Pub Date : 2024-04-26DOI: 10.1134/s0040577924040111
S. Siouane, A. Boumali, A. Guvendi
Abstract
We explore the thermal characteristics of fermionic fields with a nonminimal coupling in one, two, and three dimensions using the framework of superstatistics theory. We consider three distinct distributions: the gamma distribution, the lognormal distribution, and the F distribution. Each of these distributions is governed by a specific probability density function. To calculate the partition function, we use the Euler–Maclaurin formula, specifically in the low-energy asymptotic approximation of superstatistics. This calculation takes the remainder term into consideration. In each scenario, using the derived partition functions, we analyze the variations in entropy and specific heat with varying temperatures and the universal parameter denoted as (q). In general, we observe that increasing the value of (q) enhances all the curves. Additionally, we note that entropy values tend to increase as the temperature decreases, and tend to decrease as the parameter (q) increases.
摘要 我们利用超统计理论框架探讨了具有非最小耦合的费米子场在一维、二维和三维的热特性。我们考虑了三种不同的分布：伽马分布、对数正态分布和 F 分布。每种分布都由特定的概率密度函数支配。为了计算分区函数，我们使用了欧拉-麦克劳林公式，特别是在超统计的低能渐近近似中。这种计算方法考虑了余项。在每种情况下，我们利用推导出的分区函数，分析了熵和比热随温度和普遍参数（表示为 (q)）的变化而变化的情况。一般来说，我们发现增加 （q）的值会增强所有曲线。此外，我们注意到，熵值往往随着温度的降低而增加，随着参数 （q）的增加而降低。
{"title":"Superstatistical properties of the Dirac oscillator with gamma, lognormal, and F distributions","authors":"S. Siouane, A. Boumali, A. Guvendi","doi":"10.1134/s0040577924040111","DOIUrl":"https://doi.org/10.1134/s0040577924040111","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We explore the thermal characteristics of fermionic fields with a nonminimal coupling in one, two, and three dimensions using the framework of superstatistics theory. We consider three distinct distributions: the gamma distribution, the lognormal distribution, and the F distribution. Each of these distributions is governed by a specific probability density function. To calculate the partition function, we use the Euler–Maclaurin formula, specifically in the low-energy asymptotic approximation of superstatistics. This calculation takes the remainder term into consideration. In each scenario, using the derived partition functions, we analyze the variations in entropy and specific heat with varying temperatures and the universal parameter denoted as <span>(q)</span>. In general, we observe that increasing the value of <span>(q)</span> enhances all the curves. Additionally, we note that entropy values tend to increase as the temperature decreases, and tend to decrease as the parameter <span>(q)</span> increases. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-26DOI: 10.1134/s0040577924040032
A. Pourkia
Abstract
We construct solutions of the Yang–Baxter equation in any dimension (dge 2) by directly generalizing the previously found solutions for (d=2). We equip those solutions with unitarity and entangling properties. Being unitary, they can be turned into (2)-qudit quantum logic gates for qudit-based systems. The entangling property enables each of those solutions, together with all (1)-qudit gates, to form a universal set of quantum logic gates.
{"title":"Yang–Baxter equation in all dimensions and universal qudit gates","authors":"A. Pourkia","doi":"10.1134/s0040577924040032","DOIUrl":"https://doi.org/10.1134/s0040577924040032","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We construct solutions of the Yang–Baxter equation in any dimension <span>(dge 2)</span> by directly generalizing the previously found solutions for <span>(d=2)</span>. We equip those solutions with unitarity and entangling properties. Being unitary, they can be turned into <span>(2)</span>-qudit quantum logic gates for qudit-based systems. The entangling property enables each of those solutions, together with all <span>(1)</span>-qudit gates, to form a universal set of quantum logic gates. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140800618","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-22DOI: 10.1134/s0040577924030012
U. Jairuk, S. Yoo-Kong
Abstract
We present a new type of integrable one-dimensional many-body systems called a one-parameter Calogero–Moser system. At the discrete level, the Lax pairs with a parameter are introduced and the discrete-time equations of motion are obtained as together with the corresponding discrete-time Lagrangian. The integrability property of this new system can be expressed in terms of the discrete Lagrangian closure relation by using a connection with the temporal Lax matrices of the discrete-time Ruijsenaars–Schneider system, an exact solution, and the existence of a classical (r)-matrix. As the parameter tends to zero, the standard Calogero–Moser system is recovered in both discrete-time and continuous-time forms.
{"title":"One-parameter discrete-time Calogero–Moser system","authors":"U. Jairuk, S. Yoo-Kong","doi":"10.1134/s0040577924030012","DOIUrl":"https://doi.org/10.1134/s0040577924030012","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We present a new type of integrable one-dimensional many-body systems called a one-parameter Calogero–Moser system. At the discrete level, the Lax pairs with a parameter are introduced and the discrete-time equations of motion are obtained as together with the corresponding discrete-time Lagrangian. The integrability property of this new system can be expressed in terms of the discrete Lagrangian closure relation by using a connection with the temporal Lax matrices of the discrete-time Ruijsenaars–Schneider system, an exact solution, and the existence of a classical <span>(r)</span>-matrix. As the parameter tends to zero, the standard Calogero–Moser system is recovered in both discrete-time and continuous-time forms. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198161","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-22DOI: 10.1134/s0040577924030073
M. G. Ivanov, A. Yu. Polushkin
Abstract
To simulate quantum systems on classical or quantum computers, the continuous observables (e.g., coordinate and momentum or energy and time) must be reduced to discrete ones. In this paper, we consider the continuous observables represented in the positional systems as power series in the radix multiplied over the summands (“digits”), which turn out to be Hermitian operators with discrete spectrum. We investigate the obtained quantum mechanical operators of digits, the commutation relations between them, and the effects of the choice of a numeral system on lattices and representations. Renormalizations of diverging sums naturally occur in constructing the digital representation.
{"title":"Digital representation of continuous observables in quantum mechanics","authors":"M. G. Ivanov, A. Yu. Polushkin","doi":"10.1134/s0040577924030073","DOIUrl":"https://doi.org/10.1134/s0040577924030073","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> To simulate quantum systems on classical or quantum computers, the continuous observables (e.g., coordinate and momentum or energy and time) must be reduced to discrete ones. In this paper, we consider the continuous observables represented in the positional systems as power series in the radix multiplied over the summands (“digits”), which turn out to be Hermitian operators with discrete spectrum. We investigate the obtained quantum mechanical operators of digits, the commutation relations between them, and the effects of the choice of a numeral system on lattices and representations. Renormalizations of diverging sums naturally occur in constructing the digital representation. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198257","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-22DOI: 10.1134/s0040577924030036
Xinyue Li, Qian Bai, Qiulan Zhao
Abstract
We explore the Whitham modulation theory and one of its physical applications, the dam-breaking problem for the defocusing Hirota equation that describes the propagation of ultrashort pulses in optical fibers with third-order dispersion and self-steepening higher-order effects. By using the finite-gap integration approach, we deduce periodic solutions of the equation and discuss the degeneration of genus-one periodic solution to a soliton solution. Furthermore, the corresponding Whitham equations based on Riemann invariants are obtained, which can be used to modulate the periodic solutions with step-like initial data. These Whitham equations with the weak dispersion limit are quasilinear hyperbolic equations and elucidate the averaged dynamics of the fast oscillations referred to as dispersive shocks, which occur in the solution of the defocusing Hirota equation. We analyze the case where both characteristic velocities in genus-zero Whitham equations are equal to zero and the values of two Riemann invariants are taken as the critical case. Then by varying these two values as step-like initial data, we study the rarefaction wave and dispersive shock wave solutions of the Whitham equations. Under certain step-like initial data, the point where two genus-one dispersive shock waves begin to collide at a certain time, that is, the point where the genus-two dispersive shock wave appears, is investigated. We also discuss the dam-breaking problem as an important physical application of the Whitham modulation theory.
{"title":"Whitham modulation theory and dam-breaking problem under periodic solutions to the defocusing Hirota equation","authors":"Xinyue Li, Qian Bai, Qiulan Zhao","doi":"10.1134/s0040577924030036","DOIUrl":"https://doi.org/10.1134/s0040577924030036","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We explore the Whitham modulation theory and one of its physical applications, the dam-breaking problem for the defocusing Hirota equation that describes the propagation of ultrashort pulses in optical fibers with third-order dispersion and self-steepening higher-order effects. By using the finite-gap integration approach, we deduce periodic solutions of the equation and discuss the degeneration of genus-one periodic solution to a soliton solution. Furthermore, the corresponding Whitham equations based on Riemann invariants are obtained, which can be used to modulate the periodic solutions with step-like initial data. These Whitham equations with the weak dispersion limit are quasilinear hyperbolic equations and elucidate the averaged dynamics of the fast oscillations referred to as dispersive shocks, which occur in the solution of the defocusing Hirota equation. We analyze the case where both characteristic velocities in genus-zero Whitham equations are equal to zero and the values of two Riemann invariants are taken as the critical case. Then by varying these two values as step-like initial data, we study the rarefaction wave and dispersive shock wave solutions of the Whitham equations. Under certain step-like initial data, the point where two genus-one dispersive shock waves begin to collide at a certain time, that is, the point where the genus-two dispersive shock wave appears, is investigated. We also discuss the dam-breaking problem as an important physical application of the Whitham modulation theory. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198254","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-22DOI: 10.1134/s0040577924030061
A. A. Samokhin, A. V. Zyl, N. L. Zamarashkin
Abstract
We study the applicability of the formula that factors the trace of the diagonal part of spin operator products in the case of a relatively small number of particles of an isolated spin system. The validity of this formula for a large number of particles follows from the basic principles of quantum statistical mechanics. The spin system under consideration includes dipole–dipole interaction and the Zeeman interaction with an external magnetic field. We establish that the accuracy of this formula monotonically increases as the magnetic field increases. At the same time, the dependence on the number of particles in the range (2div10) for various configurations turns out to be sharply nonmonotone.
{"title":"On the factorization method for the quantum statistical description of dynamics of an isolated spin system","authors":"A. A. Samokhin, A. V. Zyl, N. L. Zamarashkin","doi":"10.1134/s0040577924030061","DOIUrl":"https://doi.org/10.1134/s0040577924030061","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the applicability of the formula that factors the trace of the diagonal part of spin operator products in the case of a relatively small number of particles of an isolated spin system. The validity of this formula for a large number of particles follows from the basic principles of quantum statistical mechanics. The spin system under consideration includes dipole–dipole interaction and the Zeeman interaction with an external magnetic field. We establish that the accuracy of this formula monotonically increases as the magnetic field increases. At the same time, the dependence on the number of particles in the range <span>(2div10)</span> for various configurations turns out to be sharply nonmonotone. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198458","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-22DOI: 10.1134/s0040577924030024
Xue Wang, Dianlou Du, Hui Wang
Abstract
We propose a hierarchy of the nonlocal mKdV (NmKdV) equation. Based on a constraint, we obtain nonlocal finite-dimensional integrable systems in a Lie–Poisson structure. By a coordinate transformation, the nonlocal Lie–Poisson Hamiltonian systems are reduced to nonlocal canonical Hamiltonian systems in the standard symplectic structure. Moreover, using the nonlocal finite-dimensional integrable systems, we give parametric solutions of the NmKdV equation and the generalized nonlocal nonlinear Schrödinger (NNLS) equation. According to the Hamilton–Jacobi theory, we obtain the action–angle-type coordinates and the inversion problems related to Lie–Poisson Hamiltonian systems.
{"title":"A nonlocal finite-dimensional integrable system related to the nonlocal mKdV equation","authors":"Xue Wang, Dianlou Du, Hui Wang","doi":"10.1134/s0040577924030024","DOIUrl":"https://doi.org/10.1134/s0040577924030024","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We propose a hierarchy of the nonlocal mKdV (NmKdV) equation. Based on a constraint, we obtain nonlocal finite-dimensional integrable systems in a Lie–Poisson structure. By a coordinate transformation, the nonlocal Lie–Poisson Hamiltonian systems are reduced to nonlocal canonical Hamiltonian systems in the standard symplectic structure. Moreover, using the nonlocal finite-dimensional integrable systems, we give parametric solutions of the NmKdV equation and the generalized nonlocal nonlinear Schrödinger (NNLS) equation. According to the Hamilton–Jacobi theory, we obtain the action–angle-type coordinates and the inversion problems related to Lie–Poisson Hamiltonian systems. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198258","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-22DOI: 10.1134/s0040577924030048
R. M. Iakhibbaev, D. M. Tolkachev
Abstract
We attempt to generalize the integrable Gromov–Sever models, the so-called fishchain models, which are dual to biscalar fishnets. We show that they can be derived in any dimension, at least for some integer deformation parameter of the fishnet lattice. In particular, we focus on the study of fishchain models in AdS(_7) that are dual to the six-dimensional fishnet models.
{"title":"Generalizing the holographic fishchain","authors":"R. M. Iakhibbaev, D. M. Tolkachev","doi":"10.1134/s0040577924030048","DOIUrl":"https://doi.org/10.1134/s0040577924030048","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We attempt to generalize the integrable Gromov–Sever models, the so-called fishchain models, which are dual to biscalar fishnets. We show that they can be derived in any dimension, at least for some integer deformation parameter of the fishnet lattice. In particular, we focus on the study of fishchain models in AdS<span>(_7)</span> that are dual to the six-dimensional fishnet models. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198261","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-01DOI: 10.1134/s0040577924030097
Abstract
We consider a liquid layer of a finite depth described by Euler’s equations. The ice cover is geometrically modeled by a nonlinear elastic Kirchhoff–Love plate. We determine the trajectories of liquid particles under an ice cover in the field of a nonlinear surface traveling wave rapidly decaying at infinity, namely, a solitary wave packet (a monochromatic wave under the envelope, with the wave velocity equal to the envelope velocity) of a small but finite amplitude. Our analysis is based on the use of explicit asymptotic expressions for solutions describing the wave structures at the water–ice interface of a solitary wave packet type, as well as asymptotic solutions for the velocity field generated by these waves in the depth of the liquid.
{"title":"Motion of particles in the field of nonlinear wave packets in a liquid layer under an ice cover","authors":"","doi":"10.1134/s0040577924030097","DOIUrl":"https://doi.org/10.1134/s0040577924030097","url":null,"abstract":"<span> <h3>Abstract</h3> <p> We consider a liquid layer of a finite depth described by Euler’s equations. The ice cover is geometrically modeled by a nonlinear elastic Kirchhoff–Love plate. We determine the trajectories of liquid particles under an ice cover in the field of a nonlinear surface traveling wave rapidly decaying at infinity, namely, a solitary wave packet (a monochromatic wave under the envelope, with the wave velocity equal to the envelope velocity) of a small but finite amplitude. Our analysis is based on the use of explicit asymptotic expressions for solutions describing the wave structures at the water–ice interface of a solitary wave packet type, as well as asymptotic solutions for the velocity field generated by these waves in the depth of the liquid. </p> </span>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-03-01DOI: 10.1134/s0040577924030103
Abstract
We study the time evolution of magnetic fields in various configurations of spatially inhomogeneous pseudoscalar fields that are a coherent superposition of axions. For such systems, we derive a new induction equation for the magnetic field, which takes this inhomogeneity into account. Based on this equation, we study the evolution of a pair of Chern–Simons waves interacting with a linearly decreasing pseudoscalar field. The nonzero gradient of the pseudoscalar field leads to the mixing of these waves. We then consider the problem in a compact domain in the case where the initial Chern–Simons wave is mirror symmetric. The pseudoscalar field inhomogeneity then leads to an effective change in the (alpha) dynamo parameter. Thus, the influence of a spatially inhomogeneous pseudoscalar field on the magnetic field evolution bears a strong dependence on the system geometry.
{"title":"Evolution of the magnetic field in spatially inhomogeneous axion structures","authors":"","doi":"10.1134/s0040577924030103","DOIUrl":"https://doi.org/10.1134/s0040577924030103","url":null,"abstract":"<span> <h3>Abstract</h3> <p> We study the time evolution of magnetic fields in various configurations of spatially inhomogeneous pseudoscalar fields that are a coherent superposition of axions. For such systems, we derive a new induction equation for the magnetic field, which takes this inhomogeneity into account. Based on this equation, we study the evolution of a pair of Chern–Simons waves interacting with a linearly decreasing pseudoscalar field. The nonzero gradient of the pseudoscalar field leads to the mixing of these waves. We then consider the problem in a compact domain in the case where the initial Chern–Simons wave is mirror symmetric. The pseudoscalar field inhomogeneity then leads to an effective change in the <span> <span>(alpha)</span> </span> dynamo parameter. Thus, the influence of a spatially inhomogeneous pseudoscalar field on the magnetic field evolution bears a strong dependence on the system geometry. </p> </span>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-03-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140198534","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"物理与天体物理","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
扫码关注我们
求助内容：
应助结果提醒方式：
京公网安备 11010802042870号