Jaywan Chung, Seungmin Kang, Ho-Youn Kim, Yong-Jung Kim
The Dirichlet and Neumann conditions are commonly employed as boundary conditions for the heat equation, yet their legitimacy is debatable in certain scenarios. This paper aims to demonstrate that, in fact, diffusion laws autonomously select boundary conditions. To illustrate this, we incorporate the bounded domain into a larger domain with a diffusivity parameter ϵ > 0 and examine the solution’s behavior at the interface. Our findings reveal that homogeneous Neumann or Dirichlet boundary conditions emerge as ϵ → 0, contingent upon the type of the heterogeneous diffusion.
{"title":"Emergence of boundary conditions in the heat equation","authors":"Jaywan Chung, Seungmin Kang, Ho-Youn Kim, Yong-Jung Kim","doi":"10.1063/5.0215656","DOIUrl":"https://doi.org/10.1063/5.0215656","url":null,"abstract":"The Dirichlet and Neumann conditions are commonly employed as boundary conditions for the heat equation, yet their legitimacy is debatable in certain scenarios. This paper aims to demonstrate that, in fact, diffusion laws autonomously select boundary conditions. To illustrate this, we incorporate the bounded domain into a larger domain with a diffusivity parameter ϵ > 0 and examine the solution’s behavior at the interface. Our findings reveal that homogeneous Neumann or Dirichlet boundary conditions emerge as ϵ → 0, contingent upon the type of the heterogeneous diffusion.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"21 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528543","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}
In this paper we study the fractional Schrödinger–Poisson system ε2s(−Δ)su+V(x)u=ϕ|u|2s*−3u+|u|2s*−2u,ε2s(−Δ)sϕ=|u|2s*−1,x∈R3, where s ∈ (0, 1), ɛ > 0 is a small parameter, 2s*=63−2s is the critical Sobolev exponent and V∈L32s(R3) is a nonnegative function which may be zero in some regions of R3, e.g., it is of the critical frequency case. By virtue of a new global compactness lemma, and the Lusternik–Schnirelmann category theory, we relate the number of bound state solutions with the topology of the zero set where V attains its minimum for small values of ɛ.
{"title":"Multiple bound states for a class of fractional critical Schrödinger–Poisson systems with critical frequency","authors":"Xiaoming He, Yuxi Meng, Patrick Winkert","doi":"10.1063/5.0174872","DOIUrl":"https://doi.org/10.1063/5.0174872","url":null,"abstract":"In this paper we study the fractional Schrödinger–Poisson system ε2s(−Δ)su+V(x)u=ϕ|u|2s*−3u+|u|2s*−2u,ε2s(−Δ)sϕ=|u|2s*−1,x∈R3, where s ∈ (0, 1), ɛ > 0 is a small parameter, 2s*=63−2s is the critical Sobolev exponent and V∈L32s(R3) is a nonnegative function which may be zero in some regions of R3, e.g., it is of the critical frequency case. By virtue of a new global compactness lemma, and the Lusternik–Schnirelmann category theory, we relate the number of bound state solutions with the topology of the zero set where V attains its minimum for small values of ɛ.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"27 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528541","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}
Fabian R. Lux, Tom Stoiber, Shaoyun Wang, Guoliang Huang, Emil Prodan
Frieze groups are discrete subgroups of the full group of isometries of a flat strip. We investigate here the dynamics of specific architected materials generated by acting with a frieze group on a collection of self-coupling seed resonators. We demonstrate that, under unrestricted reconfigurations of the internal structures of the seed resonators, the dynamical matrices of the materials generate the full self-adjoint sector of the stabilized group C*-algebra of the frieze group. As a consequence, in applications where the positions, orientations and internal structures of the seed resonators are adiabatically modified, the spectral bands of the dynamical matrices carry a complete set of topological invariants that are fully accounted by the K-theory of the mentioned algebra. By resolving the generators of the K-theory, we produce the model dynamical matrices that carry the elementary topological charges, which we implement with systems of plate resonators to showcase several applications in spectral engineering. The paper is written in an expository style.
楣群是平面条带等距全群的离散子群。我们在此研究了通过楣群作用于一系列自耦合种子谐振器而产生的特定结构材料的动力学。我们证明，在种子谐振器内部结构不受限制地重新配置的情况下，材料的动力学矩阵会产生楣群稳定群 C* 代数的全自结合扇形。因此，在对种子谐振器的位置、方向和内部结构进行绝热修改的应用中，动力学矩阵的谱带会携带一套完整的拓扑不变式，这些不变式完全由上述代数的 K 理论所解释。通过解析 K 理论的生成器，我们产生了携带基本拓扑电荷的模型动力矩阵，并通过板谐振器系统将其实现，从而展示了光谱工程中的若干应用。本文以说明文的形式撰写。
{"title":"Topological spectral bands with frieze groups","authors":"Fabian R. Lux, Tom Stoiber, Shaoyun Wang, Guoliang Huang, Emil Prodan","doi":"10.1063/5.0127973","DOIUrl":"https://doi.org/10.1063/5.0127973","url":null,"abstract":"Frieze groups are discrete subgroups of the full group of isometries of a flat strip. We investigate here the dynamics of specific architected materials generated by acting with a frieze group on a collection of self-coupling seed resonators. We demonstrate that, under unrestricted reconfigurations of the internal structures of the seed resonators, the dynamical matrices of the materials generate the full self-adjoint sector of the stabilized group C*-algebra of the frieze group. As a consequence, in applications where the positions, orientations and internal structures of the seed resonators are adiabatically modified, the spectral bands of the dynamical matrices carry a complete set of topological invariants that are fully accounted by the K-theory of the mentioned algebra. By resolving the generators of the K-theory, we produce the model dynamical matrices that carry the elementary topological charges, which we implement with systems of plate resonators to showcase several applications in spectral engineering. The paper is written in an expository style.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"72 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}
This paper is concerned with periodic measures of fractional stochastic complex Ginzburg–Landau equations with variable time delay on unbounded domains. We first derive the uniform estimates of solutions. Then we establish the regularity and prove the equicontinuity of solutions in probability, which is used to prove the tightness of distributions of solutions. In order to overcome the non-compactness of Sobolev embeddings on unbounded domains, we use the uniform estimates on the tails in probability. As a result, we prove the existence of periodic measures by combining Arzelà-Ascoli theorem and Krylov-Bogolyubov method.
{"title":"Existence of periodic measures of fractional stochastic delay complex Ginzburg-Landau equations on Rn","authors":"Zhiyu Li, Xiaomin Song, Gang He, Ji Shu","doi":"10.1063/5.0180975","DOIUrl":"https://doi.org/10.1063/5.0180975","url":null,"abstract":"This paper is concerned with periodic measures of fractional stochastic complex Ginzburg–Landau equations with variable time delay on unbounded domains. We first derive the uniform estimates of solutions. Then we establish the regularity and prove the equicontinuity of solutions in probability, which is used to prove the tightness of distributions of solutions. In order to overcome the non-compactness of Sobolev embeddings on unbounded domains, we use the uniform estimates on the tails in probability. As a result, we prove the existence of periodic measures by combining Arzelà-Ascoli theorem and Krylov-Bogolyubov method.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"76 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528544","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}
We show the representability of density-entropy pairs with canonical and grand-canonical states, and we provide bounds on the kinetic energy of the representing states.
我们展示了具有典型态和大典型态的密度-熵对的可表征性，并提供了表征态的动能边界。
{"title":"Mixed state representability of entropy-density pairs","authors":"Louis Garrigue","doi":"10.1063/5.0169120","DOIUrl":"https://doi.org/10.1063/5.0169120","url":null,"abstract":"We show the representability of density-entropy pairs with canonical and grand-canonical states, and we provide bounds on the kinetic energy of the representing states.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"5 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509065","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}
In this paper, we aim to discuss a class of (k1, k2)-type Hessian system with gradient terms. In the case of k1 = k2 = 1 and 2 ≤ k1, k2 ≤ n, we obtain a sufficient and necessary condition for the existence of the entire admissible sub-solution of the system according to the value range of different parameters, which is also called the generalized Keller–Osserman condition. Based on this, we also discuss the conditions of existence and non-existence of the entire sub-solution, respectively. Finally, we extend the nonlinear terms to the degenerate case and consider the condition of the existence of the positive sub-solution for the above system.
{"title":"Necessary and sufficient conditions of entire sub-solutions for a (k1, k2)-type Hessian systems with gradient terms","authors":"Chenghua Gao, Xingyue He","doi":"10.1063/5.0192926","DOIUrl":"https://doi.org/10.1063/5.0192926","url":null,"abstract":"In this paper, we aim to discuss a class of (k1, k2)-type Hessian system with gradient terms. In the case of k1 = k2 = 1 and 2 ≤ k1, k2 ≤ n, we obtain a sufficient and necessary condition for the existence of the entire admissible sub-solution of the system according to the value range of different parameters, which is also called the generalized Keller–Osserman condition. Based on this, we also discuss the conditions of existence and non-existence of the entire sub-solution, respectively. Finally, we extend the nonlinear terms to the degenerate case and consider the condition of the existence of the positive sub-solution for the above system.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"18 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509066","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}
Polar duality is a well-known concept from convex geometry and analysis. In the present paper we study a symplectically covariant versions of polar duality, having in mind their applications to quantum harmonic analysis. It makes use of the standard symplectic form on phase space and allows a precise study of the covariance matrix of a density operator.
{"title":"Symplectic and Lagrangian polar duality; applications to quantum harmonic analysis","authors":"Maurice de Gosson, Charlyne de Gosson","doi":"10.1063/5.0192334","DOIUrl":"https://doi.org/10.1063/5.0192334","url":null,"abstract":"Polar duality is a well-known concept from convex geometry and analysis. In the present paper we study a symplectically covariant versions of polar duality, having in mind their applications to quantum harmonic analysis. It makes use of the standard symplectic form on phase space and allows a precise study of the covariance matrix of a density operator.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"144 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528546","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}
We consider the spectral radius of a large random matrix X with independent, identically distributed entries. We show that its typical size is given by a precise three-term asymptotics with an optimal error term beyond the radius of the celebrated circular law. The coefficients in this asymptotics are universal but they differ from a similar asymptotics recently proved for the rightmost eigenvalue of X in Cipolloni et al., Ann. Probab. 51(6), 2192–2242 (2023). To access the more complicated spectral radius, we need to establish a new decorrelation mechanism for the low-lying singular values of X − z for different complex shift parameters z using the Dyson Brownian Motion.
我们考虑了具有独立同分布条目的大型随机矩阵 X 的频谱半径。我们证明，其典型大小是由精确的三项渐近法给出的，其最佳误差项超出了著名的圆周率半径。这个渐近中的系数是通用的，但与最近在 Cipolloni 等人的 Ann.Probab.51(6), 2192-2242 (2023).为了获得更复杂的频谱半径，我们需要利用戴森布朗运动（Dyson Brownian Motion）为不同复变参数 z 的 X - z 低层奇异值建立一种新的去相关机制。
{"title":"Precise asymptotics for the spectral radius of a large random matrix","authors":"Giorgio Cipolloni, László Erdős, Yuanyuan Xu","doi":"10.1063/5.0209705","DOIUrl":"https://doi.org/10.1063/5.0209705","url":null,"abstract":"We consider the spectral radius of a large random matrix X with independent, identically distributed entries. We show that its typical size is given by a precise three-term asymptotics with an optimal error term beyond the radius of the celebrated circular law. The coefficients in this asymptotics are universal but they differ from a similar asymptotics recently proved for the rightmost eigenvalue of X in Cipolloni et al., Ann. Probab. 51(6), 2192–2242 (2023). To access the more complicated spectral radius, we need to establish a new decorrelation mechanism for the low-lying singular values of X − z for different complex shift parameters z using the Dyson Brownian Motion.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"20 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528545","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}
In this article, we investigate the Nordström–Vlasov system in the whole space. The kinetic model is a relativistic generalization of the classical Vlasov–Poisson system in the gravitational case and describes the ensemble motion of collisionless particles interacting by means of a self-consistent scalar gravitational field. With the Fourier analysis and the smoothing effect of low velocity particles, we get a better regularity of weak solutions for the field than the one proved by Calogero and Rein [J. Differ. Equ. 204, 323 (2004)]. Meanwhile, under the additional integrability condition, we establish the energy conservation of the weak solution.
在本文中，我们研究了整个空间中的诺德斯特伦-弗拉索夫系统。该动力学模型是经典弗拉索夫-泊松系统在引力情况下的相对论广义化，通过自洽标量引力场描述了相互作用的无碰撞粒子的集合运动。通过傅立叶分析和低速粒子的平滑效应，我们得到了比 Calogero 和 Rein [J. Differ. Equ. 204, 323 (2004)]证明的更好的场弱解的正则性。同时，在附加的可整性条件下，我们建立了弱解的能量守恒。
{"title":"Properties of weak solutions to the Nordström–Vlasov system","authors":"Meixia Xiao","doi":"10.1063/5.0150177","DOIUrl":"https://doi.org/10.1063/5.0150177","url":null,"abstract":"In this article, we investigate the Nordström–Vlasov system in the whole space. The kinetic model is a relativistic generalization of the classical Vlasov–Poisson system in the gravitational case and describes the ensemble motion of collisionless particles interacting by means of a self-consistent scalar gravitational field. With the Fourier analysis and the smoothing effect of low velocity particles, we get a better regularity of weak solutions for the field than the one proved by Calogero and Rein [J. Differ. Equ. 204, 323 (2004)]. Meanwhile, under the additional integrability condition, we establish the energy conservation of the weak solution.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"214 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141528547","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}
We describe the homotopy classes of loops in the space of 2 × 2 simple (=non-degenerate) matrices with various symmetries. This turns out to be an elementary exercise in the homotopy of closed curves in R3/{0}. Since closed curves in R3/{0} can be readily visualized, no advanced tools of algebraic topology are needed. The matrices represent gapped Bloch Hamiltonians in 1D with a two dimensional Hilbert space per unit cell.
{"title":"Homotopy of periodic 2 × 2 matrices","authors":"Joseph E. Avron, Ari M. Turner","doi":"10.1063/5.0138809","DOIUrl":"https://doi.org/10.1063/5.0138809","url":null,"abstract":"We describe the homotopy classes of loops in the space of 2 × 2 simple (=non-degenerate) matrices with various symmetries. This turns out to be an elementary exercise in the homotopy of closed curves in R3/{0}. Since closed curves in R3/{0} can be readily visualized, no advanced tools of algebraic topology are needed. The matrices represent gapped Bloch Hamiltonians in 1D with a two dimensional Hilbert space per unit cell.","PeriodicalId":16174,"journal":{"name":"Journal of Mathematical Physics","volume":"65 1","pages":""},"PeriodicalIF":1.3,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141198334","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"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}
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