are +constants. The original assumed identity (40) dictates a great amount of information about these +functions, while not specifying them exactly. +What this result says is that functions which are simultaneously regular functions of an inner +and an outer variable, for all values of those variables in certain intervals, are either constant or +logarithmic. If it is known that the function is not constant, then this property of simultaneity is +equivalent to being logarithmic. +4.7 +A more realistic version +The result of Izakson and Millikan is mathematically rigorous, in the sense that if (40) holds in +some intervals as specified, i.e. if some function can be expressed exactly in terms of either an inner +or an outer variable in some overlap region, then the logarithmic properties (43) and (44) also hold +in that same region. +However, the assumption (40) probably never holds exactly in practice, so that the conclusion +is vacuous. In any concrete situation, it will be far more reasonable to assume that (40) is only +approximately satisfied. Then the suggestion, which is not yet proved, is that the functions involved +will be approximately logarithmic. +Fortunately, such a conclusion can also be rigorously established, provided that appropriate +senses are given to the two approximations (the one in the hypothesis, and the one in the conclusion). +That will be done in this section. Note also that Gill in [6] provided such a proof under more involved +assumptions and with entirely different methods. +In place of (40), we now assume that (40) holds with an additional small error term r(η, y, ǫ), +which will be written r(η, y), since ǫ can be expressed in terms of η and y. The function r is not +known exactly, but that does not prevent estimates about the accuracy of (43) and (44) being +deduced in terms of the magnitude of r (and, as it turns out, of its derivatives). Thus we start with +uo(η) + G(ǫ) = U(y) + r(η, y). +(45) +As we shall show, although (43) and (44) no longer hold, one can still find upper and lower bounds +of logarithmic type for the functions uo, U, G in terms of bounds on r and its derivatives. +Differentiating (45) as before, one obtains +ηu′ +o(η) = yU′(y) + R(η, y), +(46) +where R(η, y) = ηrη + yry, subscripts denoting derivatives. Let ρ be an upper bound for |R|, valid +for all η, y in the intervals I, J respectively. Choose any y0 ∈ J and let A = y0U′(y0). Then +setting y = y0 in (46), we get +|ηu′ +o(η) − A| ≤ ρ. +(47) +This holds for all η ∈ I. Also from (46), +yU′(y) − A = ηu′ +o − A − R, +15 + +so that by (47) +|yU′(y) − A| ≤ |ηu′ +o − A| + ρ ≤ 2ρ. +(48) +This says that when ρ is small, the quantities on the left sides of (42) are almost constant, the +deviation from constancy being no greater than ρ and 2ρ, respectively. Dividing (47) by η and +rearranging terms, we get +A − ρ +η +≤ u′ +o(η) ≤ A + ρ +η +, +and after integrating, +(A − ρ) ln η + B1 ≤ uo(η) ≤ (A + ρ) ln η + B2, +(49) +where Bi are integration constants. In this sense, uo is almost logarithmic when ρ is small. In the +same way we obtain +(A − 2ρ) ln y + C1 ≤ U(y) ≤ (A + 2ρ) ln y + C2, +(50) +with similar upper and lower bounds for G. +In short, assuming R, i.e. ρ, is small in the intervals selected, one concludes that U(y) is bounded +above and below by logarithmic expressions with coefficients of the logarithm functions which are +close to each other, the discrepancy being smaller than 2ρ. It must be admitted, however, that one +generally does not know the nature of the error term R, its magnitude ρ, or indeed the interval +of overlap, if any. If the intervals I, J, hence K are chosen to be shorter and well placed, one +might expect ρ to be smaller and therefore the logarithmic approximations to be better. There +is a tradeoff between the accuracy of the logarithmic approximation and the size of the interval +where that approximation is valid. It can be guessed that as one moves closer to the canonical +outer domain, r increases in magnitude because although the outer approximation uo(η) + G(ǫ) is +more exact, the inner expression U(y) is not a good approximation, so the two cannot be close to +each other, as (45) would seem to imply. Somewhere between the outer and the inner domains, r +would be minimal, but still not zero. This was with ǫ fixed. A natural further question, therefore, +is whether, in such fixed intervals of the space variable η, the error r and its relative R approach +0 as ǫ→0. The present argument does not answer that question, but an argument in support of a +similar conclusion in the context of wall-induced turbulence will be given below in Sec. 6.6. +5 +The mean structure of turbulent Couette flow in a channel +At this point we leave the mathematical digression about scaling and turn to the main issue of this +paper, namely the application of scaling ideas to turbulence induced by wall friction. Turbulent +Couette flow is possibly the simplest nontrivial example. From this point through Sec. 5.6, we +will be repeating well-known standard arguments but with different emphases. Newer, hence lesser +known, material will be found in sections beyond that. +Denote the components of x and u by (x, y, z) and (u, v, w) respectively. Consider a channel +bounded on top and bottom by horizontal planes {y = 0; y = 2δ}. The top plane moves with +given steady velocity in the streamwise direction x, while the lower plane remains stationary. This +causes a shear stress in the fluid between the two planes. In the mean, fluid particles which are +vertically aligned at one moment of time slide past each other horizontally. +If the velocity is sufficiently large, the resulting shear causes the flow to be turbulent. We seek +to understand the “scaling structure” of the mean velocity of the fluid, and other quantities, when +the flow has reached “equilibrium”. The scaling analysis of turbulent Couette flow described here, +was developed in [7, 8, 9, 10, 11]. Here scaling structure will mean that we will look for scaling +16 + +patches for the mean velocity and Reynolds stress profiles, or other relevant ways to scale those +functions. And the term equilibrium will refer to the state in which the mean velocity is everywhere +horizontal and depends, with Reynolds stress, only on the normal coordinate y. Although there +will be perhaps violent particle fluctuations in all directions, the prevailing (mean) motion will be +only horizontal. +The dependence only on y implies that all averaged quantities are uniform along the channel +and do not change in time. The upshot is that the averaged Navier-Stokes equations have only one +independent variable (y). +Before we get into the turbulence analysis, consider the corresponding laminar flow, in which +all fluctuation parts vanish, and in fact the velocity has only an x-component, which is simply a +linear function of y, the coefficients being adjusted to match the given velocities of the bounding +planes. This is a very simple solution, and can be verified to be a solution of the Navier-Stokes +equations (1) with 0 pressure gradient. In fact the inertia terms of the conservation equation for +the x-component of momentum vanish automatically because each velocity component is either 0 +or has x-derivative 0. For a similar reason, the viscosity terms also vanish. +The analogous problem for turbulent flow is orders of magnitude more difficult, and can only +be solved imprecisely. We will examine it in the framework of Reynolds averaging. +But a possible paradox appears. It was just brought out that the eminently simple laminar +flow, in which u is a linear function of y, is an exact solution of the Navier-Stokes equations, and +those equations govern the flow, laminar or turbulent. So if we have a solution, why do we need to +look for a “turbulent” one? The answer is because (a) as we set up the problem, there are more +solutions than just the laminar one; solutions are not unique; and (b) the laminar one happens to +be very unstable at high Reynolds numbers, therefore not seen in practice and hence unphysical. +5.1 +The differential equations +To proceed, suppose the turbulent flow to be “fully developed,” statistically stationary and two- +dimensional in the channel. All averaged quantities except pressure do not depend on x, z, or t; +just on y. Moreover, we suppose that the only nonzero component of the mean velocity will be the +x-component, which we denote by U(y). (This part is the same as the laminar case.) +The x and y components of (8) are greatly simplified, because of the independence of τ on +x, z, t: +−ν d2U +dy2 + 1 +ρ +∂P +∂x + d +dy⟨u′v′⟩ = 0, +(51) +1 +ρ +∂P +∂y − d +dy⟨(v′)2⟩ = 0. +(52) +At the two walls {y = 0, y = 2δ}, all the fluctuating parts of u are 0, so the Reynolds stress +⟨u′v′⟩ vanishes as well. At the lower wall, U = 0 and its derivative is related to the frictional stress +exerted on the wall by the fluid (or vice versa). +We have here two equations for the four unknown functions U, P, ⟨u′v′⟩ and ⟨(v′)2⟩. Actually, +Couette flow is characterized by the absence of an applied pressure gradient; the sole impetus for +the flow is the differential motion of the two walls. Therefore we set ∂P +∂x = 0 (P itself depends on +y, however, as one can see from (52)). In the case of other steady channel flows, ∂P +∂x may not be +zero, but it would turn out necessarily to be constant. +The equation (51) becomes +−ν d2U +dy2 + d +dy⟨u′v′⟩ = 0. +(53) +17 + +It is important to recognize that this is a simple balance of forces, which must occur at each point +of this steady (in the mean) flow. The two forces exerted on the fluid are (1) the friction, or viscous, +force ν d2U +dy2 , and (2) the force − d +dy⟨u′v′⟩ caused by the turbulent fluctuations, i.e. the gradient of +the appropriate Reynolds stress, by which x-momentum is transported in the y (perpendicular) +direction by the fluid’s fluctuations in both directions. An intuitive description of the two forces +can be given. (1) The derivative dU +dy measures the magnitude of the shearing motion, i.e. the rate +at which nearby x-directed particles are moving relative to one another. This shearing magnitude, +times the kinematic viscosity ν, is proportional to the force per unit area that the fluid lying above +the horizontal plane through the point under consideration is exerting on the fluid below (and vice +versa). This is called the shear stress. The difference between the shear stress at two nearby values +of y, say y1 and y2, is the net effective force per unit area experienced by the slab of fluid in the +intermediate region {y1 ≤ y ≤ y2}. Dividing this expression by y2 − y1 and passing to the limit, we +obtain the net force per unit mass acting on fluid particles at that location, and see that it equals +ν d2U +dy2 . Again, this is termed the viscous force. +(2) As shown in Sec. 2, the y-derivative +d +dy⟨u′v′⟩ is the x-component of the mean force per unit +mass exerted on particles at that location by the turbulent fluctuations. +Since there are only these two forces in balance, it can be said, contrary to some assertions, that +the viscous forces and those due to Reynolds stresses are both equal players, hence both important, +everywhere in the flow. +At this point we have reduced the problem to a single equation (53) for U and ⟨u′v′⟩. There are +still more unknowns (2) than equations (1). But we shall nevertheless be able to at least surmise +some important information just from the simple mean balance law (53) plus known boundary +conditions, in concert with educated assumptions. +5.2 +The friction velocity and boundary conditions +We start with some crucially important concepts and constants associated with the interaction of +the fluid with the wall. Let τw denote the mean stress exerted on the wall by the fluid flowing past +it. It is proportional to the viscosity µ, as well as the magnitude of the mean velocity’s shear at +the wall. It is given by +τw = µ d +dy⟨u⟩ = µ +�dU +dy + d +dy⟨u′⟩ +� += µdU +dy , +(54) +since ⟨ d +dyu′⟩ = +d +dy⟨u′⟩ = 0. At y = 0, ν dU +dy = 1 +ρ(µ dU +dy ) = 1 +ρτw > 0. Now the quantity 1 +ρτw has the +dimensions of velocity squared; it therefore defines a characteristic velocity u∗, called the friction +velocity, by +1 +ρτw = (u∗)2. +(55) +From this relation and the previous one, we can express +dU +dy = u∗2 +ν +at y = 0. +(56) +This, together with the stipulation that +U(0) = 0, +(57) +provide a pair of boundary conditions at the wall for U(y). It seems natural to treat the velocity of +the upper wall as a given quantity to be built into the mathematical formulation; but analytically +18 + +the simpler route is instead to think of the velocity u∗ (in (56)) as given, and the upper wall velocity +as to be determined. It is clear from physical considerations that either of these two velocities is a +monotone function of the other. +As noted before, the fact that all fluctuations vanish at the wall imply similar conditions for +the Reynolds stress. In fact, +⟨u′v′⟩ = d +dy⟨u′v′⟩ = d2 +dy2 ⟨u′v′⟩ = 0 +(58) +at y = 0. The first two conditions here follow from u′ = v′ = 0 at the wall. The third condition +comes about because, using subscripts y to denote derivatives, we have ⟨u′v′⟩yy = ⟨u′ +yyv′⟩+2⟨u′ +yv′ +y⟩+ +⟨u′v′ +yy⟩. The first and third terms in this expression vanish at the wall because u′ and v′ do; and +the second term also vanishes because (6) together with u′ +x = 0 implies v′ +y = 0. +Finally at the center of the channel at y = δ, we invoke the symmetry of the flow to conclude +that +d +dy⟨u′v′⟩ = 0 at y = δ, +(59) +since every vertical fluctuation v′ is matched, and canceled during the averaging process, by another +one in the opposite direction. From (53), U has an inflection point. In fact, consistent with (59), +the Reynolds stress ⟨u′v′⟩ increases in magnitude from 0 at the wall to a maximal value at the +centerline. +5.3 +Dimensionless variables in the core region +For better insight, we now seek to nondimensionalize (53). That is, we rescale the variables by +multiplying them by typical and meaningful characteristic dimensional constants so that the results +are dimensionless. There are at least two traditional and natural ways to do this. In any case, +we argue that the characteristic velocity in this turbulent flow should probably be u∗, because the +former’s magnitude should be directly related to the wall stress, which is what slows the fluid down +at the lower wall and causes it to go forward at the upper wall, and so in some sense causes the +turbulence. So we nondimensionalize U by u∗. (As brought out above, another natural, but less +convenient velocity unit would be the maximum mean velocity or the average mean velocity.) +In the interior of the channel, it is intuitive that y should probably be scaled by the channel +half-width δ, simply because that is the only immediately obvious characteristic length appropriate +to that part of the channel (further justification will be given in Sec. 6.8). And everywhere, the +Reynolds shear stress should be scaled by the shear stress at the boundary, so by (u∗)2. We therefore +define +U+ = U +u∗ , +η = y +δ , +T = −⟨u′v′⟩ +(u∗)2 . +(60) +The centerline of the flow region is at η = 1, and by symmetry considerations, the mathematical +problem can be set up in the half-channel 0 ≤ η ≤ 1. The superscript “+” in U+ is the traditional +way to signify that this variable is normalized with parameters related to what happens at the +wall: the friction velocity u∗ and/or τw. The scaled distance η is the first natural way to nondi- +mensionalize the y-coordinate, and η will be called the “outer distance variable”. This, along with +the “inner distance variable” y+ to be discussed in Sec. 5.4 below, have been standard choices for +dimensionless distance since the beginning of theoretical investigations of wall-induced turbulence. +However, the existence of scaling patches as defined in Sec. 4.1 employing these scaled distances +remains to be established. That will be done in Section 6.8. +19 + +With the normalization (60) we obtain in place of (53), +dT +dη + ν +u∗δ +d2U+ +dη2 += 0. +(61) +We also use u∗ and δ to define our Reynolds number R∗ = u∗δ +ν +and small parameter ǫ2 = (R∗)−1. +Using this notation, we get +dT +dη + (R∗)−1 d2U+ +dη2 += dT +dη + ǫ2 d2U+ +dη2 += 0. +(62) +As noted earlier, we have an underdetermined problem—a single equation for the two unknowns +T and U+. But there is even more bad news: the boundary condition at η = 0 is in trouble. Naively +assuming the formally small second term can be neglected across the whole channel, we get that +T is constant, and since it vanishes at the wall, we would obtain that T = 0 everywhere. This is +incorrect, of course, for a reason similar to that which applies to the classical example in Sec. 4.2: +a different scaling applies near the wall. +We construct now a second scaling domain, near the wall, to partially remedy this defect, as +was done for the classical example in Sec. 4.2. +5.4 +The wall layer, rescaling, and law of the wall +It is traditionally recognized that we do have at least two space scales in the channel—one of them +associated with δ and another one close to the wall; we’ll get to that shortly. We will have an “outer” +and an “inner” approximate solution. Finding out how to connect them makes an interesting story. +However, we cannot carry out a full-blown asymptotic analysis because too much is unknown. +Much of the following proceeds by “reasonable suppositions”, which can also be verified to some +extent by experiments. +It is natural to choose the inner scaling in such a way that the two terms on the left of (62) +have the same orders of magnitude. After all, those two terms represent the two forces in the fluid +which have to balance (note, that’s not what we did to get (61)). Thus we define +y+ = ηR∗ = u∗ +ν y. +(63) +Then (62) becomes +dT +dy+ + d2U+ +dy+2 = 0. +(64) +In this, there is no incompatibility with the boundary conditions at y+ = 0: +T = 0 and dU+ +dy+ = 1 at y+ = 0. +(65) +The integrated form of (64) is +T + dU+ +dy+ = 1. +(66) +To summarize, we have the traditional approximation that asserts that T is constant (but no +approximation as yet for U+) in the outer region near the channel’s centerline, and an equation +(64) or (66) relating T and U+ in the inner region. +20 + +The region next to the wall where the spatial variations (in y) have characteristic length +ν +u∗, is +call the wall layer (as opposed to boundary layer). The choice (63) of scaling in this region says +that we are treating the scaled Reynolds stress T on the same footing as the wall stress or skin +friction (R∗)−1 dU+ +dη . The former arises from the inertia terms in the Navier-Stokes equations and +the latter from the viscosity terms. +In the wall layer, then, it is reasonable to suppose that U+ and T are (approximately) functions +only of the inner variable y+. This property is called the law of the wall. +5.5 +Velocity in the core +The law of the wall is no longer valid for large R∗ as we move into the interior of the channel and +on into the core region. The outer scaling will turn out to be valid there. +But remember, at this point we only have that T is approximately constant there. U+ may be +unbounded as a function of ǫ; the traditional way to incorporate the validity of the outer coordinate +η in an expression for U+ is to postulate the defect law +U+ = U+(1) + h(η), +(67) +where h is unknown, except that h(1) = 0. +The concept of thickness of the wall and the core scaling regions should be clarified. +One +interpretation of these regions would be where the solution’s characteristic length is unity in the +appropriate scaled coordinate. +Then the thickness would be O(1) as measured in that scaled +variable, be it y+ or η. In particular, the wall region is characterized by {y+ ≤ O(1)}, and that +of the outer scaling region is characterized by η0 < η ≤ 1 for some arbitrary positive number η0 +independent of ǫ. Note that the ratio of the thickness of the wall layer to that of the core is O +� +1 +R∗ +� +; +this contrasts with the laminar hydrodynamic boundary layer (Prandtl theory), where the ratio is +O +� +Re−1/2� +. +Another interpretation of the two regions would be where one or the other of the approximate +representations of the solution, namely the law of the wall or the defect law, are valid. +This +interpretation yields, as it turns out, larger regions. +Let us assume that these two laws are correct in their respective domains; better justification +for this will be given later in Sec. 6.8. Recall +ǫ = (R∗)−1/2; +(68) +since U+(1) will depend on ǫ, we set U+(1) = G(ǫ). +In the outer region, then, we may posit the representation +U+ = G(ǫ) + uo(η). +(69) +In the inner region, on the other hand, we are using the law of the wall approximation +U+ = p(y+), +(70) +i.e. U+ is a function only of y+. +The unknown quantities here are p, G, uo. +21 + + 0 + 5 + 10 + 15 + 20 + 25 + 30 + 35 + 40 + 0.1 + 1 + 10 + 100 + 1000 + 10000 + 100000 + 1e+06 +U +y +Reτ=181 (Kawamura et al.) +Reτ=642 (Iwamoto et al.) +Reτ=1655 (Wei and Willmarth) +Reτ=11062 (McKeon et al.) +Reτ=103340 (McKeon et al.) +Reτ=534538 (McKeon et al.) +1./0.41*ln(y)+5. +Figure 1: Inner normalized mean streamwise velocity in Couette flow, pressure-driven channel flow +and pipe flow. Couette flow data are from DNS of Kawamura’s group [13, 14]. Pressure-driven +channel flow DNS data are from Iwamoto et al [15] and experimental data are from Wei and +Willmarth [16]. Pipe flow data are from superpipe data of McKeon et al [17]. +5.6 +Application of the Izakson-Millikan reasoning +A great number of analytical and semianalytical studies of turbulent mean profiles have utilized the +Izakson-Millikan observation as an essential ingredient. It has also been generalized and elaborated +upon in many ways. For example the idea of composite expansions in more traditional settings of +two-scale problems has been extended to the present scenario. We recommend the review paper +[12]. +Given the two approximations (69) and (70), they can be thought of as the inner and outer func- +tions specified in the two sides of (40).The variables p, uo, G, y+, η are analogous to U, uo, G, y, η. +Thus the left side of (40), uo(x) + G(ǫ), is analogous to the right side of (69), and the right side of +(40), U(y), is analogous to the right side of (70). +If we now make the Izakson-Millikan hypothesis that there exists a common region in which the +two expressions are almost equal, then the three functions in question must be either approximately +constant or approximately logarithmic, as in (43) and (44). +In the present context, the conclusion is that in the common region, whatever it is, the following +hold: +uo(η) ≈ A ln η + B, +p(y+) ≈ A ln y+ + C, +G(ǫ) ≈ − A ln ǫ + C − B. +(71) +Therefore the functions are approximately either constant (A = 0) or logarithmic. +This is a well-known conclusion, and indeed the mean velocity profile in wall-bounded turbulent +flows is seen to exhibit logarithmic type behavior in certain regions which can be estimated on the +basis of experimental data. Examples of the logarithmic property can be seen from the empirical +data shown in Fig. 1. The coefficients A, B, C can also be so estimated. All in all, the Isakson- +Millikan observation, in all its simplicity, should be counted as one of the great success stories of +22 + +theoretical turbulence. +Focussing on our stated objective to at least try to answer the question why?, we discuss the +given derivation of the logarithmic property in some detail. In particular, we ask whether it can +be supported by alternative trains of thought. +5.7 +Observations on the foregoing procedure +The conclusion (71) gives a surprising amount of information about the inner and outer approxi- +mations, based on what appears to be a small amount of input. +The basis for the argument rests on very little physics or fluid dynamics; it is simply an as- +sumption about inner and outer approximations agreeing somewhere. If one is willing to admit the +existence of those inner and outer approximations, what remains is simply a mathematical issue, +and could apply to any situation where there are two space scales with different but overlapping +domains representing a strictly monotone function. +Let us rephrase what has been found in terms of the more realistic conclusion corresponding to +Sec. 4.7. If the mean velocity profile is everywhere monotone and there is a region in the flow where +that profile can be expressed approximately and simultaneously as a function of the inner variable +alone and the outer variable alone (up to an additive function of ǫ alone), then these functions must +be approximately logarithmic, the degree of the latter approximation being dependent on that of +the former. +Let us take for granted the monotone part; that property of the mean velocity profile is well +known and can be rationalized by the supposition that the viscous stress is everywhere positive +and a decreasing function of distance from the wall, which is the site of the imposition of such +stress by outside means. Given the monotonicity, what other information can we use to determine +the profile, at least approximately? The needed additional information should be theoretical in +nature, because our aim is to explain the reasons for observed behavior. We know the Izakson- +Millikan implication, which, stated succinctly, says, “overlap implies logarithmic”. It is a simple +piece of reasoning which is classical and, again, well-known. Being so simple and direct, it places +the hypothesis that a particular region is an overlap region very close to the conclusion, namely +close to assuming logarithmic behavior. Either property can be substituted for the other. If this +is true, the argument is close to being circular. If available, independent arguments to determine +properties of the mean velocity and Reynolds stress profiles, not relying on the overlap hypothesis, +would be highly desirable. That will be a principal aim in the following sections. +One direction of enquiry which is suggested is to consider all possible increasing profiles which +are functions of the inner variable near the wall and of the outer variable near the centerline; +and try to use some reasonable selection criterion to at least make a good argument for what the +actual profile should be. The set of profiles which have the overlap property, hence the logarithmic +property, is only a small part of the set of all conceivable profiles, because the latter includes an +ample collection of arbitrary non-logarithmic ones. If the overlapping logarithmic ones are to be +selected, as empirical evidence suggests in some regions, then theory should be able to supply a +reason for that choice. +We should mention in passing a small cloud hanging over the search process: it seems on the +basis of examples explored in Sec. 4 that overlap regions, if they exist, are usually characterized +by the functions being constants in those regions. Given that U is nowhere constant, one wonders +whether the assumption that an overlap region exists is itself reasonable. +Another possible direction of enquiry is to ask whether a unified approach is possible, which +gives more justification to the inner and outer scalings themselves, and at the same time is capable +23 + +of handling all other parts of the profile as well. +In the next subsection we draw attention to an intuitive and vague train of thought which may +be relevant to the question of why the mean profiles might have certain features. In Section 6, +a relatively new approach to this and other questions will be explored. It involves a systematic +search for scaling patches and does not require separate assumptions about the validity of inner +and outer scales. +Apart from the above, another consideration to be kept in mind is that the overlap hypothesis +provides no theoretical basis for determining the location of any expected overlap region, nor the +accuracy of (69) in that region (neither does the scaling patch approach, for that matter). It is +expected that the size of the region is related to the accuracy of the basic hypothesis. Increas- +ing the region where the two expressions are approximately valid decreases the accuracy of the +approximation in that region. +5.8 +The uniformizing effect of turbulence and some possible implications. +The turbulent nature of the fluid motion, which is not explicit in the averaged equation under +study, has a mixing and hence uniformizing effect. This, it may be argued, tends to smooth out +or eliminate abrupt spatial changes in the average properties of the flow as one moves from one +location to another. The flow properties at one place will be similar to what they are at nearby +places. +Although sources of stress which are imposed from the outside, such as wall friction and en- +joined pressure gradients, are often also sources of turbulence, they may work in opposition to the +uniformizing action. For example the generation of stress due to the wall in channel flow acts at +a different location from an imposed pressure gradient, and this difference results in a nonuniform +distribution of Reynolds and viscous stress. +Still, the claim of uniformization seems without much doubt to be valid for some properties if +the vague terms “properties” and “similar” are given some appropriate definition. +If we assume that this principle holds for the local scaling properties of the variables, including +for the local characteristic length as it depends on location, then there should be a whole continuum +of characteristic lengths, each appropriate to a specific locale, i.e. specific distance from the wall. +Thus as we pass from the wall to the centerline of the channel, the uniformizing principle suggests +that the characteristic length encountered during the passage should change continuously from the +one for the inner scaling to the one for the outer scaling. +In fact, this is one of the main conclusions to be brought out by a different reasoning in the +next section, which introduces and develops a very different approach to the problem of determining +the nature of the profiles. It uses scaling and order of magnitude arguments in a major way, but +does not assume a priori the validity of the inner or outer approximations. Rather, it presents an +alternative criterion for “scaling patches” and shows that the inner and outer regions, among many +others, fit that criterion. In this sense, the identification of patches is derived from the assumed +criterion rather than presupposed. +6 +The search for scaling patches +The more recent approach to understanding of the scaling structure of the mean velocity and +Reynolds stress profiles presented here was introduced in [7, 8, 9, 10, 11]. It forms an alternative +to the approaches considered above, not in any sense of replacing them, but rather in the spirit of +24 + +-1 +-0.5 + 0 + 0.5 + 1 + 0 + 20 + 40 + 60 + 80 + 100 + 120 + 140 + 160 + 180 + 200 +Tβ = T+ - βy+ +y+ +Couette: T+ +β=0.1000 +β=0.0700 +β=0.0465 +β=0.0150 +β=0.0055 +β=0.0020 +β=0.0004 +Figure 2: Adjusted Reynolds stress profile for various values of β. The case β = ǫ4 corresponds +within O(ǫ2) to the genuine Reynolds stress for Couette flow and β = ǫ2 is an approximation to +that for pressure driven channel flow. The DNS data are from [18], δ+ = Reτ = 181.3 and ǫ = .074. +adding to them new information and insights. Finally, it is applicable to a wide variety of wall- +induced turbulence scenarios. For the time being, we continue to operate within the context of +turbulent Couette flow through a channel. +6.1 +Adjusted Reynolds stresses, balance exchange phenomena, and the identi- +fication of patches +Let β be a small positive number. Restrictions on it will be given later. In terms of the originally +defined dimensionless Reynolds stress T, let +T β(y+) = T(y+) − βy+. +(72) +(Note that β is a superscript not an exponent.) The function T β is simply a mathematical construct +that will be called an adjusted Reynolds stress. It satisfies +dT β +dy+ = dT +dy+ − β, +(73) +and from (64) +d2U+ +dy+2 + dT β +dy+ + β = 0. +(74) +The T β are plotted for various values of β in Fig. 2. +The genuine Reynolds stress, which +corresponds to the case β = 0, vanishes with its derivative at the wall and rises to attain a +maximum at the centerline. As β increases, the position of the maximum moves toward the wall, +eventually disappearing. The main interest is in those adjusted stress functions that exhibit local +maxima—which will be the case when β is not too large—because, as it turns out, scaling patches +Lβ with characteristic lengths with orders of magnitude O(β−1/2) (as β → 0) exist at those peak +locations. The reasoning below will justify this assertion. Part of the argument involves obtaining +25 + +an exact differential equation in rescaled variables having no explicit dependence on ǫ or β. Another +part entails the recognition that (74) expresses an approximate balance between its first two terms +(since β is small), and that this balance is necessarily broken at some point and changed to another +kind of balance, because y+ eventually attains a value such that the three terms in (74) have the +same order of magnitude. +Let us pursue this idea of balance exchange. As was brought out in Sec. 5.2 following (59), +the function T(y+) increases from being 0 at the wall to attain its maximal value at the centerline +η = 1, i.e. y+ = ǫ−2 = δ+. Assuming that β is small and positive, therefore, we see that the +adjusted stress T β has negative derivative at y+ = δ+, so must attain its maximal value Tm(β) at +a point y+ +β < δ+. Moreover, the location of this maximum decreases as β increases, because when +β increases, the zero derivative at any maximum becomes negative. +Within the inner scaling region where the law of the wall holds, for example when y+ ≤ O(1), +the two derivatives in (74) will generally have magnitudes O(1) except very near the wall in the +viscous sublayer, where they are both very small. Since β ≪ 1, those two derivatives will balance, +except for an error represented by the last term in (74). Both derivatives will therefore generally +be O(1) quantities, except as noted above. This occurs within the inner scaling region. However as +y+ increases to a neighborhood of y+ +β , this necessarily changes, because the value of dT β +dy+ decreases +to zero at y+ = y+ +β . For points near enough to that value, the second term in (74) must take on +values ≤ O(β), and therefore by (74) again, the first term does as well. It is therefore natural to +propose that there may be a scaling patch occupying that neighborhood with respect to which all +three terms of (74) have the same formal order of magnitude. To be more precise, the last term +balances the sum of the first two terms, each of which is ≤ O(β). +It turns out that it is possible to construct a candidate for such a patch. It will be centered at +the location y+ = y+ +β , where dT β +dy+ = 0. At that point, the derivatives appearing in (74) annihilate +the linear terms in the Taylor series of the functions U+ and T about y+ = y+ +β (in fact the linear +term in T is identically zero). As a result, those linear parts do not play a role in the rescaling +process, and one may work only with the remainders after those parts have been separated off. +With this in mind, we write, for some coefficients α(β), γ(β), λ(β) to be determined, +y+ = y+ +β + αˆy, +T(y+) = Tm(β) + γ ˆT(ˆy), +U+(y+) = U+(y+ +β ) + m(y+ − y+ +β ) + λ ˆU(ˆy), +(75) +where m is the slope m(β) = dU+ +dy+ (y+ +β ) of the mean velocity profile. The slope m is unknown at this +point, but will be found later in Sec. 6.4.3. The linear parts, which have been segregated in (75), are +Tm(β) and U+(y+ +β ) + m(β)(y+ − y+), respectively. They are separated off because the derivatives +appearing in (74) annihilate them and they take no part in the present calculation. That is also +why m is not determinable until later in Sec. 6.4.3 (see (93)). The new rescaled variables are ˆy, ˆT +and ˆU. Then (74) becomes +λ +α2 +d2 ˆU +dˆy2 + γ +α +d ˆT +dˆy + β = 0. +(76) +In order for the three terms to be formally equal in order of magnitude, one can specify +α = +�λ +β +�1/2 +, +γ = (λβ)1/2, +(77) +so that +y+ = y+ +β + +�λ +β +�1/2 +ˆy, +T(y+) = Tm(β) + (λβ)1/2 ˆT(ˆy), +U+ = U+(y+ +β ) + my+ + λ ˆU(ˆy), +(78) +26 + +and (74) is transformed into the parameterless equation +d2 ˆU +dˆy2 + d ˆT +dˆy + 1 = 0. +(79) +The criterion of equal formal orders of magnitude, therefore, does not by itself determine +uniquely the three scaling factors α, γ, λ; (77) leaves the factor λ undetermined. +This sug- +gests that there may be a one parameter family of potential scaling patches at the location y+ +β , the +parameter being λ. We are confronted with an extra degree of indeterminacy, because the present +line of reasoning does not offer a way to determine which of the potential patches represent actual +ones. However there is considerable evidence, to be summarized in Sec. 6.7, that the correct scaling +at this location y+ +β is given by (77) with λ = 1. The analysis to follow in this and the next two +subsections holds for other choices of λ as well. +We assume that λ(β), like α and γ, is a power of β, and define, in place of λ, the parameter +σ by λ = β−σ. (No constant coefficient is needed with this power law because we are dealing only +with orders of magnitude.) It will be shown that the case σ = 0 leads to a logarithmic-like profile +for the mean velocity U+, and when σ is positive, we get behavior like a power law with exponent +depending on σ. +For reference, the prototypical case is σ = 0, when +y+ = y+ +β + β−1/2ˆy, +T(y+) = Tm(β) + β1/2 ˆT(ˆy), +U+ = U+(y+ +β ) + m(β)(y+ − y+ +β ) + ˆU(ˆy). +(80) +No information is available at this point about the slope m, which bears on the profile of U+. +However, information about it will be found later (93). +We argue that the scaling (78), for some choice of σ, is the natural one in a neighborhood of y+ +β , +and that this neighborhood is a scaling patch. On the basis of the explanation in Sec. 4.1 together +with (75) and (77), it will follow that the characteristic length in that patch will be +ℓ(β) = +�λ +β +�1/2 += β−(σ+1)/2. +(81) +In fact not only does the scaling produce a parameter-free exact form (79) of the momentum balance +equation, but at locations in the proposed patch it can be verified that the individual derivatives +in (79) have the right order of magnitude, namely ≤ 1 with at least some of them = O(1)). For +example, at the peak location, the three terms on the left of (79) are −1, 0, 1 respectively. Leading +up to that peak, the middle term is positive but still ≤ O(1), which makes the first term also O(1), +according to (79) again. +In a scaling patch, as defined originally, all derivatives using the scaled variables are ≤ O(1), +and in the case of at least one of those variables, the magnitudes of its derivatives are not all +strictly < O(1). If that were not the case, the scaling factor α could be decreased without forcing +some of the new rescaled derivatives to be unbounded as β→0. In the present case, we have shown +that these order of magnitude relations hold for the particular derivatives appearing in (79). That +fact makes the scaled neighborhood of y+ +β with width O(1)) in ˆy a candidate for a scaling patch, +provided the correct choice of σ is taken. This will be our accepted criterion for the existence of a +patch. +Thus if the correct σ is taken, then given any suitable small number β, one concludes that there +is a corresponding scaling patch, which we shall call Lβ, with characteristic length ℓ(β) = β−(1+σ)/2 +located at the point y+ +β where T β achieves its maximum. +27 + +W=β-εt +2 +ym +β +Ι +y+ +W(y+) +Figure 3: Schematic diagram showing the role of the function W(y+), which is called P(y+) in the +text, in determining the relation between β and the location yβ +m of the corresponding scaling patch. +The interval I shown here is one of many choices of interval on which W(= P) is decreasing. See +(82). +One may question whether this characteristic length ℓ, for the correct choice of σ, is comparable +to the mixing length of Prandtl in Sec. 3 or the scaling parameter introduced by von Karman (15). +They arise from apparently different considerations; the lengths used by von Karman and Prandtl +are characteristic lengths vaguely associated with the fluctuating velocity, simply postulated to +exist, whereas the present one is related to the function T. And yet T is in essence something +which is defined in terms of those fluctuations, and so one may surmise that the two concepts are +related somehow. Because of the vagueness of mixing length concepts, that may be all that can be +said. +6.2 +The locations of the scaling patches +In the following the analysis will be done only for the case σ = 0. Analogous calculations may +be done in the other cases as well; we will simply provide some key equations in the general case, +without derivations. +At this point one should ask how to determine the range of parameter values β for which the +foregoing construction of scaling patches is possible. This is best answered at first in terms of the +known qualitative properties of the function P(y+) ≡ +dT +dy+ (y+); later a more complete answer will +be given. The function P vanishes at the wall (y+ = 0) and, by symmetry of the function T, also at +the centerline y+ = δ+ = ǫ−2. As y+ passes from the wall to the centerline, T rises to its maximum +at the latter location; during the transition, T and P are both positive. Since P = 0 at those two +locations, it must attain a positive maximum at some intermediate point; call it y+ +p . Being the +gradient of T, P is expected take on its greatest values in the inner region, where the length scale +is shortest and gradients are largest. Therefore the location y+ +p of the positive maximum of P will +be expected to lie in the inner region, so that y+ +p = O(1). +28 + +-0.01 + 0 + 0.01 + 0.02 + 0.03 + 0.04 + 0.05 + 0.06 + 0.07 + 0 + 10 + 20 + 30 + 40 + 50 + 60 + 70 + 80 + 90 + 100 +dT+/dy+ +y+ +y+=7.0 +Reτ=82 (Bech et al.) +Reτ=128 (Kawamura et al.) +Reτ=181 (Kawamura et al.) +ReΘ=1410 (Spalart) +dT+/dy+=0.007 +Figure 4: Inner normalized Reynolds stress gradient for a variety of flows. The turbulent Couette +flow data are from [19] and [18]. Also included are the turbulent channel DNS data from [20] and +turbulent boundary layer DNS from [21]. +For y+ > y+ +p , P(y+) will decrease from its maximum. Let us assume it is a decreasing function +on the entire interval (y+ +p , δ+). Given any point y∗ in that interval, let β(y∗) be the corresponding +value of P, i.e. +P(y∗) = β. +(82) +According to (73), which can be written dT β +dy+ = P(y+) − β, the function dT β +dy+ = 0 at the point y∗ +under consideration, and since P is a decreasing function, dT β +dy+ changes from being positive to being +negative as y+ increases past the point y∗. Therefore T β has a maximum, which we may call T β +m, +at that point. The point y∗ in question may therefore also be labeled y+ +β . This is illustrated in Fig. +3, with plots of typical functions P in Fig. 4. +What has been shown is that any value y∗ ∈ (y+ +p , δ+) will serve as y+ +β if β is chosen to be +P(y∗). The set of all such values of β is the set for which the above construction of a scaling patch +will work, namely the set of all values of P(y+) for y+ ∈ (y+ +p , δ+). That range in y+ provides, +then, the locations where we have succeeded in finding a scaling patch; moreover we have found +the characteristic lengths of all these patches; they are given according to (80) in the case σ = 0 +by β−1/2. These characteristic lengths increase with increasing distance from the wall, since P +decreases with distance. It will be argued in Sec. 6.4, in fact, that asymptotically as β→0, they are +proportional to distance from the wall. +6.3 +More on the locations of the patches +It will be desirable to correlate the locations y+ +β of the scaling patches with the characteristic lengths +given by (81). The above discussion accomplishes this in terms of the function P. We now ask +whether information can be obtained even if P is not known. +In order to proceed we now further exploit the facts that T β has a maximum at y+ = y+ +β , and +that (80) expresses the normal (natural) scaling near that point. In terms of the rescaled variables +given in (80), we have that ˆT has a maximum of 0 at ˆy = 0. We are going to emphasize the +29 + +variation of β and the dependence of the rescaling shown in (78) on β. Then the fact that T β(y+) +has a maximum at y+ +β implies that for all β in the allowed interval of values, +dT β(y+) +dy+ +(y+ +β ) = dT(y+) +dy+ +(y+ +β ) − β = 0. +(83) +This may be differentiated with respect to β to yield +d2T(y+) +(dy+)2 (y+ +β ) +dy+ +β +dβ − 1 = 0. +(84) +Writing the second derivative on the left in terms of the scaled variables, we find +β3/2 d2 ˆT(ˆy) +(dˆy)2 +�����(ˆy=0) +dy+ +β +dβ − 1 = 0. +(85) +In the patch Lβ, the scaled variables satisfy (79), which is parameter-independent; moreover ˆT +satisfies the two equations ˆT(0) = d ˆT +dˆy (0) = 0, which are also parameter-independent relations. One +can therefore argue that the second derivative on the left of (85), which we shall designate by +−A ≡ d2 ˆT(ˆy) +(dˆy)2 +����� +(ˆy=0) +, +(86) +should not depend in any major way on the parameter β. This is reminiscent of similarity hy- +potheses, since an assumption that A is constant is an assumption that the quantity A is invariant +under certain transformations associated with changing β. Certainly the order of magnitude of A +(with respect to β)) is O(1), hence β-independent, and we shall argue below in Sec. 6.6 that in +some regions any β-independence should vanish in the limit as ǫ→0. From (85), then, +dy+ +β +dβ = −A−1β−3/2. +(87) +Recall that y+ +β is the location, in the original inner variable, of the scaling patch with charac- +teristic length β−1/2. The equation (87), therefore, provides some insight into the dependence of +that characteristic length on location, without using much knowledge about the function P. +6.4 +The case A = constant +If A, as it depends on β, were known, the locations y+ +β could be found by solving differential +equations. It is not known, of course; but its order of magnitude is known to be unity. That +knowledge provides order of magnitude information about the profiles and about the locations y+ +β . +It is most instructive at this point to look at the calculations in the easiest case A = constant. +After that, we can see how the results so obtained are still valid in an order of magnitude sense. +6.4.1 +Characteristic length as it depends on location +As mentioned, the scaling given in (78) shows, among other things, that the characteristic length +ℓ of a patch is given by (81). We shall sometimes write relations below in terms of ℓ instead of β. +The symbol C, with or without subscript, will denote a variety of different constants, sometimes +integration constants, which depend on σ but not on β or A. +30 + +If A is constant in some given interval, then in that interval (87) may be integrated to obtain +y+ +β = −A−1 +� +β−3/2dβ = A−1C0β−1/2 + C1 = A−1C0ℓ + C, +(88) +where the final C is an integration constant. We now drop the subscript β from y+ +β and obtain the +relation +ℓ = CA(y+ − C). +(89) +This same relation in fact holds as well for other choices of σ. The left side is the characteristic +length of the patch located at y+; except for a shift in the independent variable y+, that length is +proportional to distance from the wall. +6.4.2 +The Reynolds stress +In (82) the point y∗ may be identified with the general point y+ in (89), the location of the patch +Lβ. Substituting the expression for β given in (89) into the right side of (82) and recalling the +definition of P, we obtain +dT +dy+ = β = C1A−2(y+ − C2)−2. +(90) +Integrating this provides T as a function of y+: +T(y+) = −C3A−2(y+ − C)−1 + C4, +(91) +with another integration constant C4. The relation (91) holds only under the assumption that A is +constant, and only for values of y+ in the interval where T is a decreasing function, because that +is where the scaling patches were found. This range extends up to the channel midline at y+ = δ+, +at which point the left side of (91) vanishes. This gives a relation among δ+ and the integration +constants. Since the right side of (90) cannot vanish and the left side does, the assumption that A +is constant is incorrect at least near the centerline. +6.4.3 +The mean velocity profile +From (66) and (91) we find +dU+ +dy+ = 1 + CA−2(y+ − C)−1 − C5. +(92) +We require this derivative to be small for large y+; this simply means that at the centerline U+ +must become flat as ǫ→0. Therefore it appears that C5 ≈ 1 and +dU+ +dy+ ≈ CA−2(y+ − C)−1. +(93) +Combining (93) with (89) yields an asymptotic relation between the characteristic length ℓ and +the slope of the mean velocity profile: +dU+ +dy+ (y+) ≈ CA−2ℓ−1. +(94) +For other values of σ the right side should be replaced by CA−2/(1+σ)ℓ−(1−σ)/(1+σ). In any case, +we may express this in terms of β and identify it with the slope m in (78). Note that in the case +31 + +σ = 0, m ≈ ℓ−1, which says that the characteristic length of the linear part of U+ in (78) is the +same, ℓ, as that of the nonlinear part. More generally, in any case if the latter is true, then m +should also be identified with the ratio of characteristic increments ∆U+/∆y+, which from (78) is +λ/α = β(1−σ)/2 = ℓ−(1−σ)/(1+σ). This is in agreement with (94). The conclusion is that for any +choice of σ, the characteristic slope of the U+ profile calculated by integrating (87) on the basis +of the nonlinear increments is the same as that which one would surmise by assuming that the +characteristic increments for the linear parts of (78) are the same as for the nonlinear parts. This +fact lends credence to that assumption. +Integrating (93) again gives +U+ ≈ +� +CA−2/(1+σ)(y+ − C)2σ/(σ+1) + C6, +σ > 0, +CA−2 ln (y+ − C) + C6, +σ = 0. +(95) +This expression (95) in the case σ = 0 is similar to that (71) for the mean velocity in a +hypothesized overlap region given by the Izakson–Millikan argument. In the case 0 < σ ≪ 1 it +gives a power law with small exponent; that also has been suggested in the past, but experimental +or DNS data has left the resolution of the question unclear. +In both the Izakson–Millikan argument and the present argument, questionable assumptions +lead to the derivation of (95), or at least part of it. In the I-M case, those assumptions have been +reviewed in Section 4.6. In the present case it is mainly, but not solely, the assumption that A +is constant. The crucial assumption in either case must realistically be only approximate, with +unknown error. In the present scenario, however, there is a good reason to believe that in some +regions, the error in the assumption about A approaches 0 as ǫ→0, i.e. in the limit of large Reynolds +numbers. That reasoning will be given below in Sec. 6.6. +6.5 +Relaxing the assumption that A is constant +Although the similarity assumption A ∼ 1 is true in order of magnitude, it is unlikely to be strictly +true except in interior regions at high Reynolds numbers, as shown below in Sec. 6.6. +If A = A(β) depends on β, then (87) is still valid. Moreover if A remains O(1), i.e. bounded +above and below by positive constants independent of β or ǫ, then (88) and (91) still hold in a +weakened sense. They are replaced by pairs of inequalities for some constants Ki independent of +all parameters. In the logarithmic case σ = 0, we have +K1ℓ ≤ y+ +β ≤ K2ℓ, +(96) +K3(y+ − C)−2 ≤ dT +dy+ = β ≤ K4(y+ − C)−2. +(97) +Thus in an asymptotic sense as y+→∞, the characteristic length is still proportional to distance +from the wall. +Similar transformations to pairs of inequalities are valid for (90)–(95). For example if σ = 0 +the latter becomes +K5 ln y+ ≤ U+ ≤ K6 ln y+. +(98) +In domains where A is nearer to being constant, such inequalities are valid with constants Ki +which are closer to one another. In fact the error in assuming K5 = K6, for example, can be +estimated in terms of any assumed error bound in assuming that A = constant in the domain being +considered. This is parallel to the error consideration in the Izakson-Millikan argument, which was +32 + +detailed in Sec. 4.7. In both cases the conclusion of logarithmic growth is probably never exact; +it should be considered an approximate statement, with the accuracy of the statement dependent +on the accuracy of the underlying assumption. In the I-M case, the underlying assumption is that +the outer and inner approximations are both exact in some domain; in the present case, it is that +the quantity A is constant in some domain. In both cases there is no theoretical way to gauge how +accurate the approximations are (see, however, the following subsection). +6.6 +Approximate constancy of A in interior zones +As was brought out before, the order of magnitude of the quantity A remains O(1) in β and ǫ. +In locations far away from the endpoints of the range of the continuum of scaling patches, it can +be argued that A should be almost constant. The reason is that the data we have for A, namely +the differential equation it satisfies and the known exact values of the terms in that equation, +are parameter-independent. Therefore any variation in A due to changes in β will be caused not +from those sources. This invariance as β changes suggests, by a similarity consideration, that A +is constant if it is not subject to other influences. Those would be influences from neighboring +patches, hence ultimately from locations, on either side of the continuum, where the boundary +would introduce “external” influences. Only at those places would the similarity suffer external +disruption. And the effects of that disruption would be most likely to happen near those disrupting +sites, either toward the outer or the inner zones. That leaves interior regions far away from those +zones as candidates for places where A is nearly constant. It was shown above in Sec. 6.4.3 that +those are the regions where the mean velocity profile is logarithmic-like. +The extent (in inner units) of these zones of near similarity will grow as ǫ becomes smaller, +because there will be a larger range of patches far away from the extremal patches. +6.7 +Evidence for logarithmic growth, i.e. σ = 0 +The possibility that the mean velocity profile grows in parts of the flow according to a power +law, rather than a logarithmic law, has been discussed by other authors. In either case, the actual +expression for the mean velocity will depend somewhat on the Reynolds number and is very unlikely +to be exactly logarithmic or a power function. These may represent approximations, but that is all. +What we are concerned with are trends, brought about by relations such as (98) and its analog for +power functions. They are generated by the scaling parameter λ = β−σ in (78). Here we summarize +the evidence in favor of choosing σ = 0. +• The Izakson-Millikan argument (Sec. 5.6). +• It was shown from empirical data in [7] that the increment in U+ across the mesolayer is +O(1), in fact near 1 independently of the (large) Reynolds number. The mesolayer is one +example of a scaling patch for turbulent Poiseuille flow. The analysis of that flow in Sec. 7 +follows the present analysis (which has been for Couette flow). The role of the parameter σ +in determining the increment in U+ across any scaling patch, including the mesolayer, is seen +from (75). The nonlinear part of the increment is simply O(λ) = O(β−σ), and it was brought +out following (94) that in all cases the linear part of the increment, governed by m, is the +same. Therefore the only case in which this increment is O(1), as apparently required in the +mesolayer, is the case σ = 0. +33 + +6.8 +The inner scaling patch at the wall and the outer scaling patch at the +midline +The construction of the patches given in section 6 has, as a primary ingredient, the fact that as the +peak in the adjusted Reynolds stress is approached, a region must appear in which all three terms +in the mean momentum balance equation, which in this case is (74), will have the same order of +magnitude. This is simply because the gradient dT β +dy+ approaches 0. (There will, of course, be a +smaller region encompassing the peak in which the last term on the left of (74) has smaller order +of magnitude than the others, because it vanishes at the peak.) +Curiously, a somewhat similar phenomenon happens when y+→0, since the gradient of the +actual Reynolds stress, rather the adjusted one, is zero at the wall (y+ = 0) and positive for small +values of y+ > 0. We are speaking of the first term +dT +dy+ in (64). With the inner scaling, both terms +in (64) have equal orders of magnitude, both actual and formal. This in itself provides evidence +that this wall region, together with the inner scaling used in (64), defines a scaling patch. But +there is further evidence from the boundary condition (65) for U+. That condition of course was +engineered by our very choice of inner scaling. But whatever its origin, it furnishes decisive evidence +that a scaling patch exists there. As before, the width of this patch is O(1), measured in y+. As +was brought out before, it also encompasses the crucial point y+ +p discussed in Sec. 6.2. +To summarize, at the wall, U+, T, and +dT +dy+ are all 0, but dU+ +dy+ = 1 (that is how the inner scaling +was selected). Thus all derivatives of interest in the scaled variables at that point either vanish, or +(in one case) are unity. This circumstance is an adequate criterion for the validity of that scaling. +On the other end of the continuum, where ℓ is large, we know that ℓ reaches a maximum of +ǫ−2, because that is the half-width of the channel and no larger scaling patch could fit into the +latter. According to (81), it corresponds to β = ǫ4/(1+σ). That forms a lower bound on the possible +values of β. In the case σ = 0 for example, when β = ǫ4 the existence of a scaling patch can be +ascertained by the previous argument in Sec. 6.1, which still holds true (y+ +β for that value of β must +lie a distance ≤ O(1) from the centerline). +It should be noted that this “outer” patch encompassing the centerline is not the same as the +traditional outer length scaling spoken of in Sec. 5.3, although the two ideas are compatible. The +present concept of outer patch is that of an interval in the core together with a rescaling of all +the variables, not just y+, which will produce a parameter-free version of the mean momentum +balance, namely (79), at that location. +Thus our construction of scaling patches is valid up to and including the centerline, and down +to the inner region. In all, the scaling patches cover the entire channel. +6.9 +Discussion +We have found that at each point in the Couette flow, the Reynolds stress and mean velocity profiles +have a natural scaling; in other words, a scaling patch is located at that point. In order of mag- +nitude, its characteristic length, which is essentially the width of the patch, increases continuously +from 1 (in inner units) in the inner region where the law of the wall holds, to ǫ−2 at the centerline +of the channel, where the outer scaling holds. Each patch is associated with a peak in one of the +adjusted Reynolds stresses, and with a balance exchange event that occurs there involving that +same adjusted stress. +The patches can be parameterized by their characteristic lengths, which increase monotonically +with distance from the wall. Asymptotically as y+→∞ (since y+ is limited by ǫ−2, necessarily +ǫ→0), the characteristic length is proportional to that distance, and in any case up to order of +34 + +magnitude, is given by a solution of an ordinary differential equation. +Again up to order of magnitude, the mean velocity and Reynolds stress profiles are determined. +In certain regions in the limit as the Reynolds number approaches ∞, these order of magnitude +results are replaced by explicit functions. +This is really a statement about the validity of an +approximation for large Reynolds number. +The argument can be framed as a similarity and invariance issue. There is a family of rescalings +depending on a parameter β, applied at β-dependent locations, which leave the governing rescaled +momentum balance equation and associated numerical values of the derivatives appearing in that +equation invariant as β is varied. The statement then is that another derivative, denoted by A, +is, in order of magnitude, invariant as well, and in some regions in fact approximately numerically +invariant. +These results are consistent with logarithmic growth predicted by the Izakson–Millikan argu- +ment, which by hypothesis is to hold in some overlap zone between the inner and outer regions. The +scaling patch and order of magnitude results presented here, on the other hand, are true throughout +the channel; they generally differ from the exact logarithmic law except where and if the quantity +A is constant. Finally, this constancy condition is unlikely to hold exactly anywhere, except in the +limit as ǫ→0. +7 +Turbulent Poiseuille flow in a channel +In this idealized picture, the two walls at y = 0, y = 2δ are stationary, so that their motion no +longer provides impetus for the flow; however such an impetus is provided by a given pressure +gradient streamwise along the channel. The treatment here follows that in [8, 9]. +7.1 +Differences from Couette flow +Mathematically, the differences between Couette and Poiseuille flows lie in the facts that Px is no +longer 0 in (51), and the boundary conditions at the two walls are changed. Namely, +U = ⟨u′v′⟩ = d +dy⟨u′v′⟩ = d2 +dy2 ⟨u′v′⟩ = 0 +(99) +at y = 0 and 2δ; and at the centerline y = δ by symmetry, +dU +dy = ⟨u′v′⟩ = 0. +(100) +In fact U(y) rises from 0 at y = 0 to attain a maximum at y = δ. The function is even about that +latter location, which means that for y > δ, U(y) = U(2δ − y). Similarly, ⟨u′v′⟩ is odd about that +point. These properties allow us to formulate the problem entirely on the half channel {0 ≤ y ≤ δ}. +If the solution is known in the half channel, it can be found by reflection in the other half. +A standard argument shows that the pressure gradient term in the mean momentum balance +equation (51) must be constant. Specifically, we refer back to that equation and (52). Setting +Q +ρ = P +ρ − ⟨(v′)2⟩, we see that(52) implies that Q depends only on x; however by stationarity ⟨(v′)2⟩ +does not depend on x. The pressure gradient term in (51) can be written 1 +ρ +∂Q +∂x , which as noted is +independent of y. It is also independent of x, since the other two terms in (51) are. +The friction velocity, Reynolds number R∗, small parameter ǫ = (R∗)−1/2, inner scaling (law of +the wall) and outer scaling, valid in the channel’s center, are all the same as before. Integrating +35 + +(51) across the half channel produces a global force balance, resulting in the dimensionless form ǫ2 +for the pressure gradient. The dimensionless form (64) becomes +dT +dη + 1 + ǫ2 d2U +dη2 = 0, +(101) +and that of (66) is +T + ǫ2 dU+ +dη += 1 − η. +(102) +The traditional outer approximation is +Tout = 1 − η, +0 < η ≤ 1. +(103) +The momentum balance equation with inner normalization is +d2U+ +dy+2 + dT +dy+ + ǫ2 = 0. +(104) +7.2 +Hierarchy +To exhibit a continuous family of scaling patches covering the channel flow profile, all that is needed +is to revise slightly the definition of the adjusted Reynolds stresses (72). The new one is +T β(y+) = T(y+) + ǫ2y+ − βy+. +(105) +It is remarkable that the mathematical problems for the mean velocity and Reynolds stress in these +two scenarios—pressure gradient driven and shear driven—can be almost completely transformed +one into the other by such a simple device as (105). It transforms the basic momentum balance +equation (104) into +d2U+ +dy+2 + dT β +dy+ + β = 0, +(106) +which is of the same form as (74). +Therefore, with the newly adjusted Reynolds stresses, the channel flow context is amenable +to the balance exchange processes described in Section 6.1, the construction of a continuum of +scalings with associated scaling patches Lβ in Sections 6.1 to 6.3, and (under some assumptions) +the derivation of logarithmic-like profiles in Section 6.4.3. The scaling in Lβ is still given by (78). +The mean profile calculations are given here only under the simplifying approximation A = +constant, although analogs of the more general case can be derived. The expressions (78) and (79) +are valid in the present setting as well. +As far as estimating the locations of the patches and the profiles, there are some changes. +The relation (90) becomes +dT +dy+ = 4A−2(y+ − C)−2 − ǫ2. +(107) +The constant C in (107) can be related to the location y+ = y+ +m of the maximum of the original +unadjusted function T. This is because it is required that +dT +dy+ = 0 at y+ = y+ +m. +We obtain +dT +dy+ = 4A−2 +� +y+ − y+ +m + 2 +Aǫ +�−2 +− ǫ2, +(108) +36 + +or alternatively by eliminating ǫ, +dT +dy+ = 4A−2 � +(y+ − C)−2 − (y+ +m − C)−2� +. +(109) +When one inserts this into (104) and integrates twice, using the requirement dU+ +dy+ →0 as y+→∞, +the result is +U(y+) = 4 +A2 ln +� +y+ − y+ +m + 2 +Aǫ +� ++ C2 +(110) +for another integration constant C2. +Strictly speaking, there is another required condition, due to the symmetry of the configuration +at the centerline: +dU+ +dy+ = 0 at y+ = ǫ−2. It cannot be satisfied exactly within the framework of +(110), which indicates that the approximation A = constant cannot be exact near the centerline. +The expression (110) is only expected to give a good representation of the real profile in some +regions away from both the wall and the centerline. +To reiterate, the most we can say theoretically is that this is suggestive of a logarithmic ap- +proximation to some segment of the mean velocity profile. More specifically, this is all under the +(doubtful) assumption that A is exactly constant. In the case that it is almost constant, one gets +a pair of upper and lower bounds as before, valid now for the mean velocity in channel flow for the +range of y+ constructed as before. +Note that in the case β = ǫ2, by (105) T β = T. The scaling patch in this particular case is +traditionally called the “mesolayer”, and it occurs near the peak in Reynolds stress, because for +this case T β = T. The characteristic length in that patch is O(ǫ−1), which is the geometric mean +of those in the inner (O(1)) and outer (O(ǫ−2)) regions. +7.3 +Behavior near the wall +The construction of the patches given in sections 6.1 and 7.2 by means of a balance exchange has, +as a primary ingredient, the fact that as the peak in adjusted Reynolds stress is approached, a +region must appear in which all three terms in the mean momentum balance equation, (61), will +have the same order of magnitude. This is simply because the gradient dT + +dy+ approaches 0. There +will, of course, be a smaller region encompassing the peak in which the last term on the left of (61) +has smaller order of magnitude than the others, because it vanishes at the peak. +A similar phenomenon happens when y+→0, since that same gradient is zero at the wall (y+ = +0) and positive for small values of y+ > 0. The same conclusion may therefore be deduced: in a +small region near the wall, all three terms in (61) will have the same order of magnitude. But the +argument in Section 6.1 can only partially be continued beyond this stage to produce a patch with +different scaling; in fact it is well known (see also the reason given below) that the characteristic +length scale arbitrarily near the wall remains the inner scale. The reason the argument is no longer +completely valid will now be explained. In addition, the correctly scaled mean momentum balance +very near the wall will be derived. +There have been many analytical, empirical, and computational studies of the properties of the +near wall region; we mention only [22], as our results fit particularly well with theirs. Our purpose +here is to show that a scaling patch exists there, whose derivation and description fits within the +framework of the methodology developed here and in our previous papers. +37 + +At the wall, additional constraints are imposed on the functions U+ and T +, besides the ba- +sic differential equation (61). First of all, the very definition of inner scaling requires an auto- +matic boundary condition dU+ +dy+ (0) = 1. The inner scaling was chosen just so that condition holds. +Secondly, the no-slip condition requires the boundary conditions U+(0) = T +(0) = +dT + +dy+ (0) = +d2T + +dy+2 (0) = 0. The first requirement is simply a result of our choice of normalized variables y+ and +U+, and is not a statement of any physical constraint. The other boundary conditions result from +a physical effect located at the wall. They have no analog at the mesoscaling patch, and constitute +the basic reason that the present construction is different from the mesoscale construction. +If one proceeds in the same vein as before on the basis of (75) and (76), the effect of the first +boundary condition dU+ +dy+ (0) = 1 is that the length scale in that patch is given by α = 1. This means +that ˆy = y+: the length scale in that patch is the same as that with the original inner normalized +scaling. This is of course almost a tautology. +But then the rest of the argument, following (61), in which α and γ are determined, can no +longer be carried out as it stands, since α has already been determined. However, one can proceed +after some reformulation of the problem. +At this point, the first substantial difference in method emerges between the derivation of the +patches in Sec. 6.1 and the present argument for what we shall call the wall patch. As mentioned, +it is allied with the physical no-slip constraint. In both cases, we look for scaled solutions of (74) +or its analog (104) in a neighborhood of a maximum or minimum of T + or Tβ. +In both the case in Sec. 6.1 and the present wall case, a scaling, namely a choice of α, γ, λ +in (75), (76), is sought which will render the three terms of (74) or (104) of forms which have +the same formal order of magnitude. A unique choice is only possible if one of these three factors +is specified by other means. In the previous case, empirical data having to do with the velocity +increment across the patch, and also the rate of growth of U+ in the hierarchy, was used to motivate +selection of the value λ = 1, while leaving open the additional possibility of other choices leading +to different growth rates. This serves to determine the other two factors. In the wall patch, where +this argument is not applicable, the definition of the inner scaling requires α = 1, and the equality +criterion now can be used to determine γ and λ. Namely, evaluating the three terms of (104) under +the transformation (75), (76) with α = 1 tells us that γ = λ = ǫ2. In neither case does this provide +the value of m, because the derivatives in (74) or (104) annihilate the linear terms. However, the +value of m was obtained by other means: through use of (91) and the connection T has with the +slope, in the case of patches embedded in the hierarchy, and by means of the boundary condition +giving the slope at the wall, in the present case. +The scaled version of (104) in the wall patch is +1 + d2 ˆU +dy+2 + d ˆT +dy+ = 0. +(111) +The individual terms of this equation are known only at y+ = 0; but it provides a linear relation +between the two derivatives. This equation is the analog of (79), and is in the form of a balance of +three rescaled forces. +In short, there is a scaling patch near the wall, no doubt including the traditional viscous +sublayer, in which the inner length scale is correct, but the deviations of the functions U+ and T + +from their linear parts depend on R∗ like ǫ2 ≈ (R∗)−1. +Finally, as y+ enters the patch from above, there is a balance exchange from Layer II (in the +terminology used in [7]), where the viscous and turbulent forces balance, to Layer I, where the +pressure gradient balances the viscous plus turbulence force. +38 + +The location of this exchange, and in fact the size of Layer I, is ≤ O(1) in wall units, because +that is the length scale for the parameterless (111). +In terms of the traditional buffer and logarithmic layer, we surmise that they lie outside Layer I, +which can be identified as the viscous sublayer (although at its outer edge the viscous and turbulent +forces are equal in order of magnitude). Outside that layer is approximately where the hierarchy +begins (say y+ ≈ 7), which is also where the traditionally defined buffer layer begins. The loga- +rithmic mean profile approximation associated with the hierarchy, however, does not become valid +until distances from the beginning of the hierarchy are sufficient for A in (87) to be approximately +constant. +8 +Other applications and conclusion +The analysis covered in Secs. 6 and 7 has been applied not only to Couette and Poiseuille flow, but +also to +• combined Couette-Poiseuille turbulent flow [23]; +• favorable pressure gradient boundary layers (Metzger and Fife, in preparation); +• transport of heat through turbulent channel flow [24]. +It should again be emphasized that our goal has been not so much to discover numerical values +associated with the profiles, but rather to gain theoretical understanding of why important features +occur. We look for answers to why? in preference to what? +The greatest emphasis has been placed on two approaches to this question: +• the classical Izakson-Millikan observation; and +• the search for scaling patches. +Each of these results is based on an assumption that another approximation is valid. The first +relies on the standard inner and outer approximations being simultaneously valid somewhere. The +second approach is an argument involving matters of similarity and invariance. There is a family of +rescalings depending on a parameter β, applied at β-dependent locations, which leave the governing +rescaled momentum balance equation and associated numerical values of the derivatives appearing +in that equation invariant as β is varied. The statement then is that another derivative is, in order +of magnitude, invariant as well, and in some regions is in fact approximately numerically invariant. +In neither case is there a theoretical way to gauge the error of the assumed approximation, or the +extent of the region where it is valid. Some qualitative conclusions on the latter issue are provided +in the second case. +The various approaches to answering our basic question all contribute a portion of insight. All +of them leave unanswered questions, but they add to one another. +No one of them should be +considered the final word on the subject. +Acknowledgments +The results in Secs. 6, 7, and 8 were obtained in collaboration with Joe Klewicki, Tie Wei, Meredith +Metzger, and Pat McMurtry. I thank them all for their invaluable exchange of ideas. +39 + +References +[1] O. Reynolds. On the dynamical theory of incompressible viscous fluids and the determination +of the criterion. Phil. Trans. Roy. Soc. 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Experiments in Fluids, 25:165–170, 1998. +[23] T. Wei, P. Fife, and J. Klewicki. On scaling the mean momentum balance and its solutions in +turbulent Couette-Poiseuille flow. J. Fluid Mech. to appear. +[24] T. Wei, P. Fife, J. Klewicki, and P. McMurtry. Scaling heat transfer in fully developed turbulent +channel flow. Int. Jour. Heat Mass Trans., 48:5284–5296, 2005. +41 + diff --git a/dtFAT4oBgHgl3EQf6x4A/content/tmp_files/load_file.txt b/dtFAT4oBgHgl3EQf6x4A/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..f31c6b1cf5aeba65fc06e48cd839d2c4dc8c9433 --- /dev/null +++ b/dtFAT4oBgHgl3EQf6x4A/content/tmp_files/load_file.txt @@ -0,0 +1,2202 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf,len=2201 +page_content='Scaling approaches to steady wall-induced turbulence Paul C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Fife August 12, 2006 Abstract The problem of discerning key features of steady turbulent flow adjacent to a wall has drawn the attention of some of the most noted fluid dynamicists of all time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Standard examples of such features are found in the mean velocity profiles of turbulent flow in channels, pipes or boundary layers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The aim of this review article is to expound the essence of some elementary theoretical efforts in this regard.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Possibly the best known of them, and certainly the simplest, is the argument (obtained independently) by Izakson (1937) and Millikan (1939).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' They showed that if an inner scaling and an outer scaling for the profile are valid near the wall and near the center of the flow (or the edge of the boundary layer), respectively, and if there is an overlap region where both scalings are valid, then the profile must be logarithmic in that common region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That theoretical justification has been used and expanded upon by innumerable authors for over 60 years, and at the present time is still rightly enjoying popularity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Although background discussions of several related topics are included in the present article, for example the classical ideas of Prandtl and von Karman, the main foci will be on (i) a careful examination of the Izakson-Millikan argument, together with a presentation of a better mathematical justification for its conclusion;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' and (ii) a detailed clarification of a newer approach to gaining theoretical understanding of the mean velocity and Reynolds stress profiles based on the “search for scaling patches”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The two approaches share common goals, they are both heavily involved with scaling con- cepts, and many results are similar, but the logical trains of thought are entirely different.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The first, as mentioned, dates back to the 30’s and the second was introduced in a series of recent papers by Fife et al, Wei et al, and Klewicki et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Our emphasis will be on the question of how and how well these arguments supply insight into the structure of the mean flow profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Although empirical results may initiate the search for explanations, they will be viewed simply as means to that end.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Contents 1 Introduction 2 2 The Navier-Stokes equations and Reynolds averaging 3 3 Early scaling concepts of Prandtl and von Karman 5 4 The notion of scaling 7 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 Scaling patches .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 30 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 Characteristic length as it depends on location .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' 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the midline .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 34 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='9 Discussion .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 35 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='2 Hierarchy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 36 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='3 Behavior near the wall .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 37 8 Other applications and conclusion 39 1 Introduction Every theoretical investigation of highly turbulent fluid dynamics is necessarily incomplete, because accepted accurate models such as the Navier-Stokes equations lie beyond the scope of full solution by existing methods, whether those methods be numerical or analytical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Faced with this failing, researchers have often turned to seeking theoretical information through partial analyses, incomplete models, or reasoning which is not fully based on rigorous deduction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The search for such approaches is bound to yield, and historically has yielded, some fruitful avenues leading to insightful, though perhaps tentative, conclusions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The goals of this paper are to examine, and elaborate on, some of the major elementary attempts in this vein to gain some understanding of the mechanisms behind steady (in the mean) turbulent 2 incompressible flow bounded by a wall, as well as to expound some recent developments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The prototypical examples will be rapid flow in a channel or pipe forced by an imposed pressure gradient or the differential motion of the walls, and turbulent boundary layers of various types.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' There has been an enormous amount of work along these lines, and a complete review of it will not be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Rather we look only at some efforts to gain insight into these mechanisms as they affect simple mean flow quantities, through the use of scaling concepts and averaged Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' These efforts have been diverse, and when compared one with another have sometimes relied on assumptions which appear to be incompatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Nevertheless despite any seeming incongruities, the various approaches should not necessarily be thought of as competing among themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Rather by contributing their individual insights, they collectively add to our understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' No one approach can ever rationally be viewed as the last word on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' So to summarize, the goal in these notes is to understand what phenomena relating to average events occur in wall-bounded turbulent flow and, more importantly, why they occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This means that our approach will focus on supplying theoretical explanations for observed features of the flow, as well as providing predictions of features which can then be correlated with empirical data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' When possible, we will not be content with pointing to experimental evidence for properties of the flow, but will strive to develop reasons why those properties should hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' These reasons will not generally be rigorous, and at times, of necessity, they will be partly based on empirical findings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' But to advance the goal of basic understanding, they will, as far as possible, be grounded in theory, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' in accepted mathematical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Mostly, these models will be built from the averaged Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 2 The Navier-Stokes equations and Reynolds averaging Our mathematical models use the symbols u = (u1, u2, u3), p, t and x = (x1, x2, x3) to denote the velocity, pressure, time and space variables, with µ and ρ the material constants of viscosity and fluid density.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The standard incompressible Navier-Stokes equations when there is no body force can be written as follows: ∂u ∂t + (u·∇)u = ν∇2u − 1 ρ∇p, (1) where ν = µ/ρ is the kinematic viscosity;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' and div u = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (2) The equations (1) (a vector equation which has three components) and (2) are four equations in all, for four unknown functions: the three components of u plus the pressure p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' These equations are almost universally recognized as an accurate representation, on the macro- scopic level, of flows of many standard kinds of incompressible fluids.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Generalizations exist in many forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The second (nonlinear) term on the left of (1) is called the “inertia term”, and the first term on the right is the “viscosity term”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Broad features of the solutions are typically governed by the order of magnitude of the Reynolds number, a ratio of the effects of these terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Here we start with a very common framework for studying turbulent motions called Reynolds averaging [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It is a short cut, and as such does not supply all the desired information about any given flow scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' A wealth of details about fluid motions is erased.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The modeler is left with the task of partially filling the resulting deficiency with assumptions about features of the fluid motion in its disorganized state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' These attempts are bound to be hit or miss to some extent, but as we 3 will see, there may be tests which one can apply to gauge the validity of such assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For example, two or more avenues to obtaining models may yield similar results, which would tend to corroborate both.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' A very common remedy for the underdetermined nature of the Reynolds averaged equations is to posit closure relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That approach is especially useful in numerical simulations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' But the magnitude of these endeavors precludes their inclusion here.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The goal, then, is to use the Navier-Stokes equations in their averaged form, together with scal- ing considerations, to gain insight into some basic properties of wall-induced turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Specifi- cally, we focus on features of the mean velocity and Reynolds stress profiles, as functions of distance from the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The Reynolds stress is one of the measures of turbulence activity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' There have in fact been several distinct approaches to the task of using the averaged equations to provide insight into wall-induced turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That fact, together with the apparent simplicity of the formulation within that framework, makes the latter an ideal arena in which to explore the interaction of mathematical and intuitive tools to tackle a very complex problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We now review the well-known ideas behind the process of Reynolds averaging [1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' One supposes that at each spatial location x and time t, the velocity and the other flow quantities have well defined average values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This could be an imagined “ensemble” average over many similar observations, or over some small (but not too small) region in space-time containing the point in question.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the case of “steady” fully developed turbulence, it could be the time average.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The mean (average) values may then depend on space and time, but on scales perhaps larger than those over which the average is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Thus, we can write the velocity vector as its mean plus its fluctuation about the mean: u(x, t) = U(x, t) + u′(x, t), (3) with a similar decomposition for vorticity, pressure, etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Averages will be taken of not only primary flow variables, but also of products of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The averaging operation is denoted by angle brackets— for example ⟨u⟩ = U;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' ⟨∂iu⟩ = ∂iU for any derivative ∂i = ∂ ∂xi or ∂ ∂t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' ⟨u′⟩ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' ⟨Uiu′ j⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We substitute (3) into the Navier-Stokes equations (1), (2) and then take the average of the resulting equations to obtain ∂tU − ν∇2U + (U·∇)U + ⟨u′·∇u′⟩ + 1 ρ∇P = 0, (4) ∇·U = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (5) Note also that ∇·u′ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (6) The i-th component of the 4th term in (4) is ⟨u′ j∂ju′ i⟩ = ∂j⟨u′ iu′ j⟩ (summing over the repeated index), the equality by virtue of (6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This shows that the 4th term, being a divergence, acts as a pseudo-stress gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Accordingly, we define the “Reynolds stress tensor” τ by τij = −⟨u′ iu′ j⟩, (7) which is symmetric, as any stress tensor should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Therefore (4) becomes ∂tU − ν∇2U + (U·∇)U + ∇ �1 ρP − τ � = 0, (8) 4 where the last term can also be written ∇·T , where Tij = 1 ρPδij − τij.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' An intuitive feel for the role of the tensor τ in the transport of momentum can be gained by looking closely at the case i = 1, j = 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We relabel u′ i = u′, u′ 2 = v′, x1 = x, x2 = y, and think of the coordinates x1 and x2 as being horizontal and vertical, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Then the average ⟨u′v′⟩ appearing in (7) involves the horizontal and vertical components of the fluctuating part of the velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Imagine a particle whose instantaneous velocity has both a positive x-component u′ and a positive y component v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The former says that the particle is bearing some x-momentum, and the latter says that what it bears is being transported in the vertical (y) direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The magnitude of this vertical transport of momentum is proportional to the product of the two components, u′v′.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If one or both of these two components changes sign, this proportionality property is still seen to hold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Therefore ⟨u′v′⟩ is proportional to the mean vertical transport of x-momentum (which could be positive or negative) by means of turbulent fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This vertical transport represents the mean force per unit area exerted by the fluid above the point under consideration on the fluid below that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Newton’s first law characterizes a force as a rate of change of momentum;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' in this case, that rate of change is given by vertical transport of horizontal momentum-bearing particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Finally, the y-derivative ∂ ∂y⟨u′v′⟩, being a scaled difference of forces exerted at two nearby points, is the net x-component of the mean force per unit mass exerted on particles at that location by the turbulent fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Generalizing these notions leads to an intuitive recognition of the last term in (8), −∇τ, as a force produced by turbulent fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If we knew the tensor τ, then we could use that knowledge in (8) so that (8) and (5) would constitute a closed system for the determination of U and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' But nature is not so kind, and τ is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Many attempts have been made in the past to remedy this basic defect, by writing down other equations for the determination of τ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' These models propose a “closure” mechanism to produce a closed system for the mean values of various flow quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In all cases, these other equations are at least partly ad hoc, and in most cases partly empirical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 3 Early scaling concepts of Prandtl and von Karman There have been many turbulence theories utilizing the ideas of Reynolds averaging;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' in fact the ones to be discussed in this paper are important examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Prandtl and von Karman were, at an early stage, responsible for well known models in this category.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Their concepts also included types of characteristic lengths, which are allied to the scaling theme to be developed below.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Therefore we begin with brief discussions of some of their ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In 1925, Prandtl [2] proposed a conceptual framework designed to provide insight into the mechanisms producing turbulence in shear flow, and also to provide relationships among key fluid dynamical quantities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' One aim of his was to relate the flux of momentum caused by turbulent fluctuations to the gradient of mean momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That there is such a relationship was simply an assumption springing from an analogy with conventional diffusion of quantities like heat in a stationary medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In that setting, Fick’s law asserts that the flux is proportional to the gradient of the concentration of the substance (like heat) that is being transported.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the present situation, since ⟨u′v′⟩ is (see previous section) a measure of the mean turbulent transport, in the vertical direction, of horizontal 5 momentum, the analog of the heat equation for the mean x-momentum U would be ⟨u′v′⟩ = −ǫdU dy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (9) for some pseudo-diffusion coefficient ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That coefficient is generally called the coefficient of eddy diffusion, because one could argue that the transfer of momentum is caused by many eddies in the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' To say more about ǫ, Prandtl introduced the concept of mixing length ℓ, which more or less represents the distance a fluid particle typically travels before transferring its momentum to another particle.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Various versions of Prandl’s theory have been involved with characterizing ℓ as it depends on local or global properties of the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Since we will be dealing with vague definitions, the symbol “∼” will be used to indicate a relation which is not precisely defined, but rather should be considered as part of a modeling concept.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Moreover if numerical values are given to the two sides of the relation, they will be equal only in order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Prandtl’s ideas related the left side of (9) directly to other local quantities as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Suppose that the gradient dU dy is responsible, in a linear manner, for the local characteristic mag- nitude of the velocity fluctuations: |u′| ∼ k1 ���dU dy ��� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' |v′| ∼ k2 ��� dU dy ���.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The proportionality coefficients ki in these relations have to have the dimensions of length, and the most natural local characteristic length is the mixing length ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' So we set ki ∼ ℓ to obtain |u′| ∼ ℓ ���� dU dy ���� ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' |v′| ∼ ℓ ���� dU dy ���� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (10) Then the average |⟨u′v′⟩| will be related to the quantity on the right of (10) squared: |⟨u′v′⟩| ∼ ℓ2 ���� dU dy ���� 2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (11) Now apply this to (9), at the same time ensuring that signs are chosen so that ǫ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We find that ǫ ∼ ℓ2 ���� dU dy ���� , (12) so that ⟨u′v′⟩ ∼ −ℓ2 ���� dU dy ���� dU dy .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (13) It seems intuitive that the local distance ℓ should grow shorter as one draws near the wall, because the wall exerts a constraint on the motions;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' and Prandtl sometimes proposed a linear relation ℓ ∼ y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (14) This is probably not a good approximation very near the wall, because it would predict that ⟨u′v′⟩ grows like y2 as one moves away from the wall, whereas there is abundant evidence that the growth rate is like y3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This, then, is how Prandtl and others proposed handling a truly complex phenomenon (forces caused by the turbulent transport of momentum) by relating it to a much simpler quantity (the gradient of mean velocity).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This proposal had very limited success in explaining the processes of turbulence, although there was often good correlation with experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6 Von Karman [3] proposed his similarity hypothesis in 1930;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' a prominent ingredient was the assumption that ǫ and ℓ should be characterizable in terms of local properties of the fluid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For example, (14) would not qualify because the distance y from the wall is not a local property.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' One might argue that the similarity hypothesis may approximate conditions far enough away from the walls and centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This hypothesis proceeds from rescaling considerations, the idea being that if lengths for the velocity fluctuations in a neighborhood of a point in the flow are rescaled with a characteristic length ℓ (which we can identify as essentially the mixing length above), then key hydrodynamic quantities in that neighborhood are, except for an appropriate rescaling factor, functions only of the rescaled lengths;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' not of the position in the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' One then searches for a local quantity with the dimensions of length that one could use to characterize ℓ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The possibly simplest choice would be the ratio ℓ ∼ −dU dy � d2U dy2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (15) (The minus sign comes about from noting that the ratio itself is always negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=') Note that if this and (14) are both true, one could solve for U(y) to get U(y) ∼ C1| ln y| + C2, (16) an example of the renowned logarithmic profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Other theories, as we shall see, are in partial agreement with this result and suggest other information as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In particular, the theories brought out here in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5 and 6 involve ideas reminiscent of those of Prandtl and von Karman, but the differences outweigh the similarities.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The main approaches to wall-induced turbulence that we shall examine are heavily involved with various scaling concepts, and so we make a digression here to discuss some ideas basic to that subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4 The notion of scaling Much of this section will consist of a review of scaling concepts, mostly well known, relating to the remainder of the paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It will be useful to give a formal definition of scaling patch (less well known) in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 and to formulate many of our results in terms of those patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For example, it will be important to know that the overlapping regions of the classical example sketched in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='2 and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='3 are not scaling patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The classical Izakson-Millikan observation covered in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='6 is best known in the setting of turbulent flows, but is given here in a more general mathematical context and in the form of an approximative statement (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='7) which is mathematically rigorous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The implications for wall-induced turbulent flow are discussed later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='6 and, more importantly, in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Models involving small or large parameters are commonplace in the natural sciences;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' in more cases than not there are processes making up the action which operate on more than one, often many, different space and time scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The phenomenon being studied can then most clearly and naturally be represented, in certain subdomains, in terms of functions of “rescaled variables”, or of a combination of rescaled variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Here rescaling means that new dependent and independent variables are defined, in differential form, as linear transformations of the original ones, the coef- ficients in the transformation generally being functions of the original small or large parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 7 Multiscaling refers to the event that more than one scaling are appropriately used, either in different subdomains or simultaneously in the same subdomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Our focus will be on differential equations containing parameters which are supposed to be small or large.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The ultimate goal here will be to gain some understanding of fluid motions by applying scaling concepts to the averaged Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The recognition that the dynamics of turbulent fluids operate on a great many space and time scales has been a cornerstone of well known investi- gations into those processes, including the construction of mathematical models.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Our goal is more limited, yet still daunting: to surmise information about important mean flow quantities on the basis of the averaged, rather than the original, Navier-Stokes equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Rather than positing, at each specific location in the flow domain, an array of length scales, as would be the case for the microscopic turbulent motions, the mean flow profiles will themselves have unique characteristic lengths associated with each such location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Our principal technique will be, first, to attempt to ascertain the local scaling behavior of those mean quantities by means of the averaged equations together with judicious assumptions, and then to derive further information about the mean profiles themselves.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In most of the following, there will be one independent small parameter called ǫ, although another parameter β, which will also be small, will appear in some sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The coefficients in the scaling transformations will depend on ǫ, and possibly on β, and their orders of magnitude will be of primary importance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The notation O(1) will refer to a quantity or function which generally depends on ǫ, but is bounded above and below by positive constants independent of ǫ and β as those parameters approach 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' When the lower bound is not assumed, we will generally write ≤ O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The meaning of other order relations hopefully will be clear from the context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 Scaling patches We shall be dealing with functions of a single independent variable, and so the following discussion will apply to that case only.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Moreover, the independent variable will represent a space coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Finally, all statements about magnitudes in the following are to be understood in the order of magnitude sense relative to the small or large parameter under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For example if ǫ is a small parameter, then ǫ has smaller order of magnitude than ǫ1/2 as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' To repeat what was said above, we shall use the convention that a quantity is O(1) with respect to ǫ if it is bounded, and also bounded away from zero, by positive constants independent of ǫ as ǫ → 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Consider an interval in the domain of the independent space variable, and a single rescaling (of dependent and independent variables) in that subdomain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The interval may depend on the scaling, hence on ǫ, and we assume the interval has size O(1) when measured in the rescaled distance variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' A “characteristic length” for the interval can be defined in terms of the scaling of the original space variable (call it x), which produces a new space variable (call it ˆx).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Using differentials, we have, say, dx = α dˆx, with scaling coefficient α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Then a characteristic length for that subdomain will be α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' As ˆx traverses the O(1) length of the interval, x changes by an amount O(α).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The stipulation that the size of the subdomain (called a “patch” below) be of size O(1) in the rescaled variable is designed thereby so that α is a proper definition of characteristic length.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Importantly, a patch cannot be artificially enlarged by adjoining a section in which the characteristic length is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The rescaling (including that of the dependent as well as of the independent variables) can be thought of as being “natural” for a given problem if, when the solution is expressed in terms 8 of the rescaled variables, the scaled dependent variables are seen to undergo, in that subdomain, variations which are not too large and not too small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This rate of variation could be gauged by the magnitude of the rescaled derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The requirement “not too large” then would be taken to mean that all derivatives, of orders 1 up to some appropriate order, are bounded in magnitude independently of the parameters in the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It usually happens that some of these derivatives are necessarily zero or very small in places, so the corresponding (opposite) criterion cannot be imposed to gauge the satisfaction of the requirement “not too small”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Instead, a proper criterion would be that the characteristic length α associated with the scaling under consideration cannot be decreased without the above criterion “not too large” being violated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In this, “decreased” means in the order of magnitude sense: if α is replaced by a different function of ǫ with smaller order of magnitude, so that in the newly scaled variables the derivatives are correspondingly larger, then the magnitudes of some of these derivatives must not be bounded independently of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Such an interval, together with its natural scaling, will in the following be called a scaling patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This appears to be a reasonable meaning for the concept of natural scaling in patches, but it ignores, so far, the important question, how does one find the scaling patches?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That is, how does one determine the proper scaling in the proper locations?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The search for scaling patches in wall- bounded turbulent flow will be the primary activity in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' But it is not an easy question in general, and there are few easy mathematically rigorous criteria which can be applied to determine them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Nevertheless, there are nonrigorous arguments, most easily introduced through examples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Here is a straightforward one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='2 A classical example The following is an elementary textbook model example of a problem in which scaling consider- ations, and in particular scaling patches, are very pertinent to understanding salient features of the solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The solution, it turns out, can be written down explicitly, but we choose a different approach in order to better bring forth the ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Although it is well known, this example is chosen because we want to put forward a slightly nonstandard point of view, and it bears some similarity with the much more difficult averaged wall-induced turbulence problems which will be discussed later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Both scenarios have traditional inner and outer scaling regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' On the other hand there are also striking differences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Let ǫ be a small positive parameter, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 0 < ǫ ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We wish to solve the following boundary value problem for u(η): ǫ2 d2u dη2 − u + g(η) = 0 for 0 < η < 1, (17) u(0) = α, u(1) = β, (18) where α and β are fixed numbers and g is any given smooth function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' To make things simple, we suppose that g is independent of ǫ, and that it is not a constant (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' it has nontrivial variation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The solution of course will depend on ǫ as well as on η, and the desire is to find that dependence for all small enough values of ǫ, say for 0 < ǫ ≤ ǫ0, where ǫ0 is a fixed small number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This dependence of the solution on both η and ǫ is sometimes expressed by writing u(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' As a first guess for an approximate solution, we try discarding the first term in (17), which has a small factor ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We obtain the “reduced” problem u = g(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (19) 9 The function u = g(η) approximately satisfies (17) in a formal sense, but is generally far from satisfying the two boundary conditions (18) (unless by unlikely accident g(0) = α or g(1) = β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' So we must either scrap that solution altogether, or doctor it up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The latter is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is where the search for scaling patches comes in.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Two things are clear: the locations of the trouble are at the two boundaries η = 0 and η = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' And secondly, somehow the discarded term ǫ2 d2u dη2 at those two locations must be important after all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We try constructing an “internal” patch by excluding neighborhoods of the two troublesome boundary points as follows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Take any number δ > 0 independent of ǫ, and use, for the patch, the original scaling, leading to (19), in the subdomain defined by {η : δ < η < 1−δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This interval, together with the original scaling in which the variables η and u remain unchanged, will be the first scaling patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' To lowest approximation in the parameter ǫ, the differential equation (17) becomes (19).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' An important part of our argument later will be centered around the issue of how far we can enlarge this interval by letting δ depend on ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Now let us also try to construct a scaling patch which encompasses the left boundary η = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The appearance of the differential equation can be changed by rescaling in such a way as to render the derivative term overtly important.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The most natural way is by passing to a new independent variable y by y = η ǫ , U = u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (20) Then we consider U to be a function of y in some subdomain consisting of values of η near 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We shall be conservative at first and take that subdomain as {0 ≤ y < y0}, where y0 is independent of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Later, we see about extending the interval by letting y0 depend on ǫ The rescaled differential equation is d2U dy2 − U(y) + g(ǫy) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (21) Neglecting ǫ in (21) leads to d2U dy2 − U(y) + g(0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (22) The boundary condition at η = 0 (the same as y = 0) now becomes U(0) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (23) It is proposed, then, that the scaling given by (20) in the interval {0 ≤ y < y0} be our second scaling patch, and that the solution in that patch be approximately a solution of (22) and (23).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Moreover it is reasonable to impose the condition that U be bounded when y grows large (like O(1/ǫ)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This provides a unique solution and the formal approximation U(y) ≈ g(0) �1 − e−y� + αe−y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (24) This, of course, is only an approximation which is put forward for further verification.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For one thing, notice that limy→∞ U(y) = g(0) and that the original approximation u = g(η) is also close to the value g(0) for η ≪ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Thus in a sense the approximation at the left boundary meshes smoothly with the supposed approximation in the interior of the interval.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Let us dwell on the idea of smooth meshing a little more carefully.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Let ω be any positive number, arbitrarily small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We will show that there are large values of y and small values of ǫ for which the boundary approximation U(y) and the interior approximation g(η) = g(ǫy) are both closer than ω to g(0), hence arbitrarily 10 close to each other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Let y1 be a large number, depending on ω, such that |U(y) − g(0)| < 1 2ω for all y > y1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Such a number exists, as you can see from (24).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Next, let η1 be a small number, again depending on ω, such that |g(η) − g(0)| < 1 2ω for all η < η1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Such a number η1 exists because of the continuity of the function g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Since η = ǫy, both of these statements will be valid if ǫ is chosen such that ǫ < η1 y1 , and y is in the interval y1 < y < η1/ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Since both are valid for these values of y, necessarily |U(y)−g(0)| < ω.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The conclusion is that if ǫ is small enough, depending on ω, there are places where the inner and outer approximations are closer together than ω, which can be chosen as small as desired.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is what we mean by the two approximations meshing smoothly together.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In summary, we have found (a) what appears to be a reasonable approximation for the solution u(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' ǫ) in regions of the interval [0, 1] not too close to either boundary, and also (b) a reasonable approximation in regions close to the left boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The same procedure yields a third scaling patch near the right hand boundary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The solution in it undergoes a similar exponential (in terms of a rescaled variable) transition from the value β to g(1) as we move left from that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The rescaling in that case is z = 1 − η ǫ , V (z) = u(η) = u(1 − ǫz), and the right inner solution in that patch is V (z) ≈ g(1) �1 − e−z� + βe−z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (25) In the interior, the solution is approximately a function of only η with no ǫ-dependence, so no new scaling is necessary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In a region which excludes small neighborhoods of each boundary (we will discuss how small later), the original variables are the natural ones in our adopted sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That region, together with the original unscaled variables, constitutes one scaling patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The small interval y < y0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' η < ǫy0), together with the rescaling (20), constitutes a second scaling patch, in which the solution undergoes, in an exponential manner, a transition from the imposed value α to the value g(0) associated with the first scaling patch, but extended to the forbidden boundary point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' At this point the two patches do not touch each other but, as we shall see, the ranges of validity of the corresponding approximations can be enlarged so that they overlap.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Thus through proper “scaling”, the original problem may be clarified and simplified in certain subranges of the range of independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The outlined procedure for finding scaling patches in the example problem is only heuristic, but it can be made rigorous in this and other cases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Moreover, we have only found the most basic (lowest order) approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Successive higher order approximations, with errors of orders O(ǫ), O(ǫ2), etc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=', can be found and proved to be correct in this and in a large class of similar problems, such as elliptic boundary value problems when there are several independent variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='3 Inner, outer, and overlapping regions We have constructed independent approximate representations of the solution in three subdomains: (19) for η not too close to either boundary point, (24) near the left boundary, and (25) near the right one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Traditionally, these three approximate solutions are called the outer, left inner, and right inner solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We denote the outer solution by uo(η) and the left inner by U(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the following, we shall usually disregard the right inner approximation because its analysis follows that of the left inner solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 11 We now ask about the possibility of extending those subdomains, while retaining the validity of the approximate representations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We do this by introducing a family of possible intermediate scalings parameterized by an exponent γ in the range 0 < γ < 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The independent variable η is rescaled to obtain a new variable s by η = ǫγs, y = ǫγ−1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (26) We are excluding the choice γ = 0, which would correspond to the original scaling, and γ = 1, corresponding to (20).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Set Uγ(s) = u(η) = u(ǫγs).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The rescaled differential equation (17) becomes ǫ2−2γ d2Uγ ds2 − Uγ + g(ǫγs) = 0, (27) and the boundary condition is U(0) = α.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We ask, can the region defined by {0 < s0 < s < s1}, where s0, s1 are independent of ǫ, be the location of a new scaling patch?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (The answer will turn out to be no in the strict sense, but we can try.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=') If so, the corresponding solution should, to lowest order as ǫ→0, hence ǫγ→0, satisfy (27), which to lowest order comes out to be Uγ(s) = g(0) (28) (since 0 < γ < 1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That is a very simple approximate solution indeed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Keeping that in mind, we now examine the left inner and the outer solutions (24) and (19) in the proposed patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For the left inner, we obtain the expression U �ǫγ−1s � ≈ [g(0) (1 − e−y) + αe−yg(0)]y=eγ−1s.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' To find the lowest order version of this expression, just let ǫ→0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We get, again to lowest order, the same result as (28): U � ǫγ−1s � ≈ g(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (29) In the case of the outer solution, a similar procedure yields uo (ǫγs) ≈ g(0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (30) This means that in any of the proposed intermediate scaling patches (as it will turn out, they are not true patches), the inner and outer solutions are both approximate solutions of the rescaled equation (27), so that in this sense each of them is a valid approximation to the true solution, for small enough ǫ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This verifies that the original subdomains of the inner and outer scalings can be extended to include the intermediate regions, the corresponding approximations continuing to be valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In other words, there is a region of overlap, in which both the inner and outer solutions are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In fact, there are many overlap regions, depending on the choices of γ, s0, and s1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' These are all approximate solutions, and the main effective differences among them lies in the accuracy of the approximation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Bear in mind that ǫγ is approximated by 0 better when γ is larger.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Finally, note that the approximate solutions in any of these overlap regions are not very interest- ing: they are all constant to lowest order, equal to g(0)!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This implies that the regions {s0 < s < s1}, together with the scaling (26), do not form legitimate scaling patches!' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The reason is that the solu- tion does not satisfy one of the stated criteria in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1, namely that the length scale α, which by (26) is equal to ǫγ, can certainly be reduced without the scaled dependent variable, which we have seen is constant, having unbounded derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The same procedure yields overlapping regions near the right boundary point, in which the approximate solutions are all another constant, g(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 12 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='4 A uniform approximation The separate approximations in the three specified regions, namely uo(η) = g(η) (19), U(y) (24) and V (z) (25), can be combined into a single expression, providing a uniform approximation throughout the interval [0, 1].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For this, we define F(y) = U(y) − g(0), H(z) = V (z) − g(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (31) The uniform approximation is then Uunif(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' ǫ) = uo(η) + F(y) + H(z) = uo(η) + F(η/ǫ) + H((1 − η)/ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (32) Its validity can be verified directly in each of the regions considered above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This form consists of a sum of terms, each a function of one of the scaled variables.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='5 A generalization It is natural to ask whether the outlined features in the overlap region of that classical example apply also to a wider class of functions, and indeed they do.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Rather than starting with a differential equation, we take a general class of functions expressed in the form of a sum of individual functions of the three scaled variables η, y, z separately, as in (32).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' So we consider any function in the form (32), irrespective of whether it is a solution or approximate solution of some problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Such a function may still have, depending on uo, F and H, an outer approximation and two inner approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It may also have a region of overlap between the outer and one of the inner solutions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' No generality is lost by taking H = 0, and we do that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Let ǫ ≪ 1, y = η ǫ, and uo(η) and F(y) be any functions such that uo(η) has a limit as η→0 and F has a limit as y→∞: lim η→0 uo(η) = uo(0);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' lim y→∞ F(y) = G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (33) We call η the outer variable and y the inner variable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Finally, let u(η;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' ǫ) = uo(η) + F(y) = uo(η) + F(η/ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (34) This will be the central prototypical two-scaled function which will be approximated differently in an inner and an outer region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In a region {η > δ} for any fixed δ > 0, η/ǫ→∞ uniformly as ǫ→0, so that F can be approxi- mated by G: u(η, ǫ) ≈ uo(η) + G ≡ uout(η), (35) which we call the outer representation of u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Similarly near η = 0, we replace η by 0 in the first term on the right of (34) to obtain the inner representation uin(y) ≡ uo(0) + F(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (36) Now consider a family of intermediate regions parameterized by γ ∈ (0, 1) in which s, defined by (26), is confined to the interval s0 ≤ s ≤ s1, the si being specified constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In such a region, we find the approximations uin ≈ uo(0) + F(ǫγ−1s) ≈ uo(0) + G, (37) 13 uout ≈ uo(ǫγs) + G ≈ uo(0) + G, (38) u(η, ǫ) ≈ uo(ǫγs) + F(ǫγ−1s) ≈ uo(0) + G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (39) The conclusions are, first, that the inner and outer representations are both valid in the interme- diate region, which is therefore an overlap zone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Secondly, in the overlap region, the approximation is equal to the constant uo(0) + G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' These conclusions are the same, in our generic class of examples, as those for the classical example in section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='6 The Izakson-Millikan observation In [4] and [5], Izakson and Millikan independently considered functions of the form (34) without imposing limit conditions such as (33).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Their context was the averaged equations of wall-induced turbulence, but their reasoning is valid in the present more general scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' They showed, more or less, that an assumption of the existence of an overlap region is sufficient to imply that the approximations in that region are either constant or logarithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Beginning in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6, a different approach, but with a similar objective, will be discussed in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Its method, limited to the turbulence problem, will be based on a search for scaling patches In the Izakson-Millikan argument, then, no special conditions are required at either boundary of the interval I (below) on which our functions are defined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Moreover, instead of providing functions of inner and outer variables and asking whether there is an overlap region of validity of the corresponding inner and outer approximations, we now do just the opposite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We assume that there is such a region of common validity, and ask what more, if anything, that implies about those approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The answer is surprising.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Assume, for some unknown functions uo(η), U(y) (where y = η/ǫ as always), G(ǫ) and interval I, that uo(η) + G(ǫ) = U(y) (40) for all values of η ∈ I and all values of ǫ in some interval K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Since η and ǫ can vary independently of each other in their respective domains, η and y also vary independently of each other, for η ∈ I and y ∈ J, J here being the set of all values of y = η/ǫ for η ∈ I and ǫ in its previously allocated interval of variation K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The assumed identity (40) asserts that there is a region of overlap where the outer function uo(η) (plus a correction G(ǫ)) equals the inner function U(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' All these functions are at present unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It turns out that the correction G is necessary to include, because without it, (40) is too stringent a condition to be fulfilled in general.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' First, in (40) make ǫ a fixed number in K, and of course y = η/ǫ for all η ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Differentiating (40) with respect to η and multiplying by η, we obtain ηu′ o(η) = η ǫ U′(y) = yU′(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (41) This is valid for every chosen value of ǫ ∈ K, so is true for all y ∈ J as well as η ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' But since, as remarked before, η and y vary independently of each other, each side of (41) must be a constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For example η can be allowed to vary throughout I while y is held constant, which of course implies that the right side of (41) is held constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That means the left side is also constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Therefore for some constant A, ηu′ o(η) = A;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' yU′(y) = A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (42) Integrating these two equations, we find, for some other constants B and C, uo(η) = A ln η + B, U(y) = A ln y + C = A ln η + C − A ln ǫ, (43) 14 the last equation a result of the identity y = η/ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Since uo is assumed to be a function only of η, necessarily B is a constant independent of ǫ, and the same holds true of C for a similar reason.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The function G can now be determined if we substitute the expressions (43) back into (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Doing so while noting that ln y = ln η − ln ǫ yields G(ǫ) = −A ln ǫ + C − B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (44) In short, the three functions in question must be of the form (43), (44), where A, B, C are constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The original assumed identity (40) dictates a great amount of information about these functions, while not specifying them exactly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' What this result says is that functions which are simultaneously regular functions of an inner and an outer variable, for all values of those variables in certain intervals, are either constant or logarithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If it is known that the function is not constant, then this property of simultaneity is equivalent to being logarithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='7 A more realistic version The result of Izakson and Millikan is mathematically rigorous, in the sense that if (40) holds in some intervals as specified, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' if some function can be expressed exactly in terms of either an inner or an outer variable in some overlap region, then the logarithmic properties (43) and (44) also hold in that same region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' However, the assumption (40) probably never holds exactly in practice, so that the conclusion is vacuous.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In any concrete situation, it will be far more reasonable to assume that (40) is only approximately satisfied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Then the suggestion, which is not yet proved, is that the functions involved will be approximately logarithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Fortunately, such a conclusion can also be rigorously established, provided that appropriate senses are given to the two approximations (the one in the hypothesis, and the one in the conclusion).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That will be done in this section.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Note also that Gill in [6] provided such a proof under more involved assumptions and with entirely different methods.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In place of (40), we now assume that (40) holds with an additional small error term r(η, y, ǫ), which will be written r(η, y), since ǫ can be expressed in terms of η and y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The function r is not known exactly, but that does not prevent estimates about the accuracy of (43) and (44) being deduced in terms of the magnitude of r (and, as it turns out, of its derivatives).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Thus we start with uo(η) + G(ǫ) = U(y) + r(η, y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (45) As we shall show, although (43) and (44) no longer hold, one can still find upper and lower bounds of logarithmic type for the functions uo, U, G in terms of bounds on r and its derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Differentiating (45) as before, one obtains ηu′ o(η) = yU′(y) + R(η, y), (46) where R(η, y) = ηrη + yry, subscripts denoting derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Let ρ be an upper bound for |R|, valid for all η, y in the intervals I, J respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Choose any y0 ∈ J and let A = y0U′(y0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Then setting y = y0 in (46), we get |ηu′ o(η) − A| ≤ ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (47) This holds for all η ∈ I.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Also from (46), yU′(y) − A = ηu′ o − A − R, 15 so that by (47) |yU′(y) − A| ≤ |ηu′ o − A| + ρ ≤ 2ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (48) This says that when ρ is small, the quantities on the left sides of (42) are almost constant, the deviation from constancy being no greater than ρ and 2ρ, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Dividing (47) by η and rearranging terms, we get A − ρ η ≤ u′ o(η) ≤ A + ρ η , and after integrating, (A − ρ) ln η + B1 ≤ uo(η) ≤ (A + ρ) ln η + B2, (49) where Bi are integration constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In this sense, uo is almost logarithmic when ρ is small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the same way we obtain (A − 2ρ) ln y + C1 ≤ U(y) ≤ (A + 2ρ) ln y + C2, (50) with similar upper and lower bounds for G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In short, assuming R, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' ρ, is small in the intervals selected, one concludes that U(y) is bounded above and below by logarithmic expressions with coefficients of the logarithm functions which are close to each other, the discrepancy being smaller than 2ρ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It must be admitted, however, that one generally does not know the nature of the error term R, its magnitude ρ, or indeed the interval of overlap, if any.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If the intervals I, J, hence K are chosen to be shorter and well placed, one might expect ρ to be smaller and therefore the logarithmic approximations to be better.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' There is a tradeoff between the accuracy of the logarithmic approximation and the size of the interval where that approximation is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It can be guessed that as one moves closer to the canonical outer domain, r increases in magnitude because although the outer approximation uo(η) + G(ǫ) is more exact, the inner expression U(y) is not a good approximation, so the two cannot be close to each other, as (45) would seem to imply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Somewhere between the outer and the inner domains, r would be minimal, but still not zero.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This was with ǫ fixed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' A natural further question, therefore, is whether, in such fixed intervals of the space variable η, the error r and its relative R approach 0 as ǫ→0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The present argument does not answer that question, but an argument in support of a similar conclusion in the context of wall-induced turbulence will be given below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5 The mean structure of turbulent Couette flow in a channel At this point we leave the mathematical digression about scaling and turn to the main issue of this paper, namely the application of scaling ideas to turbulence induced by wall friction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Turbulent Couette flow is possibly the simplest nontrivial example.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' From this point through Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='6, we will be repeating well-known standard arguments but with different emphases.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Newer, hence lesser known, material will be found in sections beyond that.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Denote the components of x and u by (x, y, z) and (u, v, w) respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Consider a channel bounded on top and bottom by horizontal planes {y = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' y = 2δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The top plane moves with given steady velocity in the streamwise direction x, while the lower plane remains stationary.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This causes a shear stress in the fluid between the two planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the mean, fluid particles which are vertically aligned at one moment of time slide past each other horizontally.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If the velocity is sufficiently large, the resulting shear causes the flow to be turbulent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We seek to understand the “scaling structure” of the mean velocity of the fluid, and other quantities, when the flow has reached “equilibrium”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The scaling analysis of turbulent Couette flow described here, was developed in [7, 8, 9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Here scaling structure will mean that we will look for scaling 16 patches for the mean velocity and Reynolds stress profiles, or other relevant ways to scale those functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' And the term equilibrium will refer to the state in which the mean velocity is everywhere horizontal and depends, with Reynolds stress, only on the normal coordinate y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Although there will be perhaps violent particle fluctuations in all directions, the prevailing (mean) motion will be only horizontal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The dependence only on y implies that all averaged quantities are uniform along the channel and do not change in time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The upshot is that the averaged Navier-Stokes equations have only one independent variable (y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Before we get into the turbulence analysis, consider the corresponding laminar flow, in which all fluctuation parts vanish, and in fact the velocity has only an x-component, which is simply a linear function of y, the coefficients being adjusted to match the given velocities of the bounding planes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is a very simple solution, and can be verified to be a solution of the Navier-Stokes equations (1) with 0 pressure gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In fact the inertia terms of the conservation equation for the x-component of momentum vanish automatically because each velocity component is either 0 or has x-derivative 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For a similar reason, the viscosity terms also vanish.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The analogous problem for turbulent flow is orders of magnitude more difficult, and can only be solved imprecisely.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We will examine it in the framework of Reynolds averaging.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' But a possible paradox appears.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It was just brought out that the eminently simple laminar flow, in which u is a linear function of y, is an exact solution of the Navier-Stokes equations, and those equations govern the flow, laminar or turbulent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' So if we have a solution, why do we need to look for a “turbulent” one?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The answer is because (a) as we set up the problem, there are more solutions than just the laminar one;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' solutions are not unique;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' and (b) the laminar one happens to be very unstable at high Reynolds numbers, therefore not seen in practice and hence unphysical.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 The differential equations To proceed, suppose the turbulent flow to be “fully developed,” statistically stationary and two- dimensional in the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' All averaged quantities except pressure do not depend on x, z, or t;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' just on y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Moreover, we suppose that the only nonzero component of the mean velocity will be the x-component, which we denote by U(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (This part is the same as the laminar case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=') The x and y components of (8) are greatly simplified, because of the independence of τ on x, z, t: −ν d2U dy2 + 1 ρ ∂P ∂x + d dy⟨u′v′⟩ = 0, (51) 1 ρ ∂P ∂y − d dy⟨(v′)2⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (52) At the two walls {y = 0, y = 2δ}, all the fluctuating parts of u are 0, so the Reynolds stress ⟨u′v′⟩ vanishes as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' At the lower wall, U = 0 and its derivative is related to the frictional stress exerted on the wall by the fluid (or vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We have here two equations for the four unknown functions U, P, ⟨u′v′⟩ and ⟨(v′)2⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Actually, Couette flow is characterized by the absence of an applied pressure gradient;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' the sole impetus for the flow is the differential motion of the two walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Therefore we set ∂P ∂x = 0 (P itself depends on y, however, as one can see from (52)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the case of other steady channel flows, ∂P ∂x may not be zero, but it would turn out necessarily to be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The equation (51) becomes −ν d2U dy2 + d dy⟨u′v′⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (53) 17 It is important to recognize that this is a simple balance of forces, which must occur at each point of this steady (in the mean) flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The two forces exerted on the fluid are (1) the friction, or viscous, force ν d2U dy2 , and (2) the force − d dy⟨u′v′⟩ caused by the turbulent fluctuations, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' the gradient of the appropriate Reynolds stress, by which x-momentum is transported in the y (perpendicular) direction by the fluid’s fluctuations in both directions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' An intuitive description of the two forces can be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (1) The derivative dU dy measures the magnitude of the shearing motion, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' the rate at which nearby x-directed particles are moving relative to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This shearing magnitude, times the kinematic viscosity ν, is proportional to the force per unit area that the fluid lying above the horizontal plane through the point under consideration is exerting on the fluid below (and vice versa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is called the shear stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The difference between the shear stress at two nearby values of y, say y1 and y2, is the net effective force per unit area experienced by the slab of fluid in the intermediate region {y1 ≤ y ≤ y2}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Dividing this expression by y2 − y1 and passing to the limit, we obtain the net force per unit mass acting on fluid particles at that location, and see that it equals ν d2U dy2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Again, this is termed the viscous force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (2) As shown in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 2, the y-derivative d dy⟨u′v′⟩ is the x-component of the mean force per unit mass exerted on particles at that location by the turbulent fluctuations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Since there are only these two forces in balance, it can be said, contrary to some assertions, that the viscous forces and those due to Reynolds stresses are both equal players, hence both important, everywhere in the flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' At this point we have reduced the problem to a single equation (53) for U and ⟨u′v′⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' There are still more unknowns (2) than equations (1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' But we shall nevertheless be able to at least surmise some important information just from the simple mean balance law (53) plus known boundary conditions, in concert with educated assumptions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='2 The friction velocity and boundary conditions We start with some crucially important concepts and constants associated with the interaction of the fluid with the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Let τw denote the mean stress exerted on the wall by the fluid flowing past it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It is proportional to the viscosity µ, as well as the magnitude of the mean velocity’s shear at the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It is given by τw = µ d dy⟨u⟩ = µ �dU dy + d dy⟨u′⟩ � = µdU dy , (54) since ⟨ d dyu′⟩ = d dy⟨u′⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' At y = 0, ν dU dy = 1 ρ(µ dU dy ) = 1 ρτw > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Now the quantity 1 ρτw has the dimensions of velocity squared;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' it therefore defines a characteristic velocity u∗, called the friction velocity, by 1 ρτw = (u∗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (55) From this relation and the previous one, we can express dU dy = u∗2 ν at y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (56) This, together with the stipulation that U(0) = 0, (57) provide a pair of boundary conditions at the wall for U(y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It seems natural to treat the velocity of the upper wall as a given quantity to be built into the mathematical formulation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' but analytically 18 the simpler route is instead to think of the velocity u∗ (in (56)) as given, and the upper wall velocity as to be determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It is clear from physical considerations that either of these two velocities is a monotone function of the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' As noted before, the fact that all fluctuations vanish at the wall imply similar conditions for the Reynolds stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In fact, ⟨u′v′⟩ = d dy⟨u′v′⟩ = d2 dy2 ⟨u′v′⟩ = 0 (58) at y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The first two conditions here follow from u′ = v′ = 0 at the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The third condition comes about because, using subscripts y to denote derivatives, we have ⟨u′v′⟩yy = ⟨u′ yyv′⟩+2⟨u′ yv′ y⟩+ ⟨u′v′ yy⟩.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The first and third terms in this expression vanish at the wall because u′ and v′ do;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' and the second term also vanishes because (6) together with u′ x = 0 implies v′ y = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Finally at the center of the channel at y = δ, we invoke the symmetry of the flow to conclude that d dy⟨u′v′⟩ = 0 at y = δ, (59) since every vertical fluctuation v′ is matched, and canceled during the averaging process, by another one in the opposite direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' From (53), U has an inflection point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In fact, consistent with (59), the Reynolds stress ⟨u′v′⟩ increases in magnitude from 0 at the wall to a maximal value at the centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='3 Dimensionless variables in the core region For better insight, we now seek to nondimensionalize (53).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That is, we rescale the variables by multiplying them by typical and meaningful characteristic dimensional constants so that the results are dimensionless.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' There are at least two traditional and natural ways to do this.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In any case, we argue that the characteristic velocity in this turbulent flow should probably be u∗, because the former’s magnitude should be directly related to the wall stress, which is what slows the fluid down at the lower wall and causes it to go forward at the upper wall, and so in some sense causes the turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' So we nondimensionalize U by u∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (As brought out above, another natural, but less convenient velocity unit would be the maximum mean velocity or the average mean velocity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=') In the interior of the channel, it is intuitive that y should probably be scaled by the channel half-width δ, simply because that is the only immediately obvious characteristic length appropriate to that part of the channel (further justification will be given in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='8).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' And everywhere, the Reynolds shear stress should be scaled by the shear stress at the boundary, so by (u∗)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We therefore define U+ = U u∗ , η = y δ , T = −⟨u′v′⟩ (u∗)2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (60) The centerline of the flow region is at η = 1, and by symmetry considerations, the mathematical problem can be set up in the half-channel 0 ≤ η ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The superscript “+” in U+ is the traditional way to signify that this variable is normalized with parameters related to what happens at the wall: the friction velocity u∗ and/or τw.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The scaled distance η is the first natural way to nondi- mensionalize the y-coordinate, and η will be called the “outer distance variable”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This, along with the “inner distance variable” y+ to be discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='4 below, have been standard choices for dimensionless distance since the beginning of theoretical investigations of wall-induced turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' However, the existence of scaling patches as defined in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 employing these scaled distances remains to be established.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That will be done in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 19 With the normalization (60) we obtain in place of (53), dT dη + ν u∗δ d2U+ dη2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (61) We also use u∗ and δ to define our Reynolds number R∗ = u∗δ ν and small parameter ǫ2 = (R∗)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Using this notation, we get dT dη + (R∗)−1 d2U+ dη2 = dT dη + ǫ2 d2U+ dη2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (62) As noted earlier, we have an underdetermined problem—a single equation for the two unknowns T and U+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' But there is even more bad news: the boundary condition at η = 0 is in trouble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Naively assuming the formally small second term can be neglected across the whole channel, we get that T is constant, and since it vanishes at the wall, we would obtain that T = 0 everywhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is incorrect, of course, for a reason similar to that which applies to the classical example in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='2: a different scaling applies near the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We construct now a second scaling domain, near the wall, to partially remedy this defect, as was done for the classical example in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='4 The wall layer, rescaling, and law of the wall It is traditionally recognized that we do have at least two space scales in the channel—one of them associated with δ and another one close to the wall;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' we’ll get to that shortly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We will have an “outer” and an “inner” approximate solution.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Finding out how to connect them makes an interesting story.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' However, we cannot carry out a full-blown asymptotic analysis because too much is unknown.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Much of the following proceeds by “reasonable suppositions”, which can also be verified to some extent by experiments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It is natural to choose the inner scaling in such a way that the two terms on the left of (62) have the same orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' After all, those two terms represent the two forces in the fluid which have to balance (note, that’s not what we did to get (61)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Thus we define y+ = ηR∗ = u∗ ν y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (63) Then (62) becomes dT dy+ + d2U+ dy+2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (64) In this, there is no incompatibility with the boundary conditions at y+ = 0: T = 0 and dU+ dy+ = 1 at y+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (65) The integrated form of (64) is T + dU+ dy+ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (66) To summarize, we have the traditional approximation that asserts that T is constant (but no approximation as yet for U+) in the outer region near the channel’s centerline, and an equation (64) or (66) relating T and U+ in the inner region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 20 The region next to the wall where the spatial variations (in y) have characteristic length ν u∗, is call the wall layer (as opposed to boundary layer).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The choice (63) of scaling in this region says that we are treating the scaled Reynolds stress T on the same footing as the wall stress or skin friction (R∗)−1 dU+ dη .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The former arises from the inertia terms in the Navier-Stokes equations and the latter from the viscosity terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the wall layer, then, it is reasonable to suppose that U+ and T are (approximately) functions only of the inner variable y+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This property is called the law of the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='5 Velocity in the core The law of the wall is no longer valid for large R∗ as we move into the interior of the channel and on into the core region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The outer scaling will turn out to be valid there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' But remember, at this point we only have that T is approximately constant there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' U+ may be unbounded as a function of ǫ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' the traditional way to incorporate the validity of the outer coordinate η in an expression for U+ is to postulate the defect law U+ = U+(1) + h(η), (67) where h is unknown, except that h(1) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The concept of thickness of the wall and the core scaling regions should be clarified.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' One interpretation of these regions would be where the solution’s characteristic length is unity in the appropriate scaled coordinate.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Then the thickness would be O(1) as measured in that scaled variable, be it y+ or η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In particular, the wall region is characterized by {y+ ≤ O(1)}, and that of the outer scaling region is characterized by η0 < η ≤ 1 for some arbitrary positive number η0 independent of ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Note that the ratio of the thickness of the wall layer to that of the core is O � 1 R∗ � ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' this contrasts with the laminar hydrodynamic boundary layer (Prandtl theory), where the ratio is O � Re−1/2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Another interpretation of the two regions would be where one or the other of the approximate representations of the solution, namely the law of the wall or the defect law, are valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This interpretation yields, as it turns out, larger regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Let us assume that these two laws are correct in their respective domains;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' better justification for this will be given later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Recall ǫ = (R∗)−1/2;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (68) since U+(1) will depend on ǫ, we set U+(1) = G(ǫ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the outer region, then, we may posit the representation U+ = G(ǫ) + uo(η).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (69) In the inner region, on the other hand, we are using the law of the wall approximation U+ = p(y+), (70) i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' U+ is a function only of y+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The unknown quantities here are p, G, uo.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 21 0 5 10 15 20 25 30 35 40 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 1 10 100 1000 10000 100000 1e+06 U y Reτ=181 (Kawamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=') Reτ=642 (Iwamoto et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=') Reτ=1655 (Wei and Willmarth) Reτ=11062 (McKeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=') Reτ=103340 (McKeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=') Reτ=534538 (McKeon et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=') 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='/0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='41*ln(y)+5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Figure 1: Inner normalized mean streamwise velocity in Couette flow, pressure-driven channel flow and pipe flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Couette flow data are from DNS of Kawamura’s group [13, 14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Pressure-driven channel flow DNS data are from Iwamoto et al [15] and experimental data are from Wei and Willmarth [16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Pipe flow data are from superpipe data of McKeon et al [17].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='6 Application of the Izakson-Millikan reasoning A great number of analytical and semianalytical studies of turbulent mean profiles have utilized the Izakson-Millikan observation as an essential ingredient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It has also been generalized and elaborated upon in many ways.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For example the idea of composite expansions in more traditional settings of two-scale problems has been extended to the present scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We recommend the review paper [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Given the two approximations (69) and (70), they can be thought of as the inner and outer func- tions specified in the two sides of (40).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='The variables p, uo, G, y+, η are analogous to U, uo, G, y, η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Thus the left side of (40), uo(x) + G(ǫ), is analogous to the right side of (69), and the right side of (40), U(y), is analogous to the right side of (70).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If we now make the Izakson-Millikan hypothesis that there exists a common region in which the two expressions are almost equal, then the three functions in question must be either approximately constant or approximately logarithmic, as in (43) and (44).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the present context, the conclusion is that in the common region, whatever it is, the following hold: uo(η) ≈ A ln η + B, p(y+) ≈ A ln y+ + C, G(ǫ) ≈ − A ln ǫ + C − B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (71) Therefore the functions are approximately either constant (A = 0) or logarithmic.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is a well-known conclusion, and indeed the mean velocity profile in wall-bounded turbulent flows is seen to exhibit logarithmic type behavior in certain regions which can be estimated on the basis of experimental data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Examples of the logarithmic property can be seen from the empirical data shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The coefficients A, B, C can also be so estimated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' All in all, the Isakson- Millikan observation, in all its simplicity, should be counted as one of the great success stories of 22 theoretical turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Focussing on our stated objective to at least try to answer the question why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=', we discuss the given derivation of the logarithmic property in some detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In particular, we ask whether it can be supported by alternative trains of thought.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='7 Observations on the foregoing procedure The conclusion (71) gives a surprising amount of information about the inner and outer approxi- mations, based on what appears to be a small amount of input.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The basis for the argument rests on very little physics or fluid dynamics;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' it is simply an as- sumption about inner and outer approximations agreeing somewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If one is willing to admit the existence of those inner and outer approximations, what remains is simply a mathematical issue, and could apply to any situation where there are two space scales with different but overlapping domains representing a strictly monotone function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Let us rephrase what has been found in terms of the more realistic conclusion corresponding to Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If the mean velocity profile is everywhere monotone and there is a region in the flow where that profile can be expressed approximately and simultaneously as a function of the inner variable alone and the outer variable alone (up to an additive function of ǫ alone), then these functions must be approximately logarithmic, the degree of the latter approximation being dependent on that of the former.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Let us take for granted the monotone part;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' that property of the mean velocity profile is well known and can be rationalized by the supposition that the viscous stress is everywhere positive and a decreasing function of distance from the wall, which is the site of the imposition of such stress by outside means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Given the monotonicity, what other information can we use to determine the profile, at least approximately?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The needed additional information should be theoretical in nature, because our aim is to explain the reasons for observed behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We know the Izakson- Millikan implication, which, stated succinctly, says, “overlap implies logarithmic”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It is a simple piece of reasoning which is classical and, again, well-known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Being so simple and direct, it places the hypothesis that a particular region is an overlap region very close to the conclusion, namely close to assuming logarithmic behavior.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Either property can be substituted for the other.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If this is true, the argument is close to being circular.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If available, independent arguments to determine properties of the mean velocity and Reynolds stress profiles, not relying on the overlap hypothesis, would be highly desirable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That will be a principal aim in the following sections.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' One direction of enquiry which is suggested is to consider all possible increasing profiles which are functions of the inner variable near the wall and of the outer variable near the centerline;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' and try to use some reasonable selection criterion to at least make a good argument for what the actual profile should be.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The set of profiles which have the overlap property, hence the logarithmic property, is only a small part of the set of all conceivable profiles, because the latter includes an ample collection of arbitrary non-logarithmic ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If the overlapping logarithmic ones are to be selected, as empirical evidence suggests in some regions, then theory should be able to supply a reason for that choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We should mention in passing a small cloud hanging over the search process: it seems on the basis of examples explored in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4 that overlap regions, if they exist, are usually characterized by the functions being constants in those regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Given that U is nowhere constant, one wonders whether the assumption that an overlap region exists is itself reasonable.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Another possible direction of enquiry is to ask whether a unified approach is possible, which gives more justification to the inner and outer scalings themselves, and at the same time is capable 23 of handling all other parts of the profile as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the next subsection we draw attention to an intuitive and vague train of thought which may be relevant to the question of why the mean profiles might have certain features.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In Section 6, a relatively new approach to this and other questions will be explored.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It involves a systematic search for scaling patches and does not require separate assumptions about the validity of inner and outer scales.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Apart from the above, another consideration to be kept in mind is that the overlap hypothesis provides no theoretical basis for determining the location of any expected overlap region, nor the accuracy of (69) in that region (neither does the scaling patch approach, for that matter).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It is expected that the size of the region is related to the accuracy of the basic hypothesis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Increas- ing the region where the two expressions are approximately valid decreases the accuracy of the approximation in that region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='8 The uniformizing effect of turbulence and some possible implications.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The turbulent nature of the fluid motion, which is not explicit in the averaged equation under study, has a mixing and hence uniformizing effect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This, it may be argued, tends to smooth out or eliminate abrupt spatial changes in the average properties of the flow as one moves from one location to another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The flow properties at one place will be similar to what they are at nearby places.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Although sources of stress which are imposed from the outside, such as wall friction and en- joined pressure gradients, are often also sources of turbulence, they may work in opposition to the uniformizing action.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For example the generation of stress due to the wall in channel flow acts at a different location from an imposed pressure gradient, and this difference results in a nonuniform distribution of Reynolds and viscous stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Still, the claim of uniformization seems without much doubt to be valid for some properties if the vague terms “properties” and “similar” are given some appropriate definition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If we assume that this principle holds for the local scaling properties of the variables, including for the local characteristic length as it depends on location, then there should be a whole continuum of characteristic lengths, each appropriate to a specific locale, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' specific distance from the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Thus as we pass from the wall to the centerline of the channel, the uniformizing principle suggests that the characteristic length encountered during the passage should change continuously from the one for the inner scaling to the one for the outer scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In fact, this is one of the main conclusions to be brought out by a different reasoning in the next section, which introduces and develops a very different approach to the problem of determining the nature of the profiles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It uses scaling and order of magnitude arguments in a major way, but does not assume a priori the validity of the inner or outer approximations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Rather, it presents an alternative criterion for “scaling patches” and shows that the inner and outer regions, among many others, fit that criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In this sense, the identification of patches is derived from the assumed criterion rather than presupposed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6 The search for scaling patches The more recent approach to understanding of the scaling structure of the mean velocity and Reynolds stress profiles presented here was introduced in [7, 8, 9, 10, 11].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It forms an alternative to the approaches considered above, not in any sense of replacing them, but rather in the spirit of 24 1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='5 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='5 1 0 20 40 60 80 100 120 140 160 180 200 Tβ = T+ - βy+ y+ Couette: T+ β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1000 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='0700 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='0465 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='0150 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='0055 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='0020 β=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='0004 Figure 2: Adjusted Reynolds stress profile for various values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The case β = ǫ4 corresponds within O(ǫ2) to the genuine Reynolds stress for Couette flow and β = ǫ2 is an approximation to that for pressure driven channel flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The DNS data are from [18], δ+ = Reτ = 181.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='3 and ǫ = .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='074.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' adding to them new information and insights.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Finally, it is applicable to a wide variety of wall- induced turbulence scenarios.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For the time being, we continue to operate within the context of turbulent Couette flow through a channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 Adjusted Reynolds stresses, balance exchange phenomena, and the identi- fication of patches Let β be a small positive number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Restrictions on it will be given later.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In terms of the originally defined dimensionless Reynolds stress T, let T β(y+) = T(y+) − βy+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (72) (Note that β is a superscript not an exponent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=') The function T β is simply a mathematical construct that will be called an adjusted Reynolds stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It satisfies dT β dy+ = dT dy+ − β, (73) and from (64) d2U+ dy+2 + dT β dy+ + β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (74) The T β are plotted for various values of β in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The genuine Reynolds stress, which corresponds to the case β = 0, vanishes with its derivative at the wall and rises to attain a maximum at the centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' As β increases, the position of the maximum moves toward the wall, eventually disappearing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The main interest is in those adjusted stress functions that exhibit local maxima—which will be the case when β is not too large—because, as it turns out, scaling patches Lβ with characteristic lengths with orders of magnitude O(β−1/2) (as β → 0) exist at those peak locations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The reasoning below will justify this assertion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Part of the argument involves obtaining 25 an exact differential equation in rescaled variables having no explicit dependence on ǫ or β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Another part entails the recognition that (74) expresses an approximate balance between its first two terms (since β is small), and that this balance is necessarily broken at some point and changed to another kind of balance, because y+ eventually attains a value such that the three terms in (74) have the same order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Let us pursue this idea of balance exchange.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' As was brought out in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='2 following (59), the function T(y+) increases from being 0 at the wall to attain its maximal value at the centerline η = 1, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' y+ = ǫ−2 = δ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Assuming that β is small and positive, therefore, we see that the adjusted stress T β has negative derivative at y+ = δ+, so must attain its maximal value Tm(β) at a point y+ β < δ+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Moreover, the location of this maximum decreases as β increases, because when β increases, the zero derivative at any maximum becomes negative.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Within the inner scaling region where the law of the wall holds, for example when y+ ≤ O(1), the two derivatives in (74) will generally have magnitudes O(1) except very near the wall in the viscous sublayer, where they are both very small.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Since β ≪ 1, those two derivatives will balance, except for an error represented by the last term in (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Both derivatives will therefore generally be O(1) quantities, except as noted above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This occurs within the inner scaling region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' However as y+ increases to a neighborhood of y+ β , this necessarily changes, because the value of dT β dy+ decreases to zero at y+ = y+ β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For points near enough to that value, the second term in (74) must take on values ≤ O(β), and therefore by (74) again, the first term does as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It is therefore natural to propose that there may be a scaling patch occupying that neighborhood with respect to which all three terms of (74) have the same formal order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' To be more precise, the last term balances the sum of the first two terms, each of which is ≤ O(β).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It turns out that it is possible to construct a candidate for such a patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It will be centered at the location y+ = y+ β , where dT β dy+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' At that point, the derivatives appearing in (74) annihilate the linear terms in the Taylor series of the functions U+ and T about y+ = y+ β (in fact the linear term in T is identically zero).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' As a result, those linear parts do not play a role in the rescaling process, and one may work only with the remainders after those parts have been separated off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' With this in mind, we write, for some coefficients α(β), γ(β), λ(β) to be determined, y+ = y+ β + αˆy, T(y+) = Tm(β) + γ ˆT(ˆy), U+(y+) = U+(y+ β ) + m(y+ − y+ β ) + λ ˆU(ˆy), (75) where m is the slope m(β) = dU+ dy+ (y+ β ) of the mean velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The slope m is unknown at this point, but will be found later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The linear parts, which have been segregated in (75), are Tm(β) and U+(y+ β ) + m(β)(y+ − y+), respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' They are separated off because the derivatives appearing in (74) annihilate them and they take no part in the present calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That is also why m is not determinable until later in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='3 (see (93)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The new rescaled variables are ˆy, ˆT and ˆU.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Then (74) becomes λ α2 d2 ˆU dˆy2 + γ α d ˆT dˆy + β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (76) In order for the three terms to be formally equal in order of magnitude, one can specify α = �λ β �1/2 , γ = (λβ)1/2, (77) so that y+ = y+ β + �λ β �1/2 ˆy, T(y+) = Tm(β) + (λβ)1/2 ˆT(ˆy), U+ = U+(y+ β ) + my+ + λ ˆU(ˆy), (78) 26 and (74) is transformed into the parameterless equation d2 ˆU dˆy2 + d ˆT dˆy + 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (79) The criterion of equal formal orders of magnitude, therefore, does not by itself determine uniquely the three scaling factors α, γ, λ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (77) leaves the factor λ undetermined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This sug- gests that there may be a one parameter family of potential scaling patches at the location y+ β , the parameter being λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We are confronted with an extra degree of indeterminacy, because the present line of reasoning does not offer a way to determine which of the potential patches represent actual ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' However there is considerable evidence, to be summarized in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='7, that the correct scaling at this location y+ β is given by (77) with λ = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The analysis to follow in this and the next two subsections holds for other choices of λ as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We assume that λ(β), like α and γ, is a power of β, and define, in place of λ, the parameter σ by λ = β−σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (No constant coefficient is needed with this power law because we are dealing only with orders of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=') It will be shown that the case σ = 0 leads to a logarithmic-like profile for the mean velocity U+, and when σ is positive, we get behavior like a power law with exponent depending on σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For reference, the prototypical case is σ = 0, when y+ = y+ β + β−1/2ˆy, T(y+) = Tm(β) + β1/2 ˆT(ˆy), U+ = U+(y+ β ) + m(β)(y+ − y+ β ) + ˆU(ˆy).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (80) No information is available at this point about the slope m, which bears on the profile of U+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' However, information about it will be found later (93).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We argue that the scaling (78), for some choice of σ, is the natural one in a neighborhood of y+ β , and that this neighborhood is a scaling patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' On the basis of the explanation in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 together with (75) and (77), it will follow that the characteristic length in that patch will be ℓ(β) = �λ β �1/2 = β−(σ+1)/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (81) In fact not only does the scaling produce a parameter-free exact form (79) of the momentum balance equation, but at locations in the proposed patch it can be verified that the individual derivatives in (79) have the right order of magnitude, namely ≤ 1 with at least some of them = O(1)).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For example, at the peak location, the three terms on the left of (79) are −1, 0, 1 respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Leading up to that peak, the middle term is positive but still ≤ O(1), which makes the first term also O(1), according to (79) again.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In a scaling patch, as defined originally, all derivatives using the scaled variables are ≤ O(1), and in the case of at least one of those variables, the magnitudes of its derivatives are not all strictly < O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If that were not the case, the scaling factor α could be decreased without forcing some of the new rescaled derivatives to be unbounded as β→0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the present case, we have shown that these order of magnitude relations hold for the particular derivatives appearing in (79).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That fact makes the scaled neighborhood of y+ β with width O(1)) in ˆy a candidate for a scaling patch, provided the correct choice of σ is taken.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This will be our accepted criterion for the existence of a patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Thus if the correct σ is taken, then given any suitable small number β, one concludes that there is a corresponding scaling patch, which we shall call Lβ, with characteristic length ℓ(β) = β−(1+σ)/2 located at the point y+ β where T β achieves its maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 27 W=β-εt 2 ym β Ι y+ W(y+) Figure 3: Schematic diagram showing the role of the function W(y+), which is called P(y+) in the text, in determining the relation between β and the location yβ m of the corresponding scaling patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The interval I shown here is one of many choices of interval on which W(= P) is decreasing.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' See (82).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' One may question whether this characteristic length ℓ, for the correct choice of σ, is comparable to the mixing length of Prandtl in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 3 or the scaling parameter introduced by von Karman (15).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' They arise from apparently different considerations;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' the lengths used by von Karman and Prandtl are characteristic lengths vaguely associated with the fluctuating velocity, simply postulated to exist, whereas the present one is related to the function T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' And yet T is in essence something which is defined in terms of those fluctuations, and so one may surmise that the two concepts are related somehow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Because of the vagueness of mixing length concepts, that may be all that can be said.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='2 The locations of the scaling patches In the following the analysis will be done only for the case σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Analogous calculations may be done in the other cases as well;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' we will simply provide some key equations in the general case, without derivations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' At this point one should ask how to determine the range of parameter values β for which the foregoing construction of scaling patches is possible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is best answered at first in terms of the known qualitative properties of the function P(y+) ≡ dT dy+ (y+);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' later a more complete answer will be given.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The function P vanishes at the wall (y+ = 0) and, by symmetry of the function T, also at the centerline y+ = δ+ = ǫ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' As y+ passes from the wall to the centerline, T rises to its maximum at the latter location;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' during the transition, T and P are both positive.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Since P = 0 at those two locations, it must attain a positive maximum at some intermediate point;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' call it y+ p .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Being the gradient of T, P is expected take on its greatest values in the inner region, where the length scale is shortest and gradients are largest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Therefore the location y+ p of the positive maximum of P will be expected to lie in the inner region, so that y+ p = O(1).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 28 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='01 0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='03 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='04 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='06 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='07 0 10 20 30 40 50 60 70 80 90 100 dT+/dy+ y+ y+=7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='0 Reτ=82 (Bech et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=') Reτ=128 (Kawamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=') Reτ=181 (Kawamura et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=') ReΘ=1410 (Spalart) dT+/dy+=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='007 Figure 4: Inner normalized Reynolds stress gradient for a variety of flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The turbulent Couette flow data are from [19] and [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Also included are the turbulent channel DNS data from [20] and turbulent boundary layer DNS from [21].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For y+ > y+ p , P(y+) will decrease from its maximum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Let us assume it is a decreasing function on the entire interval (y+ p , δ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Given any point y∗ in that interval, let β(y∗) be the corresponding value of P, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' P(y∗) = β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (82) According to (73), which can be written dT β dy+ = P(y+) − β, the function dT β dy+ = 0 at the point y∗ under consideration, and since P is a decreasing function, dT β dy+ changes from being positive to being negative as y+ increases past the point y∗.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Therefore T β has a maximum, which we may call T β m, at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The point y∗ in question may therefore also be labeled y+ β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is illustrated in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 3, with plots of typical functions P in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' What has been shown is that any value y∗ ∈ (y+ p , δ+) will serve as y+ β if β is chosen to be P(y∗).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The set of all such values of β is the set for which the above construction of a scaling patch will work, namely the set of all values of P(y+) for y+ ∈ (y+ p , δ+).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That range in y+ provides, then, the locations where we have succeeded in finding a scaling patch;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' moreover we have found the characteristic lengths of all these patches;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' they are given according to (80) in the case σ = 0 by β−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' These characteristic lengths increase with increasing distance from the wall, since P decreases with distance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It will be argued in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='4, in fact, that asymptotically as β→0, they are proportional to distance from the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='3 More on the locations of the patches It will be desirable to correlate the locations y+ β of the scaling patches with the characteristic lengths given by (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The above discussion accomplishes this in terms of the function P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We now ask whether information can be obtained even if P is not known.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In order to proceed we now further exploit the facts that T β has a maximum at y+ = y+ β , and that (80) expresses the normal (natural) scaling near that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In terms of the rescaled variables given in (80), we have that ˆT has a maximum of 0 at ˆy = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We are going to emphasize the 29 variation of β and the dependence of the rescaling shown in (78) on β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Then the fact that T β(y+) has a maximum at y+ β implies that for all β in the allowed interval of values, dT β(y+) dy+ (y+ β ) = dT(y+) dy+ (y+ β ) − β = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (83) This may be differentiated with respect to β to yield d2T(y+) (dy+)2 (y+ β ) dy+ β dβ − 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (84) Writing the second derivative on the left in terms of the scaled variables, we find β3/2 d2 ˆT(ˆy) (dˆy)2 �����(ˆy=0) dy+ β dβ − 1 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (85) In the patch Lβ, the scaled variables satisfy (79), which is parameter-independent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' moreover ˆT satisfies the two equations ˆT(0) = d ˆT dˆy (0) = 0, which are also parameter-independent relations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' One can therefore argue that the second derivative on the left of (85), which we shall designate by −A ≡ d2 ˆT(ˆy) (dˆy)2 ����� (ˆy=0) , (86) should not depend in any major way on the parameter β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is reminiscent of similarity hy- potheses, since an assumption that A is constant is an assumption that the quantity A is invariant under certain transformations associated with changing β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Certainly the order of magnitude of A (with respect to β)) is O(1), hence β-independent, and we shall argue below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='6 that in some regions any β-independence should vanish in the limit as ǫ→0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' From (85), then, dy+ β dβ = −A−1β−3/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (87) Recall that y+ β is the location, in the original inner variable, of the scaling patch with charac- teristic length β−1/2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The equation (87), therefore, provides some insight into the dependence of that characteristic length on location, without using much knowledge about the function P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='4 The case A = constant If A, as it depends on β, were known, the locations y+ β could be found by solving differential equations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It is not known, of course;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' but its order of magnitude is known to be unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That knowledge provides order of magnitude information about the profiles and about the locations y+ β .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It is most instructive at this point to look at the calculations in the easiest case A = constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' After that, we can see how the results so obtained are still valid in an order of magnitude sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 Characteristic length as it depends on location As mentioned, the scaling given in (78) shows, among other things, that the characteristic length ℓ of a patch is given by (81).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We shall sometimes write relations below in terms of ℓ instead of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The symbol C, with or without subscript, will denote a variety of different constants, sometimes integration constants, which depend on σ but not on β or A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 30 If A is constant in some given interval, then in that interval (87) may be integrated to obtain y+ β = −A−1 � β−3/2dβ = A−1C0β−1/2 + C1 = A−1C0ℓ + C, (88) where the final C is an integration constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We now drop the subscript β from y+ β and obtain the relation ℓ = CA(y+ − C).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (89) This same relation in fact holds as well for other choices of σ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The left side is the characteristic length of the patch located at y+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' except for a shift in the independent variable y+, that length is proportional to distance from the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='2 The Reynolds stress In (82) the point y∗ may be identified with the general point y+ in (89), the location of the patch Lβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Substituting the expression for β given in (89) into the right side of (82) and recalling the definition of P, we obtain dT dy+ = β = C1A−2(y+ − C2)−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (90) Integrating this provides T as a function of y+: T(y+) = −C3A−2(y+ − C)−1 + C4, (91) with another integration constant C4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The relation (91) holds only under the assumption that A is constant, and only for values of y+ in the interval where T is a decreasing function, because that is where the scaling patches were found.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This range extends up to the channel midline at y+ = δ+, at which point the left side of (91) vanishes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This gives a relation among δ+ and the integration constants.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Since the right side of (90) cannot vanish and the left side does, the assumption that A is constant is incorrect at least near the centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='3 The mean velocity profile From (66) and (91) we find dU+ dy+ = 1 + CA−2(y+ − C)−1 − C5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (92) We require this derivative to be small for large y+;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' this simply means that at the centerline U+ must become flat as ǫ→0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Therefore it appears that C5 ≈ 1 and dU+ dy+ ≈ CA−2(y+ − C)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (93) Combining (93) with (89) yields an asymptotic relation between the characteristic length ℓ and the slope of the mean velocity profile: dU+ dy+ (y+) ≈ CA−2ℓ−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (94) For other values of σ the right side should be replaced by CA−2/(1+σ)ℓ−(1−σ)/(1+σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In any case, we may express this in terms of β and identify it with the slope m in (78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Note that in the case 31 σ = 0, m ≈ ℓ−1, which says that the characteristic length of the linear part of U+ in (78) is the same, ℓ, as that of the nonlinear part.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' More generally, in any case if the latter is true, then m should also be identified with the ratio of characteristic increments ∆U+/∆y+, which from (78) is λ/α = β(1−σ)/2 = ℓ−(1−σ)/(1+σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is in agreement with (94).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The conclusion is that for any choice of σ, the characteristic slope of the U+ profile calculated by integrating (87) on the basis of the nonlinear increments is the same as that which one would surmise by assuming that the characteristic increments for the linear parts of (78) are the same as for the nonlinear parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This fact lends credence to that assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Integrating (93) again gives U+ ≈ � CA−2/(1+σ)(y+ − C)2σ/(σ+1) + C6, σ > 0, CA−2 ln (y+ − C) + C6, σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (95) This expression (95) in the case σ = 0 is similar to that (71) for the mean velocity in a hypothesized overlap region given by the Izakson–Millikan argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the case 0 < σ ≪ 1 it gives a power law with small exponent;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' that also has been suggested in the past, but experimental or DNS data has left the resolution of the question unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In both the Izakson–Millikan argument and the present argument, questionable assumptions lead to the derivation of (95), or at least part of it.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the I-M case, those assumptions have been reviewed in Section 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the present case it is mainly, but not solely, the assumption that A is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The crucial assumption in either case must realistically be only approximate, with unknown error.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the present scenario, however, there is a good reason to believe that in some regions, the error in the assumption about A approaches 0 as ǫ→0, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' in the limit of large Reynolds numbers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That reasoning will be given below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='5 Relaxing the assumption that A is constant Although the similarity assumption A ∼ 1 is true in order of magnitude, it is unlikely to be strictly true except in interior regions at high Reynolds numbers, as shown below in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If A = A(β) depends on β, then (87) is still valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Moreover if A remains O(1), i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' bounded above and below by positive constants independent of β or ǫ, then (88) and (91) still hold in a weakened sense.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' They are replaced by pairs of inequalities for some constants Ki independent of all parameters.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the logarithmic case σ = 0, we have K1ℓ ≤ y+ β ≤ K2ℓ, (96) K3(y+ − C)−2 ≤ dT dy+ = β ≤ K4(y+ − C)−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (97) Thus in an asymptotic sense as y+→∞, the characteristic length is still proportional to distance from the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Similar transformations to pairs of inequalities are valid for (90)–(95).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' For example if σ = 0 the latter becomes K5 ln y+ ≤ U+ ≤ K6 ln y+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (98) In domains where A is nearer to being constant, such inequalities are valid with constants Ki which are closer to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In fact the error in assuming K5 = K6, for example, can be estimated in terms of any assumed error bound in assuming that A = constant in the domain being considered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is parallel to the error consideration in the Izakson-Millikan argument, which was 32 detailed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In both cases the conclusion of logarithmic growth is probably never exact;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' it should be considered an approximate statement, with the accuracy of the statement dependent on the accuracy of the underlying assumption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the I-M case, the underlying assumption is that the outer and inner approximations are both exact in some domain;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' in the present case, it is that the quantity A is constant in some domain.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In both cases there is no theoretical way to gauge how accurate the approximations are (see, however, the following subsection).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='6 Approximate constancy of A in interior zones As was brought out before, the order of magnitude of the quantity A remains O(1) in β and ǫ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In locations far away from the endpoints of the range of the continuum of scaling patches, it can be argued that A should be almost constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The reason is that the data we have for A, namely the differential equation it satisfies and the known exact values of the terms in that equation, are parameter-independent.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Therefore any variation in A due to changes in β will be caused not from those sources.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This invariance as β changes suggests, by a similarity consideration, that A is constant if it is not subject to other influences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Those would be influences from neighboring patches, hence ultimately from locations, on either side of the continuum, where the boundary would introduce “external” influences.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Only at those places would the similarity suffer external disruption.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' And the effects of that disruption would be most likely to happen near those disrupting sites, either toward the outer or the inner zones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That leaves interior regions far away from those zones as candidates for places where A is nearly constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It was shown above in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='3 that those are the regions where the mean velocity profile is logarithmic-like.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The extent (in inner units) of these zones of near similarity will grow as ǫ becomes smaller, because there will be a larger range of patches far away from the extremal patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='7 Evidence for logarithmic growth, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' σ = 0 The possibility that the mean velocity profile grows in parts of the flow according to a power law, rather than a logarithmic law, has been discussed by other authors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In either case, the actual expression for the mean velocity will depend somewhat on the Reynolds number and is very unlikely to be exactly logarithmic or a power function.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' These may represent approximations, but that is all.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' What we are concerned with are trends, brought about by relations such as (98) and its analog for power functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' They are generated by the scaling parameter λ = β−σ in (78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Here we summarize the evidence in favor of choosing σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The Izakson-Millikan argument (Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='6).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It was shown from empirical data in [7] that the increment in U+ across the mesolayer is O(1), in fact near 1 independently of the (large) Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The mesolayer is one example of a scaling patch for turbulent Poiseuille flow.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The analysis of that flow in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 7 follows the present analysis (which has been for Couette flow).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The role of the parameter σ in determining the increment in U+ across any scaling patch, including the mesolayer, is seen from (75).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The nonlinear part of the increment is simply O(λ) = O(β−σ), and it was brought out following (94) that in all cases the linear part of the increment, governed by m, is the same.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Therefore the only case in which this increment is O(1), as apparently required in the mesolayer, is the case σ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 33 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='8 The inner scaling patch at the wall and the outer scaling patch at the midline The construction of the patches given in section 6 has, as a primary ingredient, the fact that as the peak in the adjusted Reynolds stress is approached, a region must appear in which all three terms in the mean momentum balance equation, which in this case is (74), will have the same order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is simply because the gradient dT β dy+ approaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (There will, of course, be a smaller region encompassing the peak in which the last term on the left of (74) has smaller order of magnitude than the others, because it vanishes at the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=') Curiously, a somewhat similar phenomenon happens when y+→0, since the gradient of the actual Reynolds stress, rather the adjusted one, is zero at the wall (y+ = 0) and positive for small values of y+ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We are speaking of the first term dT dy+ in (64).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' With the inner scaling, both terms in (64) have equal orders of magnitude, both actual and formal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This in itself provides evidence that this wall region, together with the inner scaling used in (64), defines a scaling patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' But there is further evidence from the boundary condition (65) for U+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That condition of course was engineered by our very choice of inner scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' But whatever its origin, it furnishes decisive evidence that a scaling patch exists there.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' As before, the width of this patch is O(1), measured in y+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' As was brought out before, it also encompasses the crucial point y+ p discussed in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' To summarize, at the wall, U+, T, and dT dy+ are all 0, but dU+ dy+ = 1 (that is how the inner scaling was selected).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Thus all derivatives of interest in the scaled variables at that point either vanish, or (in one case) are unity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This circumstance is an adequate criterion for the validity of that scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' On the other end of the continuum, where ℓ is large, we know that ℓ reaches a maximum of ǫ−2, because that is the half-width of the channel and no larger scaling patch could fit into the latter.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' According to (81), it corresponds to β = ǫ4/(1+σ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' That forms a lower bound on the possible values of β.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the case σ = 0 for example, when β = ǫ4 the existence of a scaling patch can be ascertained by the previous argument in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1, which still holds true (y+ β for that value of β must lie a distance ≤ O(1) from the centerline).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It should be noted that this “outer” patch encompassing the centerline is not the same as the traditional outer length scaling spoken of in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='3, although the two ideas are compatible.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The present concept of outer patch is that of an interval in the core together with a rescaling of all the variables, not just y+, which will produce a parameter-free version of the mean momentum balance, namely (79), at that location.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Thus our construction of scaling patches is valid up to and including the centerline, and down to the inner region.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In all, the scaling patches cover the entire channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='9 Discussion We have found that at each point in the Couette flow, the Reynolds stress and mean velocity profiles have a natural scaling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' in other words, a scaling patch is located at that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In order of mag- nitude, its characteristic length, which is essentially the width of the patch, increases continuously from 1 (in inner units) in the inner region where the law of the wall holds, to ǫ−2 at the centerline of the channel, where the outer scaling holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Each patch is associated with a peak in one of the adjusted Reynolds stresses, and with a balance exchange event that occurs there involving that same adjusted stress.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The patches can be parameterized by their characteristic lengths, which increase monotonically with distance from the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Asymptotically as y+→∞ (since y+ is limited by ǫ−2, necessarily ǫ→0), the characteristic length is proportional to that distance, and in any case up to order of 34 magnitude, is given by a solution of an ordinary differential equation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Again up to order of magnitude, the mean velocity and Reynolds stress profiles are determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In certain regions in the limit as the Reynolds number approaches ∞, these order of magnitude results are replaced by explicit functions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is really a statement about the validity of an approximation for large Reynolds number.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The argument can be framed as a similarity and invariance issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' There is a family of rescalings depending on a parameter β, applied at β-dependent locations, which leave the governing rescaled momentum balance equation and associated numerical values of the derivatives appearing in that equation invariant as β is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The statement then is that another derivative, denoted by A, is, in order of magnitude, invariant as well, and in some regions in fact approximately numerically invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' These results are consistent with logarithmic growth predicted by the Izakson–Millikan argu- ment, which by hypothesis is to hold in some overlap zone between the inner and outer regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The scaling patch and order of magnitude results presented here, on the other hand, are true throughout the channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' they generally differ from the exact logarithmic law except where and if the quantity A is constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Finally, this constancy condition is unlikely to hold exactly anywhere, except in the limit as ǫ→0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 7 Turbulent Poiseuille flow in a channel In this idealized picture, the two walls at y = 0, y = 2δ are stationary, so that their motion no longer provides impetus for the flow;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' however such an impetus is provided by a given pressure gradient streamwise along the channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The treatment here follows that in [8, 9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 Differences from Couette flow Mathematically, the differences between Couette and Poiseuille flows lie in the facts that Px is no longer 0 in (51), and the boundary conditions at the two walls are changed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Namely, U = ⟨u′v′⟩ = d dy⟨u′v′⟩ = d2 dy2 ⟨u′v′⟩ = 0 (99) at y = 0 and 2δ;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' and at the centerline y = δ by symmetry, dU dy = ⟨u′v′⟩ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (100) In fact U(y) rises from 0 at y = 0 to attain a maximum at y = δ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The function is even about that latter location, which means that for y > δ, U(y) = U(2δ − y).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Similarly, ⟨u′v′⟩ is odd about that point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' These properties allow us to formulate the problem entirely on the half channel {0 ≤ y ≤ δ}.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If the solution is known in the half channel, it can be found by reflection in the other half.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' A standard argument shows that the pressure gradient term in the mean momentum balance equation (51) must be constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Specifically, we refer back to that equation and (52).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Setting Q ρ = P ρ − ⟨(v′)2⟩, we see that(52) implies that Q depends only on x;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' however by stationarity ⟨(v′)2⟩ does not depend on x.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The pressure gradient term in (51) can be written 1 ρ ∂Q ∂x , which as noted is independent of y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It is also independent of x, since the other two terms in (51) are.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The friction velocity, Reynolds number R∗, small parameter ǫ = (R∗)−1/2, inner scaling (law of the wall) and outer scaling, valid in the channel’s center, are all the same as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Integrating 35 (51) across the half channel produces a global force balance, resulting in the dimensionless form ǫ2 for the pressure gradient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The dimensionless form (64) becomes dT dη + 1 + ǫ2 d2U dη2 = 0, (101) and that of (66) is T + ǫ2 dU+ dη = 1 − η.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (102) The traditional outer approximation is Tout = 1 − η, 0 < η ≤ 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (103) The momentum balance equation with inner normalization is d2U+ dy+2 + dT dy+ + ǫ2 = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (104) 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='2 Hierarchy To exhibit a continuous family of scaling patches covering the channel flow profile, all that is needed is to revise slightly the definition of the adjusted Reynolds stresses (72).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The new one is T β(y+) = T(y+) + ǫ2y+ − βy+.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (105) It is remarkable that the mathematical problems for the mean velocity and Reynolds stress in these two scenarios—pressure gradient driven and shear driven—can be almost completely transformed one into the other by such a simple device as (105).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It transforms the basic momentum balance equation (104) into d2U+ dy+2 + dT β dy+ + β = 0, (106) which is of the same form as (74).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Therefore, with the newly adjusted Reynolds stresses, the channel flow context is amenable to the balance exchange processes described in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1, the construction of a continuum of scalings with associated scaling patches Lβ in Sections 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 to 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='3, and (under some assumptions) the derivation of logarithmic-like profiles in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The scaling in Lβ is still given by (78).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The mean profile calculations are given here only under the simplifying approximation A = constant, although analogs of the more general case can be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The expressions (78) and (79) are valid in the present setting as well.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' As far as estimating the locations of the patches and the profiles, there are some changes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The relation (90) becomes dT dy+ = 4A−2(y+ − C)−2 − ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (107) The constant C in (107) can be related to the location y+ = y+ m of the maximum of the original unadjusted function T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is because it is required that dT dy+ = 0 at y+ = y+ m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We obtain dT dy+ = 4A−2 � y+ − y+ m + 2 Aǫ �−2 − ǫ2, (108) 36 or alternatively by eliminating ǫ, dT dy+ = 4A−2 � (y+ − C)−2 − (y+ m − C)−2� .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (109) When one inserts this into (104) and integrates twice, using the requirement dU+ dy+ →0 as y+→∞, the result is U(y+) = 4 A2 ln � y+ − y+ m + 2 Aǫ � + C2 (110) for another integration constant C2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Strictly speaking, there is another required condition, due to the symmetry of the configuration at the centerline: dU+ dy+ = 0 at y+ = ǫ−2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It cannot be satisfied exactly within the framework of (110), which indicates that the approximation A = constant cannot be exact near the centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The expression (110) is only expected to give a good representation of the real profile in some regions away from both the wall and the centerline.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' To reiterate, the most we can say theoretically is that this is suggestive of a logarithmic ap- proximation to some segment of the mean velocity profile.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' More specifically, this is all under the (doubtful) assumption that A is exactly constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the case that it is almost constant, one gets a pair of upper and lower bounds as before, valid now for the mean velocity in channel flow for the range of y+ constructed as before.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Note that in the case β = ǫ2, by (105) T β = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The scaling patch in this particular case is traditionally called the “mesolayer”, and it occurs near the peak in Reynolds stress, because for this case T β = T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The characteristic length in that patch is O(ǫ−1), which is the geometric mean of those in the inner (O(1)) and outer (O(ǫ−2)) regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='3 Behavior near the wall The construction of the patches given in sections 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 and 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='2 by means of a balance exchange has, as a primary ingredient, the fact that as the peak in adjusted Reynolds stress is approached, a region must appear in which all three terms in the mean momentum balance equation, (61), will have the same order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is simply because the gradient dT + dy+ approaches 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' There will, of course, be a smaller region encompassing the peak in which the last term on the left of (61) has smaller order of magnitude than the others, because it vanishes at the peak.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' A similar phenomenon happens when y+→0, since that same gradient is zero at the wall (y+ = 0) and positive for small values of y+ > 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The same conclusion may therefore be deduced: in a small region near the wall, all three terms in (61) will have the same order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' But the argument in Section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 can only partially be continued beyond this stage to produce a patch with different scaling;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' in fact it is well known (see also the reason given below) that the characteristic length scale arbitrarily near the wall remains the inner scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The reason the argument is no longer completely valid will now be explained.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In addition, the correctly scaled mean momentum balance very near the wall will be derived.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' There have been many analytical, empirical, and computational studies of the properties of the near wall region;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' we mention only [22], as our results fit particularly well with theirs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Our purpose here is to show that a scaling patch exists there, whose derivation and description fits within the framework of the methodology developed here and in our previous papers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 37 At the wall, additional constraints are imposed on the functions U+ and T +, besides the ba- sic differential equation (61).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' First of all, the very definition of inner scaling requires an auto- matic boundary condition dU+ dy+ (0) = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The inner scaling was chosen just so that condition holds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Secondly, the no-slip condition requires the boundary conditions U+(0) = T +(0) = dT + dy+ (0) = d2T + dy+2 (0) = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The first requirement is simply a result of our choice of normalized variables y+ and U+, and is not a statement of any physical constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The other boundary conditions result from a physical effect located at the wall.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' They have no analog at the mesoscaling patch, and constitute the basic reason that the present construction is different from the mesoscale construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' If one proceeds in the same vein as before on the basis of (75) and (76), the effect of the first boundary condition dU+ dy+ (0) = 1 is that the length scale in that patch is given by α = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This means that ˆy = y+: the length scale in that patch is the same as that with the original inner normalized scaling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This is of course almost a tautology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' But then the rest of the argument, following (61), in which α and γ are determined, can no longer be carried out as it stands, since α has already been determined.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' However, one can proceed after some reformulation of the problem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' At this point, the first substantial difference in method emerges between the derivation of the patches in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 and the present argument for what we shall call the wall patch.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' As mentioned, it is allied with the physical no-slip constraint.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In both cases, we look for scaled solutions of (74) or its analog (104) in a neighborhood of a maximum or minimum of T + or Tβ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In both the case in Sec.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content='1 and the present wall case, a scaling, namely a choice of α, γ, λ in (75), (76), is sought which will render the three terms of (74) or (104) of forms which have the same formal order of magnitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' A unique choice is only possible if one of these three factors is specified by other means.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the previous case, empirical data having to do with the velocity increment across the patch, and also the rate of growth of U+ in the hierarchy, was used to motivate selection of the value λ = 1, while leaving open the additional possibility of other choices leading to different growth rates.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This serves to determine the other two factors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In the wall patch, where this argument is not applicable, the definition of the inner scaling requires α = 1, and the equality criterion now can be used to determine γ and λ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Namely, evaluating the three terms of (104) under the transformation (75), (76) with α = 1 tells us that γ = λ = ǫ2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In neither case does this provide the value of m, because the derivatives in (74) or (104) annihilate the linear terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' However, the value of m was obtained by other means: through use of (91) and the connection T has with the slope, in the case of patches embedded in the hierarchy, and by means of the boundary condition giving the slope at the wall, in the present case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The scaled version of (104) in the wall patch is 1 + d2 ˆU dy+2 + d ˆT dy+ = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' (111) The individual terms of this equation are known only at y+ = 0;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' but it provides a linear relation between the two derivatives.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' This equation is the analog of (79), and is in the form of a balance of three rescaled forces.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In short, there is a scaling patch near the wall, no doubt including the traditional viscous sublayer, in which the inner length scale is correct, but the deviations of the functions U+ and T + from their linear parts depend on R∗ like ǫ2 ≈ (R∗)−1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Finally, as y+ enters the patch from above, there is a balance exchange from Layer II (in the terminology used in [7]), where the viscous and turbulent forces balance, to Layer I, where the pressure gradient balances the viscous plus turbulence force.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 38 The location of this exchange, and in fact the size of Layer I, is ≤ O(1) in wall units, because that is the length scale for the parameterless (111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In terms of the traditional buffer and logarithmic layer, we surmise that they lie outside Layer I, which can be identified as the viscous sublayer (although at its outer edge the viscous and turbulent forces are equal in order of magnitude).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Outside that layer is approximately where the hierarchy begins (say y+ ≈ 7), which is also where the traditionally defined buffer layer begins.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The loga- rithmic mean profile approximation associated with the hierarchy, however, does not become valid until distances from the beginning of the hierarchy are sufficient for A in (87) to be approximately constant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 8 Other applications and conclusion The analysis covered in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6 and 7 has been applied not only to Couette and Poiseuille flow, but also to combined Couette-Poiseuille turbulent flow [23];' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' favorable pressure gradient boundary layers (Metzger and Fife, in preparation);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' transport of heat through turbulent channel flow [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' It should again be emphasized that our goal has been not so much to discover numerical values associated with the profiles, but rather to gain theoretical understanding of why important features occur.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' We look for answers to why?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' in preference to what?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The greatest emphasis has been placed on two approaches to this question: the classical Izakson-Millikan observation;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' and the search for scaling patches.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Each of these results is based on an assumption that another approximation is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The first relies on the standard inner and outer approximations being simultaneously valid somewhere.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The second approach is an argument involving matters of similarity and invariance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' There is a family of rescalings depending on a parameter β, applied at β-dependent locations, which leave the governing rescaled momentum balance equation and associated numerical values of the derivatives appearing in that equation invariant as β is varied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The statement then is that another derivative is, in order of magnitude, invariant as well, and in some regions is in fact approximately numerically invariant.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In neither case is there a theoretical way to gauge the error of the assumed approximation, or the extent of the region where it is valid.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Some qualitative conclusions on the latter issue are provided in the second case.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The various approaches to answering our basic question all contribute a portion of insight.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' All of them leave unanswered questions, but they add to one another.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' No one of them should be considered the final word on the subject.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Acknowledgments The results in Secs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 6, 7, and 8 were obtained in collaboration with Joe Klewicki, Tie Wei, Meredith Metzger, and Pat McMurtry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' I thank them all for their invaluable exchange of ideas.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' 39 References [1] O.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Reynolds.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' On the dynamical theory of incompressible viscous fluids and the determination of the criterion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Phil.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Trans.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Roy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Soc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' London, 186:123–161, 1894.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' [2] L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Prandtl.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Bericht uber die Entstehung der Turbulenz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Z.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Angew.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Math.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=', 5:136–139, 1925.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' [3] Von Karman.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Mechanische Ahnlichkeit und Turbulenz.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Nachr.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Ges.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Wiss.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Gottingen, Math- Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Klasse, pages 58–76, 1930.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' [4] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Izakson.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' On the formula for the velocity distribution near walls.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Tech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' U.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=', IV, 2:155, 1937.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' [5] C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Millikan.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' A critical discussion of turbulent flows in channel and circular tubes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Den Hartog and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Peters, editors, Proceedings of the Fifth International Congress of applied Mechanics, pages 386–392.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Wiley, New York, 1939.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' [6] A.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' E.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Gill.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' The Reynolds number similarity argument.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Journal of Mathematics and Physics, 47:437–441, 1968.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' [7] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Wei, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Fife, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Klewicki, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' McMurtry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Properties of the mean momentum balance in turbulent boundary layer, pipe and channel flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=', 522:303–327, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' [8] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Fife, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Wei, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Klewicki, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' McMurtry.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Stress gradient balance layers and scale hierarchies in wall-bounded turbulent flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Fluid Mech.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=', 532:165–189, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' [9] P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Fife, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Klewicki, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' McMurtry, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Multiscaling in the presence of indeterminacy: wall-induced turbulence.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Multiscale Modeling and Simulation, 4:936–959, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' [10] T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Wei, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' McMurtry, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Klewicki, and P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Fife.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Meso scaling of Reynolds shear stress in turbulent channel and pipe flows.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' AIAA Jour.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=', 43:2350–2353, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' [11] J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Klewicki, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' McMurtry, P.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Fife, and T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Wei.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' A physical model of the turbulent boundary layer consonant with the structure of the mean momentum balance.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In 15th Australasian Fluid Mechanics Conference, The University of Sydney, Sydney, Australia, 2004.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' [12] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Panton.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Review of wall turbulence as described by composite expansions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Applied Mechanics Review, 2005.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' To appear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' [13] R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Kawamura, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Abe, and K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Shingai.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' DNS of turbulence and heat transport in a channel flow with different Reynolds and Prandtl numbers and boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' In Turbulence, Heat and Mass Transfer 3 (Proc.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' of the 3rd International Symposium of Turbulence, Heat and Mass Transfer), pages 15–32, 2000.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' [14] K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Shingai, H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/dtFAT4oBgHgl3EQf6x4A/content/2301.08740v1.pdf'} +page_content=' Kawamura, and Y.' 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0000000000000000000000000000000000000000..04919142da8cfcf22a93d4d4733950abb9a850af --- /dev/null +++ b/edE4T4oBgHgl3EQfQwxW/content/tmp_files/2301.04984v1.pdf.txt @@ -0,0 +1,934 @@ +Pressure-induced coevolution of transport properties and lattice +stability in CaK(Fe1-xNix)4As4 (x= 0.04 and 0) superconductors with +and without spin-vortex crystal state +Pengyu Wang1,2*, Chang Liu1,2*, Run Yang1*, Shu Cai1,3, Tao Xie 1.2, Jing Guo1,5, Jinyu +Zhao1,2, Jinyu Han1,2, Sijin Long1,2, Yazhou Zhou1, Yanchun Li3, Xiaodong Li3, Huiqian +Luo1,2,5, Shiliang Li1,2,5, Qi Wu1, Xianggang Qiu1,2,5, Tao Xiang1,2, and Liling Sun1,2,3,5† +1Institute of Physics, Chinese Academy of Sciences, Beijing 100190, China +2University of Chinese Academy of Sciences, Beijing 100190, China +3Center for High Pressure Science & Technology Advanced Research, 100094 Beijing, China +4Institute of High Energy Physics, Chinese Academy of Science, Beijing 100049, China +5Songshan Lake Materials Laboratory, Dongguan, Guangdong 523808, China +Here we report the first investigation on correlation between the transport +properties +and +the +corresponding +stability +of +the +lattice +structure +for +CaK(Fe1-xNix)4As4 (x=0.04, 0), a new type of putative topological superconductors, +with and without a spin-vortex crystal (SVC) state in a wide pressure range involving +superconducting to non-superconducting transition and the half- to full-collapse of +tetragonal (h-cT and f-cT) phases, by the complementary +measurements of +high-pressure resistance, Hall coefficient and synchrotron X-ray diffraction. We +identify the three critical pressures: Pchon that is the turn-on critical pressure of the +h-cT phase transition and it coincides with the critical pressure for the sign change of +Hall coefficient from positive to negative, a manifestation of the Fermi surface +reconstruction; Pchoff that is the turn-off pressures of the h-cT phase transition; and Pcf +that is the critical pressure of the f-cT phase transition. By comparing the +high-pressure results measured from the two kinds of samples, we find a distinct +“left-shift” of the Pchon for the doped sample, at the pressure of which its SVC state is +fully suppressed, however the Pchoff and the Pcf +remain the same as that of the +undoped one. Our results not only provide a consistent understanding on the results +reported before, but also demonstrate the importance of the Fe-As bonding in +stabilizing the superconductivity of the iron pnictide superconductors through the +“pressure window”. + +The discovery of the high-Tc iron-pnictide and iron-selenide superconductors [1, +2] provides a new platform to specify the essential ingredients existed in cuprate +superconductors, and an opportunity to achieve a better understanding on the +mechanism of high-Tc superconductivity, a ‘holy grail’ in the field of the +contemporary condensed matter physics and material sciences [3-5]. Now, several +types of iron-pnictide superconductors have been found [6-9], all of which have the +tetrahedra structure with the Fe-As +layers +stacking alternatively with other +intermediary layers/atoms. Growing evidence from experiments indicates that the +Fe-As layers play a key role in developing and stabilizing superconductivity [9]. In +2016, a new member of the iron pnictide superconductors, AeAFe4As4 (Ae=Ca, Sr +and A= K, Rb, Cs), has been found [10-26]. Structurally, it can be regarded as the +hybrid phases between AeFe2As2 and AFe2As2 with the stoichiometric composition, +while is different from most of the other known high-Tc superconductors whose +superconductivity are induced by chemical doping. As a result, these stoichiometric +superconductors have no chemical substitution-induced inhomogeneity on the lattice, +and significantly reduced the complexity of the local structure for understanding the +physics behind. Since the Ae2+ and A1+ atoms are inserted alternately across the +Fe2As2 layers, the AeAFe4As4 superconductors host two different sites of As ions in a +unit cell (As(1) and As(2), respectively) in the tetragonal unit cell with group space +P4/mmm [10]. Upon cooling, they show a superconducting transition at the +temperature (Tc) varying from 31 to 36 K. Partial substituting Fe with Ni, Co or Mn + +reduces the Tc value by more than 10 K due to the existence of a spin-vortex crystal +(SVC) +state, +with +Fe +spins +lying +in-plane +and +stacking +along +c +axis +antiferromagnetically, that competes with the superconducting (SC) state [22-24]. +Remarkably, +recent +angle-resolved +photoemission +spectroscopy +and +scanning +tunneling microscopy/spectroscopy experiments find the evidence of the Dirac +surface state and Majorana zero mode in the CaKFe4As4 superconductor [21], +implying that this type of superconductors may host some nontrivial quantum states, +such +as +topological +superconductivity. Moreover, inelastic +neutron +scattering +experiments find that the spin resonance of the CaK(Fe1-xNix)4As4 superconductor +displays both odd and even modes along the L direction [17,18], similar to the +resonance observed in bilayer cuprate superconductors [27]. These findings attract +additional research interest on them. +Pressure tuning is an effective and clean way to manipulate the crystal and +electronic structures without changing the chemistry, often providing significant +information for understanding the underlying physics of the exotic state emerging +from ambient pressure materials, through investigating the coevolution of electronic +states +and +crystal +structure. +Some +high-pressure +studies +on +the AeAFe4As4 +superconductors have been performed, and interesting results have been achieved +from both experimental and theoretical sides [12-16, 26], including the transition of +bulk-to-percolating superconducting state [12], half-to-full collapse of the tetragonal +phase [14, 15, 26]. These results provide the fundamental knowledge that the +superconductivity and the lattice structure of these superconductors are intimately + +correlated and sensitive to the external pressure. However, the reported investigations +of transport properties on Ni-doped and undoped CaKFe4As4 superconductors are +performed below the pressure of ~ 6 GPa. What happens under the pressure above 6 +GPa remains unclear. In addition, the experimental evidence for the change of +electronic state around the critical pressure of the half collapse of the tetragonal (h-cT) +phase is still lacking. In this work, we perform the high- pressure studies on the doped +and undoped sample up to the pressure where the superconductivity is fully +suppressed and the f-cT phase forms. +High-quality single crystals of CaK(Fe1-xNix)4As4 (x=0.04 and 0) were grown +using the self-flux method [17, 18, 25]. The ambient-pressure values of Tcs of the +doped- and undoped-samples were determined to be 21 K and 35 K, respectively, and +the SVC transition for the x=0.04 sample is about TN=44 K [24]. +High-pressure resistance measurements were performed in a diamond-anvil cell +(DAC), in which diamond anvils with 300 μm culets (flat area of the diamond anvil) +and a nonmagnetic rhenium gasket with 100-μm-diameter hole were adopted. The +standard four-probe electrodes were applied on the cleavage plane of the CaK(Fe1-x +Nix)4As4 (x=0.04 and 0) crystals. To obtain a quasi-hydrostatic pressure environment +for the samples, NaCl powder was employed as the pressure medium. High-pressure +Hall coefficient measurements were carried out by the van der Pauw method. To keep +the sample in the same pressure environment as that in the resistance measurements, +NaCl powder was also employed as the pressure medium. High-pressure X-ray +diffraction (XRD) measurements on CaK(Fe0.096Ni0.04)4As4 were performed at room + +temperature on beamline 4W2 at the Beijing Synchrotron Radiation Facility. +Diamonds +with +low +birefringence +were +selected +for +the +measurements. +A +monochromatic X-ray beam with a wavelength of 0.6199 Å was employed. The +pressure for all measurements was determined by the ruby fluorescence method [28]. +We first performed temperature-dependent resistance measurements on the single +crystal of CaK(Fe0.096Ni0.04)4As4 in a DAC. To investigate the doping effect on the +transport properties, a parallel measurement was conducted on the undoped +CaKFe4As4 sample. As shown in Fig.1a and 1c, the superconductivity of both samples +with and without the SVC state is sensitive to the pressure applied. For the sample +with the SVC state, the ambient-pressure Tc is much lower than that of the sample +without the SVC state. It shows that application of pressure renders Tc decreased, no +matter whether they host the SVC state or not. Intriguingly, the superconductivity of +these two samples is fully suppressed at almost the same critical pressure (~ 11 GPa), +as shown in Fig.1b and 1d, suggesting that the initial SVC state has little influence on +the critical pressure for destroying the superconducting state. We repeat the +measurements with several new samples and obtain the reproducible results (see SI - +Ref.[29]). Moreover, we investigate the pressure effect on the onset transition +temperature (TN) of the SVC state for our CaK(Fe0.96 Ni0.04)4As4 samples and find that +TN exhibits a monotonous decrease with the increment of pressure (see Ref.[29]), +similar to +what +is +observed +in +CaK(Fe1-xNix)4As4 (x=0.033 +and +0.05) +and +CaK(Fe1-xMnx)4As4 (x=0.024) samples [13, 16]. + +To +identify +the +similarity +and +peculiarity +in +the +transport +properties, +superconductivity and the electronic state in CaK(Fe0.096Ni0.04)4As4 and CaKFe4As4 +samples, we performed high-pressure measurements on Hall resistance (Rxy) by +sweeping the magnetic field (H) applied perpendicular to the ab plane of the samples, +from 0 T to 6 T at 30 K for CaK(Fe0.096Ni0.04)4As4 (Fig.2a and 2b) and at 40 K for +CaKFe4As4 (Fig.2c and 2d). We find that Rxy(H) is positive below 2.4 GPa (the +average value of the two independent runs - 2.33 GPa+2.46 GPa)/2=2.4 GPa) for the +CaK(Fe0.096Ni0.04)4As4 samples, and 3.88 GPa for the CaKFe4As4 samples (also the +average value - (3.90 GPa+3.85 GPa)/2=3.88 GPa). The plots of Rxy(H) from our +samples indicate that a hole-electron carrier balance (Rxy(H)=0) occurs at a critical +pressure (Pc). Although the value of the Pc is different for the samples with and +without the SVC state, the behavior of sign change in RH (P) is the same - below the +Pc, the sample is dominated by hole carriers, while above the Pc, the sample is +dominated by electron carriers. The observation of the pressure-induced sign change +of the Hall coefficient at the Pc provides important experimental evidence for the +dramatic change of electronic structure - the reconstruction of the Fermi surface from +a hole dominated to an electron dominated ones [30, 31]. +Since the structural stability is one of the key issues for understanding the +phenomena found in the pressure range of our experiments, we perform the high +pressure X-ray diffraction (XRD) measurements on the CaK(Fe0.096Ni0.04)4As4 sample +for the first time. The XRD patterns collected at different pressures are shown in +Fig. 3a. It is seen that all peaks measured under pressure up to 39.3 GPa can be well + +indexed by the tetragonal phase in the P/4mmm space group, indicating that no +structure phase transition occurs in the pressure range investigated. In Fig.3b, we +illustrate the crystal structure of the CaK(Fe0.096Ni0.04)4As4 sample and define the As +ions adjacent to the Ca layers as As(1) and the K layers as As(2). +However, the lattice parameter a extracted from our XRD data shows an increase +starting at ~ 2.5 GPa and reaching the maximum at ~ 5 GPa, meanwhile the lattice +parameter c displays a rapid decrease in this pressure range (Fig.4a). These results +lead us to propose that the transition of the half-collapse of the tetragonal (h-cT) phase +turns on and off at ~ 2.5 GPa and ~5 GPa, respectively (Here we define these critical +pressures as Pchon and Pchoff), and the initial tetragonal (T) phase and the h-cT phase +coexist in this pressure range. These results are similar to what have been observed in +the pressurized CaKFe4As4 samples, in which the lattice parameter a begins to +increase at ~3.5 GPa and reaches a maximum at ~ 4.7 GPa [12]. Upon further +compression to ~11 GPa, the second collapse occurs - the lattice parameter a and c +also appear noticeable changes (Here we define this critical pressure as Pcf), implying +that the T phase fully collapses (due to lack of more experimental information on the +change of the f-cT phase, we are not able to identify the critical pressure that turns off +the +f-cT +phase). +The +pressure-induced +two +collapses +in +the +tetragonal +CaK(Fe0.096Ni0.04)4As4 sample are in accordance with the theoretical calculations and +experimental results obtained from the measurements on the CaRbFe4As4 and +Cs/RbEuFe4As4 samples [14, 15, 26]. +We summarize our high-pressure results obtained from the measurements on + +CaK(Fe0.096Ni0.04)4As4 in the pressure-Tc phase diagram (Fig.4b). The four distinct +regions defined by the lattice structure can be seen in the diagram: (1) Low-pressure T +phase region below Pchon, in which the SVC state coexists with the SC state. When +pressure is applied, both of TN and Tc decrease with increasing pressure until the +pressure reaches to the Pchon. (2) The coexisted T and h-cT phase region that lies in the +range of Pchon and Pchoff. At the Pchon, the SVC state is entirely suppressed and the +sample starts the transition from a T phase to h-cT phase (Fig. 4a). Just at this pressure, +the Hall coefficient (RH) changes its sign from the positive to the negative (Fig. 4c). +These results demonstrate that the pressure drives a reconstruction of Fermi surface +which is associated to the transition from the T phase to the h-cT phase in a pressure +range below ~5 GPa (Fig.4a). (3) The h-cT phase region lied in the range of Pchoff and +Pcf, in which the Tc decreases continuously and disappears at the pressure of ~ 11 GPa. +(4) The f-cT phase region above the critical pressure of Pcf, in which the h-cT phase +fully converts to the f-cT phase, and the corresponding electronic state of the h-cT +phase is taken over by that of the non-superconducting f-cT phase. +To clarify the doping effect on the high-pressure behavior found in the sample +with the SVC state, we compare its high-pressure experimental results with that +measured from the undoped CaKFe4As4 sample. As shown in Fig.4d-4f, CaKFe4As4 +bears the similar high-pressure behavior to that of CaK(Fe0.096Ni0.04)4As4: Tc +monotonically declines with the increment of pressure (Fig.4e), in agreement with the +results reported in [12]. At ~3.88 GPa (Pchon), the Hall confident (RH) also displays a +sign change from the positive to the negative, indicating that the ambient-pressure T + +phase of the sample begins its half collapse at this pressure [12,14]. A sudden increase +of the lattice parameter a and the volume drop are also observed at ~ 3.5 GPa by +Kaluarachchi et al [12], very close to the Pchon determined by our Hall coefficient +measurements (Fig.4f). Since the observed Pchon (~3.88) and Pchoff (~ 4.9 GPa) by our +Hall coefficient measurements is close to these (~3.5 GPa and ~4.7 GPa) determined +by the high pressure and low temperature XRD measurements on the same sample +[12], we define the pressures of 3.88 GPa and 4.9 GPa as the Pchon and Pchoff of our +undoped +sample. +On +further +increasing +pressure, +Tc +measured +from +both +CaK(Fe0.096Ni0.04)4As4 and CaKFe4As4 samples shows a monotonously decrease till +the pressure around Pcf =11 GPa (Fig.4b a and 4e), at the pressure of which doped- +and undoped-samples undergo a transition from the h-cT phase to the f-cT phase [12, +26] and lose their superconductivity. This is the first report on the observation of the +pressure-induced SC-to-NSC transition and the identification on that the critical +pressure of the SC-NSC transition coincides with the pressure of the lattice transition +from the h-cT phase to the f-cT phase in the CaK(Fe1-xNix)4As4 (x=0.04 and 0) +superconductors. +Theoretical calculations on the undoped system find that the formation of the +As(1)-As(1) bond across the Ca layers and the formation of the As(2)-As(2) bond +across the K layer are responsible for the presence of the h-cT and the f-cT phases [12, +26,32], and propose that the formed As-As bond weakens the Fe-As bonding [32-34], +which, in turn, greatly affects the stability of superconductivity [32]. Our findings in +the CaK(Fe1-xNix)4As4 (x=0.04 and 0) superconductors experimentally provides a + +strong support for the prediction that Fe-As bonding is one of the essential ingredients +for the presence of superconductivity in this kind of iron pnictides. +By comparing the high-pressure behavior of the CaK(Fe0.096Ni0.04)4As4 and +CaKFe4As4 samples, we find that the substitution of Ni on Fe site shifts the Pchon to +lower pressure, while it has no obvious effects on the Pchoff and the Pcf.. Thus, +understanding why the existence of the SVC state renders the Pchon to shift to lower +pressure should be one of the key issues to reveal the underlying physics of the lattice +half collapse. We suggest that the left shift of the Pchoff in the Ni-doped sample may be +attributed to the interplay between the SVC state and the Fe-As hybridized state (the +hybridization between the Fe 3d orbital and As 4p orbital electrons), which makes the +hybridized band unstable and benefits the formation of the As(1)-As(1) bonding. +These results imply that the Pchon is sensitive to the existence of the competing order +introduced by the chemical doping. While the unchanged Pchoff and Pcf is possibly +related to the stability of the lattice that is still governed by the matrix of the initial T +phase. +In conclusion, the coevolution of the SVC state, superconductivity, dominated +carriers and stability of lattice structure in CaK(Fe1-xNix)4As4 (x=0.04 and 0) +superconductors has been investigated by the complementary measurements of +high-pressure resistance, Hall coefficient and synchrotron X-ray diffraction in a wide +pressure range involving superconducting to non-superconducting transition, and the +corresponding half- to full-collapse of tetragonal phase transition, for the first time. +We identified the three critical pressures though the comprehensive analysis of our + +results: Pchon and Pchoff that are the turn-on and -off pressure of the h-cT phase +transition, respectively, and Pcf that is the critical pressure of the f-cT phase transition. +Our results demonstrate that the formation of the As(1)-As(1) bond that is evidence by +the transition from the T phase to the h-cT phase changes the sign of RH , a +manifestation for the reconstruction of the Fermi surface. While, the formation of the +As(2)-As(2) bond characterized by the f-cT phase transition terminates the +superconductivity. These results achieved in this study not only provide a consistent +understanding on the results reported before, but also demonstrate the importance of +the Fe-As +bonding in stabilizing the superconductivity of the iron-pnictide +superconductors through the “pressure window”. +Acknowledgements +This work was supported by the National Key Research and Development +Program +of +China +(Grant +Nos. +2021YFA1401800, +2022YFA1403900, +2018YFA0704200), the National Natural Science Foundation of China (Grant Nos. +U2032214, 12104487, 12122414, 12004419, 11822411 and 11961160699), and the +Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant +No. XDB25000000). J. G., S.C. and H. L. are grateful for supports from the Youth +Innovation Promotion Association of the Chinese Academy of Sciences (Grant Nos. +2019008, Y202001) and the China Postdoctoral Science Foundation (E0BK111). +These authors with star (*) contributed equally to this work. + +Correspondence and requests for materials should be addressed to L.S. +(llsun@iphy.ac.cn) +References +[1] Y. Kamihara, T. Watanabe, M. Hirano, et al. Iron-based layered superconductor +La[O1-xFx]FeAs (x = 0.05-0.12) with Tc = 26 K. J. Am. Chem. Soc. 130, +3296-3297 (2008). +[2] F.C. Hsu, J.Y. Luo, K.W. Yeh, et al. Superconductivity in the PbO-type structure +α-FeSe. Proc. Natl. Acad. Sci. 105, 14262-14264 (2008). +[3] C.W. Chu, L.Z. Deng, B. Lv. Hole-doped cuprate high temperature +superconductors. Physica C 514, 290-313 (2015). +[4] B. Keimer, S.A. Kivelson, M.R. Norman, et al. From quantum matter to +high-temperature superconductivity in copper oxides. Nature 518, 179-186 +(2015). +[5] J. Zaanen. 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Calibration of the ruby pressure gauge to 800 kbar +under quasi-hydrostatic conditions. J. Geophys. Res. 91, 4673 (1986). +[29]See Supplementary Information for the extended resistance measurements, the +determination on the onset transition temperature of the spin-vertex crystal (SVC) +state and the pressure dependence of the volume for CaK(Fe1-x Nix)4As4 (x=0 and +0.04); also see Refs.[12,13,17,24]. +[30]G. Lin, J. Guo, Y. Zhu, et al. Correlation between Fermi surface reconstruction +and superconductivity in pressurized +FeTe0.55Se0.45. Phys. Rev. B 101, 214525 +(2020). +[31]D. Kang, Y. Zhou, W. Yi, et al. Superconductivity emerging from a suppressed +large magnetoresistant state in tungsten ditelluride. Nat. Commun. 6, 7804 +(2015). +[32]R. Yang, C. Le, L. Zhang, et al. Formation of As-As bond and its effect on +absence of superconductivity in the collapsed tetragonal phase of +Ca0.86Pr0.14Fe2As2: An optical spectroscopy study. Phys. Rev. B 91, 224507 (2015) +[33]T. Yildirim. Strong coupling of the Fe-spin state and the As-As hybridization in +iron-pnictide superconductors from first-principle calculations. Phys. Rev. Lett. +102, 037003 (2009). + +[34]J. Diehl, S. Backes, D. Guterding, et al. Correlation effects in the tetragonal and +collapsed-tetragonal phase of CaFe2As2. Phys. Rev. B 90, 085110 (2014). +Figure 1 The results of resistance measurements on the CaK(Fe1-xNix)4As4 +(x=0.04 and 0) at high pressures. (a) Temperature dependence of the resistance in +the +pressure +range +of +0.76 +GPa–37.32 +GPa +for +the +sample +#1 +of +the +CaK(Fe0.96Ni0.04)4As4 single crystal. (b) Enlarged views of the resistance-temperature +curves at different pressures for the sample #1 of CaK(Fe0.96Ni0.04)4As4. (c) Resistance +as a function of temperature for pressures ranging from 1.56 GPa to 15.5 GPa for the +sample #A of CaKFe4As4 single crystal. (d) Resistance-temperature curves at different +pressures for the sample #A of CaKFe4As4. + +(a) +(b) +0.7 +P/GPa +S#1 +CaK(Fe. +0.76 +10.92 +21.89 +1.5 +1.38 +11.80 +24.98 +0.6 +S#1 +2.07 +12.71 +27.40 +3.05 +13.68 +29.65 +0.5 +1.2 +3.66 +14.68 +32.28 +4.41 +16.13 +34.36 +0.4 +nce +5.36 +18.46 +37.32 +0.9 +6.17 +19.99 +sistan +0.3 R +7.08 +20.50 +7.91 +Resi +0.6 +8.57 +0.2 +9.51 +0.1 +0.3 +0.0 +CaK(Fe. +As +SC +Ni +90 +0.0 +0.96 +0.04/ +60 +50 +150 +250 +(K) +0 +100 +200 +300 +0 +2 +4 +30 +6 +8 +10 +Temperature (K) +12 +14 +Pressure (GPa) +0 +(d) +0.07 +(c) +P/GPa +S#A +CaKFe,As +1.56 +8.86 +0.06 +2.35 +9.77 +S#A +3.26 +10.66 +0.20 +4.07 +11.72 +0.05 +5.19 +12.57 +5.89 +13.76 +0.04 +istance +0.15 +6.76 +15.50 +7.59 +0.03 R +0.10 +Resi +0.02 +0.01 +0.05 +0.00 +CaKFe. +0.00 +90 +SC +60 +150 +(K) +0 +50 +100 +200 +250 +300 +0 +2 +30 +4 +6 +8 +10 +Temperature (K) +12 +14 +Pressure (GPa) +0Figure 2 Hall resistance (Rxy) as a function of magnetic field (H) for the +CaK(Fe1-xNix)4As4 (x=0.04 and 0) single crystals. Plots of Rxy versus H for the +CaK(Fe0.96Ni0.04)4As4 single crystals measured at 30 K in the pressure range of (a) +0.89-15.5 GPa, (b) 1.34-14.9 GPa. Rxy versus H for the CaKFe4As4 single crystals +measured at 40 K in the pressure range of (c) 1.56 – 15.5 GPa, (d) 2.24-16.43 GPa. +The dashed line indicates Rxy(B)=0 where the average pressures for the sample #2 and +#3 of the CaK(Fe0.96Ni0.04)4As4 and the sample #A and #B of the CaKFe4As4 single +crystals are estimated to be ~ 2.4 GPa and 3.88 GPa, respectively. + +(a) +(b) +LT=30K +T=30K +1 +S#2 +S#3 +0 +(uw) +-1 +P/GPa +-2 +1.34 +7.85 +Rxy +P/GPa +1.84 +8.50 +0.89 +7.74 +-4 +-3 +2.41 +9.13 +1.62 +8.81 +2.88 +9.85 +2.34 +9.15 +3.55 +10.64 +3.10 +10.33 +-4 +4.15 +11.64 +-6 +3.92 +11.41 +12.88 +4.66 +4.47 +12.94 +-5 +5.47 +13.87 +5.58 +14.11 +CaK(Fe。 +As +CaK(Fe. +,Ni. +).As +15.55 +0.96 +6.71 +14.90 +6.52 +0.96 +0.04/4 +-8 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +0 +5 +6 +H (T) +H (T) +(d) +C +LT=40K +T=40K +S#A +S#B +3 +2 +(sw) +0 +0 +P/GPa +P/GPa +1.56 +2.35 +2.24 +8.86 +8.38 +3.26 +3.08 +9.55 +9.77 +-4 +10.66 +3.81 +10.86 +4.07 +-6 +4.81 +12.68 +5.19 +11.72 +13.74 +5.89 +12.57 +5.77 +-6 +6.53 +15.02 +6.76 +13.76 +CaKFe.As4 +CaKFe,As +-9 +7.60 +16.43 +7.59 +15.50 +0 +1 +2 +3 +4 +5 +6 +0 +1 +2 +3 +4 +5 +6 +H (T) +H (T)Figure 3. X-ray diffraction results of the CaK(Fe0.96Ni0.04)4As4 sample collected at +high pressure and crystallographic illustration. (a) X-ray diffraction patterns +measured at different pressures, showing no crystal structure phase transition in the +experimental pressure range up to 39.32 GPa. (b) Schematic crystal structure of the +CaK(Fe0.96Ni0.04)4As4 sample. + +b +0.04/4 +39.32(GPa)^ +36.93 +33.38 +28.98 +24.53 +22.95 +21.56 +20.16 +18.6 +Intensity (a.u.) +17.43 +16.41 +15.18 +14.01 +13.01 +11.91 +8.98 +7.58 +AS(2)C +5.93 +5.17 +4.63 + As(2)0 +4.08 +Fe/Ni +3.61 +As(1) +3.21 +2.76 +AsC +2.29 +人1.68 +1.03 +(002) +01 +113) +14) +13 +30 +(20 +1 +1 +1 +1 +5 +10 +15 +20 +25 +20 (deg.)Figure 4. Structure information, pressure-temperature phase diagram and Hall +coefficient (RH) of the CaK(Fe1-xNix)4As4 (x=0.04 and 0) samples at different +pressures. (a) and (d) Lattice parameters a and c versus pressure for the doped- and +undoped- samples. (b) and (e) Pressure-Temperature phase diagrams, displaying the +evolution +of +the +spin-vortex +crystal +(SVC), +superconducting +(SC) +and +non-superconducting (NSC) states upon increasing pressure for the doped- and +undoped-samples. (c) and (f) Pressure dependence of Hall coefficient (RH) for the +CaK(Fe0.96Ni0.04)4As4 and CaKFe4As4 samples. T, h-cT and f-cT represent the +tetragonal phase, half-collapsed tetragonal phase and full-collapsed tetragonal phase, +respectively. TN stands for the onset transition temperature of the SVC state. SCh and +SCe represent the superconducting state with the dominance of hole-carriers and the + +CaKFe,As, +Ni. +(a) +(d) +T +h-cT +f-cT +f-cT +T +h-cT +3.88 +12.5 +13.0 +3.88 +3.86 +3.87 +12.0 +12.0 +3.86 +3.84 +11.0 +a +a +11.5 +3.85 +h-cl +h-cT +3.82 +10.0 +3.84 +11.0 +Ref.[12] +80 +80 +(b) +(e) +S#1-T. 0 +Tc +口 +Ref.[13] +S#1-T~ +S#B +TN +60 +60 +S#2-Tc +S#A Ref.[12] +V +S#2-Tn +S#B +S#3-Tc +svc +S#3-Tn +20 +20 +8 +NSC +NSC +SC. +SC +SC +SC +1.0 +1.0 +(0/g-0L) +T=30K +T=40K +0.0 +0.0 +Poff +Poff +ch +ch +-1.0 +-1.0 +RH +RH +Pon +ch +Pon +ch +-2.0 +-2.0 +5 +10 +15 +20 +25 +5 +10 +15 +20 +25 +0 +0 +Pressure (GPa) +Pressure (GPa)superconducting state with the dominance of electron-carriers. The Pchon, Pchoff +represent the turn-on and turn-off pressures for the transition from the T phase to the +h-cT phase. The Pcf stands for the critical pressure for the transition from the h-cT +phase to f-cT phase. The data in the figure (d) are taken from the Ref. [12]. + diff --git a/edE4T4oBgHgl3EQfQwxW/content/tmp_files/load_file.txt b/edE4T4oBgHgl3EQfQwxW/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..99cfe6784c6962735cb4335672bd21515a2e9ea3 --- /dev/null +++ b/edE4T4oBgHgl3EQfQwxW/content/tmp_files/load_file.txt @@ -0,0 +1,794 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf,len=793 +page_content='Pressure-induced coevolution of transport properties and lattice stability in CaK(Fe1-xNix)4As4 (x= 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04 and 0) superconductors with and without spin-vortex crystal state Pengyu Wang1,2*, Chang Liu1,2*, Run Yang1*, Shu Cai1,3, Tao Xie 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Jing Guo1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Jinyu Zhao1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Jinyu Han1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Sijin Long1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Yazhou Zhou1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Yanchun Li3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Xiaodong Li3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Huiqian Luo1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Shiliang Li1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Qi Wu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Xianggang Qiu1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Tao Xiang1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' and Liling Sun1,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='2,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='3,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5† 1Institute of Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' China 2University of Chinese Academy of Sciences,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Beijing 100190,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' China 3Center for High Pressure Science & Technology Advanced Research,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' 100094 Beijing,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' China 4Institute of High Energy Physics,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Chinese Academy of Science,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Beijing 100049,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' China 5Songshan Lake Materials Laboratory,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Dongguan,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Guangdong 523808,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' China Here we report the first investigation on correlation between the transport properties and the corresponding stability of the lattice structure for CaK(Fe1-xNix)4As4 (x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04, 0), a new type of putative topological superconductors, with and without a spin-vortex crystal (SVC) state in a wide pressure range involving superconducting to non-superconducting transition and the half- to full-collapse of tetragonal (h-cT and f-cT) phases, by the complementary measurements of high-pressure resistance, Hall coefficient and synchrotron X-ray diffraction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' We identify the three critical pressures: Pchon that is the turn-on critical pressure of the h-cT phase transition and it coincides with the critical pressure for the sign change of Hall coefficient from positive to negative, a manifestation of the Fermi surface reconstruction;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Pchoff that is the turn-off pressures of the h-cT phase transition;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' and Pcf that is the critical pressure of the f-cT phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' By comparing the high-pressure results measured from the two kinds of samples, we find a distinct “left-shift” of the Pchon for the doped sample, at the pressure of which its SVC state is fully suppressed, however the Pchoff and the Pcf remain the same as that of the undoped one.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Our results not only provide a consistent understanding on the results reported before, but also demonstrate the importance of the Fe-As bonding in stabilizing the superconductivity of the iron pnictide superconductors through the “pressure window”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' The discovery of the high-Tc iron-pnictide and iron-selenide superconductors [1, 2] provides a new platform to specify the essential ingredients existed in cuprate superconductors, and an opportunity to achieve a better understanding on the mechanism of high-Tc superconductivity, a ‘holy grail’ in the field of the contemporary condensed matter physics and material sciences [3-5].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Now, several types of iron-pnictide superconductors have been found [6-9], all of which have the tetrahedra structure with the Fe-As layers stacking alternatively with other intermediary layers/atoms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Growing evidence from experiments indicates that the Fe-As layers play a key role in developing and stabilizing superconductivity [9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' In 2016, a new member of the iron pnictide superconductors, AeAFe4As4 (Ae=Ca, Sr and A= K, Rb, Cs), has been found [10-26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Structurally, it can be regarded as the hybrid phases between AeFe2As2 and AFe2As2 with the stoichiometric composition, while is different from most of the other known high-Tc superconductors whose superconductivity are induced by chemical doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' As a result, these stoichiometric superconductors have no chemical substitution-induced inhomogeneity on the lattice, and significantly reduced the complexity of the local structure for understanding the physics behind.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Since the Ae2+ and A1+ atoms are inserted alternately across the Fe2As2 layers, the AeAFe4As4 superconductors host two different sites of As ions in a unit cell (As(1) and As(2), respectively) in the tetragonal unit cell with group space P4/mmm [10].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Upon cooling, they show a superconducting transition at the temperature (Tc) varying from 31 to 36 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Partial substituting Fe with Ni, Co or Mn reduces the Tc value by more than 10 K due to the existence of a spin-vortex crystal (SVC) state, with Fe spins lying in-plane and stacking along c axis antiferromagnetically, that competes with the superconducting (SC) state [22-24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Remarkably, recent angle-resolved photoemission spectroscopy and scanning tunneling microscopy/spectroscopy experiments find the evidence of the Dirac surface state and Majorana zero mode in the CaKFe4As4 superconductor [21], implying that this type of superconductors may host some nontrivial quantum states, such as topological superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Moreover, inelastic neutron scattering experiments find that the spin resonance of the CaK(Fe1-xNix)4As4 superconductor displays both odd and even modes along the L direction [17,18], similar to the resonance observed in bilayer cuprate superconductors [27].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' These findings attract additional research interest on them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Pressure tuning is an effective and clean way to manipulate the crystal and electronic structures without changing the chemistry, often providing significant information for understanding the underlying physics of the exotic state emerging from ambient pressure materials, through investigating the coevolution of electronic states and crystal structure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Some high-pressure studies on the AeAFe4As4 superconductors have been performed, and interesting results have been achieved from both experimental and theoretical sides [12-16, 26], including the transition of bulk-to-percolating superconducting state [12], half-to-full collapse of the tetragonal phase [14, 15, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' These results provide the fundamental knowledge that the superconductivity and the lattice structure of these superconductors are intimately correlated and sensitive to the external pressure.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' However, the reported investigations of transport properties on Ni-doped and undoped CaKFe4As4 superconductors are performed below the pressure of ~ 6 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' What happens under the pressure above 6 GPa remains unclear.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' In addition, the experimental evidence for the change of electronic state around the critical pressure of the half collapse of the tetragonal (h-cT) phase is still lacking.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' In this work, we perform the high- pressure studies on the doped and undoped sample up to the pressure where the superconductivity is fully suppressed and the f-cT phase forms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' High-quality single crystals of CaK(Fe1-xNix)4As4 (x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04 and 0) were grown using the self-flux method [17, 18, 25].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' The ambient-pressure values of Tcs of the doped- and undoped-samples were determined to be 21 K and 35 K, respectively, and the SVC transition for the x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04 sample is about TN=44 K [24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' High-pressure resistance measurements were performed in a diamond-anvil cell (DAC), in which diamond anvils with 300 μm culets (flat area of the diamond anvil) and a nonmagnetic rhenium gasket with 100-μm-diameter hole were adopted.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' The standard four-probe electrodes were applied on the cleavage plane of the CaK(Fe1-x Nix)4As4 (x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04 and 0) crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' To obtain a quasi-hydrostatic pressure environment for the samples, NaCl powder was employed as the pressure medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' High-pressure Hall coefficient measurements were carried out by the van der Pauw method.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' To keep the sample in the same pressure environment as that in the resistance measurements, NaCl powder was also employed as the pressure medium.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' High-pressure X-ray diffraction (XRD) measurements on CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='096Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 were performed at room temperature on beamline 4W2 at the Beijing Synchrotron Radiation Facility.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Diamonds with low birefringence were selected for the measurements.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' A monochromatic X-ray beam with a wavelength of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='6199 Å was employed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' The pressure for all measurements was determined by the ruby fluorescence method [28].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' We first performed temperature-dependent resistance measurements on the single crystal of CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='096Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 in a DAC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' To investigate the doping effect on the transport properties, a parallel measurement was conducted on the undoped CaKFe4As4 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='1a and 1c, the superconductivity of both samples with and without the SVC state is sensitive to the pressure applied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' For the sample with the SVC state, the ambient-pressure Tc is much lower than that of the sample without the SVC state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' It shows that application of pressure renders Tc decreased, no matter whether they host the SVC state or not.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Intriguingly, the superconductivity of these two samples is fully suppressed at almost the same critical pressure (~ 11 GPa), as shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='1b and 1d, suggesting that the initial SVC state has little influence on the critical pressure for destroying the superconducting state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' We repeat the measurements with several new samples and obtain the reproducible results (see SI - Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='[29]).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Moreover, we investigate the pressure effect on the onset transition temperature (TN) of the SVC state for our CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='96 Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 samples and find that TN exhibits a monotonous decrease with the increment of pressure (see Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' [29]), similar to what is observed in CaK(Fe1-xNix)4As4 (x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='033 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='05) and CaK(Fe1-xMnx)4As4 (x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='024) samples [13, 16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' To identify the similarity and peculiarity in the transport properties, superconductivity and the electronic state in CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='096Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 and CaKFe4As4 samples, we performed high-pressure measurements on Hall resistance (Rxy) by sweeping the magnetic field (H) applied perpendicular to the ab plane of the samples, from 0 T to 6 T at 30 K for CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='096Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='2a and 2b) and at 40 K for CaKFe4As4 (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='2c and 2d).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' We find that Rxy(H) is positive below 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='4 GPa (the average value of the two independent runs - 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='33 GPa+2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='46 GPa)/2=2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='4 GPa) for the CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='096Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 samples, and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='88 GPa for the CaKFe4As4 samples (also the average value - (3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='90 GPa+3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='85 GPa)/2=3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='88 GPa).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' The plots of Rxy(H) from our samples indicate that a hole-electron carrier balance (Rxy(H)=0) occurs at a critical pressure (Pc).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Although the value of the Pc is different for the samples with and without the SVC state, the behavior of sign change in RH (P) is the same - below the Pc, the sample is dominated by hole carriers, while above the Pc, the sample is dominated by electron carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' The observation of the pressure-induced sign change of the Hall coefficient at the Pc provides important experimental evidence for the dramatic change of electronic structure - the reconstruction of the Fermi surface from a hole dominated to an electron dominated ones [30, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Since the structural stability is one of the key issues for understanding the phenomena found in the pressure range of our experiments, we perform the high pressure X-ray diffraction (XRD) measurements on the CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='096Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 sample for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' The XRD patterns collected at different pressures are shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' 3a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' It is seen that all peaks measured under pressure up to 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='3 GPa can be well indexed by the tetragonal phase in the P/4mmm space group, indicating that no structure phase transition occurs in the pressure range investigated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' In Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='3b, we illustrate the crystal structure of the CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='096Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 sample and define the As ions adjacent to the Ca layers as As(1) and the K layers as As(2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' However, the lattice parameter a extracted from our XRD data shows an increase starting at ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5 GPa and reaching the maximum at ~ 5 GPa, meanwhile the lattice parameter c displays a rapid decrease in this pressure range (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' These results lead us to propose that the transition of the half-collapse of the tetragonal (h-cT) phase turns on and off at ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5 GPa and ~5 GPa, respectively (Here we define these critical pressures as Pchon and Pchoff), and the initial tetragonal (T) phase and the h-cT phase coexist in this pressure range.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' These results are similar to what have been observed in the pressurized CaKFe4As4 samples, in which the lattice parameter a begins to increase at ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5 GPa and reaches a maximum at ~ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='7 GPa [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Upon further compression to ~11 GPa, the second collapse occurs - the lattice parameter a and c also appear noticeable changes (Here we define this critical pressure as Pcf), implying that the T phase fully collapses (due to lack of more experimental information on the change of the f-cT phase, we are not able to identify the critical pressure that turns off the f-cT phase).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' The pressure-induced two collapses in the tetragonal CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='096Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 sample are in accordance with the theoretical calculations and experimental results obtained from the measurements on the CaRbFe4As4 and Cs/RbEuFe4As4 samples [14, 15, 26].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' We summarize our high-pressure results obtained from the measurements on CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='096Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 in the pressure-Tc phase diagram (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='4b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' The four distinct regions defined by the lattice structure can be seen in the diagram: (1) Low-pressure T phase region below Pchon, in which the SVC state coexists with the SC state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' When pressure is applied, both of TN and Tc decrease with increasing pressure until the pressure reaches to the Pchon.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (2) The coexisted T and h-cT phase region that lies in the range of Pchon and Pchoff.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' At the Pchon, the SVC state is entirely suppressed and the sample starts the transition from a T phase to h-cT phase (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' 4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Just at this pressure, the Hall coefficient (RH) changes its sign from the positive to the negative (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' 4c).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' These results demonstrate that the pressure drives a reconstruction of Fermi surface which is associated to the transition from the T phase to the h-cT phase in a pressure range below ~5 GPa (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='4a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (3) The h-cT phase region lied in the range of Pchoff and Pcf, in which the Tc decreases continuously and disappears at the pressure of ~ 11 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (4) The f-cT phase region above the critical pressure of Pcf, in which the h-cT phase fully converts to the f-cT phase, and the corresponding electronic state of the h-cT phase is taken over by that of the non-superconducting f-cT phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' To clarify the doping effect on the high-pressure behavior found in the sample with the SVC state, we compare its high-pressure experimental results with that measured from the undoped CaKFe4As4 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' As shown in Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='4d-4f, CaKFe4As4 bears the similar high-pressure behavior to that of CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='096Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4: Tc monotonically declines with the increment of pressure (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='4e), in agreement with the results reported in [12].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' At ~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='88 GPa (Pchon), the Hall confident (RH) also displays a sign change from the positive to the negative, indicating that the ambient-pressure T phase of the sample begins its half collapse at this pressure [12,14].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' A sudden increase of the lattice parameter a and the volume drop are also observed at ~ 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5 GPa by Kaluarachchi et al [12], very close to the Pchon determined by our Hall coefficient measurements (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='4f).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Since the observed Pchon (~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='88) and Pchoff (~ 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='9 GPa) by our Hall coefficient measurements is close to these (~3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5 GPa and ~4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='7 GPa) determined by the high pressure and low temperature XRD measurements on the same sample [12], we define the pressures of 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='88 GPa and 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='9 GPa as the Pchon and Pchoff of our undoped sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' On further increasing pressure, Tc measured from both CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='096Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 and CaKFe4As4 samples shows a monotonously decrease till the pressure around Pcf =11 GPa (Fig.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='4b a and 4e), at the pressure of which doped- and undoped-samples undergo a transition from the h-cT phase to the f-cT phase [12, 26] and lose their superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' This is the first report on the observation of the pressure-induced SC-to-NSC transition and the identification on that the critical pressure of the SC-NSC transition coincides with the pressure of the lattice transition from the h-cT phase to the f-cT phase in the CaK(Fe1-xNix)4As4 (x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04 and 0) superconductors.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Theoretical calculations on the undoped system find that the formation of the As(1)-As(1) bond across the Ca layers and the formation of the As(2)-As(2) bond across the K layer are responsible for the presence of the h-cT and the f-cT phases [12, 26,32], and propose that the formed As-As bond weakens the Fe-As bonding [32-34], which, in turn, greatly affects the stability of superconductivity [32].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Our findings in the CaK(Fe1-xNix)4As4 (x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04 and 0) superconductors experimentally provides a strong support for the prediction that Fe-As bonding is one of the essential ingredients for the presence of superconductivity in this kind of iron pnictides.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' By comparing the high-pressure behavior of the CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='096Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 and CaKFe4As4 samples, we find that the substitution of Ni on Fe site shifts the Pchon to lower pressure, while it has no obvious effects on the Pchoff and the Pcf.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='. Thus, understanding why the existence of the SVC state renders the Pchon to shift to lower pressure should be one of the key issues to reveal the underlying physics of the lattice half collapse.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' We suggest that the left shift of the Pchoff in the Ni-doped sample may be attributed to the interplay between the SVC state and the Fe-As hybridized state (the hybridization between the Fe 3d orbital and As 4p orbital electrons), which makes the hybridized band unstable and benefits the formation of the As(1)-As(1) bonding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' These results imply that the Pchon is sensitive to the existence of the competing order introduced by the chemical doping.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' While the unchanged Pchoff and Pcf is possibly related to the stability of the lattice that is still governed by the matrix of the initial T phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' In conclusion, the coevolution of the SVC state, superconductivity, dominated carriers and stability of lattice structure in CaK(Fe1-xNix)4As4 (x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04 and 0) superconductors has been investigated by the complementary measurements of high-pressure resistance, Hall coefficient and synchrotron X-ray diffraction in a wide pressure range involving superconducting to non-superconducting transition, and the corresponding half- to full-collapse of tetragonal phase transition, for the first time.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' We identified the three critical pressures though the comprehensive analysis of our results: Pchon and Pchoff that are the turn-on and -off pressure of the h-cT phase transition, respectively, and Pcf that is the critical pressure of the f-cT phase transition.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Our results demonstrate that the formation of the As(1)-As(1) bond that is evidence by the transition from the T phase to the h-cT phase changes the sign of RH , a manifestation for the reconstruction of the Fermi surface.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' While, the formation of the As(2)-As(2) bond characterized by the f-cT phase transition terminates the superconductivity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' These results achieved in this study not only provide a consistent understanding on the results reported before, but also demonstrate the importance of the Fe-As bonding in stabilizing the superconductivity of the iron-pnictide superconductors through the “pressure window”.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Acknowledgements This work was supported by the National Key Research and Development Program of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' 2021YFA1401800, 2022YFA1403900, 2018YFA0704200), the National Natural Science Foundation of China (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' U2032214, 12104487, 12122414, 12004419, 11822411 and 11961160699), and the Strategic Priority Research Program (B) of the Chinese Academy of Sciences (Grant No.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' XDB25000000).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=', S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' and H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' are grateful for supports from the Youth Innovation Promotion Association of the Chinese Academy of Sciences (Grant Nos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' 2019008, Y202001) and the China Postdoctoral Science Foundation (E0BK111).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' These authors with star (*) contributed equally to this work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Correspondence and requests for materials should be addressed to L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (llsun@iphy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='cn) References [1] Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Kamihara, T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Watanabe, M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Hirano, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Iron-based layered superconductor La[O1-xFx]FeAs (x = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='05-0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='12) with Tc = 26 K.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' J.' metadata={'source': 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metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Two resonant magnetic modes in an overdoped high Tc superconductor.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' 91, 237002 (2003).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' [28]H.' metadata={'source': 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'/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Geophys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Res.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' 91, 4673 (1986).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' [29]See Supplementary Information for the extended resistance measurements, the determination on the onset transition temperature of the spin-vertex crystal (SVC) state and the pressure dependence of the volume for CaK(Fe1-x Nix)4As4 (x=0 and 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04);' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' also see Refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' [12,13,17,24].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' [30]G.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Lin, J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Guo, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Zhu, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Correlation between Fermi surface reconstruction and superconductivity in pressurized FeTe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='55Se0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='45.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' B 101, 214525 (2020).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' [31]D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Kang, Y.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Zhou, W.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Yi, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Superconductivity emerging from a suppressed large magnetoresistant state in tungsten ditelluride.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Nat.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Commun.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' 6, 7804 (2015).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' [32]R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Yang, C.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Le, L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Zhang, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Formation of As-As bond and its effect on absence of superconductivity in the collapsed tetragonal phase of Ca0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='86Pr0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='14Fe2As2: An optical spectroscopy study.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' B 91, 224507 (2015) [33]T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Yildirim.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Strong coupling of the Fe-spin state and the As-As hybridization in iron-pnictide superconductors from first-principle calculations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Lett.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' 102, 037003 (2009).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' [34]J.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Diehl, S.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Backes, D.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Guterding, et al.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Correlation effects in the tetragonal and collapsed-tetragonal phase of CaFe2As2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Phys.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Rev.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' B 90, 085110 (2014).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Figure 1 The results of resistance measurements on the CaK(Fe1-xNix)4As4 (x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04 and 0) at high pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (a) Temperature dependence of the resistance in the pressure range of 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='76 GPa–37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='32 GPa for the sample #1 of the CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='96Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (b) Enlarged views of the resistance-temperature curves at different pressures for the sample #1 of CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='96Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (c) Resistance as a function of temperature for pressures ranging from 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='56 GPa to 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5 GPa for the sample #A of CaKFe4As4 single crystal.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (d) Resistance-temperature curves at different pressures for the sample #A of CaKFe4As4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (a) (b) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='7 P/GPa S#1 CaK(Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='76 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='92 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='89 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='38 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='80 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='98 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='6 S#1 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='07 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='71 27.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='40 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='05 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='68 29.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='65 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='2 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='66 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='68 32.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='28 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='41 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='13 34.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='36 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='4 nce 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='36 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='46 37.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='32 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='9 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='17 19.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='99 sistan 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='3 R 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='08 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='91 Resi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='6 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='57 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='2 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='51 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='1 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 CaK(Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' As SC Ni 90 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04/ 60 50 150 250 (K) 0 100 200 300 0 2 4 30 6 8 10 Temperature (K) 12 14 Pressure (GPa) 0 (d) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='07 (c) P/GPa S#A CaKFe,As 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='56 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='86 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='06 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='35 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='77 S#A 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='26 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='66 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='20 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='07 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='72 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='05 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='19 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='57 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='89 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='76 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04 istance 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='15 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='76 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='50 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='59 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='03 R 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='10 Resi 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='02 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='01 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='05 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='00 CaKFe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='00 90 SC 60 150 (K) 0 50 100 200 250 300 0 2 30 4 6 8 10 Temperature (K) 12 14 Pressure (GPa) 0Figure 2 Hall resistance (Rxy) as a function of magnetic field (H) for the CaK(Fe1-xNix)4As4 (x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04 and 0) single crystals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Plots of Rxy versus H for the CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='96Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 single crystals measured at 30 K in the pressure range of (a) 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='89-15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5 GPa, (b) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='34-14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='9 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Rxy versus H for the CaKFe4As4 single crystals measured at 40 K in the pressure range of (c) 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='56 – 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5 GPa, (d) 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='24-16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='43 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' The dashed line indicates Rxy(B)=0 where the average pressures for the sample #2 and #3 of the CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='96Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 and the sample #A and #B of the CaKFe4As4 single crystals are estimated to be ~ 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='4 GPa and 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='88 GPa, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (a) (b) LT=30K T=30K 1 S#2 S#3 0 (uw) 1 P/GPa 2 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='34 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='85 Rxy P/GPa 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='84 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='50 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='89 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='74 4 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='41 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='13 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='62 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='81 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='88 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='85 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='34 9.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='15 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='55 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='64 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='10 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='33 4 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='15 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='64 6 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='92 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='41 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='88 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='66 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='47 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='94 5 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='47 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='87 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='58 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='11 CaK(Fe。' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' As CaK(Fe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' ,Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' ).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='As 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='55 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='96 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='71 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='90 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='52 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='96 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04/4 8 1 2 3 4 5 6 0 1 2 3 4 0 5 6 H (T) H (T) (d) C LT=40K T=40K S#A S#B 3 2 (sw) 0 0 P/GPa P/GPa 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='56 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='35 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='24 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='86 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='38 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='26 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} 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12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='68 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='19 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='72 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='74 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='89 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='57 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='77 6 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='53 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='02 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='76 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='76 CaKFe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='As4 CaKFe,As 9 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='60 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='43 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='59 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='50 0 1 2 3 4 5 6 0 1 2 3 4 5 6 H (T) H (T)Figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' X-ray diffraction results of the CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='96Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 sample collected at high pressure and crystallographic illustration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (a) X-ray diffraction patterns measured at different pressures, showing no crystal structure phase transition in the experimental pressure range up to 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='32 GPa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (b) Schematic crystal structure of the CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='96Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 sample.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' b 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04/4 39.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='32(GPa)^ 36.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='93 33.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='38 28.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='98 24.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='53 22.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='95 21.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='56 20.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='16 18.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='6 Intensity (a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='u.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=') 17.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='43 16.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='41 15.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='18 14.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='01 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='01 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='91 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='98 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='58 AS(2)C 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='93 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='17 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='63 As(2)0 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='08 Fe/Ni 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='61 As(1) 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='21 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='76 AsC 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='29 人1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='68 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='03 (002) 01 113) 14) 13 30 (20 1 1 1 1 5 10 15 20 25 20 (deg.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' )Figure 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' Structure information, pressure-temperature phase diagram and Hall coefficient (RH) of the CaK(Fe1-xNix)4As4 (x=0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04 and 0) samples at different pressures.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (a) and (d) Lattice parameters a and c versus pressure for the doped- and undoped- samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (b) and (e) Pressure-Temperature phase diagrams, displaying the evolution of the spin-vortex crystal (SVC), superconducting (SC) and non-superconducting (NSC) states upon increasing pressure for the doped- and undoped-samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (c) and (f) Pressure dependence of Hall coefficient (RH) for the CaK(Fe0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='96Ni0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='04)4As4 and CaKFe4As4 samples.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' T, h-cT and f-cT represent the tetragonal phase, half-collapsed tetragonal phase and full-collapsed tetragonal phase, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' TN stands for the onset transition temperature of the SVC state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' SCh and SCe represent the superconducting state with the dominance of hole-carriers and the CaKFe,As, Ni.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' (a) (d) T h-cT f-cT f-cT T h-cT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='88 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5 13.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='88 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='86 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='87 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 12.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='86 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='84 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 a a 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='5 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='85 h-cl h-cT 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='82 10.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='84 11.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' [12] 80 80 (b) (e) S#1-T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' 0 Tc 口 Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' [13] S#1-T~ S#B TN 60 60 S#2-Tc S#A Ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' [12] V S#2-Tn S#B S#3-Tc svc S#3-Tn 20 20 8 NSC NSC SC.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' SC SC SC 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 (0/g-0L) T=30K T=40K 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 Poff Poff ch ch 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 RH RH Pon ch Pon ch 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content='0 5 10 15 20 25 5 10 15 20 25 0 0 Pressure (GPa) Pressure (GPa)superconducting state with the dominance of electron-carriers.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' The Pchon, Pchoff represent the turn-on and turn-off pressures for the transition from the T phase to the h-cT phase.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/edE4T4oBgHgl3EQfQwxW/content/2301.04984v1.pdf'} +page_content=' The Pcf stands for the critical pressure for the transition from the h-cT phase to f-cT phase.' metadata={'source': 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b/gNE4T4oBgHgl3EQfqw2j/content/tmp_files/2301.05203v1.pdf.txt @@ -0,0 +1,7101 @@ +Entanglement negativity in de Sitter biverse +from Stringy Axionic Bell pair: An analysis using +Bunch-Davies vacuum +Sayantan Choudhury1‡, §,. +1Centre For Cosmology and Science Popularization (CCSP), SGT University, Gurugram, +Delhi-NCR, Haryana- 122505, India. +Abstract +In this work, we study the signatures of quantum entanglement by computing entan- +glement negativity between two causally unrelated regions in 3 + 1 dimensional global de +Sitter space. We investigate a bipartite quantum field theoretic setup for this purpose, +driven by an axionic Bell pair resulting from Type IIB string compactification on a Calabi- +Yau three fold. We take into account a spherical surface that divides the spatial slice of +the global de Sitter space into exterior and interior causally unrelated sub regions. For +the computational purpose we use the simplest possible initial choice of quantum vacuum, +which is Bunch-Davies state. The quantitative quantum information theoretic measure for +entanglement negativity turns out be consistent with the results obtained for entanglement +entropy, even we have to say it is better than that from quantum information theoretic +point of view. We design the problem in a hyperbolic open chart where one of the causally +unrelated observers remains constrained and the scale dependence enters to the correspond- +ing quantum information theoretic entanglement measure for axionic Bell pair. We find +from our analysis that in the large scales initially maximally entangled Bunch-Davies state +turns out to be strongly entangled or weakly entangled depending on the axionic decay +constant and the supersymmetry breaking scale. We also find that at the small scales the +initial entanglement can be perfectly recovered. We also discuss the possibility of having +a biverse picture, which is a mini version of the multiverse in the present theoretical set +up. Last but not the least, we provide the necessary criteria for generating non vanishing +quantum entanglement measures within the framework of quantum field theory of global +de Sitter space as well as well as in primordial cosmology due to the axion derived from +string theory. +Keywords: De-Sitter vacua, Quantum Entanglement, Cosmology of Theories beyond +the SM, Quantum Information Theory aspects of Gravity, String Cosmology. +‡ Corresponding author, E-mail : sayantan ccsp@sgtuniversity.org, sayanphysicsisi@gmail.com +§ NOTE: This project is the part of the non-profit virtual international research consortium “Quantum +Aspects of Space-Time & Matter” (QASTM) . +arXiv:2301.05203v1 [hep-th] 31 Dec 2022 + +Contents +1 +Introduction +1 +2 +Basics of entanglement negativity and logarithmic negativity from Quan- +tum Information Theory +7 +3 +Computational strategy of negativity between two causally unrelated +open charts of global de Sitter space with axion +14 +3.1 +Quantum structure of open chart +14 +3.2 +Geometric structure of open chart +16 +3.3 +Mode function and wave function of axion in an open chart +19 +3.4 +Construction of reduced density matrix in open chart +41 +4 +Entanglement negativity and logarithmic negativity in open chart +43 +5 +Comparison with entanglement entropy in open chart +50 +6 +Logarithmic negativity between two causally unrelated patches of open +chart +53 +6.1 +Computational set up and construction of maximally entangled states +55 +6.2 +Construction of excited quantum state for single oscillator +57 +6.3 +Construction of reduced density matrix at the inside observer +62 +6.4 +Partial transpose operation and the comment on the negative eigenvalues +65 +6.5 +Computation of logarithmic negativity: Numerical study +68 +6.6 +Small scale limit of logarithmic negativity: Analytical study +76 +6.7 +Massless limit of logarithmic negativity: Analytical study +78 +7 +Conclusion +81 +References +85 +i + +1 +Introduction +In the present day research, different quantum information theoretic measures of quantum +entanglement is a remarkable probe in theoretical physics which helps us to distinguish +the various type of long range correlated quantum mechanical states. In this connection, +study of the explicit role of the long range quantum correlations in the framework of +quantum field theory is extremely significant, which is a fascinating topic of research itself. +See refs [1–31] for more details. The key ingredient of this study is the initial quantum +mechanical vacuum states, which are Chernikov-Tagirov, Bunch-Davies, Hartle-Hawking, +α and Motta-Allen vacua [14–16, 19, 32–37]. Quantum entanglement is treated as one of +the remarkable outcomes of the foundational theoretical aspects of quantum mechanics. +The prime reason of this thought is, a local measurement in quantum mechanics may +instantaneously put a significant impact of the outcome of the measurement beyond the +physical light cone. This is theoretically interpreted as Einstein-Podolsky-Rosen (EPR) +paradox where the concept of causality violation is explicitly demonstrated [31, 38–41]. +Amongst various types of information theoretic measures entanglement entropy is con- +sidered to be a very useful quantitative as well as qualitative probe of quantum entangle- +ment and this concept is commonly used in the framework of condensed matter physics, +quantum information theory and high energy physics. But in this connection it is im- +portant to note that the technical computation of entanglement entropy is very difficult +to perform in the context of quantum field theory. Ryu and Takayanagi in refs. [42, 43] +first did the theoretically consistent computation of entanglement entropy for a strongly +coupled quantum field theory having a gravitational dual counterpart using the underlying +physical principles of AdS/CFT (or holographic gravitational dual prescription) [44]. +In order to properly understand the direct implications of the previously proposed +AdS/CFT (or holographic gravitational dual prescription) in the technical computation +of entanglement entropy in presence of standard Bunch-Davies quantum vacuum state +within the framework of quantum field theory of de Sitter space, Maldacena and Pimentel +prescribed an extremely useful computational strategy in ref. [1] using free massive scalar +quantum field. After this work the proposed methodology was generalised for the same +problem in presence of non standard α vacua in refs [13, 19, 45–47]. Further in refs. [14– +16], these underlying physical concepts of the computation of entanglement entropy was +applied within the framework of axion quantum field theory, described in terms a specific +type of effective interaction potential originated from Type II string theory compactifi- +cation [48–50] in presence of both standard Bunch-Davies and non standard α quantum +vacua. The underlying concept of quantum entanglement could applicable to the frame- +work of cosmology, specifically beyond the Hubble horizon scale. Most importantly, in +this prescription if a particle pair is created in a casually connected Hubble horizon scale +then it is naturally expected to be dissociated as an outcome of the de Sitter cosmological +expansion. In this particular case, the findings from this prescription pointing towards the +1 + +fact that two causally unrelated patches in the de Sitter cosmological space-time has to +be entangled quantum mechanically. As a result, the corresponding observable quantum +vacuum fluctuations associated with our own universe is entangled with the other part of +the open patch of global de Sitter space, which is commonly identified to be the multiverse +in the present context of discussion. To demonstrate this underlying physical picture the +reduced density matrix formalism play a very significant role, using which it was explicitly +obtained that the quantum mechanical entanglement directly put impact on the cosmo- +logical power spectrum on the large scales and it is quantitatively similar to or larger than +the curvature radius scale connected to the current issue [18]. +It is hugely expected that the present and the upcoming observational probes may detect +the imprints and explicit effects of quantum mechanical entanglement in cosmological +paradigm, which we strongly believe will going to be extremely useful to understand a lot +of unknown mysterious fascinating facts of our own universe. In the past, there are lot of +efforts have been made to study the impacts of quantum mechanical entanglement in the +theoretical framework of cosmology. See refs. [16, 18, 21, 23, 51–58] for more details on +these aspects. +The motivation of the background physical thought of the present paper are appended +below point-wise: +1. It was first discovered in a number of refs.[59–61] to support the claim that the +physical frame of the bubble nucleation process is observer dependent and is actually +dictated by the rest frame of the observer. The bubble nucleation process being +dependent on the observer is hence only expected. The detailed study of such type +of observer dependence in the framework of quantum mechanical entanglement is one +of the prime motivations of the present work. If one can able to address this issue in +detail, then various unexplored features of quantum cosmology can be explored with +proper physical understanding. +2. Second, the explicit function of the Bunch Davies vacuum can be investigated in the +context of quantum entanglement originating from an axion field embedded in an +open patch of the global de Sitter space. This will be crucial because it allows one +to directly examine whether it is possible to find the signs of quantum entanglement +in the current multiverse [18, 62–67] motivated theoretical set up.using current or +future observational instruments. +3. Thirdly, it was noted in the refs. [14–16, 68–70] that the string theory-originated +axion can be viewed as the ideal component to build the Bell pair, which is neces- +sary to violate Bell’s inequality within the context of primordial cosmology. Within +the widely accepted framework of primordial cosmology, it is virtually difficult to +break Bell’s inequality, and it is for this reason that string theory and the axion +play the most important roles. It is significant to note that, in this context, it is +2 + +nearly impossible to test the explicit role of quantum mechanical entanglement in +observational probes without violating Bell’s inequality in the context of primordial +cosmology, which is necessary to break the degeneracy in the shape cosmological two- +point function, also known as the primordial power spectrum obtained from various +effective potentials from fundamental physical principles. The current framework +with the string theory-originated axion provides a perfect setup that is pointing to- +wards a new physics coming from a non-standard cosmology because the generation +of Bell’s inequality violating pairs, also known as the Bell pair, is practically im- +possible within the framework of the standard primordial cosmological paradigm. It +is explicitly pointed in refs.[68–70] that the prime signature of the Bell’s inequality +violation in non standard primordial cosmology is coming from the existence of one +point function from axion, which is absent in the well known standard cosmological +paradigm. This result has a great impact of producing quantum entanglement in the +present scenario. Also our derived results can be extremely useful to directly verify +along with the observational probes the applicability as well as the justifiability of +the string theory originated axion to address the issue of quantum entanglement in +the present set up. See refs. [14–16, 68–77] for more details on the Bell’s inequality +and its violation in various contexts including cosmology. +4. Finally, the motivation comes from the multiverse prescription appearing in the string +theory landscape scenario, which states that our universe may not be the single entity +of the space-time but the part of a bigger size multiverse [18, 62–67]. This is obviously +a fascinating fact which one needs to study in detail. +However, in the past the +underlying physical concept of the multiverse has been criticised in various contexts +as a hypothetical philosophical concept which is the outcome of complete theoretical +imagination and cannot be tested via cosmological observation. But the actual truth +is fay beyond all of these criticism, which is the detectable signatures can be obtained +from the multiverse set up in presence of quantum mechanical entanglement. It really +helps us to produce at least two causally separated de sitter universe, commonly +known as the de Sitter bubbles in the present context. But the numbers are not +restricted in two and most importantly this is actually the starting point of multiverse +which allows many more causally unrelated de Sitter bubbles. +Next, we explicitly write down the underlying assumptions of the present work, which +are appended below point-wise: +1. To model the present multiverse motivated scenario [18, 62–67] , we consider two +causally separated patches on the global de Sitter space by assuming that initially +they are in a maximally entangled pure quantum mechanical state. this particular +set up we identify as the biverse picture which is the mini version of the original +multiverse picture. +3 + +2. In this theoretical set up further we introduce two observers whose actual purpose is +to determine the quantum entanglement of a de Sitter universe. We also assume that +one of the observers is placed inside the a de Sitter bubble and want to determine +how the signatures of quantum mechanical entanglement with the other de Sitter +bubble can be visualized by the inside observer. Now the issue is that, the inside +observer can’t able to see outside region of their own de Sitter bubble. For this reason +instead of using the total density matrix, one needs to take the partial trace over +the causally unrelated outside region, which finally give rise to the reduced density +matrix of the system in the present context. But as an outcome some information +will be lost during this process and to describe the quantum mechanical state of the +observer instead of using a pure state one needs to explicitly use a mixed state. This +scenario is completely different for an observer who is sitting on the other causally +separated de Sitter bubble because in his frame of reference the quantum aspects +is described by the pure quantum states. For this reason we need study the effects +and outcomes of the quantum entanglement between pure and mixed states in the +present context of discussion. +3. To perform the computation for a single global de Sitter space picture as well in the +case of the entangled two de Sitter space, which is the biverse picture we need to +properly understand the factorization of the Hilbert space in terms of the individual +constituents which span the subspaces by forming complete orthonormal sets. The +geometrical structure of the global de Sitter space demands that in the hyperbolic +open chart one can symmetrically factorize in the left region and in a right region, +which we have tagged as region L and R during performing the technical computation +in this paper. Since we have symmetrically factorize the total Hilbert space in the +region L and R, it is viable to formulate the reduced description either in terms of +the degrees of freedom appearing in the region L or region R. In our computation we +take the partial trace over the degrees of freedom appearing in the region R, which +forms a reduced density matrix in the region L. +4. If the Hilbert space factorization performed correctly then one can further blindly +trust the mode decomposition in the region L and R using which we have con- +structed the Bunch Davies quantum vacuum states in terms of harmonic oscillator +modes which forms a complete orthonormal basis states. Estimation of the quan- +tum entanglement in terms of various quantum information theoretic measures are +also based on this mode decomposition and hence one can rely on the tracing out +the unwanted information from the bipartite quantum field theoretic set up under +consideration to construct the reduced density matrix. +5. To study the outcome of quantum mechanical entanglement we have blindly trusted +the above mentioned factorization for the second de Sitter space which is spanned +4 + +by the well known Bunch Davies vacuum state. For the other, which is the first +de Sitter bubble we have assumed that the detailed factorization structure is not +explicitly needed for the computation we are interested to perform in this paper. +This is because of the fact that we need to take the partial trace operation over the +first de Sitter vacuum which will remove all unwanted degrees of freedom from the +quantum information theoretic measures of entanglement we want to compute to +confirm the existence of quantum entanglement in the biverse picture. Here one can +do the computation in other way as well where one needs to take the partial trace +operation with respect to the all degrees of freedom appearing in second de Sitter +vacuum and properly factorize the first de Sitter vacuum in terms of L and R modes. +Due to having the symmetrical structure of the set up that we are considering it is +expected that for both the cases we will have the same physical outcome at the end +of the computation. We just have taken the first possibility as a choice. +6. During the biverse construction we have first constructed the maximally entangled +state which is one of the key ingredients to violate the Bell’s CHSH inequality within +the present framework. This is a very challenging to construct within the framework +of de Sitter space time as well as in primordial cosmology set up. We have constructed +the maximally entangled state which suffice the purpose. +But this could not be +possible just having usual interaction in the scalar field theory embedded in the global +patch of the de Sitter space. The Bell’s CHSH inequality violating pair is created +in the region L and R with the help of specific type of interaction in the string +theory originated axion driven scalar field theory. The structure of the interaction +is controlled by the time dependent axion decay constant and the supersymmetry +breaking scale. We have assumed that Bell pair is created and originated from the +string theory originated axionic scalar field theory. It might be possible to create +such pair from some other theoretical sector as well which we have not considered in +this paper for the time being. +7. In the present context the global coordinates can be treated as the closed slicing +from the point of view of FLRW cosmology. One usually do primordial cosmology, +particularly inflation [78–116] in the planar coordinate slicing of the de Sitter space. +By performing coordinate coordinate transformation one can transform the global to +static and then static to the flat slicing in the planer patch of de Sitter space. For this +reason whatever results we have obtained in this paper for the global patch of the de +Sitter space can be directly translated in the planer patch of the de Sitter space, which +further implies that our derived results hold good for primordial cosmological set up +constructed out of the effective interaction studied for the string theory originated +axion scalar field theory. This is obviously an interesting aspect from our computation +which shows the applicability of our derived results in the vast theoretical framework. +5 + +8. The von Neumann and Renyi entropies [14, 15, 117] are the good quantum informa- +tion theoretic measures of quantum entanglement in the context of bipartite quantum +field theory designed in terms of pure state. But with the help of mixed state if we +want to understand the features of an arbitrary bipartite quantum field theoretic +system then instead of using entropic measures, negativity or its logarithmic version +would be perfect information theoretic measure of quantum entanglement. Using +this specific measure we compute the imprints of the quantum entanglement from +the point of view of two causally separated observers in the open chart of global de +Sitter space. +-4 +-3 +-2 +-1 +0 +1 +2 +3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.5 +1.0 +1.5 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +For single de Sitter Universe +For two de Sitter Universe (biverse) +CY3 +A +AB+XicbZDLSsNAFIZP6q3W9Slm8EiuCqJVnRZ7MZlBXuRtpbJdNIOnUzCzKRQt7EjQtF3Pom7nwbp2kW2vrDwMd/zuGc+b2IM6Ud59sqrK +1vbG4Vt0s7u3v7B/bhUuFsS0SUIeyo6HFeVM0KZmtNOJCkOPE7b3qQ+r7enVCoWigc9i2g/wCPBfEawNtbAtpOe56P6Y/qUwWU6sMtOxc +mEVsHNoQy5GgP7qzcMSRxQoQnHSnVdJ9L9BEvNCKdpqRcrGmEywSPaNShwQFU/yS5P0ZlxhsgPpXlCo8z9PZHgQKlZ4JnOAOuxWq7Nzf9q3V +j7N/2EiSjWVJDFIj/mSIdoHgMaMkmJ5jMDmEhmbkVkjCUm2oRVMiG4y19ehdZFxa1Wru6r5dptHkcRTuAUzsGFa6jBHTSgCQSm8Ayv8GYl1 +ov1bn0sWgtWPnMf2R9/gCPH5L4 +Penrose +diagram +Geometry of +de Sitter +space ++ ++ +Measure +of quantum +entanglement +Measure +of quantum +entanglement +String Theory +Axion +Outcome 1 +Outcome 2 +Figure 1.1: Representative diagram of the overall quantum entanglement computation +and all possible outcomes for single de Sitter universe and two de Sitter universe (biverse) +obtained from our our analysis. +In this work, we compute the negativitity measure of quantum entanglement from an +axionic effective potential which is obtained from Type IIB string theory compactification +on a Calabi-Yau three fold in presence of NS5 brane. Earlier this model have been studied +6 + +T +=0 +二t, = const. +A +B +L +R +C +Open chartin the framework of inflationary cosmology [118–121]. Since this axions can be treated +as Bell pair, we technically compute the expression for entanglement negativity from two +causally unrelated patches of the open chart of the global de Sitter space. +From the +quantitative results obtained from our computation we perform further a consistency check +as well as the comparison of the results obtained in previous refs.[14–16]. Furthermore, to +understand various unknown fascinating facts regarding the multiverse scenario we extend +our computation in presence of axionic Bell pair in presence of de Sitter space along +with the maximally entangled quantum Bunch Davies vacuum state. We find from our +analysis that in the large scales initially maximally entangled Bunch-Davies state turns +out to be strongly entangled or weakly entangled depending on the axionic decay constant +and the supersymmetry breaking scale. We also find the at the small scales the initial +entanglement can be perfectly recovered. Last but not the least, we provide the necessary +criteria for generating non vanishing quantum entanglement measure within the framework +of primordial cosmology due to the string axion. In figure (1.1) representative diagram +of the overall quantum entanglement computation and all possible outcomes for single de +Sitter universe and two de Sitter universe (biverse) obtained from our our analysis. +The organization of this paper is as follows: +• In Section 2 we briefly review the tools and techniques required to compute nega- +tivity and logarithmic negativity along with some easy demonstrations for the better +understanding purpose. +• In Section 3 we mention the computational strategy for negativity in the hyperbolic +open chart of the global de Sitter space. +• In Section 4 we compute negativity by following aforementioned strategy for a spe- +cific effective potential, Type IIB string compactification originated axion model. +• In Section 5 we compute entanglement in terms of negativity from two causally +unrelated de Sitter bubbles, which is determined from the point of view of two ob- +servers introduced during the calculation. In this section we discuss the possibility +of having biverse, which is the mini version of the multiverse scenario. +• Finally in Section 6 we summarize our all findings in this paper along with some +future prospects of the computations performed in this paper. +2 +Basics of entanglement negativity and logarithmic negativity +from Quantum Information Theory +Within the framework quantum mechanics, particularly in quantum information theory to +study the imprints and the underlying physical aspects of entanglement various measure +have been proposed till date. Entanglement negativity and its logarithmic version, which +7 + +is commonly known as logarithmic negativity are the very useful quantum information +theoretic measures of quantum mechanical entanglement. These measures are derived by +making use of the positive partial transpose criterion for the separability. +The Peres- +Horodecki criterion [31, 122–124] is a prerequisite for the separation of the joint density +matrix of two quantum mechanical systems A and B. The phrase is sometimes referred +to as the PPT criterion, which stands for positive partial transpose. From the studies +it was found that in the 2 × 2 and 2 × 3 dimensional cases the condition is sufficient. +Particularly this is used to decide the separability of mixed quantum states, where the +well known Schmidt decomposition does not apply. It is important to note that, in higher +dimensions, this specific concept gives inconclusive result, and one should perform more +advanced tests, like entanglement witnesses, which describes a functional that distinguishes +a specific entangled state from the separable ones. In refs. [31, 122–124], i t was explicitly +shown that this measures are entanglement monotone and hence treated to be a proper +measure of quantum entanglement. In the following we give a brief overview on the topic +of entanglement negativity and logarithmic negativity. +Now let’s take a look at a quantum system that can be described by A and B. The +direct products of the Hilbert spaces of the subsystems A and B define the corresponding +total Hilbert space of the system i.e. H = HA ⊗ HB. Here H represents the Hilbert space +of the total system, HA and HB represent the Hilbert spaces of the subsystems A and B +respectively. +Further consider a pure quantum mechanical state, by applying the Schmidt decompo- +sition one can write: +|Ψ⟩ = +� +m +� +λm|m⟩A ⊗ |m⟩B, +(2.1) +where λm corresponds to the observed probability of the any general pure m-th state and +satisfy the following constraint condition: +� +m +λm = 1, +(2.2) +which physically implies that the total observed probability of the process, which is ob- +tained by summing over all possible pure states has to be conserved in the present context +of discussion. +Now, it is important to note if we are interested in the mixed states then to technically +describe the behaviour of the subsystem inclusion of the concept of reduced density matrix +is extremely important and this is actually described by the reduced density operator of +the subsystem. In the present description we consider two subsystems A and B, which are +equally likely and both of them carry same weight in the present construction. For this +reason to describe the subsystem either we will talk about the description in terms of A +or B. Let us say for the time being that we are interested to know about the underlying +physics of the subsystem A which can be obtained by taking the partial trace operation +8 + +on the information of the subsystem B. This process will finally give rise to the following +reduced density matrix of the subsystem A, which is given by: +ρA = TrB|Ψ⟩⟨Ψ| = +� +m +λm|m⟩AA⟨m|. +(2.3) +Further, utilizing this reduced density matrix of the subsystem A one can explicitly com- +pute the expression for the von Neumann entropy, which is given by: +S = −Tr [ρA ln ρA] = − +� +m +λm ln λm. +(2.4) +Technically the above mentioned equation was introduced to describe the first entangle- +ment measure in quantum information theory. In the specific situation, when there is no +quantum entanglement, then in terms of the probability we can write: +λm = +� +� +� +� +� +� +� +� +� +1 +if m = 1 +0 +if m ̸= 1 +(2.5) +In the above mentioned both the cases the measure of entanglement entropy vanish ex- +plicitly. If such a situation appears in a particular physical systems, then we can comment +that there would be no quantum mechanical entanglement for that type of systems. On the +other hand, if we find some physical systems where the probability lie within the window +0 < λm < 1∀m then the corresponding entanglement entropy measure is non-zero and it is +treated to be good measure within the framework of quantum information theory. But one +word of caution is that, the entanglement entropy measure sometimes gives inconclusive +result. For an example, if we are interested to describe the strictly classical correlation in +terms of mixed state then the entanglement entropy measure gives non-zero result, which +is itself a surprising fact. In this specific type of situation entanglement entropy mea- +sure within the framework of quantum information theory fails to distinguish between the +impacts of quantum mechanical correlations and classical correlations. +Then obviously a natural question comes to a physicists mind that what is the way +out of this situation? Is it possible to define more powerful entanglement measure in the +present framework? Answer to all of these questions are Yes, it is possible to define such +measures which can conclusively distinguish the quantum and classical effects in the corre- +lation functions computed from the underlying physical theory. Based on the separability +criterion entanglement negativity and logarithmic negativity measures are defined which +actually serve the purpose. +Let us now give a brief outline on the connection between separability and quantum +entanglement, and its usefulness in the present context of discussion. A quantum mechan- +9 + +ical state is considered to be separable iff the density matrix of the total system can be +expressed in terms of a tensor or outer products of the individual density matrices belong +to the each subsystems under consideration. Technically this statement can be written as: +ρ = +� +m +λm +� +ρA +m ⊗ ρB +m +� +where +λm ≥ 0 +∀m. +(2.6) +Here ρ represents the density matrix of the total system under consideration. For the m-th +state corresponding to the subsystems A and B the density matrices are defined as: +ρA +m := |m⟩AA⟨m| +and +ρB +m := |m⟩BB⟨m|. +(2.7) +Hence in terms of the information coming from the subsystems A and B, the density +matrix for the total system can be recast as: +ρ = +� +m +λm (|m⟩AA⟨m| ⊗ |m⟩BB⟨m|) +where +λm ≥ 0 +∀m. +(2.8) +One can further generalize the structure of the density matrix by considering both the +contributions from entangled and non-entangled states, which is given by the following +expression: +ρ = +� +m +� +n +� +p +� +q +Dmnpq (|m⟩AA⟨n| ⊗ |p⟩BB⟨q|) +where +Dmnpq ≥ 0 +∀m, n, p, q, (2.9) +where Dmnpq represents a more general coefficient which capture both the effects from +entangled and non-entangled quantum states. +Let’s investigate a partial transposition +operation with respect to the subsystem A in light of the new generation definition of the +total density matrix, which results in the following expression: +ρTA = +� +m +� +n +� +p +� +q +Dmnpq +� +(|m⟩AA⟨n|)TA ⊗ |p⟩BB⟨q| +� += +� +m +� +n +� +p +� +q +Dmnpq (|n⟩AA⟨m| ⊗ |p⟩BB⟨q|) . +(2.10) +Following a partial transpose operation with regard to the subsystem A, we obtain the +following results for the non-entangled quantum state: +ρTA = +� +m +λm +� +(|m⟩AA⟨m|)TA ⊗ |m⟩BB⟨m| +� += +� +m +λm (|m⟩AA⟨m| ⊗ |m⟩BB⟨m|) += ρ, +(2.11) +10 + +which corresponds to the fact that the total density matrix remains unchanged. Here since +for the non-entangled state λm∀m, this directly implies that ρTA ≥ 0. This also confirms +that if after performing a partial transpose operation on the total system has at least one +negative eigenvalue then the system cannot be described in terms of the above mentioned +form stated in eqn (2.8) and consequently the underlying quantum state considered in this +discussion has to be entangled. This is actually very interesting outcome coming from the +present computation in support of quantum entanglement. +Utilizing these facts one step forward the definition of an quantum information theoretic +entanglement measure, entanglement negativity is proposed, which is given by: +N = +� +λm<0 +|λm|, +(2.12) +where in this definition summation over all negative eigenvalues are explicitly taken into +account. From this definition, we can clearly see that when we have N = 0, there is no +quantum entanglement in the system. Apart from having a very good physical thought +behind the construction of the entanglement negativity measure, unfortunately it turns +out that this is not an additive measure and most importantly not at all suitable for the +description of many body or multi-subsystem appearing within the framework of quantum +field theory. Hence, to give a more physically applicable quantum information theoretic +measure another powerful quantity has been introduced, which is known as logarithmic +negativity and is treated to be most improved version of the entanglement negativity mea- +sure in the present context of discussion. +Let us now discuss in detail that how one can technically define the logarithmic negativity +by making use of the background physical facts discussed in the present context. To define +this quantity, first of all let us introduce the trace norm of the partial transposed version +of the total density matrix over the subsystem A, which is described by the following +expression: +||ρTA|| = Tr +� +(ρTA)† ρTA += +� +m +|λm| += +� � +λm>0 +|λm| + +� +λm<0 +|λm| +� += +� +N + +� +λm>0 +|λm| +� += (2N + 1) . +(2.13) +Here in the list line of the above expression to write down corresponding trace norm in +11 + +terms of the quantum entanglement negativity measure we have utilized the following sets +of useful constraints: +Tr (ρ) = 1, +Tr +� +ρTA� += 1, +� +m +λm = 1. +(2.14) +Then the logarithm of the trace norm of the of the total density matrix over the subsystem +A is identified to be logarithmic negativity, which is given by the following expression: +LN = ln +� +||ρTA|| +� += ln (2N + 1) . +(2.15) +This implies the fact that, when N ̸= 0, then LN ̸= 0 and the corresponding quantum +state is considered to be entangled. +Let us now consider a special case where we are dealing with the pure quantum mechan- +ical state. In this specific case our next objective is to explicitly compute the expression +for the logarithmic negativity. To serve this purpose we use the well known Schmidt de- +composition technique for pure quantum state, using which we can write: +|Ψ⟩ = +� +m +� +λm (|m⟩A ⊗ |m⟩B) . +(2.16) +Then using this representation of the pure quantum state one can further define the expres- +sion for the density matrix for the total system, which is given by the following expression: +ρ = |Ψ⟩⟨Ψ| = +� +m +� +n +� +λmλn ((|m⟩A ⊗ |m⟩B) (A⟨n| ⊗ B⟨n|)) . +(2.17) +Further, taking the partial transpose operation on the above mentioned density matrix of +the total system over the subsystem A, we get the following expression: +ρTA = +� +m +� +n +� +λmλn ((|m⟩A ⊗ |m⟩B) (A⟨n| ⊗ B⟨n|))TA += +� +m +� +n +� +λmλn ((|n⟩AA⟨m|) ⊗ (|n⟩AA⟨m|)) += +� +m +� +n,m=n +� +λmλn ((|n⟩AA⟨m|) ⊗ (|n⟩AA⟨m|)) ++ +� +m +� +n,m̸=n +� +λmλn ((|n⟩AA⟨m|) ⊗ (|n⟩AA⟨m|)) += +� +m +λm ((|m⟩AA⟨m|) ⊗ (|m⟩AA⟨m|)) ++ +� +m +� +n,m̸=n +� +λmλn ((|n⟩AA⟨m|) ⊗ (|n⟩AA⟨m|)) . +(2.18) +12 + +For the sake of simplicity, to diagonalize the structure of the total density matrix we +further introduce a new basis for the pure quantum states, in which two new state vectors +are defined by the following expressions: +|Ψ+ +mn⟩ : = +1 +√ +2 [(|m⟩A ⊗ |n⟩B) + (|n⟩A ⊗ |m⟩B)] +for +m < n, +(2.19) +|Ψ− +mn⟩ : = +1 +√ +2 [(|m⟩A ⊗ |n⟩B) − (|n⟩A ⊗ |m⟩B)] +for +m < n. +(2.20) +It is very easy to check the following two constraints are always satisfied by the new basis +state vectors |Ψ+ +mn⟩ and |Ψ− +mn⟩, which are given by: +(A⟨m| ⊗ B⟨m|) |Ψ+ +mn⟩ = +1 +√ +2 +� +�(A⟨m| ⊗ B⟨m|) (|m⟩A ⊗ |n⟩B) +� +�� +� +=0 ++ (A⟨m| ⊗ B⟨m|) (|n⟩A ⊗ |m⟩B) +� +�� +� +=0 +� +� += 0, +(2.21) +(A⟨m| ⊗ B⟨m|) |Ψ− +mn⟩ = +1 +√ +2 +� +�(A⟨m| ⊗ B⟨m|) (|m⟩A ⊗ |n⟩B) +� +�� +� +=0 +− (A⟨m| ⊗ B⟨m|) (|n⟩A ⊗ |m⟩B) +� +�� +� +=0 +� +� += 0. +(2.22) +This implies the fact that both the new basis state vectors |Ψ+ +mn⟩ and |Ψ− +mn⟩ are orthogonal +to the state |m⟩A ⊗ |m⟩B, which is obviously a very helpful information for the further +construction. Then the partial transpose with respect to subsystem A for the total density +matrix as stated in eqn (2.18) can be written in the diagonalized basis in terms of the two +state vectors |Ψ+ +mn⟩ and |Ψ− +mn⟩ as: +ρTA = +� +m +λm|m⟩AA⟨m| + +� +m +� +n,m 0 for this +construction. +2. The part |Ψ+ +mn⟩⟩Ψ+ +mn| has the eigenvalues λmλn for all values of m and n. +Here +λm > 0 and λn > 0, then λmλn > 0 for this construction. +3. The part |Ψ− +mn⟩⟩Ψ− +mn| has the eigenvalues −λmλn for all values of m and n. Here +λm > 0 and λn > 0, then −λmλn < 0 for this construction. So the negative eigenval- +ues appear in this computation, which is clearly the indication of having quantum +mechanical entanglement in the present framework. It is important to note that, if +13 + +in these sets of eigenvalues at least two of the λm∀m > 2 ̸= 0, then the negative +eigenvalues always appear in this computation. +Then, in the new diagonalized basis the trace norm can be further computed as: +||ρTA|| = +�� +m +λm + 2 +� +m +� +n,m 0 +Figure 3.3: Graphical behaviour of the axion effective potential with respect to the field +obtained from Type IIB String Theory compactification for stringy parameter b < 0 and +b > 0. +22 + += µ3fa +�� φ +fa +� ++ b cos +� φ +fa +�� +. +(3.19) +In figure (3.3(a)) and figure (3.3(b)) we have plotted the behaviour of the dimensionless +axion effective potential V (φ)/µ3fa with respect to the dimensionless axion field variable +φ/fa for the stringy parameter b > 0 and b > 0. The precise definition of the stringy +parameter b is written in the later part of this paper explicitly. Both the behaviour are +useful to understand the contribution of the linear and non-perterbative periodic part in +the total axion effective potential. +In this effective potential µ3 represents the coupling parameter of linear interaction +which is associated with the underlying theoretical scale, can be expressed as: +Scale of effective potential : +µ3 = +1 +faα +′2gs +exp(4A0) + R2m4 +SUSY +faα +′L4 +exp(2A0), (3.20) +where exp(A0) represents the warp factor of the lower portion of the Klebanov-Strassler +throat geometry, R is the radius which stabilized the 5 brane and antibrane in the cor- +responding string theory construction, the only mass scale involved in this construction +is mSUSY , which actually represents the underlying supersymmetry breaking scale in this +specific set up, the Regge slope is α +′ which is proportional to the inverse string tension, +string coupling is gs and finally L6 represents the world volume. Here first part of the +effective potential breaks the shift symmetry and the rest of the part preserves the sym- +metry φ → φ + 2πfa. Here fa quantifies the axionic decay parameter, which is in general +conformal time dependent (τ) and we have chosen the following useful profile: +Axion decay constant profile : +fa = +� +� +� +�100 − +80 +1 + +� +ln τ +τc +�2 H, +(3.21) +which was used in refs. [68–70] to validate Bell’s CHSH inequality violation in early universe +cosmology. Here H is the Hubble parameter, and τc is the characteristic time scale at which +we have fa = 2 +√ +5H, which is almost a constant in the background geometrical set up we +are considering. Now since the whole problem we are going to solve in terms of the physical +time variable t in both the regions R and L it is necessarily to find out the relationship +between the conformal time scale τ and the physical time scale t. Here we find the following +connecting relationship: +Dimensionless conformal time scale : +τ +τc += 1 + +ln +� +tanh +� t +2 +� +tanh +� tc +2 +� +� +ln +� +tanh +�tc +2 +��, +(3.22) +23 + +0 +2 +4 +6 +8 +10 +5 +6 +7 +8 +9 +10 +(a) For conformal time +1 +2 +3 +4 +5 +6 +7 +8 +9 +(b) For physical time +Figure 3.4: Schematic behaviour of the dimensionless axion decay constant with the +conformal and physical time scale. +24 + +where τc is given by the following expression: +Characteristic time scale : +τc = H ln +� +tanh +�tc +2 +�� +. +(3.23) +For this reason the given profile of the axion decay parameter can be further written in +terms of the physical time t as: +Axion decay constant profile : +fa = +� +� +� +� +�100 − +80 +1 + +� +ln +� +ln[tanh( t +2)] +ln[tanh( tc +2 )] +��2 H, +(3.24) +where tc represents the characteristic time scale in the physical time scale which basically +τc in the conformal time scale. In figure (3.4(a)) and figure (3.4(b)) we have plotted the +behaviour of the dimensionless axion decay parameter fa/H with respect to conformal as +well as the physical time scale. Both the plots almost depict similar feature in both the +time scales. For this reason the chosen profile is extremely useful for our analysis as it is +not changing by changing the definition of the associated time scales. +Additionally, to write down the effective potential in an simplest form we further intro- +duce a new dimensionless quantity, b, which is defined as: +New dimensionless parameter : +b = Λ4 +G +µ3fa +. +(3.25) +To define the new quantity b, a characteristic scale has been introduced, ΛG, which is given +by: +Characteristic scale : +ΛG = +� +mSUSY L3 +√ +α +′gs +exp (−cSinst) +� +�� +� +Instantonic decay +. +(3.26) +Here Sinst represents the instantonic action which finally give rise present structure of the +effective potential within the framework of string theory, the instanton coupling parameter +c ∼ O(1) which is actually treated to be constant term in this computation. Finally one +can able fix the form of the warp factor in terms of all the stringy parameters, which is +given by: +Warp factor : +exp(A0) = +� +ΛG +mSUSY +�2 L +R +� +α +′gs = +L4 +mSUSY R +� +α +′ +gs +exp (−cSinst) +� +�� +� +Instantonic decay +,(3.27) +which further corresponds to the following expression for the coupling parameter µ3, which +25 + +is given by: +Scale in terms of instantonic decay : +µ3 = g2 +s +fa +� +ΛG +mSUSY +�8 �L +R +�4 ++ α +′g2 +sR2m4 +SUSY +faL4 +� +ΛG +mSUSY +�4 �L +R +�2 += +1 +fag3 +s +L16 +m4 +SUSY R4 exp (−4cSinst) +� +�� +� +Faster decay ++m2 +SUSY L4 +fags +exp (−2cSinst) +� +�� +� +Slower decay +, +(3.28) +which is obviously a necessary input to fix the corresponding overall scale of effective +potential derived from string theory. +Last but not the least, one can further able to +compute the expression for the string scale associated with the problem, in terms of the +other stringy parameters as: +0 +2 +4 +6 +8 +10 +0 +2 +4 +6 +8 +10 +12 +14 +Figure 3.5: Schematic behaviour of the approximated axion effective potential obtained +from Case A. +String scale : +Ms = +1 +√ +α +′ exp(A0) = +� +ΛG +mSUSY +�2 L +R +√gs = +L4 +mSUSY R√gs +exp (−cSinst) +� +�� +� +Instantonic decay +(3.29) +The representative action along with the effective potential is a very important input for +the rest of the computation performed in this paper. +Here the following two physical approximation can be used to simplify the problem in +a very simpler language: +26 + +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(a) For b < 0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) For b > 0 +Figure 3.6: Schematic behaviour of the truncated axion effective potential obtained from +Case B with stringy parameter b < 0 and b > 0. +27 + +1. Case A: +This is the specific situation where we only consider the part of the effective potential +which breaks the shift symmetry φ → φ + 2πfa, which is given by: +Shift symmetry breaking effective potential : V (φ) ≈ µ3fa +� φ +fa +� +. (3.30) +In the present computational purpose, particularly in the field equation the above +mentioned potential contributes as a source term in terms of µ3 which basically fix +the overall energy scale in terms of the stringy model parameters. In figure (3.5), +the corresponding shift symmetry breaking part of the axion effective potential is +plotted, which shows linear behaviour. +2. Case B: +In this specific case we consider the small field limiting approximation in which the +dimensionless field variable φ +fa +≪ 1. For this reason one can approximate the shift +symmetry φ → φ + 2πfa preserving non-perturbative contribution as: +cos +� φ +fa +� +≈ 1 − 1 +2 +� φ +fa +�2 +, +(3.31) +where due to having the truncation at the quadratic order term at the end t.he +previously mentioned shift symmetry is broken. In prince one can take the full non- +perturbative term, but at the level of eqn of motion and later deling with such terms +are extremely difficult in the field theory language. In the present computational +purpose the higher order terms can be neglected due to having small field limit with- +out loosing any generality, on the cost of breaking the shift symmetry. Consequently +in the present case the effective potential can be approximated as: +Truncated effective potential : +V (φ) ≈ Λ4 +G + µ3fa +� φ +fa +� +− Λ4 +G +2 +� φ +fa +�2 += µ3fa +� +b + +� φ +fa +�� +− m2 +eff +2 +� φ +fa +�2 += µ3fa +� +b + +� φ +fa +� +− b +2 +� φ +fa +�2� +, (3.32) +where we define the effective mass in terms of stringy parameters by the following +expression: +Effective mass : +m2 +eff = µ3bfa = Λ4 +G = +�mSUSY L3 +√ +α +′gs +�2 +exp (−4cSinst) +� +�� +� +Instantonic decay +. +(3.33) +28 + +In figure (3.6(a)) and figure (3.6(b)), the corresponding trucated axion effective po- +tential is plotted, which shows deviation from the linear behaviour for both the +signatures of stringy parameter b < 0 and b > 0 for small field limit where φ ≪ fa +approximation works perfectly well. +We are now interested in the field equations of axion which can be obtained by varying +the effective action stated in equation (3.18) with respect to the axion field itself, give rise +to the following expressions for the above mentioned two cases: +For Case A : +� +1 +a3(t)∂t +� +a3(t)∂t +� +− +1 +H2a2(t) +ˆL2 +H3 +� +φ = µ3, +(3.34) +For Case B : +�� +1 +a3(t)∂t +� +a3(t)∂t +� +− +1 +H2a2(t) +ˆL2 +H3 +� ++ m2 +eff +� +φ = µ3 = m2 +efffa +b +, +(3.35) +where the scale factor a(t) for global de Sitter space is given by the following expression: +Scale factor +a(t) = 1 +H sinh t +where +t = +� +tR(in R), tL(in L) +� +. (3.36) +In this context we introduce a Laplacian operator ˆL2 +H3 in hyperbolic slice H3 which is +defined as: +Laplacian operator : +ˆL2 +H3 = +1 +sinh2 r +� +∂r +� +sinh2 r ∂r +� ++ +1 +sin θ∂θ (sin θ ∂θ) + +1 +sin2 θ∂2 +Φ +� +. +(3.37) +which has the following properties: +1. Laplacian operator ˆL2 +H3 satisfies the following eigenvalue equation: +Eigenvalue equation : +ˆL2 +H3Yplm(r, θ, Φ) = λpYplm(r, θ, Φ), +(3.38) +with quantum number p dependent eigenvalue: +Eigenvalue : +λp = −(1 + p2). +(3.39) +2. Eigenfunction of the Laplacian operator ˆL2 +H3 is Yplm(r, θ, Φ) which is defined as: +Eigenfunction : +Yplm(r, θ, Φ) = Γ (ip + l + 1) +Γ (ip + 1) +p +√ +sinh r +P +−(l+ 1 +2) +(ip− 1 +2) (cosh r) Ylm(θ, Φ), +(3.40) +29 + +where p, l and m are three quantum numbers associated with the above mentioned +eigenfunction. Here Ylm(θ, Φ) is the well known spherical harmonics which is de- +pendent on two quantum numbers l and m and on two angular coordinates as it +defined in S2 and last but not the least the radial solution is characterized by the +function, P +−(l+ 1 +2) +(ip− 1 +2) (cosh r), which is the well known associated Legendre polynomial +in this context. +After quantization the classical solution obtained from the field equation is promoted in +terms of the quantum operator and by following the well known canonical quantization +technique the corresponding quantum operator can be written in terms of the creation +and annihilation operators along with basis Bunch Davies mode function, which is nothing +but the classical counterpart of the solution of the field equation. The total quantum +solution for the axion field operator for both the Case A and Case B can be written in +the following compact form: +Quantum Mode function : +�φ(t, r, θ, Φ) = +� ∞ +0 +dp +� +σ=±1 +p−1 +� +l=0 ++l +� +m=−l +[aσplmUσplm(t, r, θ, Φ) ++a† +σplmU ∗ +σplm(t, r, θ, Φ) +� +∀ t = (tR, tL). +(3.41) +In this context the Bunch-Davies vacuum is defined by the following expression: +Bunch − Davies vacuum : aσplm|BD⟩ = 0 +∀σ = (+1, −1); 0 < p < ∞; +l = 0, · · · , p − 1, m = −l, · · · , +l. +(3.42) +Here Uσplm(t, r, θ, Φ) represents the classical solution of the field equation for the axion +for both the Case A and Case B which forms the complete basis. After quantization +this basis functions, which is sometimes referred as the mode functions are tagged by the +three quantum numbers, p, l and m, which are appearing as an outcome of the canonical +quantization of the modes in the present context of discussion. The solution of the mode +functions can be obtained by solving the corresponding axion field equations, which are +basically solving partial differential equations using the well known method of separation +of variables for the Case A and Case B. This finally give rise to the following expression +for both the mentioned possibilities: +Bunch − Davies Mode function : Uσplm(t, r, θ, Φ) = +1 +a(t)χp,σ(t)Yplm(r, θ, Φ) += +H +sinh tχp,σ(t)Yplm(r, θ, Φ). (3.43) +Here it is important to note that the time dependent part of the mode function χp,σ(t) +30 + +only works for the positive frequencies and hence forms a complete set in the present +theoretical set up. This part of the solution is dependent on the momentum p which is +actually the wave number and in the quantum mechanical picture it is playing the role of +a quantum number as clearly mentioned earlier. Particularly this time dependent part of +the wave function is extremely significant for the present discussion as we are interested in +the dynamical behaviour of the mode function in the R and L region of the open chart of +global de Sitter space time. This particular part will going to control the behaviour of the +quantum entanglement measure in the mentioned space time. If we can able to extract the +hidden features from the time dependent part of the field equations from the Case A and +Case B then half of the computational job is done. Now since in both the cases we are +dealing with inhomogeneous second order differential equations the total solution can be +written as the sum of complementary part (χ(c) +p,σ(t)) and particular integral part (χ(p) +p,σ(t)) +i.e. +Total time dependent solution : χp,σ(t) = +χ(C) +p,σ (t) +� �� � +Complementary part ++ +χ(P) +p,σ (t) +� �� � +Particular integral part +. +(3.44) +Here the complementary part (χ(c) +p,σ(t)) of the time dependent solution of the mode function +satisfy the homogeneous part of the field equation for the Case A and Case B can be +written as: +Complementary part of the field equation : +0 = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +∂2 +t + 3 coth t ∂t + (1 + p2) +sinh2 t +� +χ(C) +p,σ (t) +for Case A +� +∂2 +t + 3 coth t ∂t + (1 + p2) +sinh2 t + m2 +eff +H2 +� +χ(C) +p,σ (t) +for Case B. +(3.45) +The solution of the above equations for the Case A and Case B combiningly can be +written as: +Complementary solution : +31 + +χ(c) +p,σ(t) = +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +� +1 +2 sinh πp +� +(eπp − iσ e−iπν) +Γ +� +ν + 1 +2 + ip +� Pip +(ν− 1 +2)(cosh tR) +−(e−πp − iσ e−iπν) +Γ +� +ν + 1 +2 − ip +� P−ip +(ν− 1 +2)(cosh tR) +�� +σ=±1 +for R +� +σ +2 sinh πp +� +(eπp − iσ e−iπν) +Γ +� +ν + 1 +2 + ip +� Pip +(ν− 1 +2)(cosh tL) +−(e−πp − iσ e−iπν) +Γ +� +ν + 1 +2 − ip +� P−ip +(ν− 1 +2)(cosh tL) +�� +σ=±1 +for L, +(3.46) +where we introduce a new parameter ν, which is known as mass parameter are defined for +Case A and Case B as: +Mass parameter : ν = +� +� +� +� +� +� +� +� +� +� +� +3 +2 +for Case A +� +9 +4 − m2 +eff +H2 += +� +9 +4 − µ3b +faH2 = +� +9 +4 − +Λ4 +G +f 2 +aH2 +for Case B. +(3.47) +Here the solution has following properties: +1. Here σ = ±1 for R and L regions. +2. In the Case B if we consider meff ≪ H, the mass parameter is approximated as +ν = 3/2 which is exactly the Case A. +3. In the Case B if we consider meff = +√ +2H, then the mass parameter is given by +ν = 1/2 and this is the case of conformally coupling. +4. In the Case B if we consider meff < +√ +2H, the mass parameter is lying within the +range, 1/2 < ν < 3/2, which is the low mass region. +5. In the Case B if we consider meff = H, then the mass parameter is given by, +ν = 5/2, which is the intermediate mass region. +6. In the Case B if we consider meff ≫ H, then the mass parameter is given by, +ν = i +� +m2 +eff +H2 − 9 +4 ≈ imeff +H , which is the high mass region. +7. In the Case B if we consider +√ +2H < meff < 3H/2, then the mass parameter ν is +lying within the range, 0 < ν < 1/2. +8. Complementary part of the solution satisfy, χ(C) +p,σ (t) = χ(C) +−p,σ(t). +32 + +9. Here one can define the following Klien-Gordon inner product in terms of the com- +plementary part of the time dependent fields equation: +Klien − Gordon product : +�� +χ(C) +p,σ (t), χ(C) +p,σ′(t) +�� +KG = Npσδσσ′, +(3.48) +where Npσ is the normalization constant, which is given by: +Normalization : +Npσ = 4 +π +� +cosh πp − σ cos +� +ν − 1 +2 +�� +|Γ +� +ν + 1 +2 + ip +� +|2 +∀σ = ±1. +(3.49) +This will going to be extremely useful to further fix the overall normalization factor +of the complementary part of the time dependent contribution of the corresponding +mode function in the present context of discussion. +On the other hand the particular integral part satisfy the following time dependent part +of the field equation: +Particular part of the field equation : +� +� +� +� +� +� +� +� +� +� +∂2 +t + 3 coth t ∂t + (1 + p2) +sinh2 t +� +χ(P) +p,σ (t) = µ3 +for Case A +� +∂2 +t + 3 coth t ∂t + (1 + p2) +sinh2 t + m2 +eff +H2 +� +χ(P) +p,σ (t) = m2 +efffa +b +for Case B. +(3.50) +Here in both the cases we have inhomogeneous differential equations and the inhomoge- +neous contributions in both the cases playing the role of source terms in the present context +of discussion. Since we have chosen a specific time dependent profile of the axion decay +parameter to prepare Bell CHSH inequality violating pair for the Case B the source term +in this particular case will be time dependent in a very specific fashion. +It is a well known fact that apart from having any type of structure of the source con- +tribution one can able to solve the inhomogeneous differential equation using the Green’s +function method. This leads to the following solution of the particular integral part for +both the cases considered for our analysis in this paper: +Particular solution : χ(P) +p,σ (t) = +� +� +� +� +� +� +� +� +� +� +dt +′ Gσ(t, t +′) µ3 +for Case A +� +dt +′ Gσ(t, t +′) m2 +efffa(t +′) +b +for Case B. +(3.51) +where Gσ(t, t +′) is the Green’s function for axion field, which is given by the following +33 + +general expression: +Green′s function : Gσ(t, t +′) = sinh2 t +∞ +� +n=0 +1 +(p2 − p2 +n)χ(C) +pn,σ(t)χ(C) +pn,σ(t +′) +where +σ = ±1. +(3.52) +Further, we use the following new notations to express the total solution of the time +dependent part of the field equation: +New notation : Pq = Pip +(ν− 1 +2)(cosh tq), +Pq,n = Pipn +(ν− 1 +2)(cosh tq) +where +q = (R, L) . +(3.53) +As a result, the solution’s entire time-dependent component can be reduced to the +following: +χp,σ(t) = +� +q=R,L +� +� +� +� +� +� +� +� +� +1 +Np +� +ασ +q Pq + βσ +q Pq∗� +� +�� +� +Complementary solution ++ +∞ +� +n=0 +1 +Npn (p2 − p2 +n) +� +¯ασ +q,n ¯Pq,n + ¯βσ +q,n ¯P∗q,n� +� +�� +� +Particular solution +� +� +� +� +� +� +� +� +� +∀σ = ±1 , +(3.54) +where we use two new redefined symbol for the further simplification purpose in the present +computation: +P +q,n = sinh2 t Pq,n × +� +� +� +� +� +� +� +� +� +� +dt +′ χ(C) +pn,σ,q(t +′) µ3 +for Case A +� +dt +′ χ(C) +pn,σ,q(t +′) m2 +efffa(t +′) +b +for Case B. +(3.55) +Np = 2 sinh πp +� +Npσ = 4 sinh πp +�� +cosh πp − σ cos +� +ν − 1 +2 +�� +π|Γ +� +ν + 1 +2 + ip +� +|2 +∀σ = ±1, q = (R, L) +(3.56) +Npn = 2 sinh πpn +� +Npnσ = 4 sinh πpn +�� +cosh πpn − σ cos +� +ν − 1 +2 +�� +π|Γ +� +ν + 1 +2 + ipn +� +|2 +∀σ = ±1, q = (R, L) +(3.57) +In the above mentioned solution as mentioned in equation (3.54), we define few expansion +coefficients, which are given by: +Expansion coefficients : +ασ +R = 1 +σασ +L = (eπp − iσe−iπν) +Γ +� +ν + 1 +2 + ip +� , ασ +R,n = 1 +σασ +L,n = (eπpn − iσe−iπν) +Γ +� +ν + 1 +2 + ipn +� +(3.58) +34 + +βσ +R = 1 +σβσ +L = −(e−πp − iσe−iπν) +Γ +� +ν + 1 +2 − ip +� ., βσ +R,n = 1 +σβσ +L,n = −(e−πpn − iσe−iπν) +Γ +� +ν + 1 +2 − ipn +� . +(3.59) +Further the solution written in equation (3.54) can be written in matrix notation for +the further simplification: +Matrix solution : +χI = +1 +Np +MI +JPJ +� +�� +� +Complementary solution ++ +∞ +� +n=0 +1 +Np,(n) +� +M(n) +�I +J PJ +(n) +� +�� +� +Particular solution +(3.60) +where we define two square matrices for the complementary and particular part as: +MI +J = +� +� +� +� +� +ασ +q +βσ +q +βσ∗ +q +ασ∗ +q +� +� +� +� +� , +� +M(n) +�I +J = +� +� +� +� +� +¯ασ +q,n +¯βσ +q,n +¯βσ∗ +q,n +¯ασ∗ +q,n +� +� +� +� +� , +σ = ±1, q = (R, L), (I, J) = 1, 2, 3, 4. (3.61) +and similarly the useful column matrices can be expressed as: +PJ +(n) = +� +� +� +� +� +Pq,n +Pq∗,n +� +� +� +� +� , +χI = +� +� +� +� +� +χσ(t) +χ∗ +σ(t), +� +� +� +� +� , +PJ = +� +� +� +� +� +Pq +Pq∗, +� +� +� +� +� +σ = ±1, q = (R, L), (I, J) = 1, 2, 3, 4. +(3.62) +Also we introduce another new symbol for the normalization factor Np,(n) as obtained for +the particular part of the solution, which is given by: +Np,(n) = 2 sinh πpn +� +Npnσ +� +p2 − p2 +n +� += 4 sinh πpn +� +p2 − p2 +n +� +�� +cosh πpn − σ cos +� +ν − 1 +2 +�� +π|Γ +� +ν + 1 +2 + ipn +� +|2 +∀σ = ±1, q = (R, L). (3.63) +Hence the Bunch-Davies mode function can be rewritten as: +H +sinh taIχI = +H +sinh taI +� +1 +Np +MI +JPJ + +∞ +� +n=0 +1 +Np,(n) +� +M(n) +�I +J PJ +(n) +� +, +where aI = (aσ, a† +σ). (3.64) +Further we define: +bJ = a(C) +I +MI +J, +bJ(n) = a(P) +I(n) +� +M(n) +�I +J , +where a(C) +I += (a(C) +σ , a(C)† +σ +), a(P) +I(n) = (a(P) +σ,n, a(P)† +σ,n ). +(3.65) +35 + +This implies that the operator ansatz is following: +aI = +� +a(c) +I ++ +∞ +� +n=0 +a(p) +I(n) +� +, a(c) +I += bJ +� +M−1�I +J , +a(p) +I(n) = bJ(n) +� +M−1 +(n) +�I +J , +(3.66) +where inverse square matrices are defined as: +� +M−1�I +J = +� +� +� +� +� +γσq +δσq +δ∗ +σq +γ∗ +σq +� +� +� +� +� , +� +M−1 +(n) +�I +J = +� +� +� +� +� +γσq,n +δσq,n +δ +∗ +σq,n +γ∗ +σq,n +� +� +� +� +� , +(3.67) +The individual components of these matrices are given by: +γjσ = Γ +� +ν + 1 +2 + ip +� +eπp+iπ(ν+ 1 +2) +4 sinh πp +� +� +� +� +� +1 +eπp+iπ(ν+ 1 +2) + 1 +1 +eπp+iπ(ν+ 1 +2) − 1 +1 +eπp+iπ(ν+ 1 +2) + 1 +− +1 +eπp+iπ(ν+ 1 +2) − 1 +� +� +� +� +� +(3.68) +δ∗ +jσ = Γ +� +ν + 1 +2 − ip +� +eiπ(ν+ 1 +2) +4 sinh πp +� +� +� +� +� +1 +eπp + eiπ(ν+ 1 +2) +− +1 +eπp − eiπ(ν+ 1 +2) +1 +eπp + eiπ(ν+ 1 +2) +1 +eπp − eπp+iπ(ν+ 1 +2) +� +� +� +� +� +(3.69) +γjσ,n = Γ +� +ν + 1 +2 + ipn +� +eπpn+iπ(ν+ 1 +2) +4 sinh πpn +� +� +� +� +� +1 +eπpn+iπ(ν+ 1 +2) + 1 +1 +eπpn+iπ(ν+ 1 +2) − 1 +1 +eπpn+iπ(ν+ 1 +2) + 1 +− +1 +eπpn+iπ(ν+ 1 +2) − 1 +� +� +� +� +�(3.70) +δ +∗ +jσ,n = Γ +� +ν + 1 +2 − ipn +� +eiπ(ν+ 1 +2) +4 sinh πpn +� +� +� +� +� +1 +eπpn + eiπ(ν+ 1 +2) +− +1 +eπpn − eiπ(ν+ 1 +2) +1 +eπpn + eiπ(ν+ 1 +2) +1 +eπpn − eπpn+iπ(ν+ 1 +2) +� +� +� +� +� +(3.71) +Additionally, we have following two constraints: +a(C) +I +� ∞ +� +n=0 +1 +Np,(n) +� +M(n) +�I +J PJ +(n) +� +� +�� +� +Particular solution += 0, +a(P) +I(n) +� 1 +Np +MI +JPJ +� +� +�� +� +Complementary solution += 0. +(3.72) +Then in terms of the previously mentioned matrix elements the annihilation and creation +36 + +operators are explicitly defined as: +Annihilation operator : aσ = +� +q=R,L +� +� +γqσbq + δ∗ +qσb† +q +� ++ +∞ +� +n=0 +� +γqσ,nbq,n + δ +∗ +qσ,nb +† +q,n +�� +∀σ = ±1, (3.73) +Creation operator : a† +σ = +� +q=R,L +� +� +γ∗ +qσb† +q + δqσbq +� ++ +∞ +� +n=0 +� +γ∗ +qσ,nb +† +q,n + δqσ,nbq,n +�� +∀σ = ±1. (3.74) +Now the Bunch-Davies quantum vacuum state can be written in terms of the tensor product +of R and L vacua by making use of the following Bogoliubov transformation: +|BD⟩ = exp +� +�K +� � +|R⟩ ⊗ |L⟩ +� +for +HBD := HR ⊗ HL, +(3.75) +where the new quantum Bogoliubov operator ˆK can be expressed as: +Bogoliubov operator I : +�K = +� +1 +2 +� +i,j=R,L +mij b† +i b† +j +� +�� +� +Complementary part ++ 1 +2 +� +i,j=R,L +∞ +� +n=0 +mij,n b +† +i,n b +† +j,n +� +�� +� +Particular integral part +� +,(3.76) +where our objective is to determine the coefficients mij and ¯mij,n in this work. Also the R +and L vacuum states are defined as: +Factorization of states : |R⟩ = +� +|R⟩(C) + +∞ +� +n=0 +|R⟩(P),n +� +, +|L⟩ = +� +|L⟩(C) + +∞ +� +n=0 +|L⟩(P),n +� +, +(3.77) +which satisfy the following constraints: +bL|L⟩(C) = 0, bR|R⟩(C) = 0, +(3.78) +bL,n|L⟩(P) = 0, bR,n|R⟩(P) = 0. +(3.79) +which satisfy the following commutation algebra: +� +bi, b† +j +� += δij, +[bi, bj] = 0 = +� +b† +i, b† +j +� +. +(3.80) +� +bi,n, b +† +j,m +� += δijδnm, +� +bi,n, bj,m +� += 0 = +� +b +† +i,m, b +† +j,m +� +. +(3.81) +37 + +This implies the following relation: +� +(mijγjσ + δ∗ +iσ) b† +i +� +�� +� +Complementary part ++ +∞ +� +n=0 +� +mij,nγjσ,n + δ +∗ +iσ,n +� +b +† +i,n +� +�� +� +Particular integral part +� +(|R⟩ ⊗ |L⟩) = 0. +(3.82) +which further correspond to the following conditions: +(mijγjσ + δ∗ +iσ) = 0, +(3.83) +� +¯mij,n¯γjσ,n + ¯δ∗ +iσ,n +� += 0 +∀n, +(3.84) +using which we define the following mass matrices, which are given by: +mij = −δ∗ +iσ +� +γ−1� +σj ≡ +� +� +� +� +� +mRR +mRL +mLR +mLL +� +� +� +� +� ≈ +eiθ√ +2 e−pπ +√cosh 2πp + cosh 2πν +� +� +� +� +� +cos πν +i sinh pπ +i sinh pπ +cos πν +� +� +� +� +� . +(3.85) +¯mij,n = −δ +∗ +iσ,n +� +γ−1� +σj,n ≡ +� +� +� +� +� +mRR,n +mRL,n +mLR,n +mLL,n +� +� +� +� +� ≈ +eiθ√ +2 e−pnπ +√cosh 2πpn + cosh 2πν +� +� +� +� +� +cos πν +i sinh pnπ +i sinh pnπ +cos πν +� +� +� +� +� . +(3.86) +The eigenvalues of the solutions are given by: +λ± = +� +mRR ± mRL +� += eiθ +√ +2 e−pπ (cos πν ± i sinh pπ) +√cosh 2πp + cos 2πν +, +(3.87) +λ±,n = +� +¯mRR,n ± ¯mRL,n +� += eiθ +√ +2 e−pnπ (cos πν ± i sinh pnπ) +√cosh 2πpn + cos 2πν +. +(3.88) +But this extremely complicated to take the partial trace operation from the contributions +obtained from R and L. For this reason it is unsuitable basis for our calculation. +To perform the above mentioned operation we need to perform another Bogoliubov +transformation in terms of the following suitable basis, where the new quantum operators +are defined as: +cR = +� +u bR + v b† +R +� +, +CR,n = +� +Un bR,n + Vn b† +R,n +� +. +(3.89) +cL = +� +u bL + v b† +L +� +, +CL,n = +� +U n bL,n + V n b† +L,n +� +, +(3.90) +38 + +which satisfy the following normalization constraints on the mode function: +� +|u|2 − |v|2 +� += 1, +� +|Un|2 − |Vn|2 +� += 1. +(3.91) +� +|¯u|2 − |¯v|2 +� += 1, +� +| ¯Un|2 − | ¯Vn|2 +� += 1. +(3.92) +Using this information Bunch-Davies vacuum state can be expressed in the newly Bogoli- +ubov transformed basis as: +|BD⟩ = 1 +Np +exp +� +� +W +� � +|R +′⟩ ⊗ |L +′⟩ +� +where +Np ≈ +1 +� +� +� +� +� +1 − +� +|γp|2 + +∞ +� +n=0 +|Γp,n|2 +��, (3.93) +where |R +′⟩ and |L +′⟩ are new basis operators in HBD := HR ⊗ HL. +Additionally, we +introduce a new quantum operator � +W, which is defined as: +Bogoliubov operator II : � +W = +� +γp c† +R c† +L +� +�� +� +Complementary part ++ +∞ +� +n=0 +Γp,n C† +R,n C† +L,n +� +�� +� +Particular integral part +� +,(3.94) +where our prime objective is to determine γp and Γp,n from the present calculation. +The new oscillator algebra is given by: +� +ci, c† +j +� += δij, +[ci, cj] = 0 = +� +c† +i, c† +j +� +. +(3.95) +� +Ci,n, C† +j,m +� += δijδnm, +[Ci,n, Cj,m] = 0 = +� +C† +i,m, C† +j,m +� +. +(3.96) +Here the new operators are defined as: +cR|BD⟩ = γp c† +L|BD⟩, +cL|BD⟩ = γp c† +R|BD⟩, +(3.97) +CR,n|BD⟩ = Γp,n C† +L,n|BD⟩, +CL,n|BD⟩ = Γp,n C† +R,n|BD⟩. +(3.98) +In the new basis we have the following expressions: +cJ = bIGI +J, CJ(n) = ¯bJ(n) +� +G(n) +�I +J where GI +J = +� +� +� +� +� +Uq +V ∗ +q +Vq +U ∗ +q +� +� +� +� +� , +� +G(n) +�I +J = +� +� +� +� +� +U q,n +V +∗ +σq,n +V q,n +U +∗ +q,n +� +� +� +� +� , +(3.99) +39 + +where the components of the new matrices are given by: +Uq ≡ diag (u, u) , +Vq ≡ diag (v, v) , +U q,n ≡ diag +� +Un, U n +� +, +V q,n ≡ diag +� +Vn, V n +� +. (3.100) +Finally we derive the following sets of homogeneous algebraic equations: +mRRu + v − γpmRLv∗ = 0, +(3.101) +mRRu + v − γpmRLv∗ = 0, +(3.102) +mRLu − γpu∗ − γpmRRv∗ = 0, +(3.103) +mRLu − γpu∗ − γpmRRv∗ = 0, +(3.104) +¯mRR,nUn + Vn − Γp,nmRL,nV +∗ +n = 0, +(3.105) +mRR,nU n + V n − Γp,nmRL,nV ∗ +n = 0, +(3.106) +mRL,nUn − Γp,nU +∗ +n − Γp,nmRR,nV +∗ +n = 0, +(3.107) +mRL,nU n − Γp,nU ∗ +n − Γp,nmRR,nV ∗ +n = 0, +(3.108) +Here we have the following properties of the above mentioned equations: +1. Property I: +mRR = mLL = m∗ +RR = ω = +√ +2 e−pπ cos πν +√cosh 2πp + cos 2πν , +(3.109) +mRL = mLR = −m∗ +RL = ζ = ei π +2 +√ +2 e−pπ sinh pπ +√cosh 2πp + cos 2πν , +(3.110) +2. Property II: +¯mRR,n = ¯mLL,n = ¯m∗ +RR,n = ωn = +√ +2 e−pnπ cos πν +√cosh 2πpn + cos 2πν , +(3.111) +¯mRL,n = ¯mLR,n = − ¯m∗ +RL,n = ζn = ei π +2 +√ +2 e−pnπ sinh pnπ +√cosh 2πpn + cos 2πν . +(3.112) +3. Property III: +If we have the following two conditions: +γ∗ +p = −γp, +Γ∗ +p,n = −Γp,n, +(3.113) +then we can fix the following constraint: +v∗ = v, +u∗ = u, +V ∗ +n = V n, +U ∗ +n = U n. +(3.114) +As a result, we have two normalization conditions instead of two, which are given +40 + +by: +� +|u|2 − |v|2 +� += 1, +� +|Un|2 − |Vn|2 +� += 1. +(3.115) +Finally, we have the following expressions: +γp = +1 +2mRL +�� +1 + m2 +RL − m2 +RR +� +− +� +(1 + m2 +RL − m2 +RR)2 − 4m2 +RL +� += i +√ +2 +√cosh 2πp + cos 2πν + √cosh 2πp + cos 2πν + 2. +(3.116) +Γp,n = +1 +2 ¯mRL,n +�� +1 + ¯m2 +RL,n − ¯m2 +RR,n +� +− +�� +1 + ¯m2 +RL,n − ¯m2 +RR,n +�2 − 4 ¯m2 +RL,n +� += i +√ +2 +√cosh 2πpn + cos 2πν + √cosh 2πpn + cos 2πν + 2. +(3.117) +Here it is important to note that we have taken the negative signature in front of the square +root contribution to strictly satisfy the constraint, |γp| < 1 and |Γp,n| < 1. For the other +signature, which is appearing from other branch of solution these mentioned constraints +are not satisfied at all and for this reason the other solutions are not physical in the +present context of discussion. Also the above equations satisfy the following normalization +conditions: +� +|u|2 − |v|2 +� += 1, +� +|U n|2 − |V n|2 +� += 1. +(3.118) +general solutions of these equations can be expressed as: +u = +1 − γpζ +� +|1 − γpζ|2 − |ω|2 = u∗ = u, +v = +ω +� +|1 − γpζ|2 − |ω|2 = v∗ = v, +(3.119) +U n = +1 − Γpnζn +� +|1 − Γpnζn|2 − |ω|2 = U ∗ +n = Un +V n = +ωn +� +|1 − Γpnζn|2 − |ω|2 = V ∗ +n = Vn.(3.120) +Here we have two additional constraints which are satisfied during this computation: +ω∗ = ω, +ζ∗ = −ζ, +γ∗ +p = −γp, +Γ∗ +p,n = −Γp,n. +(3.121) +These derived expressions will be used further to derive the rest of the results in this paper. +3.4 +Construction of reduced density matrix in open chart +In this portion our job is construct the expression for reduced density operator correspond- +ing to the previously defined Bunch Davies state. After quantization the corresponding +quantum state is characterized in terms of the three important quantum numbers p, l and +41 + +m. As we have already mentioned before that the contributions coming from region R and +region L are exactly same and symmetric in nature. Because of this reason we are going +to derive the expression for the reduced density operator in the region L by taking partial +trace over all the contributions coming from region R. This leads to the following result +for the corresponding density operator: +ρL;p,l,m = TrR +� +|BD⟩⟨BD| +� += +� +(1 − |γp|2) +(1 + fp) +∞ +� +k=0 +|γp|2k|k; p, l, m⟩L′ L′⟨k; p, l, m| ++ +f 2 +p +(1 + fp) +∞ +� +n=0 +∞ +� +r=0 +|Γp,n|2r|n, r; p, l, m⟩L′ L′⟨n, r; p, l, m| +� +,(3.122) +where γp and Γp,n we have computed in the previous subsection and the new normalization +factor f p is defined as: +fp = +1 +� ∞ +� +n=0 +1 +1 − |Γp,n|2 +�. +(3.123) +In the definition of the density matrix the quantum mechanical states |k; p, l, m⟩L′ and +|n, r; p, l, m⟩L′ can be further written in terms of creation operators in the new basis |L +′⟩ +as: +|k; p, l, m⟩L′ = +1 +√ +k! +(c† +L)k|L +′⟩, +(3.124) +|n, r; p, l, m⟩L′ = +1 +√ +r! +(C† +L,n)r|L +′⟩. +(3.125) +In the new representative basis the reduced density operator takes the following diagonal +form: +ρL = +� +� +1 − |γp|2� +diag +� +1, |γp|2, |γp|4, |γp|6 · · · +� +� +�� +� +Complementary part ++ f 2 +p +∞ +� +n=0 +diag +� +1, |Γp,n|2, |Γp,n|4, |Γp,n|6 · · · +� +� +�� +� +Particular integral part +� +, +(3.126) +42 + +Here it is important to note that: +1 +(1 + fp)Tr +�� +1 − |γp|2� +diag +� +1, |γp|2, |γp|4, |γp|6 · · · +�� += (1 − |γp|2) +(1 + fp) +∞ +� +k=0 +|γp|2k = +1 +(1 + fp), +(3.127) +f 2 +p +(1 + fp)Tr +� ∞ +� +n=0 +diag +� +1, |Γp,n|2, |Γp,n|4, |Γp,n|6 · · · +� +� += +f 2 +p +(1 + fp) +∞ +� +n=0 +∞ +� +r=0 +|Γp,n|2r = +fp +(1 + fp), (3.128) +where we have used two following facts by assuming γp ≪ 1 and Γp,n ≪ 1: +∞ +� +k=0 +|γp|2k = +1 +(1 − |γp|2), +(3.129) +∞ +� +n=0 +∞ +� +r=0 +|Γp,n|2r = +� ∞ +� +n=0 +1 +1 − |Γp,n|2 +� += 1 +fp +. +(3.130) +As a result finally we have: +Tr (ρL) = +� +1 +(1 + fp) + +fp +(1 + fp) +� += (1 + fp) +(1 + fp) = 1, +(3.131) +It suggests that the reduced density operator utilised in this paper has the proper normal- +isation in its structure. The remainder of the computation carried out in this study will +benefit greatly from the current derived structure of the reduced density operator for the +region L. +4 +Entanglement negativity and logarithmic negativity in open +chart +Our main goal in this part is to formulate equations for the entanglement negativity and +the logarithmic negativity between the region of R and L in the open chart of global de +Sitter space time. Both of the aforementioned sections will be treated as causally unrelated +during this computation. +The Bunch Davies quantum vacuum state can be factored according to the contributions +made by the complementary and particular integral parts of the solution in terms of the +quantum numbers p, l, and m as follows: +|BD⟩ = +�� +(1 − |γp|2) +(1 + fp) +∞ +� +k=0 +|γp|k +� +|k; p, l, m⟩R′ ⊗ |k; p, l, m⟩L′ +� ++ +fp +� +(1 + fp) +∞ +� +n=0 +∞ +� +r=0 +|Γp,n|r +� +|n, r; p, l, m⟩R′ ⊗ |n, r; p, l, m⟩L′ +�� +, +(4.1) +43 + +One can directly compute the expression for the eigenvalues, which is given by the following +expression, by further employing the fundamental physical idea of Schmidt decomposition +for a pure quantum state as mentioned in the earlier section of this paper: +� +λk = +�� +(1 − |γp|2) +(1 + fp) |γp|k + +fp +� +(1 + fp) +∞ +� +n=0 +|Γp,n|k +� +∀k = [0, ∞]. +(4.2) +Then the logarithmic negativity from the present theoretical set up can be computed as: +LN(p, ν) = 2 ln +� +∞ +� +k=0 +λk +� += 2 ln +� +∞ +� +k=0 +�� +(1 − |γp|2) +(1 + fp) |γp|k + +fp +� +(1 + fp) +∞ +� +n=0 +|Γp,n|k +�� += 2 ln +�� +(1 − |γp|2) +(1 + fp) +∞ +� +k=0 +|γp|k + +fp +� +(1 + fp) +∞ +� +n=0 +∞ +� +k=0 +|Γp,n|k +� += ln +� +1 +(1 + fp) +�� +(1 + |γp|) +(1 − |γp|) + fp +f p +�2� +. +(4.3) +where we introduce a new notation f p, which is defined as: +f p = +1 +� ∞ +� +n=0 +1 +1 − |Γp,n| +� +(4.4) +In the last step we have used the following results to compute the summations: +∞ +� +k=0 +|γp|k = +1 +(1 − |γp|), +(4.5) +∞ +� +k=0 +|Γp,n|k = +1 +(1 − |Γp,n|). +(4.6) +Hence, the entanglement negativity can be further computed in terms of the expression +derived for the logarithmic negativity as: +N(p, ν) = 1 +2 +� +exp (LN(p, ν)) − 1 +� += 1 +2 +� +1 +(1 + fp) +�� +(1 + |γp|) +(1 − |γp|) + fp +f p +�2 +− 1 +� +(4.7) +44 + +Now one would anticipate that the two causally unrelated areas, R and L, are quantum +mechanically entangled with one another under the current framework for any finite values +of p, which is basically the direct outcome of having non vanishing contribution from both +|γp| and |Γp,n|. +Now after integrating over p in presence of appropriate contribution from the density +of quantum mechanical states under consideration in open chart we finally obtain the +following expression for the logarithmic negativity in the volume of a hyperboloid: +LN(ν) = V reg +H3 +� ∞ +0 +dp D(p) LN(p, ν), +(4.8) +where the quantity D(p) represents the density of quantum states corresponding to the +radial contribution on the hyperboloid H3, which is given by: +D(p) = +1 +2π2p2. +(4.9) +Additionally, in the above mentioned expression the regularized finite part of the volume +of the hyperboloid H3 is given by: +V reg +H3 = 1 +2VS2 = 1 +2 × 4π = 2π. +(4.10) +Consequently, equation (4.8) can be further expressed in terms of the following simplified +form: +LN(ν) = 1 +π +� ∞ +0 +dp p2 LN(p, ν). +(4.11) +However, in most of the physical problem in the upper limit of the above mentioned +integration the integrand becomes divergent. For this reason, one need to introduce a +regulator Λ is the upper limit of the integration instead of strictly putting this to be infinity. +However, for the computational purpose we fix the value of Λ to be very large number. In +this specific problem this cut-off is physically treated to be the Ultra Violet (UV) cut-off. +In quantum field theoretic prescription sometimes this UV cut-off is physically interpreted +as the manifestation of lattice regulator for the type of computation we are performing +in this paper. +On the other hand, it is important to note that, in the lower limit of +integration the integrand becomes convergent in most of the interesting physical situation. +In the technical language this lower limit corresponds to the Infra Red (IR) which is safe +for the particular problem we are doing in this paper. Considering all of these facts stated +above one can further recast equation (4.11) to quantify the regularized version of the +45 + +-1 +0 +1 +2 +3 +0.70 +0.75 +0.80 +0.85 +0.90 +0.95 +1.00 +1.05 +(a) For fp = 0. +-1 +0 +1 +2 +3 +0.85 +0.90 +0.95 +1.00 +(b) For small fp ̸= 0. +Figure +4.1: +Graphical +behaviour +of +the +normalized +logarithmic +negativity +(LN(ν)/LN(ν = 1/2)) with mass parameter squared (ν2) for both fp = 0 and small +fp ̸= 0. Vertical dashed lines are drawn for ν = 1/2 (conformally coupled) and ν = 3/2 +(massless case). +In both the plots reference scale corresponds to the scale at which +LN(ν)/LN(ν = 1/2) = 1. +46 + +-4 +-3 +-2 +-1 +0 +1 +2 +3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(a) For fp = 0. +-4 +-3 +-2 +-1 +0 +1 +2 +3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) For small fp ̸= 0. +Figure +4.2: +Graphical +behaviour +of +the +normalized +entanglement +negativity +(N(ν)/N(ν = 1/2)) with mass parameter squared (ν2) for both fp = 0 and small +fp ̸= 0. Vertical dashed lines are drawn for ν = 1/2 (conformally coupled) and ν = 3/2 +(massless case). +In both the plots reference scale corresponds to the scale at which +N(ν)/N(ν = 1/2) = 1. +47 + +logarithmic negativity in the following modified format: +LN(ν) = 1 +π +� Λ +0 +dp p2 LN(p, ν) += 1 +π +� Λ +0 +dp p2 ln +� +1 +(1 + fp) +�� +(1 + |γp|) +(1 − |γp|) + fp +f p +�2� +. +(4.12) +Also, the regularized version of the entanglement negativity can be further computed using +equation (4.12) as: +N(ν) = 1 +2 +� +exp (LN(ν)) − 1 +� += 1 +2 +� +exp +� +1 +π +� Λ +0 +dp p2 ln +� +1 +(1 + fp) +�� +(1 + |γp|) +(1 − |γp|) + fp +f p +�2�� +− 1 +� +. +(4.13) +Further, if we substitute each of the components of the above equation from what we have +derived in the previous section, then one can clearly see from the complicated structure of +this equation that the final result is not analytically computable. For this specific reason +we have analysed the above expression for the fp = 0 and small fp ̸= 0 numerically in this +paper. It basically covers both Case A and Case B solutions in the present context. +In figure (4.1(a)) and figure (4.1(b)), we have explicitly depicted the behaviour of nor- +malized logarithmic negativity (LN(ν)/LN(ν = 1/2)) with mass parameter squared (ν2) +for both fp = 0 and small fp ̸= 0. Similarly, in figure (4.2(a)) and figure (4.2(b)), we have +explicitly plotted the behaviour of normalized entanglement negativity (N(ν)/N(ν = 1/2)) +with mass parameter squared (ν2) for both fp = 0 and small fp ̸= 0. Vertical dashed lines +are drawn for ν = 1/2 and ν = 3/2 (Case A) cases. The outcomes and the physical +interpretation of these plots are very interesting which we are writing point-wise in the +following: +1. From both of these plots is is clearly observed that the normalized logarithmic nega- +tivity obtained from the small fp ̸= 0 is larger than the fp = 0. The normalization has +been been done with respect to value of the logarithmic negativity at ν = 1/2, which +is the conformal coupled result for the theory under consideration for the present +computational framework. +2. From both of these plots we found that when the mass parameter squared ν2 < 0 +i.e. ν = −i|ν| then we are dealing with heavy effective mass where we get falling +behaviour of normalized logarithmic negativity LN(ν)/LN(ν = 1/2) and normalized +entanglement negativity N(ν)/N(ν = 1/2). This implies that in the heavy region we +have less quantum correlation and the effect of quantum mechanical entanglement +is falling as as increase the mass in the definition of mass parameter. Here it is +48 + +important to note that we have not considered the possibility from ν = i|ν|, because +this will give rise to exponentially divergent contribution as a Boltzmann contribution +which cannot be possible to handle in the present computation. So to get a correct +measure of quantum mechanical entanglement here in this analysis we have restricted +our discussion to ν = −i|ν| branch of solutions only. This possibility corresponds to +Case B in the present context. +3. We also found that at the conformal coupling limit ν = 1/2 the normalized log- +arithmic negativity LN(ν)/LN(ν = 1/2) = 1 and normalized entanglement neg- +ativity N(ν)/N(ν = 1/2) = 1. +The same thing happened in the massless limit +ν = 3/2 where from the numerical plots we found that the normalized logarith- +mic negativity and entanglement negativity again LN(ν)/LN(ν = 1/2) = 1 and +N(ν)/N(ν = 1/2) = 1. +At both this points, ν = 1/2 and ν = 3/2 (Case A) +we get the maximum effect from quantum mechanical entanglement in the present +computation. This further physically implies that at these points we get the max- +imum contribution from quantum correlations. In both the plots we have drawn a +horizon line at LN(ν)/LN(ν = 1/2) = 1 and N(ν)/N(ν = 1/2) = 1 to indicate the +reference level of our computation. +4. Additionally we have found that in the small mass region ν2 > 0 we get oscillatory +type of feature where the period of oscillation is increasing as we increase the value +of ν2 in the positive axis. This possibility correspond to Case A and Case B in the +present context. +5. Also from both of these plots we infer that effect of quantum mechanical entangle- +ment is larger in the small mass region ν2 > 0 compared to the contribution from +heavy mass region, where we have ν2 < 0. So small mass or the massless cases are +more favourable than the heavy mass profile if we want to achieve more quantum +mechanical effects from the entanglement measure. +6. Last but not the least, we now comment on the comparison among the outcomes +obtained from the plots for normalized logarithmic negativity LN(ν)/LN(ν = 1/2) +and normalized entanglement negativity N(ν)/N(ν = 1/2) for both fp = 0 and +small fp ̸= 0. Normalized entanglement negativity gives better understanding re- +garding the information content compared to the normalized logarithmic negativity +for both the cases fp = 0 and small fp ̸= 0. It seems like overall features obtained +from normalized logarithmic negativity and normalized entanglement negativity are +almost same. But due to having differences in the definitions of the corresponding +quantum information theoretic measures the outcomes are more prominent in nor- +malized entanglement negativity. Though it is important to note that both of them +are connected to each other mathematically. The effect of heavy mass for ν2 < 0 is +49 + +more prominently observed in the case of normalized entanglement negativity com- +pared to the normalized logarithmic negativity. In the asymptotic limit of ν2 < 0 we +see that for normalized entanglement negativity the quantum entanglement and the +corresponding correlation saturates to a constant value. On the other hand, we found +that there is sharp fall in the case of normalized logarithmic negativity which further +implies there is no asymptotic value at which it becomes constant. However, apart +from having this significant difference in the large mass limit, in the small mass and +massless limiting cases we have found out exactly same behaviour from the presented +plots. +5 +Comparison with entanglement entropy in open chart +In this section we give a clear comparison between entanglement entropy and logarithmic +negativity, where we know both of them is used to describe the long range quantum +correlation and entanglement effects in the present information theory motivated quantum +field theoretic picture. +Let us start our discussion with the entanglement entropy which is described by Von +Neumann measure by the following expression [14]: +S(p, ν) = −Tr [ρL ln ρL] += +� +− +� +1 + +fp +1 + fp +� � +ln +� +1 − |γp|2� ++ +|γp|2 +(1 − |γp|2) ln +� +|γp|2�� +− (1 − fp) ln (1 + fp) +� +, (5.1) +where we have computed each explicit parameter in the earlier section of this study, to- +gether with the explicit expression for the reduced density matrix. +Now by following the same prescription stated before the regularized part of the entan- +glement entropy can be further expressed by the following simplified expression: +S(ν) = V reg +H3 +� ∞ +0 +dp D(p) S(p, ν) = 1 +π +� ∞ +0 +dp p2 S(p, ν), +(5.2) +where D(p) and V reg +H3 are defined earlier. Further applying the same logical argument with +the UV cut off for the computational purpose we further use the following UV cut off +regulated version of Von Neumann entropy: +S(ν) = 1 +π +� Λ +0 +dp p2 S(p, ν). +(5.3) +Our next job is to numerically compute the expressions for entanglement entropy from the +50 + +-4 +-3 +-2 +-1 +0 +1 +2 +3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(a) For fp = 0. +-4 +-3 +-2 +-1 +0 +1 +2 +3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) For small fp ̸= 0. +Figure 5.1: Graphical behaviour of the normalized entanglement entropy from Von Neu- +mann measure (S(ν)/S(ν = 1/2)) with mass parameter squared (ν2) for both fp = 0 +and small fp ̸= 0. Vertical dashed lines are drawn for ν = 1/2 (conformally coupled) and +ν = 3/2 (massless case). In both the plots reference scale corresponds to the scale at which +S(ν)/S(ν = 1/2) = 1. +51 + +-4 +-3 +-2 +-1 +0 +1 +2 +3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(a) For fp = 0. +-4 +-3 +-2 +-1 +0 +1 +2 +3 +0.0 +0.2 +0.4 +0.6 +0.8 +1.0 +(b) For small fp ̸= 0. +Figure 5.2: Graphical comparison between the behaviour of the normalized entanglement +entropy from Von Neumann measure (S(ν)/S(ν = 1/2)) and normalized entanglement +negativity (N(ν)/N(ν = 1/2)) with mass parameter squared (ν2) for both fp = 0 and +small fp ̸= 0. Vertical dashed lines are drawn for ν = 1/2 (conformally coupled) and +ν = 3/2 (massless case). In both the plots reference scale corresponds to the scale at +which N(ν)/N(ν = 1/2) = 1 and S(ν)/S(ν = 1/2) = 1. +52 + +above mentioned UV regulated Von Neumann measure which will going to be extremely +useful for the further comparison purpose with the results obtained from logarithmic neg- +ativity and entanglement negativity measures. +In figure (5.1(a)) and figure (5.1(b)), we have explicitly depicted the behaviour of nor- +malized entanglement entropy from Von Neumann measure (S(ν)/S(ν = 1/2)) with mass +parameter squared (ν2) for both fp = 0 and small fp ̸= 0. Similarly, in figure (5.2(a)) +and figure (5.2(b)), we have explicitly plotted comparative behaviour between normalized +entanglement negativity (N(ν)/N(ν = 1/2)) and normalized entanglement entropy from +Von Neumann measure (S(ν)/S(ν = 1/2)) with mass parameter squared (ν2) for both +fp = 0 and small fp ̸= 0. Vertical dashed lines are drawn for ν = 1/2 and ν = 3/2 (Case +A) cases. In both the plots we have drawn a horizon line at N(ν)/N(ν = 1/2) = 1 and +S(ν)/S(ν = 1/2) = 1 to indicate the reference level of our computation. The outcomes +and the physical interpretation of these plots are very interesting which we are writing +point-wise in the following: +1. From all of these plots we found that compared entanglement entropy computed +from Von Neumann measure entanglement negativity give more information from +the same system under consideration. The reason behind this statement is both in +the heavy mass ν2 < 0 and small mass ν2 > 0 regions we get more amplitude from +the normalized version of the information content. +2. In the heavy mass ν2 < 0 region we found that in the asymptotic limit entanglement +entropy gives vanishing contribution, but in the same limit entanglement negativity +gives small but significantly non vanishing contribution. On the other hand, in the +small mass ν2 > 0 region the amplitude of fluctuation of normalized entanglement +entropy is larger amplitude than the normalized entanglement negativity. +3. Only at ν = 1/2 (conformally coupled) and ν = 3/2 (massless) we get exactly same +contribution from both N(ν)/N(ν = 1/2) and S(ν)/S(ν = 1/2), which is exactly +N(ν)/N(ν = 1/2) = S(ν)/S(ν = 1/2) = 1. +This is obviously a very interesting finding from our analysis that in case of the present +quantum field theoretic set up entanglement negativity captures better information regard- +ing quantum mechanical correlations and quantum entanglement than the Von Neumann +measure of entanglement entropy. +6 +Logarithmic negativity between two causally unrelated patches +of open chart +Our main goal in this part is to calculate and estimate the logarithmic negativity between +two patches of the open chart of the global de Sitter space that are not causally related. +53 + +We provide information on two physical observers whose role it is to explicitly determine +and estimate the quantum mechanical entanglement in the current theoretical setup in +order to serve the goal. For the purpose of computational simplification, we additionally +assume that the quantum states corresponding to these two observers constitute an initial +pure state within the multiverse and are maximally entangled to one another.Since we are +dealing with two observers, the corresponding framework in a more simpler language can be +interpreted as biverse which can be easily further generalise to a general multiverse scenario. +From the starting point of the present construction we have explicitly mentioned that the +sub regions R and L in the penrose diagram as well as in the Hilbert space construction +is taken to be completely symmetric. For this reason to extend this theoretical tool to +compute the entanglement negativity from the corresponding biverse set up we consider +that the region L is in the inside of the de Sitter bubble. One can do the similar thing by +making use of the region R as well. Another important assumption we have considered +during this computation is that there is no bubble wall exists in this framework for the +open chart of the global de Sitter space. Further in this biverse set up we place another +observer in the other open chart of the global de Sitter bubble. From the present theoretical +computation our objective is to explicitly find the role of the inside observer to detect the +quantum mechanical signatures of entanglement between two de Sitter bubbles using the +well known Bunch Davies initial quantum states for the biverse. +Now in the present +quantum field theoretic set up causality demands that the region R has to be causally +unrelated from the other region L. This is because of the fact that there is no access to the +region R during this computation. For this specific reason one needs to consider the partial +trace operation over the inaccessible region R, which further give rise to the information +loss regarding this specific region. As a consequence the corresponding quantum mechanical +state which describe the observer would be a mixed quantum state for this computation. +In this set up the quantum mechanical state associated with the other observer becomes +a pure quantum state which belongs to the other causally unrelated patch of the open +chart of the global de Sitter bubble. This implies in the present computation we need to +consider the entanglement between a mixed and pure quantum mechanical states which +belong to two causally unrelated de Sitter bubbles. If in the computation of quantum +entanglement from two subsystems only pure quantum states are involved then in such +a system Von Neumann measure is the best measure to quantify entanglement entropy. +But if any of the subsystem is described by mixed quantum state then for such a situation +Von Neumann measure is not the appropriate quantum information theoretic measure. In +that case entanglement negativity or the logarithmic negativity can give better measure of +quantum entanglement, which is somewhat clear to us from the comparison that we have +already drawn from our previous analysis. He strongly believe that this new measure will +going to explain various unexplored underlying physics involved in the present theoretical +set up. +54 + +6.1 +Computational set up and construction of maximally entangled states +Starting with the total quantum vacuum state, which is actually represented by the product +of the quantum vacuum states for each oscillator that we found computed explicitly in the +previous section of this paper, we begin to study the effects from entanglement negativity +between two causally unrelated patches of open chart. It’s crucial to remember that each +oscillator’s quantum mechanical state is identified by one of the three quantum numbers +p, l, or m for future calculation purposes. For this reason, one must take product over p +in the final expression of total quantum. The final Bunch Davies quantum vacuum state +in this configuration is stated as: +|0⟩BD = +� +p +|0p⟩BD, +(6.1) +where we define the Bunch Davies states for each mode as: +|0p⟩BD = +�� +(1 − |γp|2) +(1 + fp) +∞ +� +k=0 +|γp|k +� +|kp⟩R′ ⊗ |kp⟩L′ +� ++ +fp +� +(1 + fp) +∞ +� +n=0 +∞ +� +r=0 +|Γp,n|r +� +|n, rp⟩R′ ⊗ |n, rp⟩L′ +�� +. +(6.2) +Here it is important to note that, in the above expression for the simplification in the +writing purpose we have removed the tags of l and m on the individual direct product states +defined in the region R and L in the Bogoliubov transformed basis. This simplification +will going to further help us to deal with cumbersome expressions. +In this specific computation we are actually considering smaller version of the multiverse +where many causally unrelated patches in the open chart of the de Sitter bubbles forms +a maximally entangled state which is necessary ingredient for the present calculation. In +this calculation it is assumed that each of the causally unrelated patches corresponds to +a Bunch Davies quantum vacuum states which will finally form a maximally entangled +state out of all possible Bunch Davies states. However this is not a strict assumption +to extract the required outcome from the present computational set up. One can think +about a scenario in which some quantum vacuum states are distinct from Bunch Davies +vacuum and are entangled with one another in a more general version of the computation. +This is quite an interesting possibility in this context. Now we don’t want to complicate +our understanding at this level of computation. For this reason we are going to restrict +our analysis by considering the two Bunch Davies quantum vacuum states which belong +to two causally unrelated patches of the open chart of the de Sitter bubbles. +Now to +theoretically model the present computational set up let us first consider two momentum +modes having momenta p = pin and p = pout of the axionic field theory described by +Case A (massless) and Case B (massive) respectively. With this knowledge, it is possible +55 + +to further express the maximally entangled state in terms of the unique contributions of +Bunch Davies vacuum to the two causally independent regions of the de Sitter bubbles as +follows: +|Ψ⟩ME : = +1 +√ +2 +� +i=0,1 +� +|ipout⟩BD1 ⊗ |ipin⟩BD2 +� += +1 +√ +2 +� +|0pout⟩BD1 ⊗ |0pin⟩BD2 + |1pout⟩BD1 ⊗ |1pin⟩BD2 +� +. +(6.3) +In the above equation, |0pout⟩BD1 and |1pout⟩BD1 represent the ground and first single +HBD1 +AB/HicdVBNS8NAEN3Ur1q/oj16WSyCp5KUW +tbqR56rGBtoSlhs920SzebsLsRQoh/xYsHRbz6Q7z5b9y0FVT0wcDjvRlm5 +nkRo1JZ1odRWFvf2Nwqbpd2dvf2D8zDo1sZxgKTPg5ZKIYekoRTvqKkaG +kSAo8BgZePL3B/cESFpyG9UEpFxgKac+hQjpSXLKcORgx2Mzd1PB92rlw +7c82KVbU0Gg2YE7tp2Zq0Ws1arQXthWVZFbBCzXfnUmI4BwhRmScmRbkR +qnSCiKGclKTixJhPAcTclIU4CIsfp4vgMnmplAv1Q6OIKLtTvEykKpEwCT3 +cGSM3kby8X/JGsfKb45TyKFaE4+UiP2ZQhTBPAk6oIFixRBOEBdW3QjxDA +mGl8yrpEL4+hf+T21rVrlfPr+uVdmcVRxEcgxNwBmxwAdqgC3qgDzBIwAN4 +As/GvfFovBivy9aCsZopgx8w3j4B4J6USw= +⌦ +AB7nicdVDLSsNAF +L3xWeur6tLNYBFchSR9uiu6cVnBPqANZTKdtkMnkzAzEUr +oR7hxoYhbv8edf+OkraCiBy4czrmXe+8JYs6UdpwPa219Y +3NrO7eT393bPzgsHB23VZRIQlsk4pHsBlhRzgRtaY57ca +S4jDgtBNMrzO/c0+lYpG407OY+iEeCzZiBGsjdfqRZiFV +g0LRsS/rVa/iIcd2nJpXqmbEq5W9EnKNkqEIKzQHhf+MC +JSIUmHCvVc51Y+ymWmhFO5/l+omiMyRSPac9Qgc0SP12c +O0fnRhmiUSRNCY0W6veJFIdKzcLAdIZYT9RvLxP/8nqJHt +X9lIk40VSQ5aJRwpGOUPY7GjJieYzQzCRzNyKyARLTLR +JKG9C+PoU/U/anu2W7cptudi4WsWRg1M4gwtwoQYNuIEmt +IDAFB7gCZ6t2Hq0XqzXZeuatZo5gR+w3j4B62yP+Q= +1st de Sitter universe +2nd de Sitter universe +Axionic Biverse +HBD2 = +✓ +HL ⌦ HR +◆ +ACMXicdVBNSwJBGJ6 +1L7Mvq2OXIQnsIrtipodArIOHDhaZgrs+OsDs5+MDMbyLJ/qUv/JLp4KJrf6JZNcioBwYenud5ed9nJBRIXV9qmVWVtfWN7Kbua3tnd29/P7BvQgijkHByzgPQcJwqhPOpJKRnoh +J8hzGOk648vU7z4QLmjg38lJSCwPDX3qUoykux8KzYxYrCV2LHpuLB5ZeTC9Ohw2Fx2blOzEBSjwi4rN8ms/SpnS/oJV2hWoUpMWq6oUi9XiuX69CYWbpeAu07fyzOQhw5BFfYoaE6 +Bt6K0YcUkxI0nOjAQJER6jIekr6iO124pnFyfwRCkD6AZcPV/CmfpzIkaeEBPUkPyZH47aXiX14/km7NiqkfRpL4eL7IjRiUAUzrgwPKCZsogjCnKq/QjxCHGpSs6pEr4vhf+T+3 +LJqJTObiqFRnNRxYcgWNQBAY4Bw3QAm3QARg8ghfwCt60J2qvWsf82hGW8wcgiVon18IXamP +H := +✓ +HBD1 ⌦ HBD2 +◆ += +✓ +HBD1 ⌦ +✓ +HL ⌦ HR +◆◆ +ACh3ichVFbS8MwFE7r1DlvUx9CQ5h+jDbOmcnCDp98MEHFTeFdZQ0S7uw9EKSCqP0r/ijfP +PfmN3AKx5I+PJ950tOzvESRoU0jHdNXygsLi0XV0qra+sbm+Wt7Y6IU45JG8cs5s8eEoTRiLQlYw8J5yg0G +PkyRtejfWnF8IFjaNHOUpIL0RBRH2KkVSUW37NHIwYvMnPzh2PBkF1fnYzx/Nh69o1cyeWNCTih2LlE8vBv +87f9Nu5CL/yD7NLp7tbrhg1Q0WjAcfAtA1TgWbTtqwmNCeSYVTALO7c8pvTj3EakhihoTomkYiexnikmJG8 +pKTCpIgPEQB6SoYIVBL5v0MYf7iulDP+ZqRJO2M+ODIVCjEJPZYZIDsR3bUz+pnVT6du9jEZJKkmEpw/5K +YMyhuOhwD7lBEs2UgBhTlWtEA8QR1iq0ZVUE+Y/hX+DjlUz67WT+3rlojVrRxHsgj1QBSY4BRfgBtyBNsBa +QTvUjrW6vqIf6Q3dnqbq2syzA76EfvkBJtPC4Q= +Hilbert space factorization +CY3 +AB+XicbZDLSsNAFIZP6q3W9Slm8EiuCqJVnRZ7MZlBXuRtpbJdNIOnUzCzKRQt7EjQtF3Pom7nw +bp2kW2vrDwMd/zuGc+b2IM6Ud59sqrK1vbG4Vt0s7u3v7B/bhUuFsS0SUIeyo6HFeVM0KZmtNOJCkOPE7b3qQ ++r7enVCoWigc9i2g/wCPBfEawNtbAtpOe56P6Y/qUwWU6sMtOxcmEVsHNoQy5GgP7qzcMSRxQoQnHSnVdJ9L9BE +vNCKdpqRcrGmEywSPaNShwQFU/yS5P0ZlxhsgPpXlCo8z9PZHgQKlZ4JnOAOuxWq7Nzf9q3Vj7N/2EiSjWVJDFI +j/mSIdoHgMaMkmJ5jMDmEhmbkVkjCUm2oRVMiG4y19ehdZFxa1Wru6r5dptHkcRTuAUzsGFa6jBHTSgCQSm8Ayv +8GYl1ov1bn0sWgtWPnMf2R9/gCPH5L4 +Axion +CY3 +AB+XicbZDLSsNAFIZP6q3W9Slm8EiuCqJ +VnRZ7MZlBXuRtpbJdNIOnUzCzKRQt7EjQtF3Pom7nwbp2kW2vrDwMd/zuGc+b2IM6Ud59sqrK1vbG4Vt0s7u3v7B/bhUuFsS0SUIeyo6HFeVM0KZmtNOJCkOPE7b3qQ+r7enVCoWigc9i2g/wCPBfEawNtbAtpOe56P6Y/qUwW +U6sMtOxcmEVsHNoQy5GgP7qzcMSRxQoQnHSnVdJ9L9BEvNCKdpqRcrGmEywSPaNShwQFU/yS5P0ZlxhsgPpXlCo8z9PZHgQKlZ4JnOAOuxWq7Nzf9q3Vj7N/2EiSjWVJDFIj/mSIdoHgMaMkmJ5jMDmEhmbkVkjCUm2oRVMiG4y19eh +dZFxa1Wru6r5dptHkcRTuAUzsGFa6jBHTSgCQSm8Ayv8GYl1ov1bn0sWgtWPnMf2R9/gCPH5L4 +Axion +Figure 6.1: Representative diagram of the Hilbert space factorization of the axionic +biverse which is constructed out of two de Sitter vacua. Each of the vacua is spanned in +terms of the complete set of Bunch Davies states. +particle excited quantum state characterized by the momentum mode pout in the first +Bunch Davies vacuum obtained from the first patch of the open chart of global the de +Sitter space. Similarly, |0pin⟩BD2 and |1pin⟩BD2 represent the ground and first single particle +56 + +T +=0 +二excited quantum state characterized by the momentum mode pin in the second Bunch +Davies vacuum obtained from the second patch of the open chart of global the de Sitter +space. For the further computational simplification purpose we assume that outside and +inside observers are associated with two detectors having mode momenta pout and pin +respectively. +It is also crucial to note that, given the current quantum field theoretic +framework, the complete Hilbert space can be factored as follows: +H := +� +HBD1 ⊗ HBD2 +� += +� � +HL ⊗ HR +� +BD1 +� +�� +� +For first subspace +⊗ +� +HL ⊗ HR +� +BD2 +� +�� +� +For second subspace +� +, +(6.4) +where the first and the second Bunch Davies vacuum states are associated with two sub- +space, whose corresponding product Hilbert space can be further factorized in terms Hilbert +spaces associated with region R and L as: +HBD1 = +� +HL ⊗ HR +� +BD1 +, +(6.5) +HBD2 = +� +HL ⊗ HR +� +BD2 +. +(6.6) +In the multiverse picture one can further generalize this result y considering many Bunch +Davies states and different quantum vacua than Bunch Davies, such as α vacua, Motta +Allen vacua etc. In figure (6.1) representative diagram of the Hilbert space factorization +of the axionic biverse which is constructed out of two de Sitter vacua. Each of the vacua is +spanned in terms of the complete set of Bunch Davies states which we have pointed clearly +in this diagram. +6.2 +Construction of excited quantum state for single oscillator +Our main task in this paragraph is to create the excited quantum state for a single oscil- +lator, which will also be helpful to create the overall maximally entangled state required +for the specific computation. Let’s start with the characteristic matrix equation for the +oscillators in the recently announced Bogoliubov transformed basis to demonstrate this +issue in more detail: +cJ = bIGI +J, +CJ(n) = ¯bJ(n) +� +G(n) +�I +J +where +cJ = (cq, c† +q), CJ(n) = (Cq(n), C† +q(n)), +(6.7) +57 + +where we define the two square matrices as appearing in the above expressions by the +following equations: +GI +J = +� +� +� +� +� +Uq +V ∗ +q +Vq +U ∗ +q +� +� +� +� +� , +� +G(n) +�I +J = +� +� +� +� +� +U q,n +V +∗ +σq,n +V q,n +U +∗ +q,n +� +� +� +� +� , +(6.8) +where the components of the new matrices are given by: +Uq ≡ diag (u, u) , +(6.9) +Vq ≡ diag (v, v) , +(6.10) +U q,n ≡ diag +� +Un, U n +� +, +(6.11) +V q,n ≡ diag +� +Vn, V n +� +. +(6.12) +Hence one can able to find the relationship between the a-type of oscillators with c-type +of oscillators, which are given by: +a(c) +J += bJ +� +M−1�I +J = cK +� +G−1�K +I +� +M−1�I +J , +(6.13) +a(p) +J(n) = bJ(n) +� +M−1 +(n) +�I +J = CK(n) +� +G−1 +(n) +�K +I +� +M−1 +(n) +�I +J , +(6.14) +which further form the following quantity: +aI = +� +a(c) +I ++ +∞ +� +n=0 +a(p) +I(n) +� += +� +cK +� +G−1�K +I +� +M−1�I +J +� +�� +� +Complementary part ++ +∞ +� +n=0 +CK(n) +� +G−1 +(n) +�K +I +� +M−1 +(n) +�I +J +� +�� +� +Particular integral part +� +. +(6.15) +Here the the product of two inverse matrices are parametrized by another (4 × 4) square +matrix, which is given by: +� +G−1�K +I +� +M−1�I +J = +� +� +� +� +� +Qσq +R∗ +σq +Rσq +Q∗ +σq +� +� +� +� +� , +(6.16) +� +G−1 +(n) +�K +I +� +M−1 +(n) +�I +J = +� +� +� +� +� +Qσq,n +R +∗ +σq,n +Rσq,n +Q +∗ +σq,n +� +� +� +� +� , +(6.17) +58 + +where the elements of both the matrices are itself (2 × 2) matrices, which are provided by +the subsequent expressions: +Qσq = +� +� +� +� +� +�Au +− �Bu + �D∗v +− �Bu + �D∗v +�Au +� +� +� +� +� , +(6.18) +Rσq = +� +� +� +� +� +− �Av +�Bv − �D∗u +�Bv − �D∗u +− �Av +� +� +� +� +� , +(6.19) +Qσq,n = +� +� +� +� +� +�AnUn +− �BnUn + �D∗ +nVn +− �BnUn + �D∗ +nVn +�AnUn +� +� +� +� +� , +(6.20) +Rσq,n = +� +� +� +� +� +− �AnVn +�BnVn − �D∗ +nUn +�BnVn − �D∗ +nUn +− �AnVn +� +� +� +� +� . +(6.21) +Here the coefficients ( �A, �B, �D) and ( �An, �Bn, �Dn) aare provided by the subsequent expres- +sions: +�A = +√πp +��Γ +� +ν + 1 +2 + ip +��� +exp +� πp +2 +� +√cosh 2πp + cos 2πν , +(6.22) +�B = �A +cos πν +i sinh πp += +√πp +��Γ +� +ν + 1 +2 + ip +��� +exp +� πp +2 +� +√cosh 2πp + cos 2πν +cos πν +i sinh πp, +(6.23) +�D = − �A cos(ip + ν)π +i sinh πp +exp (−πp) Γ +� +ν + 1 +2 + ip +� +Γ +� +ν + 1 +2 − ip +� += − +√πp +��Γ +� +ν + 1 +2 + ip +��� +exp +� +− πp +2 +� +√cosh 2πp + cos 2πν +cos(ip + ν)π +i sinh πp +Γ +� +ν + 1 +2 + ip +� +Γ +� +ν + 1 +2 − ip +�, +(6.24) +and +�An = +√πpn +��Γ +� +ν + 1 +2 + ipn +��� +exp +� πpn +2 +� +√cosh 2πpn + cos 2πν , +(6.25) +�Bn = �An +cos πν +i sinh πpn +59 + += +√πpn +��Γ +� +ν + 1 +2 + ipn +��� +exp +� πpn +2 +� +√cosh 2πpn + cos 2πν +cos πν +i sinh πpn +, +(6.26) +�Dn = − �An +cos(ipn + ν)π +i sinh πpn +exp (−πpn) Γ +� +ν + 1 +2 + ipn +� +Γ +� +ν + 1 +2 − ipn +� += − +√πpn +��Γ +� +ν + 1 +2 + ipn +��� +exp +� +− πpn +2 +� +√cosh 2πpn + cos 2πν +cos(ipn + ν)π +i sinh πpn +Γ +� +ν + 1 +2 + ipn +� +Γ +� +ν + 1 +2 − ipn +�.(6.27) +From the structure of the above mentioned matrices we have found the following charac- +teristics: +1. First of all we have found that for each of the above mentioned matrices the diag- +onal and off-diagonal elements are same, which implies each of them are symmetric +matrices under the matrix transpose operation i.e. +Qqσ = QT +σq = Qσq, +(6.28) +Rqσ = RT +σq,n = Rσq,n, +(6.29) +Qqσ,n = QT +σq,n = Qσq,n, +(6.30) +Rqσ,n = RT +σq,n = Rσq,n. +(6.31) +2. In the above mentioned expressions we have also used the following relationships to +write down the final result in a more simplified and compact format: +�A∗ = �A, +�B∗ = − �B, +u∗ = u = u, +v∗ = v = v, +(6.32) +�A∗ +n = �An, +�B∗ +n = − �Bn, +U ∗ +n = Un = U n, +V ∗ +n = Vn = V n. +(6.33) +3. If we set B = 0 and �B = 0 along with v = 0 and Vn = 0 correspond to case of +conformal coupling (ν = 1/2) and the massless theory (ν = 3/2). +The following formulas in the region L provide the creation and annihilation operators +for oscillators of the a type, and fixing this will fix their explicit mathematical structures: +a† +L : = +� +�Auc† +L − �AvcL + +� +�Bu + �Dv +� +c† +R − +� +�Bv + �Du +� +cR +� +� +�� +� +Complementary part ++ +∞ +� +n=0 +� +�AnUnC† +L(n) − �AnVnCL(n) + +� +�BnUn + �DnVn +� +C† +R(n) − +� +�BnVn + �DnUn +� +CR(n) +� +� +�� +� +Particular integral part +, +(6.34) +60 + +aL : = +� +�AucL − �Avc† +L + +� +− �Bu + �D∗v +� +cR − +� +− �Bv + �D∗u +� +c† +R +� +� +�� +� +Complementary part ++ +∞ +� +n=0 +� +�AnUnCL(n) − �AnVnC† +L(n) + +� +− �BnUn + �D† +nVn +� +CR(n) − +� +− �BnVn + �D† +nUn +� +C† +R(n) +� +� +�� +� +Particular integral part +, +(6.35) +The following expression represents the excited quantum state for a single oscillator and +corresponds to the quantum state of the inside observer: +|1pin⟩BD2 = a† +L|0pin⟩BD2 += +� � +�Auc† +L − �AvcL + +� +�Bu + �Dv +� +c† +R − +� +�Bv + �Du +� +cR +� +� +�� +� +Complementary part ++ +∞ +� +n=0 +� +�AnUnC† +L(n) − �AnVnCL(n) + +� +�BnUn + �DnVn +� +C† +R(n) − +� +�BnVn + �DnUn +� +CR(n) +� +� +�� +� +Particular integral part +��� +(1 − |γpin|2) +(1 + fpin) +∞ +� +k=0 +|γpin|k +� +|kpin⟩R′ ⊗ |kpin⟩L′ +� ++ +fpin +� +(1 + fpin) +∞ +� +n=0 +∞ +� +r=0 +|Γpin,n|r +� +|n, rpin⟩R′ ⊗ |n, rpin⟩L′ +�� += +�� +(1 − |γpin|2) +(1 + fpin) +� +∆1 +∞ +� +k=0 +|γpin|k√ +k + 1 +� +|kpin⟩R′ ⊗ |(k + 1)pin⟩L′ +� ++∆2 +∞ +� +k=0 +|γpin|k√ +k + 1 +� +|(k + 1)pin⟩R′ ⊗ |kpin⟩L′ +�� ++ +fpin +� +(1 + fpin) +� +∞ +� +n=0 +∆3,n +∞ +� +r=0 +|Γpin,n|r√ +r + 1 +� +|n, rpin⟩R′ ⊗ |n, (r + 1)pin⟩L′ +� ++ +∞ +� +n=0 +∆4,n +∞ +� +r=0 +|Γpin,n|r√ +r + 1 +� +|n, (r + 1)pin⟩R′ ⊗ |n, rpin⟩L′ +��� +. +(6.36) +In the above expression we introduce four symbols ∆1, ∆2, ∆3,n and ∆4,n which are defined +61 + +by the following expressions: +∆1 = +� +�Au − ( �Bv + �Du)γpin +� +, +(6.37) +∆2 = +� +− �Avγpin + ( �Bu + �Dv) +� +, +(6.38) +∆3,n = +� +�AnUn − +� +�BnVn + �DnUn +� � +, +(6.39) +∆4,n = +� +− �AnVnΓpin,n + +� +�BnUn + �DnVn +� � +. +(6.40) +To compute the expression for the excited quantum state for the single oscillator we have +used the usual harmonic oscillator algebra in terms of the quantum states. +We must perform a partial trace operation over all of the degrees of freedom in the +region R since the building of the current theoretical framework requires that the inside +observer be located at the region L of one of the open charts of the global de Sitter space. +This will ultimately result in a density matrix, which we have precisely computed in the +following subsection. Due to this, we suggest the following ansatz for factorising the entire +Hilbert space used in the computation here: +H := +� +HBD1 ⊗ HBD2 +� += +� +HBD1 ⊗ +� +HL ⊗ HR +� +BD2 +� +. +(6.41) +Though the similar factorization exists in the first Bunch Davies quantum vacuum state, +for the time being to perform the present computation we don’t need the factorization +details of this subspace. +The prime for this particular choice is because to construct +the density matrix the details of the subspace which belongs to the first Bunch Davies +quantum vacuum state is not explicitly required. This will help us to compute the rest of +the computations of this paper in a very simplified fashion. In the next subsection we will +discuss the technical details of this construction to formulate the maximal entangled state +and hence the reduced density matrix. +6.3 +Construction of reduced density matrix at the inside observer +Our primary objective in this section is to derive the equation for the reduced density +matrix in the region L by performing a partial trace operation over the contributions from +the region R. The matching maximally entangled state, denoted by the following phrase, +62 + +must first be created: +|Ψ⟩ME : = +1 +√ +2 +� +|0pout⟩BD1 ⊗ +�� +(1 − |γpin|2) +(1 + fpin) +∞ +� +k=0 +|γpin|k +� +|kpin⟩R′ ⊗ |kpin⟩L′ +� ++ +fpin +� +(1 + fpin) +∞ +� +n=0 +∞ +� +r=0 +|Γpin,n|r +� +|n, rpin⟩R′ ⊗ |n, rpin⟩L′ +�� ++|1pout⟩BD1 ⊗ +�� +(1 − |γpin|2) +(1 + fpin) +� +∆1 +∞ +� +k=0 +|γpin|k√ +k + 1 +� +|kpin⟩R′ ⊗ |(k + 1)pin⟩L′ +� ++∆2 +∞ +� +k=0 +|γpin|k√ +k + 1 +� +|(k + 1)pin⟩R′ ⊗ |kpin⟩L′ +�� ++ +fpin +� +(1 + fpin) +� +∞ +� +n=0 +∆3,n +∞ +� +r=0 +|Γpin,n|r√ +r + 1 +� +|n, rpin⟩R′ ⊗ |n, (r + 1)pin⟩L′ +� ++ +∞ +� +n=0 +∆4,n +∞ +� +r=0 +|Γpin,n|r√ +r + 1 +� +|n, (r + 1)pin⟩R′ ⊗ |n, rpin⟩L′ +���� +. (6.42) +From the above mentioned detailed structure of the maximal entangled state constructed +in the present set up it is observed that the scale dependence in this state comes through +the quantities, γpin, Γpin,n, ∆1, ∆2, ∆3,n and ∆4,n appearing in the computation. This +particular fact is the direct outcome of the factorization of the inside observer’s subspace +in to two symmetric subspaces R and L. Our further job is to to study the imprints of this +scale dependence on the physical outcomes of the systems to explore the unknown facts +from the theoretical set up under consideration. +We now need to take a partial trace over the degrees of freedom of region R, because +we already know that the inside observer’s subspace does not get any information content +from this region. +This will allow us to create the reduced density matrix out of this +configuration. Due to the fact that the above-mentioned newly built maximally entangled +quantum state, which is actually a mixed state in the current prescription, will be taken +into account throughout this computation, we must be mindful of this fact. As a result, +the reduced density matrix can be expressed simply as follows: +ρreduced : = TrR′ [|Ψ⟩ME +ME⟨Ψ|] += +∞ +� +mpin=0 +R′⟨mpin|Ψ⟩MEME⟨Ψ|m⟩R′ + +∞ +� +s=0 +∞ +� +mpin=0 +R′⟨s, mpin|Ψ⟩MEME⟨Ψ|s, mpin⟩R′ += +� +∞ +� +mpin=0 +ρmpin + +∞ +� +mpin=0 +∞ +� +s=0 +ρmpin,s +� +, +(6.43) +63 + +where we define ρmpin and ρmpin,s by the following expressions: +ρm = (1 − |γpin|2) +2 (1 + fpin) |γpin|2mpin +� +|0pout⟩BD1|mpin⟩L′ BD1⟨0|L′⟨mpin| ++∆∗ +2γpin +� +mpin + 1|0pout⟩BD1|mpin + 1⟩L′ BD1⟨1|L′⟨mpin| ++∆2γ∗ +pin +� +mpin + 1|1pout⟩BD1|mpin⟩L′ BD1⟨0|L′⟨mpin + 1| ++|∆2|2(mpin + 1)|1pout⟩BD1|mpin⟩L′ BD1⟨1|L′⟨mpin| ++∆∗ +1 +� +mpin + 1|0pout⟩BD1|mpin⟩L′ BD1⟨1|L′⟨mpin + 1| ++∆1 +� +mpin + 1|1pout⟩BD1|mpin + 1⟩L′ BD1⟨0|L′⟨mpin| ++∆∗ +1∆2γ∗ +pin +� +(mpin + 1)(mpin + 2)|1pout⟩BD1|mpin⟩L′ BD1⟨1|L′⟨mpin + 2| ++∆1∆∗ +2γpin +� +(mpin + 1)(mpin + 2)|1pout⟩BD1|mpin + 2⟩L′ BD1⟨1|L′⟨mpin| ++|∆1|2(mpin + 1)|1pout⟩BD1|mpin + 1⟩L′ BD1⟨1|L′⟨mpin + 1| +� +, +(6.44) +and +ρm,s = +f 2 +pin +2 (1 + fpin)|Γpin,s|2mpin +� +|0pout⟩BD1|s, mpin⟩L′ BD1⟨0|L′⟨s, mpin| ++∆∗ +4,sΓpin,s +� +mpin + 1|0pout⟩BD1|s, (mpin + 1)⟩L′ BD1⟨1|L′⟨s, mpin| ++∆4,sΓ∗ +pin,s +� +mpin + 1|1pout⟩BD1|s, mpin⟩L′ BD1⟨0|L′⟨s, (mpin + 1)| ++|∆4,s|2(mpin + 1)|1pout⟩BD1|s, mpin⟩L′ BD1⟨1|L′⟨s, mpin| ++∆∗ +3,s +� +mpin + 1|0pout⟩BD1|s, mpin⟩L′ BD1⟨1|L′⟨s, (mpin + 1)| ++∆3,s +� +mpin + 1|1pout⟩BD1|s, mpin + 1⟩L′ BD1⟨0|L′⟨s, mpin| ++∆∗ +1∆2Γ∗ +pin,s +� +(mpin + 1)(mpin + 2)|1pout⟩BD1|s, mpin⟩L′ BD1⟨1|L′⟨s, (mpin + 2)| ++∆3,s∆∗ +4,sΓpin,s +� +(mpin + 1)(mpin + 2)|1pout⟩BD1|s, (mpin + 2)⟩L′ BD1⟨1|L′⟨s, mpin| ++|∆3,s|2(mpin + 1)|1pout⟩BD1|s, (mpin + 1)⟩L′ BD1⟨1|L′⟨s, (mpin + 1)| +� +. +(6.45) +The internal observer is essentially described by the quantum mechanical state that emerges +in this situation for both the complementary and specific integral parts, where the corre- +sponding observer is positioned in one of the regions of the open chart of the global de +Sitter space. Furthermore, it is significant to remember that all mode eigen values for +the complementary and specific integral parts are identical and marked with the symbol +mpin. This is because of the fact that in the particular integral part the index s which is +appearing due to putting source term is not going to effect the eigen values of the mode +64 + +function at the end of the day. Though one can tag the corresponding quantum states +of the particular integral part with s and mpin, to make a distinction from the quantum +modes of the complementary part which is tagged by only the quantum number mpin. This +is a very crucial point which is very important to mention at this stage of computation to +avoid all further unnecessary confusion. +6.4 +Partial transpose operation and the comment on the negative eigenvalues +Identifying the negative eigenvalues of the obtained formula for the reduced density matrix +discussed before is the main objective of this section. In order to perform this computa- +tion, we separate the contributions from the complementary section and the particular +integral component. Here, we focus on the partial transpose operation with respect to the +component that corresponds to the first quantum vacuum state of Bunch Davies, which is +given by the following expressions: +ρT,BD1 +m += (1 − |γpin|2) +2 (1 + fpin) |γpin|2mpin +� +|0pout⟩BD1|mpin⟩L′ BD1⟨0|L′⟨mpin| ++∆∗ +2γpin +� +mpin + 1|1pout⟩BD1|mpin + 1⟩L′ BD1⟨0|L′⟨mpin| ++∆2γ∗ +pin +� +mpin + 1|0pout⟩BD1|mpin⟩L′ BD1⟨1|L′⟨mpin + 1| ++|∆2|2(mpin + 1)|1pout⟩BD1|mpin⟩L′ BD1⟨1|L′⟨mpin| ++∆∗ +1 +� +mpin + 1|1pout⟩BD1|mpin⟩L′ BD1⟨0|L′⟨mpin + 1| ++∆1 +� +mpin + 1|0pout⟩BD1|mpin + 1⟩L′ BD1⟨1|L′⟨mpin| ++∆∗ +1∆2γ∗ +pin +� +(mpin + 1)(mpin + 2)|1pout⟩BD1|mpin⟩L′ BD1⟨1|L′⟨mpin + 2| ++∆1∆∗ +2γpin +� +(mpin + 1)(mpin + 2)|1pout⟩BD1|mpin + 2⟩L′ BD1⟨1|L′⟨mpin| ++|∆1|2(mpin + 1)|1pout⟩BD1|mpin + 1⟩L′ BD1⟨1|L′⟨mpin + 1| +� +, +(6.46) +and +ρT,BD1 +m,s += +f 2 +pin +2 (1 + fpin)|Γpin,s|2mpin +� +|0pout⟩BD1|s, mpin⟩L′ BD1⟨0|L′⟨s, mpin| ++∆∗ +4,sΓpin,s +� +mpin + 1|1pout⟩BD1|s, (mpin + 1)⟩L′ BD1⟨0|L′⟨s, mpin| ++∆4,sΓ∗ +pin,s +� +mpin + 1|0pout⟩BD1|s, mpin⟩L′ BD1⟨1|L′⟨s, (mpin + 1)| ++|∆4,s|2(mpin + 1)|1pout⟩BD1|s, mpin⟩L′ BD1⟨1|L′⟨s, mpin| ++∆∗ +3,s +� +mpin + 1|1pout⟩BD1|s, mpin⟩L′ BD1⟨0|L′⟨s, (mpin + 1)| ++∆3,s +� +mpin + 1|0pout⟩BD1|s, mpin + 1⟩L′ BD1⟨1|L′⟨s, mpin| ++∆∗ +3,s∆4,sΓ∗ +pin,s +� +(mpin + 1)(mpin + 2)|1pout⟩BD1|s, mpin⟩L′ BD1⟨1|L′⟨s, (mpin + 2)| +65 + ++∆3,s∆∗ +4,sΓpin,s +� +(mpin + 1)(mpin + 2)|1pout⟩BD1|s, (mpin + 2)⟩L′ BD1⟨1|L′⟨s, mpin| ++|∆3,s|2(mpin + 1)|1pout⟩BD1|s, (mpin + 1)⟩L′ BD1⟨1|L′⟨s, (mpin + 1)| +� +. +(6.47) +Now if from the above mentioned partial transposed version of the reduced density matrices +computed from the complementary and particular integral part after taking addition and +summing over s if we found that at least one eigenvalue is negative, then we can conclude +from our theoretical set up that quantum mechanical states corresponding to the inside +and outside observers are entangled. +Let’s first express the transposed version of the reduced density matrices derived from +the complementary and particular integral component in square matrix form before con- +tinuing with the computation, which is given by the following expressions: +ρT,BD1 +m += (1 − |γpin|2) +2 (1 + fpin) |γpin|2mpin +� +� +� +� +� +� +� +� +� +Ampin +Bmpin +Cmpin +B∗ +mpin +Dmpin +0 +C∗ +mpin +0 +0 +� +� +� +� +� +� +� +� +� +, +(6.48) +ρT,BD1 +mpin,s = +f 2 +pin +2 (1 + fpin)|Γpin,s|2mpin +� +� +� +� +� +� +� +� +� +Ampin,s +Bmpin,s +Cmpin,s +B∗ +mpin,s +Dmpin,s +0 +C∗ +mpin,s +0 +0 +� +� +� +� +� +� +� +� +� +, +(6.49) +where we define each of entries of the above mentioned square matrices by the following +expressions: +Ampin = 1 + |∆2|2(mpin + 1), +(6.50) +Bmpin = +� +mpin + 1 +� +∆2γ∗ +pin + ∆∗ +1 +� +, +(6.51) +Cmpin = +� +(mpin + 1)(mpin + 2) ∆∗ +1∆2γ∗ +pin, +(6.52) +Dmpin = |∆1|2(mpin + 1), +(6.53) +and +Ampin,s = 1 + |∆4,s|2(mpin + 1), +(6.54) +Bmpin,s = +� +mpin + 1 +� +∆4,sΓ∗ +pin,s + ∆∗ +3,s +� +, +(6.55) +66 + +Cmpin,s = +� +(mpin + 1)(mpin + 2) ∆∗ +3,s∆4,sΓ∗ +pin,s, +(6.56) +Dmpin,s = |∆3,s|2(mpin + 1). +(6.57) +Our next job is to compute the eigenvalue equations from the total partial transposed +matrix, which is given by the following expression: +�λ3 +mpin − Ampin�λ2 +mpin + Bmpin�λmpin + Cmpin = 0. +(6.58) +where in the above mentioned two expressions we have introduced some shorthand rede- +fined symbols which are given by the following expressions: +Ampin = +1 +2 (1 + fpin) +� +|γpin|2mpin � +1 − |γpin|2� � +Ampin + Dmpin +� ++f 2 +pin +∞ +� +s=0 +|Γpin,s|2m � +Ampin,s + Dmpin,s +� +� +, +(6.59) +Bmpin = +1 +4 (1 + fpin)2 +� +|γpin|4mpin � +1 − |γpin|2�2 � +AmpinDmpin − +� +|Bmpin|2 + |Cmpin|2�� ++f 4 +pin +∞ +� +s=0 +|Γpin,s|4mpin � +Ampin,sDmpin,s − +� +|Bmpin,s|2 + |Cmpin,s|2�� +� +, (6.60) +Cmpin = +1 +8 (1 + fpin)3 +� +|γpin|6mpin � +1 − |γpin|2�3 |Cmpin|2Dmpin ++f 6 +pin +∞ +� +s=0 +|Γpin,s|6mpin|Cmpin,s|2Dmpin,s +� +. +(6.61) +The real root of the eigenvalue computed from the (mpin, mpin + 1) block is given by: +�λmpin = 1 +3 +� +Ampin + f(Ampin, Bmpin, Cmpin) +3√ +2 +− +3√ +2 +� +3Bmpin − A +2 +mpin +� +f(Ampin, Bmpin, Cmpin) +� +. +(6.62) +where we define the newly defined function f(Ampin, Bmpin, Cmpin) which is defined as: +f(Ampin, Bmpin, Cmpin) : = +� +2A +3 +mpin − 9AmpinBmpin − 27Cmpin ++3 +√ +3 +� +18AmpinBmpinCmpin + 4B +3 +mpin + 27C +2 +mpin +67 + +−4A +3 +mpinCmpin − A +2 +mpinB +2 +mpin +�1 +2 +�1 +3 +. +(6.63) +Then the logarithmic negativity from the present set up can be further computed as: +LN = ln +� +� +�2 +� +�λmpin <0 +�λmpin + 1 +� +� +� += ln +� +2 +3 +� +Ampin + f(Ampin, Bmpin, Cmpin) +3√ +2 +− +3√ +2 +� +3Bmpin − A +2 +mpin +� +f(Ampin, Bmpin, Cmpin) +� ++ 1 +� +. +(6.64) +In the next subsection we have studied this possibility numerically to extract the unknown +facts from the present set up. +6.5 +Computation of logarithmic negativity: Numerical study +In figure (6.2(a)) and (6.2(b)), we have depicted the representative 3D plot of the eignen- +value of the partial transposed matrix with the mass parameter and the corresponding +momentum mode associated with the computation. In these plots we have considered two +possibilities, vanishing fpin = 0 and a very small value but fpin ̸= 0. Here the partial +transpose operation is taken with respect to the quantum vacuum state of the first Bunch +Davies state which is characterizing the corresponding open chart of the global de Sitter +space. The outcomes and the physical interpretation of these plots are very interesting +which we are writing point-wise in the following: +1. In these plots we can clearly observe that the corresponding eignevalue that we +have plotted for the mode having mpin = 0 gives negative contributions for both +figure (6.2(a)) and (6.2(b)). This is a clear signature of having quantum mechanical +entanglement in the biverse as well as well as the multiverse picture that we have +theoretically constructed in this paper. +2. The other higher modes mpin > 0 gives positive contribution to the eigenvalue for +which we have not incorporated those plots in this paper. Most importantly such con- +tributions will not be in support of quantum entanglement in the present theoretical +picture, so those solutions are not interesting in the present context. +3. Comparing both of these plots we can also observe that for fpin = 0 we get less +quantum entanglement compared to the case that we have studied for small fpin ̸= 0. +4. Also we have found that for heavy mass field which is governed by ν = −i|ν| or +ν2 < 0 the corresponding eigenvalue of the partial transposed version of the reduced +68 + +(a) For fpin = 0. +(b) For small fpin ̸= 0. +Figure 6.2: Representative 3D plot of the eignenvalue of the partial transposed matrix +with the mass parameter and the corresponding momentum mode associated with the +computation. Here the partial transpose operation is taken with respect to the quantum +vacuum state of the first Bunch Davies state which is characterizing the corresponding +open chart of the global de Sitter space. +69 + +0.5 +1.0 +1.5 +-0.15 +-0.10 +-0.05 +(a) For fpin = 0. +0.5 +1.0 +1.5 +-0.18 +-0.16 +-0.14 +-0.12 +-0.10 +-0.08 +-0.06 +-0.04 +(b) For small fpin ̸= 0. +Figure 6.3: Graphical behaviour of the of the eignenvalue of the partial transposed matrix +with the mass parameter for given value of the momentum mode associated with the +computation. +70 + +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +0.0 +(a) For fpin = 0. +0.2 +0.3 +0.4 +0.5 +0.6 +0.7 +-0.5 +-0.4 +-0.3 +-0.2 +-0.1 +(b) For small fpin ̸= 0. +Figure 6.4: Graphical behaviour of the of the eignenvalue of the partial transposed matrix +with the momentum mode associated with the computation for the given value of mass +parameter. +71 + +0.5 +1.0 +1.5 +0.05 +0.10 +0.15 +0.20 +0.25 +0.30 +(a) For fpin = 0. +0.5 +1.0 +1.5 +0.10 +0.15 +0.20 +0.25 +0.30 +(b) For small fpin ̸= 0. +Figure 6.5: Graphical behaviour of the of the logarithmic negativity computed from +the negative eigenvalues of the partial transposed reduced density matrix with the mass +parameter associated with the computation for the given value of momentum mode. +72 + +0.05 +0.10 +0.20 +0.50 +0.001 +0.010 +0.100 +1 +(a) For fpin = 0. +0.05 +0.10 +0.20 +0.50 +10-4 +0.001 +0.010 +0.100 +1 +(b) For small fpin ̸= 0. +Figure 6.6: Graphical behaviour of the of the logarithmic negativity computed from the +negative eigenvalues of the partial transposed reduced density matrix with the momentum +mode associated with the computation for the given value of mass parameter. +73 + +density matrix is highly positive. Since this fact is not the desirable one, we have not +included such plots in this paper. But it is strictly confirmed that to get high amount +of entanglement heavy field is not desirable in the present prescribed framework. For +this reason we have only shown the plots for massless or partially massless fields +which can be studied using the solution for the branch ν > 0. +5. From both of these plots we can clearly observe that for the values of the mass +parameter ν = 1/2 (conformal coupling) and ν = 3/2 (massless case) there are two +dips in the eigenvalue spectrum, which correspond to maximum entanglement from +the theoretical set up under consideration. On the other hand, in both of these plots +we see that at ν = 1 there is minimum contribution from the quantum entanglement +compared to the previously mentioned two values of the mass parameter ν. +6. For any other values of the mass parameter lying within the window 0 < ν < 2 we +get the intermediate amount of quantum mechanical entanglement. Particularly, it +is important to note that for ν = 0 the amount of entanglement is larger than the +amount of entanglement obtained for the value ν = 2. +7. It seems like that the eigenvalue spectrum is distributed symmetrically around the +value of the mass parameter ν = 1. +But actually this is not the case. +Crucial +observation suggests that the hight of the spectrum is lesser at ν = 0 compared to +ν = 2. +8. In both of these plots we have restricted our parameter space within the region +0 < pin < 0.8 and 0 < ν < 2, which we found the most suitable parameter space to +get the negative contribution from the eigenvalue spectrum for the mode mpin = 0. +In figure (6.3(a)) and (6.3(b)), we have depicted the representative graphical behaviour +of the of the eignenvalue of the partial transposed matrix with the mass parameter for +given value of the momentum mode associated with the computation. In these plots we +have considered two possibilities, vanishing fpin = 0 and a very small value but fpin ̸= 0. +In both of these plots we have fixed the value of the momentum mode within the region +0.2 < pin < 0.5 and have studied the behaviour of the eigenvalue spectrum with respect +to mass parameter within the range 0 < ν < 2. In our computation we found that this is +the most desirable window of parameters within which one can get maximum contribution +from the quantum entanglement in terms of getting maximum negative contribution from +the eigenvalue. +Further, in figure (6.4(a)) and (6.4(b)), we have depicted the representative graphical +graphical behaviour of the of the eignenvalue of the partial transposed matrix with the +momentum mode associated with the computation for the given value of mass parameter. +In these plots we have considered two possibilities, vanishing fpin = 0 and a very small +value but fpin ̸= 0. In both of these plots we have fixed the value of the mass parameter +74 + +within the region 1 < ν < 7/4 and have studied the behaviour of the eigenvalue spectrum +with respect to momentum mode within the range 0.1 < pin < 0.8. We have found that +for very small value of momentum mode with the mass parameter in the region ν < 1 is +not desirable in the present context as it gives very large positive eigenvalue. Also from +this analysis we have found that for very large value of the momentum mode for any +values of the mass parameter ν the corresponding eigenvalue spectrum will saturate to a +negative value. After comparing figure (6.4(a)) and (6.4(b)), we also have found that this +asymptotic saturation value for small fpin ̸= 0 is larger compared to result obtained from +fpin = 0 case. +Next, in figure (6.5(a)) and (6.5(b)), we have depicted the representative graphical +graphical behaviour of the the logarithmic negativity computed from the negative eigen- +values of the partial transposed reduced density matrix with the mass parameter associated +with the computation for the given value of momentum mode. Finally, in figure (6.6(a)) +and (6.6(b)), we have depicted the representative graphical graphical behaviour of the the +logarithmic negativity with the momentum mode for the given value of mass parameter. +In these plots we have considered two possibilities, vanishing fpin = 0 and a very small +value but fpin ̸= 0. The outcomes and the physical interpretation of these plots are very +interesting which we are writing point-wise in the following: +1. From the figure (6.5(a)) and (6.5(b)), we have found that for fpin = 0 logarithmic +negativity vanishes at ν = 1, but for small fpin ̸= 0 it is non zero but very small. +It further implies that at the value of the mass parameter ν = 1 we have almost +negligible contribution from the quantum mechanical entanglement on large scales. +Additionally it is important to note that, this particular outcome is obtained for a +specific value of the momentum mode, pin = 0.2. +2. On the other hand, we have found that at ν = 1/2 and ν = 3/2 the obtained value +of the logarithmic negativity from the present theoretical set up reach the maximum +value, which further correspond to the maximum quantum mechanical entanglement +or maximum correlation. This outcome is true for the momentum mode, pin = 0.2. +One important thing here to mention that the large scale limit corresponds to the +small value of the momentum mode pin in our present computation. +3. For the other values of momentum mode lying within the window 0.2 < pin < 0.5 we +have found that the variation with respect to the mass parameter ν is less compared +to the case that we have studied for the momentum mode pin = 0.2. Comparing +all the outcomes obtained for the different momentum modes within the mentioned +range we have found that if we increase the value of pin then the corresponding +variation is reduced and we get intermediate values of negativity, which corresponds +to the intermediate amount of quantum mechanical entanglement for the system +under consideration. This further implies the fact that lower values of the momentum +75 + +mode for the prescribed analysis is more desirable as it is giving higher amplitude of +logarithmic negativity, hence higher amount of quantum mechanical entanglement. +4. We have also found that the underlying quantum mechanical entanglement directly +put impact on the shape of the spectrum of logarithmic negativity at the large scales +which nearly equals or exceeds the mass parameter’s value ν ∼ 3/2. +5. Further, in figure (6.6(a)) and (6.6(b)), we have found out that for large values of +the momentum mode pin > 0.5 the corresponding logarithmic negativity computed +from the prescribed theoretical set up saturates to a constant non zero positive non +negligible value. This further implies constant amount of quantum entanglement +for any arbitrary positive real value of the mass parameter ν. +However, in this +asymptotic limit one cannot distinguish the individual effect of the mass parameter +in the present computation. This also implies that the low momentum modes are +more desirable for the present analysis to distinguish the individual effects of the +mass parameter ν. +6. Last but not least, we have also found that except ν = 3/2 for all other values of +the mass parameter ν we get a sharp changing behaviour in the specific value of the +momentum mode. +6.6 +Small scale limit of logarithmic negativity: Analytical study +In the small scale limit one needs to take the limit pin → ∞ to compute the logarithmic +negativity from the present computational set up. In this limit we have: +∆1 = ∆∗ +1 → 1, +∆2 = ∆∗ +2 → 0, +∆3,s = ∆∗ +3,s → 1, +∆4,s = ∆∗ +4,s → 0. +(6.65) +The following simplified formulas in the small scale limit are obtained after taking par- +tial transposition with regard to the subsystem corresponding to the first Bunch Davies +quantum vacuum state: +ρT,BD1 +mpin += (1 − |γpin|2) +2 (1 + fpin) |γpin|2mpin +� +|0pout⟩BD1|mpin⟩L′ BD1⟨0|L′⟨mpin| ++ +� +mpin + 1|1pout⟩BD1|mpin⟩L′ BD1⟨0|L′⟨mpin + 1| ++ +� +mpin + 1|0pout⟩BD1|mpin + 1⟩L′ BD1⟨1|L′⟨mpin| ++(mpin + 1)|1pout⟩BD1|mpin + 1⟩L′ BD1⟨1|L′⟨mpin + 1| +� +, +(6.66) +76 + +and +ρT,BD1 +mpin,s = +f 2 +pin +2 (1 + fpin)|Γpin,s|2mpin +� +|0pout⟩BD1|s, mpin⟩L′ BD1⟨0|L′⟨s, mpin| ++ +� +mpin + 1|1pout⟩BD1|s, mpin⟩L′ BD1⟨0|L′⟨s, (mpin + 1)| ++ +� +mpin + 1|0pout⟩BD1|s, mpin + 1⟩L′ BD1⟨1|L′⟨s, mpin| ++(mpin + 1)|1pout⟩BD1|s, (mpin + 1)⟩L′ BD1⟨1|L′⟨s, (mpin + 1)| +� +. +(6.67) +Let us express the above mentioned transposed version of the reduced density matrices +computed from the complementary and particular integral part in square matrix form, +which are given by the following expressions: +ρT,BD1 +mpin += (1 − |γpin|2) +2 (1 + fpin) |γpin|2mpin +� +� +� +� +� +� +� +� +� +1 +�mpin + 1 +0 +�mpin + 1 +(mpin + 1) +0 +0 +0 +0 +� +� +� +� +� +� +� +� +� +, +(6.68) +ρT,BD1 +mpin,s = +f 2 +pin +2 (1 + fpin)|Γpin,s|2mpin +� +� +� +� +� +� +� +� +� +1 +�mpin + 1 +0 +�mpin + 1 +(mpin + 1) +0 +0 +0 +0 +� +� +� +� +� +� +� +� +� +, +(6.69) +Next we compute the eigenvalue equation from the total partial transposed matrix after +summing over source mode s, which is given by the following expression: +�λ2 +mpin +� +�λmpin − (1 − |γpin|2) +2 (1 + fpin) |γpin|2mpin(mpin + 2) − +f 2 +pin +2 (1 + fpin)(mpin + 2)gpin +� += 0. (6.70) +The non trivial root of the eigenvalue computed from the (mpin, mpin + 1) block in the +small scale limit is given by: +�λmpin = (mpin + 2) +2 (1 + fpin) +� +� +1 − |γpin|2� +|γpin|2mpin + f 2 +pingpin +� +, +(6.71) +77 + +where we define: +gpin := +∞ +� +s=0 +|Γpin,s|2mpin. +(6.72) +Then the logarithmic negativity from the present set up can be further computed as: +LN = ln +� +� +�2 +� +�λmpin <0 +�λmpin + 1 +� +� +� += ln +� +(mpin + 2) +(1 + fpin) +� +� +1 − |γpin|2� +|γpin|2mpin + f 2 +pingpin +� ++ 1 +� +. +(6.73) +We get a negative contribution from the eigenvalue if the following condition is satisfied +for the small scale limiting situation: +� +|γpin|2mpin � +1 − |γpin|2� ++ f 2 +pingpin +� +< 0. +(6.74) +Now from the previous analysis we have already found that mpin = 0 mode is most desirable +one to obtain the negative contribution from the eigenvalue spectrum. In such a case using +the well known Riemann zeta function regularization we can write the following result for +mpin = 0: +gpin := +∞ +� +s=0 +1 = 1 + +∞ +� +s=1 +1 = 1 + (1 + 1 + 1 + · · · ) = 1 + ζ(0) = 1 − 1 +2 = 1 +2. +(6.75) +which also suggests the following restriction on the small scale limiting circumstance: +|γpin| > +� +� +� +� +� +1 + f 2 +pin +2 +� +. +(6.76) +6.7 +Massless limit of logarithmic negativity: Analytical study +In the massless limit one needs to take the limit ν = 3/2 (exact masslessness) or ν = 1/2 +(conformal invariance) to compute the logarithmic negativity from the present computa- +tional set up. In this limit we have: +∆1 → +� +�A − �Dγpin +� +u, +∆2 → 0, +∆3,s → +� +�An − �DnΓpin,n +� +Un, +∆4,s → 0. +(6.77) +78 + +The partial transposition operation for the subsystem corresponding to the first Bunch +Davies quantum vacuum state is given by the following formulas for the massless limiting +situation: +ρT,BD1 +mpin += (1 − |γpin|2) +2 (1 + fpin) |γpin|2mpin +� +|0pout⟩BD1|mpin⟩L′ BD1⟨0|L′⟨mpin| ++∆∗ +1 +� +mpin + 1|1pout⟩BD1|mpin⟩L′ BD1⟨0|L′⟨mpin + 1| ++∆1 +� +mpin + 1|0pout⟩BD1|mpin + 1⟩L′ BD1⟨1|L′⟨mpin| ++|∆1|2(mpin + 1)|1pout⟩BD1|mpin + 1⟩L′ BD1⟨1|L′⟨mpin + 1| +� +, +(6.78) +and +ρT,BD1 +mpin,s = +f 2 +pin +2 (1 + fpin)|Γpin,s|2mpin +� +|0pout⟩BD1|s, mpin⟩L′ BD1⟨0|L′⟨s, mpin| ++∆∗ +3,s +� +mpin + 1|1pout⟩BD1|s, mpin⟩L′ BD1⟨0|L′⟨s, (mpin + 1)| ++∆3,s +� +mpin + 1|0pout⟩BD1|s, mpin + 1⟩L′ BD1⟨1|L′⟨s, mpin| ++|∆3,s|2(mpin + 1)|1pout⟩BD1|s, (mpin + 1)⟩L′ BD1⟨1|L′⟨s, (mpin + 1)| +� +.(6.79) +In this special case the factors ∆1 and ∆3,s can be further simplified as: +∆1 = +� +�A − �Dγpin +� +u += +1 +sinh πpin +� +exp(πpin) − i exp(−πpin)(1 + pin) +(1 − pin) +Γ(ipin) +Γ(−ipin) +� +, +(6.80) +∆3,s = +� +�As − �DsΓpin,s +� +Us += +1 +sinh πpin,s +� +exp(πpin,s) − i exp(−πpin,s)(1 + pin,s) +(1 − pin,s) +Γ(ipin,s) +Γ(−ipin,s) +� +. +(6.81) +Let us express the above mentioned transposed version of the reduced density matrices +computed from the complementary and particular integral part in square matrix form, +79 + +which are given by the following expressions for the massless case: +ρT,BD1 +mpin += (1 − |γpin|2) +2 (1 + fpin) |γpin|2mpin +� +� +� +� +� +� +� +� +� +1 +�mpin + 1∆∗ +1 +0 +�mpin + 1∆1 +0 +0 +0 +0 +0 +� +� +� +� +� +� +� +� +� +, +(6.82) +ρT,BD1 +mpin,s = +f 2 +pin +2 (1 + fpin)|Γpin,s|2mpin +� +� +� +� +� +� +� +� +� +1 +�mpin + 1∆∗ +3,s +0 +�mpin + 1∆3,s +0 +0 +0 +0 +0 +� +� +� +� +� +� +� +� +� +. (6.83) +Next we compute the eigenvalue equation from the total partial transposed matrix after +summing over source mode s, which is given by the following expression: +�λmpin +� +�λ2 +mpin − Ampin�λmpin + Bmpin +� += 0. +(6.84) +where in the above mentioned expression we have introduced some shorthand redefined +symbols which are given by the following expressions: +Am = +1 +2 (1 + fpin) +� +� +1 − |γpin|2� +|γpin|2m + f 2 +pingpin +� +, +(6.85) +Bm = − (mpin + 1) +4 (1 + fpin)2 +� +� +1 − |γpin|2�2 |γpin|4m|∆1|2 + f 4 +pin +∞ +� +s=0 +|Γpin,s|4m|∆3,s|2 +� +. +(6.86) +The non trivial roots of the eigenvalue computed from the (mpin, mpin + 1) block are given +by: +�λ± +mpin = 1 +2 +� +Ampin ± +� +A +2 +mpin − 4Bmpin +� +. +(6.87) +Then the logarithmic negativity computed from the negative eigenvalue from the present +set up can be written for the massless limit as: +LN = ln +� +� +�2 +� +�λmpin <0 +�λmpin + 1 +� +� +� = ln +� +Ampin + 1 − +� +A +2 +mpin − 4Bmpin +� +. +(6.88) +80 + +Here it is important to note that the eigenvalue and the associated logarithmic negativity +computed in this particular massless limiting situation with fpin = 0 is similar to the case +that is obtained for explaining the entanglement between an inertial and a non-inertial +frame of reference for a free massless scalar degree of freedom in Minkowski flat space +time as discussed in the reference [125]. Small difference appearing due to having separate +thermal behaviour of the Minkowski space time and global de Sitter space time. Now this +difference in the result is more prominent and significant once we consider the effect of +source term in the effective axion potential with small fpin ̸= 0. +7 +Conclusion +We conclude our discussion with the following points which we have found from our analysis +performed in this paper: +• Firstly, we have started with the basic discussion regarding the entanglement neg- +ativity and logarithmic negativity for a general quantum mechanical set up which +is appearing in the context of quantum information theory. We have provided the +technical details for the related computations from a general quantum mechanical +set up. Further we have provided a proper physical justification that why the above +mentioned two measures are physically relevant and significant for the computation +we want to perform for the open chart of the global de Sitter space. +• Then we have given a detailed justification of the factorization of the total Hilbert +space in open chart to associate our computation in the region L and R, which is +necessarily required to construct the reduced density matrix and to compute the +above mentioned entanglement measures from the system under consideration. +• Additionally, we have fully covered the specifics of the geometrical arrangement of +the open chart of the de Sitter space, which is the platform on which we intend to +carry out the remaining calculation. We have independently determined the metric’s +structure in the area between L and R, which is a crucial piece of knowledge for +determining how scalar modes would behave based on our calculations. +• Next, we computed the explicit equation for the mode function using the string +theory-derived axionic effective interaction, with which we created the Bunch Davies +vacuum state and subsequently the expression for the reduced density matrix. +• Further, we have computed the expressions for the entanglement negativity and +logarithmic negativity for the mentioned axionic effective interaction in open chart. +We have found that, the newly studied quantum information theoretic measures are +more significant compared to the Von Neumann measure of entanglement entropy, +which is commonly used to describe the impact of quantum entanglement. +81 + +• The result that offers the most promise is that it enables us to compute and estimate +the quantum entanglement between the inside and outside of a de Sitter bubble +without the need for a boundary. +• We also found that for the large mass the amount of the quantum entanglement +decays exponential in the entanglement negativity vs mass parameter squared plots. +On the other hand this decaying behaviour is slightly different when we consider the +logarithmic negativity measure in this context. +• We also have found that for the case of conformal coupling ν = 1/2 and massless +case ν = 3/2 in the logarithmic negativity and entanglement negativity spectrum +there are two consecutive peaks appear of equal hights. This is really an interesting +feature we have found from the prescribed theoretical set up studied in this paper. +Additionally we have found that apart from having two prominent peaks for the +mentioned values of the mass parameter we have oscillation in the spectrum due to +having small mass parameter of the axion field during the de Sitter expansion in +the global coordinates. This oscillation becomes more rapid if the mass parameter +is very small. On the other hand, the period of oscillation is larger and less rapid if +the mass parameter is large. +• Then, using the Bunch Davies quantum vacuum state, we expanded our computation +to compute the formula for the logarithmic negativity between two causally indepen- +dent patches of the open chart of the global de Sitter space. This is a scaled-down +version of the well-known multiverse scenario and is known as the biverse picture. +In order to perform this calculation, we presupposed a correct factorization of the +Hilbert spaces in the two subspaces that are currently spanned by the modes of the +Bunch Davies quantum vacuum states. We don’t need to be aware of the explicit +content of the first subspace in order to apply our methodology to compute the quan- +tum entanglement between two de Sitter spaces in the global coordinates. This is +necessary because we need to take a partial trace over the first de Sitter space’s entire +subspace content. However, the explicit sub factorization of the second subspace is +crucial in this situation since it significantly affects the explicit content of the modes +from the regions R and L. +• The most important aspect of this particular computation is the maximally entangled +state, which we used to build the reduced density matrix by extracting all the data +from the initial Bunch Davies vacuum state. We have further used this result to +compute the expression for the partial transposed version of the reduced density +matrix. We have next found that the mode corresponding to mpin = 0 (ground state) +correspond to the negative eigen value spectrum, using which we have numerically +studied the behaviour of logarithmic negativity from the prescribed theoretical set +up. We have found from our analysis that the mode corresponding to mpin = 0 can +82 + +produce large measure of quantum entanglement due to having negative eigen value +spectrum. For the other values of mpin we have found that the corresponding eigen +values are largely positive which is not desirable to construct a biverse which can +produce large effect of quantum mechanical entanglement at the end. +• In the biverse construction we also have found conformally coupled case with ν = +1/2 and the massless case with ν = 3/2 are the two very special points in the +entanglement spectrum where the amount of the quantum correlation is equal and +high in amplitude. On the other hand, we have found that for the mass parameter +ν = 1 the amount of quantum correlation estimated from the corresponding picture +reaches its minimum value. This is obviously a promising information which we have +obtained after performing our analysis on the biverse picture and it is expected that +it can be generalized to any multiverse scenario constructed out of the specific model +that we have studied in this paper. +• In the present context the global coordinates can be treated as the closed slicing +from the point of view of FLRW cosmology. By performing coordinate coordinate +transformation one can transform the global to static and then static to the flat +slicing in the planer patch of de Sitter space. For this reason whatever results we +have obtained in this paper for the global patch of the de Sitter space can be directly +translated in the planer patch of the de Sitter space, which further implies that our +derived results hold good for primordial cosmology. +Here we have some interesting immediate future direction on which one can extend our +analysis: +• We have restricted our analysis for the estimation of entanglement negativity and +logarithmic negativity by considering the Bunch Davies quantum vacuum state. Im- +mediately one can extend our analysis for a general non Bunch Davies vacua, such +as α vacua. It is expected to have many interesting outcome if we extend our anal- +ysis for non Bunch Davies vacua because the quantum correlations and its various +unknown applications will be known from this type of future analysis. +• Since the global coordinates and planar coordinates of de Sitter space are connected +via coordinate transformation, it is good to explicitly know how the present results +can be explained within the framework of primordial cosmology. This possibility one +can seriously think for future work to connect with observation. +• One can further compute various other quantum information theoretic measures, +like quantum discord, fidelity and many more interesting quantities from the present +theoretical set up. +83 + +• The direct connection between the higher point quantum correlations with the all of +these possible quantum information theoretic measures, imprints of quantum entan- +glement in quantum correlations computed in the quantum field theory of de Sitter +space and the primordial cosmological set ups are another interesting possibilities +which one can study in future from the present set up that we have constructed in +this paper. +• Extending the present computation to study role quantum mechanical decoherence +[25, 26, 29, 126, 127] and quantum diffusion [128] might be very useful for the future +study which can explain various unknown facts from the present set up in global as +well as in the planer patch of de Sitter space. +• The construction of squeezed quantum mechanical states and its consequences is a +common area of research in cosmological set up [33, 129–141]. It would be really good +if we can able to construct a squeezed quantum state out of the present theoretical set +up that we are considering in this paper. This will going to help to figure out various +quantum information theoretic measures and its applications in various contexts. If +the squeezed state construction is not possible then also one can study various other +possibilities out of the present set up [142–145]. +• Till now the computation is restricted to a quantum system which is completely +adiabatic in nature, because we are considering closed quantum system. It would be +really good if we can study the open quantum system version of the present set up +within the framework of quantum field theory of de Sitter space [24, 34, 117, 146–156]. +Acknowledgements +SC would like to thank the work friendly environment of The Thanu Padmanabhan Cen- +tre For Cosmology and Science Popularization (CCSP), Shree Guru Gobind Singh Tri- +centenary (SGT) University, Gurugram, Delhi-NCR for providing tremendous support in +research and offer the Assistant Professor (Senior Grade) position. SC also thanks all +the members of our newly formed virtual international non-profit consortium Quantum +Aspects of the SpaceTime & Matter (QASTM) for elaborative discussions. Last but not +least, we would like to acknowledge our debt to the people belonging to the various parts +of the world for their generous and steady support for research in natural sciences. +84 + +References +[1] J. Maldacena and G. L. 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Tasinato, “Open EFTs, IR effects \& late-time +resummations: systematic corrections in stochastic inflation,” JHEP 01 (2016) 153, +arXiv:1512.00169 [gr-qc]. +94 + diff --git a/gNE4T4oBgHgl3EQfqw2j/content/tmp_files/load_file.txt b/gNE4T4oBgHgl3EQfqw2j/content/tmp_files/load_file.txt new file mode 100644 index 0000000000000000000000000000000000000000..702dd55e54022250cf219b0cc7f42d8ea23d252c --- /dev/null +++ b/gNE4T4oBgHgl3EQfqw2j/content/tmp_files/load_file.txt @@ -0,0 +1,2675 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf,len=2674 +page_content='Entanglement negativity in de Sitter biverse from Stringy Axionic Bell pair: An analysis using Bunch-Davies vacuum Sayantan Choudhury1‡, §,.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' 1Centre For Cosmology and Science Popularization (CCSP), SGT University, Gurugram, Delhi-NCR, Haryana- 122505, India.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Abstract In this work, we study the signatures of quantum entanglement by computing entan- glement negativity between two causally unrelated regions in 3 + 1 dimensional global de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' We investigate a bipartite quantum field theoretic setup for this purpose, driven by an axionic Bell pair resulting from Type IIB string compactification on a Calabi- Yau three fold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' We take into account a spherical surface that divides the spatial slice of the global de Sitter space into exterior and interior causally unrelated sub regions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' For the computational purpose we use the simplest possible initial choice of quantum vacuum, which is Bunch-Davies state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The quantitative quantum information theoretic measure for entanglement negativity turns out be consistent with the results obtained for entanglement entropy, even we have to say it is better than that from quantum information theoretic point of view.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' We design the problem in a hyperbolic open chart where one of the causally unrelated observers remains constrained and the scale dependence enters to the correspond- ing quantum information theoretic entanglement measure for axionic Bell pair.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' We find from our analysis that in the large scales initially maximally entangled Bunch-Davies state turns out to be strongly entangled or weakly entangled depending on the axionic decay constant and the supersymmetry breaking scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' We also find that at the small scales the initial entanglement can be perfectly recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' We also discuss the possibility of having a biverse picture, which is a mini version of the multiverse in the present theoretical set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Last but not the least, we provide the necessary criteria for generating non vanishing quantum entanglement measures within the framework of quantum field theory of global de Sitter space as well as well as in primordial cosmology due to the axion derived from string theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Keywords: De-Sitter vacua, Quantum Entanglement, Cosmology of Theories beyond the SM, Quantum Information Theory aspects of Gravity, String Cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' ‡ Corresponding author, E-mail : sayantan ccsp@sgtuniversity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='org, sayanphysicsisi@gmail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='com § NOTE: This project is the part of the non-profit virtual international research consortium “Quantum Aspects of Space-Time & Matter” (QASTM) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='05203v1 [hep-th] 31 Dec 2022 Contents 1 Introduction 1 2 Basics of entanglement negativity and logarithmic negativity from Quan- tum Information Theory 7 3 Computational strategy of negativity between two causally unrelated open charts of global de Sitter space with axion 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='1 Quantum structure of open chart 14 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='2 Geometric structure of open chart 16 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='3 Mode function and wave function of axion in an open chart 19 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='4 Construction of reduced density matrix in open chart 41 4 Entanglement negativity and logarithmic negativity in open chart 43 5 Comparison with entanglement entropy in open chart 50 6 Logarithmic negativity between two causally unrelated patches of open chart 53 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='1 Computational set up and construction of maximally entangled states 55 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='2 Construction of excited quantum state for single oscillator 57 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='3 Construction of reduced density matrix at the inside observer 62 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='4 Partial transpose operation and the comment on the negative eigenvalues 65 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='5 Computation of logarithmic negativity: Numerical study 68 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='6 Small scale limit of logarithmic negativity: Analytical study 76 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='7 Massless limit of logarithmic negativity: Analytical study 78 7 Conclusion 81 References 85 i 1 Introduction In the present day research, different quantum information theoretic measures of quantum entanglement is a remarkable probe in theoretical physics which helps us to distinguish the various type of long range correlated quantum mechanical states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In this connection, study of the explicit role of the long range quantum correlations in the framework of quantum field theory is extremely significant, which is a fascinating topic of research itself.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' See refs [1–31] for more details.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The key ingredient of this study is the initial quantum mechanical vacuum states, which are Chernikov-Tagirov, Bunch-Davies, Hartle-Hawking, α and Motta-Allen vacua [14–16, 19, 32–37].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Quantum entanglement is treated as one of the remarkable outcomes of the foundational theoretical aspects of quantum mechanics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The prime reason of this thought is, a local measurement in quantum mechanics may instantaneously put a significant impact of the outcome of the measurement beyond the physical light cone.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' This is theoretically interpreted as Einstein-Podolsky-Rosen (EPR) paradox where the concept of causality violation is explicitly demonstrated [31, 38–41].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Amongst various types of information theoretic measures entanglement entropy is con- sidered to be a very useful quantitative as well as qualitative probe of quantum entangle- ment and this concept is commonly used in the framework of condensed matter physics, quantum information theory and high energy physics.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' But in this connection it is im- portant to note that the technical computation of entanglement entropy is very difficult to perform in the context of quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Ryu and Takayanagi in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' [42, 43] first did the theoretically consistent computation of entanglement entropy for a strongly coupled quantum field theory having a gravitational dual counterpart using the underlying physical principles of AdS/CFT (or holographic gravitational dual prescription) [44].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In order to properly understand the direct implications of the previously proposed AdS/CFT (or holographic gravitational dual prescription) in the technical computation of entanglement entropy in presence of standard Bunch-Davies quantum vacuum state within the framework of quantum field theory of de Sitter space, Maldacena and Pimentel prescribed an extremely useful computational strategy in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' [1] using free massive scalar quantum field.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' After this work the proposed methodology was generalised for the same problem in presence of non standard α vacua in refs [13, 19, 45–47].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Further in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' [14– 16], these underlying physical concepts of the computation of entanglement entropy was applied within the framework of axion quantum field theory, described in terms a specific type of effective interaction potential originated from Type II string theory compactifi- cation [48–50] in presence of both standard Bunch-Davies and non standard α quantum vacua.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The underlying concept of quantum entanglement could applicable to the frame- work of cosmology, specifically beyond the Hubble horizon scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Most importantly, in this prescription if a particle pair is created in a casually connected Hubble horizon scale then it is naturally expected to be dissociated as an outcome of the de Sitter cosmological expansion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In this particular case, the findings from this prescription pointing towards the 1 fact that two causally unrelated patches in the de Sitter cosmological space-time has to be entangled quantum mechanically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' As a result, the corresponding observable quantum vacuum fluctuations associated with our own universe is entangled with the other part of the open patch of global de Sitter space, which is commonly identified to be the multiverse in the present context of discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' To demonstrate this underlying physical picture the reduced density matrix formalism play a very significant role, using which it was explicitly obtained that the quantum mechanical entanglement directly put impact on the cosmo- logical power spectrum on the large scales and it is quantitatively similar to or larger than the curvature radius scale connected to the current issue [18].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' It is hugely expected that the present and the upcoming observational probes may detect the imprints and explicit effects of quantum mechanical entanglement in cosmological paradigm, which we strongly believe will going to be extremely useful to understand a lot of unknown mysterious fascinating facts of our own universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In the past, there are lot of efforts have been made to study the impacts of quantum mechanical entanglement in the theoretical framework of cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' See refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' [16, 18, 21, 23, 51–58] for more details on these aspects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The motivation of the background physical thought of the present paper are appended below point-wise: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' It was first discovered in a number of refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' [59–61] to support the claim that the physical frame of the bubble nucleation process is observer dependent and is actually dictated by the rest frame of the observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The bubble nucleation process being dependent on the observer is hence only expected.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The detailed study of such type of observer dependence in the framework of quantum mechanical entanglement is one of the prime motivations of the present work.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' If one can able to address this issue in detail, then various unexplored features of quantum cosmology can be explored with proper physical understanding.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Second, the explicit function of the Bunch Davies vacuum can be investigated in the context of quantum entanglement originating from an axion field embedded in an open patch of the global de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' This will be crucial because it allows one to directly examine whether it is possible to find the signs of quantum entanglement in the current multiverse [18, 62–67] motivated theoretical set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='using current or future observational instruments.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Thirdly, it was noted in the refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' [14–16, 68–70] that the string theory-originated axion can be viewed as the ideal component to build the Bell pair, which is neces- sary to violate Bell’s inequality within the context of primordial cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Within the widely accepted framework of primordial cosmology, it is virtually difficult to break Bell’s inequality, and it is for this reason that string theory and the axion play the most important roles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' It is significant to note that, in this context, it is 2 nearly impossible to test the explicit role of quantum mechanical entanglement in observational probes without violating Bell’s inequality in the context of primordial cosmology, which is necessary to break the degeneracy in the shape cosmological two- point function, also known as the primordial power spectrum obtained from various effective potentials from fundamental physical principles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The current framework with the string theory-originated axion provides a perfect setup that is pointing to- wards a new physics coming from a non-standard cosmology because the generation of Bell’s inequality violating pairs, also known as the Bell pair, is practically im- possible within the framework of the standard primordial cosmological paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' It is explicitly pointed in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' [68–70] that the prime signature of the Bell’s inequality violation in non standard primordial cosmology is coming from the existence of one point function from axion, which is absent in the well known standard cosmological paradigm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' This result has a great impact of producing quantum entanglement in the present scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Also our derived results can be extremely useful to directly verify along with the observational probes the applicability as well as the justifiability of the string theory originated axion to address the issue of quantum entanglement in the present set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' See refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' [14–16, 68–77] for more details on the Bell’s inequality and its violation in various contexts including cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Finally, the motivation comes from the multiverse prescription appearing in the string theory landscape scenario, which states that our universe may not be the single entity of the space-time but the part of a bigger size multiverse [18, 62–67].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' This is obviously a fascinating fact which one needs to study in detail.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' However, in the past the underlying physical concept of the multiverse has been criticised in various contexts as a hypothetical philosophical concept which is the outcome of complete theoretical imagination and cannot be tested via cosmological observation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' But the actual truth is fay beyond all of these criticism, which is the detectable signatures can be obtained from the multiverse set up in presence of quantum mechanical entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' It really helps us to produce at least two causally separated de sitter universe, commonly known as the de Sitter bubbles in the present context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' But the numbers are not restricted in two and most importantly this is actually the starting point of multiverse which allows many more causally unrelated de Sitter bubbles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Next, we explicitly write down the underlying assumptions of the present work, which are appended below point-wise: 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' To model the present multiverse motivated scenario [18, 62–67] , we consider two causally separated patches on the global de Sitter space by assuming that initially they are in a maximally entangled pure quantum mechanical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' this particular set up we identify as the biverse picture which is the mini version of the original multiverse picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' 3 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In this theoretical set up further we introduce two observers whose actual purpose is to determine the quantum entanglement of a de Sitter universe.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' We also assume that one of the observers is placed inside the a de Sitter bubble and want to determine how the signatures of quantum mechanical entanglement with the other de Sitter bubble can be visualized by the inside observer.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Now the issue is that, the inside observer can’t able to see outside region of their own de Sitter bubble.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' For this reason instead of using the total density matrix, one needs to take the partial trace over the causally unrelated outside region, which finally give rise to the reduced density matrix of the system in the present context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' But as an outcome some information will be lost during this process and to describe the quantum mechanical state of the observer instead of using a pure state one needs to explicitly use a mixed state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' This scenario is completely different for an observer who is sitting on the other causally separated de Sitter bubble because in his frame of reference the quantum aspects is described by the pure quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' For this reason we need study the effects and outcomes of the quantum entanglement between pure and mixed states in the present context of discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' To perform the computation for a single global de Sitter space picture as well in the case of the entangled two de Sitter space, which is the biverse picture we need to properly understand the factorization of the Hilbert space in terms of the individual constituents which span the subspaces by forming complete orthonormal sets.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The geometrical structure of the global de Sitter space demands that in the hyperbolic open chart one can symmetrically factorize in the left region and in a right region, which we have tagged as region L and R during performing the technical computation in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Since we have symmetrically factorize the total Hilbert space in the region L and R, it is viable to formulate the reduced description either in terms of the degrees of freedom appearing in the region L or region R.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In our computation we take the partial trace over the degrees of freedom appearing in the region R, which forms a reduced density matrix in the region L.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' If the Hilbert space factorization performed correctly then one can further blindly trust the mode decomposition in the region L and R using which we have con- structed the Bunch Davies quantum vacuum states in terms of harmonic oscillator modes which forms a complete orthonormal basis states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Estimation of the quan- tum entanglement in terms of various quantum information theoretic measures are also based on this mode decomposition and hence one can rely on the tracing out the unwanted information from the bipartite quantum field theoretic set up under consideration to construct the reduced density matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' To study the outcome of quantum mechanical entanglement we have blindly trusted the above mentioned factorization for the second de Sitter space which is spanned 4 by the well known Bunch Davies vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' For the other, which is the first de Sitter bubble we have assumed that the detailed factorization structure is not explicitly needed for the computation we are interested to perform in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' This is because of the fact that we need to take the partial trace operation over the first de Sitter vacuum which will remove all unwanted degrees of freedom from the quantum information theoretic measures of entanglement we want to compute to confirm the existence of quantum entanglement in the biverse picture.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Here one can do the computation in other way as well where one needs to take the partial trace operation with respect to the all degrees of freedom appearing in second de Sitter vacuum and properly factorize the first de Sitter vacuum in terms of L and R modes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Due to having the symmetrical structure of the set up that we are considering it is expected that for both the cases we will have the same physical outcome at the end of the computation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' We just have taken the first possibility as a choice.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' During the biverse construction we have first constructed the maximally entangled state which is one of the key ingredients to violate the Bell’s CHSH inequality within the present framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' This is a very challenging to construct within the framework of de Sitter space time as well as in primordial cosmology set up.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' We have constructed the maximally entangled state which suffice the purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' But this could not be possible just having usual interaction in the scalar field theory embedded in the global patch of the de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The Bell’s CHSH inequality violating pair is created in the region L and R with the help of specific type of interaction in the string theory originated axion driven scalar field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The structure of the interaction is controlled by the time dependent axion decay constant and the supersymmetry breaking scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' We have assumed that Bell pair is created and originated from the string theory originated axionic scalar field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' It might be possible to create such pair from some other theoretical sector as well which we have not considered in this paper for the time being.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' 7.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In the present context the global coordinates can be treated as the closed slicing from the point of view of FLRW cosmology.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' One usually do primordial cosmology, particularly inflation [78–116] in the planar coordinate slicing of the de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' By performing coordinate coordinate transformation one can transform the global to static and then static to the flat slicing in the planer patch of de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' For this reason whatever results we have obtained in this paper for the global patch of the de Sitter space can be directly translated in the planer patch of the de Sitter space, which further implies that our derived results hold good for primordial cosmological set up constructed out of the effective interaction studied for the string theory originated axion scalar field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' This is obviously an interesting aspect from our computation which shows the applicability of our derived results in the vast theoretical framework.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' 5 8.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The von Neumann and Renyi entropies [14, 15, 117] are the good quantum informa- tion theoretic measures of quantum entanglement in the context of bipartite quantum field theory designed in terms of pure state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' But with the help of mixed state if we want to understand the features of an arbitrary bipartite quantum field theoretic system then instead of using entropic measures, negativity or its logarithmic version would be perfect information theoretic measure of quantum entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Using this specific measure we compute the imprints of the quantum entanglement from the point of view of two causally separated observers in the open chart of global de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' 4 3 2 1 0 1 2 3 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='0 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='2 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='4 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='6 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='8 1.' metadata={'source': 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+page_content='A ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='AB+XicbZDLSsNAFIZP6q3W9Slm8EiuCqJVnRZ7MZlBXuRtpbJdNIOnUzCzKRQt7EjQtF3Pom7nwbp2kW2vrDwMd/zuGc+b2IM6Ud59sqrK ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='1vbG4Vt0s7u3v7B/bhUuFsS0SUIeyo6HFeVM0KZmtNOJCkOPE7b3qQ+r7enVCoWigc9i2g/wCPBfEawNtbAtpOe56P6Y/qUwWU6sMtOxc ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='mEVsHNoQy5GgP7qzcMSRxQoQnHSnVdJ9L9BEvNCKdpqRcrGmEywSPaNShwQFU/yS5P0ZlxhsgPpXlCo8z9PZHgQKlZ4JnOAOuxWq7Nzf9q3V ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='j7N/2EiSjWVJDFIj/mSIdoHgMaMkmJ5jMDmEhmbkVkjCUm2oRVMiG4y19ehdZFxa1Wru6r5dptHkcRTuAUzsGFa6jBHTSgCQSm8Ayv8GYl1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='ov1bn0sWgtWPnMf2R9/gCPH5L4 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='Penrose ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='diagram ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='Geometry of ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='de Sitter ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='space ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='+ ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='Measure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='of quantum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='entanglement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='Measure ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='of quantum ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='entanglement ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='String Theory ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='Axion ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='Outcome 1 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='Outcome 2 ' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='Figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='1: Representative diagram of the overall quantum entanglement computation and all possible outcomes for single de Sitter universe and two de Sitter universe (biverse) obtained from our our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In this work, we compute the negativitity measure of quantum entanglement from an axionic effective potential which is obtained from Type IIB string theory compactification on a Calabi-Yau three fold in presence of NS5 brane.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Earlier this model have been studied 6 T =0 二t, = const.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' A B L R C Open chartin the framework of inflationary cosmology [118–121].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Since this axions can be treated as Bell pair, we technically compute the expression for entanglement negativity from two causally unrelated patches of the open chart of the global de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' From the quantitative results obtained from our computation we perform further a consistency check as well as the comparison of the results obtained in previous refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='[14–16].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Furthermore, to understand various unknown fascinating facts regarding the multiverse scenario we extend our computation in presence of axionic Bell pair in presence of de Sitter space along with the maximally entangled quantum Bunch Davies vacuum state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' We find from our analysis that in the large scales initially maximally entangled Bunch-Davies state turns out to be strongly entangled or weakly entangled depending on the axionic decay constant and the supersymmetry breaking scale.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' We also find the at the small scales the initial entanglement can be perfectly recovered.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Last but not the least, we provide the necessary criteria for generating non vanishing quantum entanglement measure within the framework of primordial cosmology due to the string axion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In figure (1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='1) representative diagram of the overall quantum entanglement computation and all possible outcomes for single de Sitter universe and two de Sitter universe (biverse) obtained from our our analysis.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The organization of this paper is as follows: In Section 2 we briefly review the tools and techniques required to compute nega- tivity and logarithmic negativity along with some easy demonstrations for the better understanding purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In Section 3 we mention the computational strategy for negativity in the hyperbolic open chart of the global de Sitter space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In Section 4 we compute negativity by following aforementioned strategy for a spe- cific effective potential, Type IIB string compactification originated axion model.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In Section 5 we compute entanglement in terms of negativity from two causally unrelated de Sitter bubbles, which is determined from the point of view of two ob- servers introduced during the calculation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In this section we discuss the possibility of having biverse, which is the mini version of the multiverse scenario.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Finally in Section 6 we summarize our all findings in this paper along with some future prospects of the computations performed in this paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' 2 Basics of entanglement negativity and logarithmic negativity from Quantum Information Theory Within the framework quantum mechanics, particularly in quantum information theory to study the imprints and the underlying physical aspects of entanglement various measure have been proposed till date.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Entanglement negativity and its logarithmic version, which 7 is commonly known as logarithmic negativity are the very useful quantum information theoretic measures of quantum mechanical entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' These measures are derived by making use of the positive partial transpose criterion for the separability.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The Peres- Horodecki criterion [31, 122–124] is a prerequisite for the separation of the joint density matrix of two quantum mechanical systems A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The phrase is sometimes referred to as the PPT criterion, which stands for positive partial transpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' From the studies it was found that in the 2 × 2 and 2 × 3 dimensional cases the condition is sufficient.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Particularly this is used to decide the separability of mixed quantum states, where the well known Schmidt decomposition does not apply.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' It is important to note that, in higher dimensions, this specific concept gives inconclusive result, and one should perform more advanced tests, like entanglement witnesses, which describes a functional that distinguishes a specific entangled state from the separable ones.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' [31, 122–124], i t was explicitly shown that this measures are entanglement monotone and hence treated to be a proper measure of quantum entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In the following we give a brief overview on the topic of entanglement negativity and logarithmic negativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Now let’s take a look at a quantum system that can be described by A and B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' The direct products of the Hilbert spaces of the subsystems A and B define the corresponding total Hilbert space of the system i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' H = HA ⊗ HB.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Here H represents the Hilbert space of the total system, HA and HB represent the Hilbert spaces of the subsystems A and B respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Further consider a pure quantum mechanical state, by applying the Schmidt decompo- sition one can write: |Ψ⟩ = � m � λm|m⟩A ⊗ |m⟩B, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='1) where λm corresponds to the observed probability of the any general pure m-th state and satisfy the following constraint condition: � m λm = 1, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='2) which physically implies that the total observed probability of the process, which is ob- tained by summing over all possible pure states has to be conserved in the present context of discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Now, it is important to note if we are interested in the mixed states then to technically describe the behaviour of the subsystem inclusion of the concept of reduced density matrix is extremely important and this is actually described by the reduced density operator of the subsystem.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In the present description we consider two subsystems A and B, which are equally likely and both of them carry same weight in the present construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' For this reason to describe the subsystem either we will talk about the description in terms of A or B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Let us say for the time being that we are interested to know about the underlying physics of the subsystem A which can be obtained by taking the partial trace operation 8 on the information of the subsystem B.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' This process will finally give rise to the following reduced density matrix of the subsystem A, which is given by: ρA = TrB|Ψ⟩⟨Ψ| = � m λm|m⟩AA⟨m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='3) Further, utilizing this reduced density matrix of the subsystem A one can explicitly com- pute the expression for the von Neumann entropy, which is given by: S = −Tr [ρA ln ρA] = − � m λm ln λm.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='4) Technically the above mentioned equation was introduced to describe the first entangle- ment measure in quantum information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In the specific situation, when there is no quantum entanglement, then in terms of the probability we can write: λm = � � � � � � � � � 1 if m = 1 0 if m ̸= 1 (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='5) In the above mentioned both the cases the measure of entanglement entropy vanish ex- plicitly.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' If such a situation appears in a particular physical systems, then we can comment that there would be no quantum mechanical entanglement for that type of systems.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' On the other hand, if we find some physical systems where the probability lie within the window 0 < λm < 1∀m then the corresponding entanglement entropy measure is non-zero and it is treated to be good measure within the framework of quantum information theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' But one word of caution is that, the entanglement entropy measure sometimes gives inconclusive result.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' For an example, if we are interested to describe the strictly classical correlation in terms of mixed state then the entanglement entropy measure gives non-zero result, which is itself a surprising fact.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In this specific type of situation entanglement entropy mea- sure within the framework of quantum information theory fails to distinguish between the impacts of quantum mechanical correlations and classical correlations.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Then obviously a natural question comes to a physicists mind that what is the way out of this situation?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Is it possible to define more powerful entanglement measure in the present framework?' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Answer to all of these questions are Yes, it is possible to define such measures which can conclusively distinguish the quantum and classical effects in the corre- lation functions computed from the underlying physical theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Based on the separability criterion entanglement negativity and logarithmic negativity measures are defined which actually serve the purpose.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Let us now give a brief outline on the connection between separability and quantum entanglement, and its usefulness in the present context of discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' A quantum mechan- 9 ical state is considered to be separable iff the density matrix of the total system can be expressed in terms of a tensor or outer products of the individual density matrices belong to the each subsystems under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Technically this statement can be written as: ρ = � m λm � ρA m ⊗ ρB m � where λm ≥ 0 ∀m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='6) Here ρ represents the density matrix of the total system under consideration.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' For the m-th state corresponding to the subsystems A and B the density matrices are defined as: ρA m := |m⟩AA⟨m| and ρB m := |m⟩BB⟨m|.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='7) Hence in terms of the information coming from the subsystems A and B, the density matrix for the total system can be recast as: ρ = � m λm (|m⟩AA⟨m| ⊗ |m⟩BB⟨m|) where λm ≥ 0 ∀m.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='8) One can further generalize the structure of the density matrix by considering both the contributions from entangled and non-entangled states, which is given by the following expression: ρ = � m � n � p � q Dmnpq (|m⟩AA⟨n| ⊗ |p⟩BB⟨q|) where Dmnpq ≥ 0 ∀m, n, p, q, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='9) where Dmnpq represents a more general coefficient which capture both the effects from entangled and non-entangled quantum states.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Let’s investigate a partial transposition operation with respect to the subsystem A in light of the new generation definition of the total density matrix, which results in the following expression: ρTA = � m � n � p � q Dmnpq � (|m⟩AA⟨n|)TA ⊗ |p⟩BB⟨q| � = � m � n � p � q Dmnpq (|n⟩AA⟨m| ⊗ |p⟩BB⟨q|) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='10) Following a partial transpose operation with regard to the subsystem A, we obtain the following results for the non-entangled quantum state: ρTA = � m λm � (|m⟩AA⟨m|)TA ⊗ |m⟩BB⟨m| � = � m λm (|m⟩AA⟨m| ⊗ |m⟩BB⟨m|) = ρ, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='11) 10 which corresponds to the fact that the total density matrix remains unchanged.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Here since for the non-entangled state λm∀m, this directly implies that ρTA ≥ 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' This also confirms that if after performing a partial transpose operation on the total system has at least one negative eigenvalue then the system cannot be described in terms of the above mentioned form stated in eqn (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='8) and consequently the underlying quantum state considered in this discussion has to be entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' This is actually very interesting outcome coming from the present computation in support of quantum entanglement.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Utilizing these facts one step forward the definition of an quantum information theoretic entanglement measure, entanglement negativity is proposed, which is given by: N = � λm<0 |λm|, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='12) where in this definition summation over all negative eigenvalues are explicitly taken into account.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' From this definition, we can clearly see that when we have N = 0, there is no quantum entanglement in the system.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Apart from having a very good physical thought behind the construction of the entanglement negativity measure, unfortunately it turns out that this is not an additive measure and most importantly not at all suitable for the description of many body or multi-subsystem appearing within the framework of quantum field theory.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Hence, to give a more physically applicable quantum information theoretic measure another powerful quantity has been introduced, which is known as logarithmic negativity and is treated to be most improved version of the entanglement negativity mea- sure in the present context of discussion.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Let us now discuss in detail that how one can technically define the logarithmic negativity by making use of the background physical facts discussed in the present context.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' To define this quantity, first of all let us introduce the trace norm of the partial transposed version of the total density matrix over the subsystem A, which is described by the following expression: ||ρTA|| = Tr � (ρTA)† ρTA = � m |λm| = � � λm>0 |λm| + � λm<0 |λm| � = � N + � λm>0 |λm| � = (2N + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='13) Here in the list line of the above expression to write down corresponding trace norm in 11 terms of the quantum entanglement negativity measure we have utilized the following sets of useful constraints: Tr (ρ) = 1, Tr � ρTA� = 1, � m λm = 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='14) Then the logarithm of the trace norm of the of the total density matrix over the subsystem A is identified to be logarithmic negativity, which is given by the following expression: LN = ln � ||ρTA|| � = ln (2N + 1) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='15) This implies the fact that, when N ̸= 0, then LN ̸= 0 and the corresponding quantum state is considered to be entangled.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Let us now consider a special case where we are dealing with the pure quantum mechan- ical state.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' In this specific case our next objective is to explicitly compute the expression for the logarithmic negativity.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' To serve this purpose we use the well known Schmidt de- composition technique for pure quantum state, using which we can write: |Ψ⟩ = � m � λm (|m⟩A ⊗ |m⟩B) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='16) Then using this representation of the pure quantum state one can further define the expres- sion for the density matrix for the total system, which is given by the following expression: ρ = |Ψ⟩⟨Ψ| = � m � n � λmλn ((|m⟩A ⊗ |m⟩B) (A⟨n| ⊗ B⟨n|)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='17) Further, taking the partial transpose operation on the above mentioned density matrix of the total system over the subsystem A, we get the following expression: ρTA = � m � n � λmλn ((|m⟩A ⊗ |m⟩B) (A⟨n| ⊗ B⟨n|))TA = � m � n � λmλn ((|n⟩AA⟨m|) ⊗ (|n⟩AA⟨m|)) = � m � n,m=n � λmλn ((|n⟩AA⟨m|) ⊗ (|n⟩AA⟨m|)) + � m � n,m̸=n � λmλn ((|n⟩AA⟨m|) ⊗ (|n⟩AA⟨m|)) = � m λm ((|m⟩AA⟨m|) ⊗ (|m⟩AA⟨m|)) + � m � n,m̸=n � λmλn ((|n⟩AA⟨m|) ⊗ (|n⟩AA⟨m|)) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='18) 12 For the sake of simplicity, to diagonalize the structure of the total density matrix we further introduce a new basis for the pure quantum states, in which two new state vectors are defined by the following expressions: |Ψ+ mn⟩ : = 1 √ 2 [(|m⟩A ⊗ |n⟩B) + (|n⟩A ⊗ |m⟩B)] for m < n, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='19) |Ψ− mn⟩ : = 1 √ 2 [(|m⟩A ⊗ |n⟩B) − (|n⟩A ⊗ |m⟩B)] for m < n.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='20) It is very easy to check the following two constraints are always satisfied by the new basis state vectors |Ψ+ mn⟩ and |Ψ− mn⟩, which are given by: (A⟨m| ⊗ B⟨m|) |Ψ+ mn⟩ = 1 √ 2 � �(A⟨m| ⊗ B⟨m|) (|m⟩A ⊗ |n⟩B) � �� � =0 + (A⟨m| ⊗ B⟨m|) (|n⟩A ⊗ |m⟩B) � �� � =0 � � = 0, (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='21) (A⟨m| ⊗ B⟨m|) |Ψ− mn⟩ = 1 √ 2 � �(A⟨m| ⊗ B⟨m|) (|m⟩A ⊗ |n⟩B) � �� � =0 − (A⟨m| ⊗ B⟨m|) (|n⟩A ⊗ |m⟩B) � �� � =0 � � = 0.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='22) This implies the fact that both the new basis state vectors |Ψ+ mn⟩ and |Ψ− mn⟩ are orthogonal to the state |m⟩A ⊗ |m⟩B, which is obviously a very helpful information for the further construction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content=' Then the partial transpose with respect to subsystem A for the total density matrix as stated in eqn (2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/gNE4T4oBgHgl3EQfqw2j/content/2301.05203v1.pdf'} +page_content='18) can be written in the diagonalized basis in terms of the two state vectors |Ψ+ mn⟩ and |Ψ− mn⟩ as: ρTA = � m λm|m⟩AA⟨m| + � m � n,m