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The 16th Hilbert Problem for Discontinuous Piecewise Linear Differential Systems Separated by the Algebraic Curve (y=x^{n}) 用代数曲线分离的不连续分段线性微分系统的第16 Hilbert问题 (y=x^{n})
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-09-19 DOI: 10.1007/s11040-023-09467-4
Jaume Llibre, Claudia Valls
{"title":"The 16th Hilbert Problem for Discontinuous Piecewise Linear Differential Systems Separated by the Algebraic Curve (y=x^{n})","authors":"Jaume Llibre,&nbsp;Claudia Valls","doi":"10.1007/s11040-023-09467-4","DOIUrl":"10.1007/s11040-023-09467-4","url":null,"abstract":"<div><p>We consider planar piecewise discontinuous differential systems formed by either linear centers or linear Hamiltonian saddles and separated by the algebraic curve <span>(y=x^n)</span> with <span>(n ge 2)</span>. We provide in a very short way an upper bound of the number of limit cycles that these differential systems can have in terms of <i>n</i>, proving the extended 16th Hilbert problem in this case. In particular, we show that for <span>(n=2)</span> this bound can be reached.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 4","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-09-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-023-09467-4.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"7183673","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Existence and Uniqueness of Solutions to Backward 2D and 3D Stochastic Convective Brinkman–Forchheimer Equations Forced by Lévy Noise lsamvy噪声强迫下的倒向二维和三维随机对流Brinkman-Forchheimer方程解的存在唯一性
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-07-03 DOI: 10.1007/s11040-023-09458-5
Manil T. Mohan
{"title":"Existence and Uniqueness of Solutions to Backward 2D and 3D Stochastic Convective Brinkman–Forchheimer Equations Forced by Lévy Noise","authors":"Manil T. Mohan","doi":"10.1007/s11040-023-09458-5","DOIUrl":"10.1007/s11040-023-09458-5","url":null,"abstract":"<div><p>The two- and three-dimensional incompressible backward stochastic convective Brinkman–Forchheimer (BSCBF) equations on a torus driven by Lévy noise are considered in this paper. A-priori estimates for adapted solutions of the finite-dimensional approximation of 2D and 3D BSCBF equations are obtained. For a given terminal data, the existence and uniqueness of pathwise adapted strong solutions is proved by using a standard Galerkin (or spectral) approximation technique and exploiting the monotonicity arguments. We also establish the continuity of the adapted solutions with respect to the terminal data. The above results are obtained for the absorption exponent <span>(rin [1,infty ))</span> for <span>(d=2)</span> and <span>(rin [3,infty ))</span> for <span>(d=3)</span>, and any Brinkman coefficient <span>(mu &gt;0)</span>, Forchheimer coefficient <span>(beta &gt;0)</span>, and hence the 3D critical case (<span>(r=3)</span>) is also handled successfully. We deduce analogous results for 2D backward stochastic Navier–Stokes equations perturbed by Lévy noise also.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 3","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-023-09458-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4125172","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Self-Adjointness of a Class of Multi-Spin–Boson Models with Ultraviolet Divergences 一类具有紫外发散的多自旋玻色子模型的自伴随性
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-06-12 DOI: 10.1007/s11040-023-09457-6
Davide Lonigro
{"title":"Self-Adjointness of a Class of Multi-Spin–Boson Models with Ultraviolet Divergences","authors":"Davide Lonigro","doi":"10.1007/s11040-023-09457-6","DOIUrl":"10.1007/s11040-023-09457-6","url":null,"abstract":"<div><p>We study a class of quantum Hamiltonian models describing a family of <i>N</i> two-level systems (spins) coupled with a structured boson field of positive mass, with a rotating-wave coupling mediated by form factors possibly exhibiting ultraviolet divergences. Spin–spin interactions which do not modify the total number of excitations are also included. Generalizing previous results in the single-spin case, we provide explicit expressions for the self-adjointness domain and the resolvent of these models, both of them carrying an intricate dependence on the spin–field and spin–spin coupling via a family of concatenated propagators. This construction is also shown to be stable, in the norm resolvent sense, under approximations of the form factors via normalizable ones, for example an ultraviolet cutoff.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-023-09457-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4801137","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
On Quantum Optimal Transport 关于量子最优输运
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-06-06 DOI: 10.1007/s11040-023-09456-7
Sam Cole, Michał Eckstein, Shmuel Friedland, Karol Życzkowski
{"title":"On Quantum Optimal Transport","authors":"Sam Cole,&nbsp;Michał Eckstein,&nbsp;Shmuel Friedland,&nbsp;Karol Życzkowski","doi":"10.1007/s11040-023-09456-7","DOIUrl":"10.1007/s11040-023-09456-7","url":null,"abstract":"<div><p>We analyze a quantum version of the Monge–Kantorovich optimal transport problem. The quantum transport cost related to a Hermitian cost matrix <i>C</i> is minimized over the set of all bipartite coupling states <span>(rho ^{AB})</span> with fixed reduced density matrices <span>(rho ^A)</span> and <span>(rho ^B)</span> of size <i>m</i> and <i>n</i>. The minimum quantum optimal transport cost <span>(textrm{T}^Q_{C}(rho ^A,rho ^B))</span> can be efficiently computed using semidefinite programming. In the case <span>(m=n)</span> the cost <span>(textrm{T}^Q_{C})</span> gives a semidistance if and only if <i>C</i> is positive semidefinite and vanishes exactly on the subspace of symmetric matrices. Furthermore, if <i>C</i> satisfies the above conditions, then <span>(sqrt{textrm{T}^Q_{C}})</span> induces a quantum analogue of the Wasserstein-2 distance. Taking the quantum cost matrix <span>(C^Q)</span> to be the projector on the antisymmetric subspace, we provide a semi-analytic expression for <span>(textrm{T}^Q_{C^Q})</span> for any pair of single-qubit states and show that its square root yields a transport distance on the Bloch ball. Numerical simulations suggest that this property holds also in higher dimensions. Assuming that the cost matrix suffers decoherence and that the density matrices become diagonal, we study the quantum-to-classical transition of the Monge–Kantorovich distance, propose a continuous family of interpolating distances, and demonstrate that the quantum transport is cheaper than the classical one. Furthermore, we introduce a related quantity—the SWAP-fidelity—and compare its properties with the standard Uhlmann–Jozsa fidelity. We also discuss the quantum optimal transport for general <i>d</i>-partite systems.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-06-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-023-09456-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4259382","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Braided Hopf Algebras and Gauge Transformations II: (*)-Structures and Examples 编织Hopf代数和规范变换II: (*) -结构和例子
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-05-11 DOI: 10.1007/s11040-023-09454-9
Paolo Aschieri, Giovanni Landi, Chiara Pagani
{"title":"Braided Hopf Algebras and Gauge Transformations II: (*)-Structures and Examples","authors":"Paolo Aschieri,&nbsp;Giovanni Landi,&nbsp;Chiara Pagani","doi":"10.1007/s11040-023-09454-9","DOIUrl":"10.1007/s11040-023-09454-9","url":null,"abstract":"<div><p>We consider noncommutative principal bundles which are equivariant under a triangular Hopf algebra. We present explicit examples of infinite dimensional braided Lie and Hopf algebras of infinitesimal gauge transformations of bundles on noncommutative spheres. The braiding of these algebras is implemented by the triangular structure of the symmetry Hopf algebra. We present a systematic analysis of compatible <span>(*)</span>-structures, encompassing the quasitriangular case.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-05-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-023-09454-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4474601","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Consistency of the Bäcklund Transformation for the Spin Calogero–Moser System 自旋卡罗热-莫泽体系Bäcklund变换的一致性
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-05-08 DOI: 10.1007/s11040-023-09450-z
Bjorn K. Berntson
{"title":"Consistency of the Bäcklund Transformation for the Spin Calogero–Moser System","authors":"Bjorn K. Berntson","doi":"10.1007/s11040-023-09450-z","DOIUrl":"10.1007/s11040-023-09450-z","url":null,"abstract":"<div><p>We prove the consistency of the Bäcklund transformation (BT) for the spin Calogero–Moser (sCM) system in the rational, trigonometric, and hyperbolic cases. The BT for the sCM system consists of an overdetermined system of ordinary differential equations; to establish our result, we construct and analyze certain functions that measure the departure of this overdetermined system from consistency. We show that these functions are identically zero and that this allows for a unique solution to the initial value problem for the overdetermined system.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-023-09450-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4353637","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Hydrodynamical Behavior for the Symmetric Simple Partial Exclusion with Open Boundary 开边界对称简单部分不相容的流体力学行为
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-05-08 DOI: 10.1007/s11040-023-09446-9
C. Franceschini, P. Gonçalves, B. Salvador
{"title":"Hydrodynamical Behavior for the Symmetric Simple Partial Exclusion with Open Boundary","authors":"C. Franceschini,&nbsp;P. Gonçalves,&nbsp;B. Salvador","doi":"10.1007/s11040-023-09446-9","DOIUrl":"10.1007/s11040-023-09446-9","url":null,"abstract":"<div><p>We analyze the symmetric simple partial exclusion process, which allows at most <span>(alpha )</span> particles per site, and we put it in contact with stochastic reservoirs whose strength is regulated by a parameter <span>(theta in {mathbb {R}})</span>. We prove that the hydrodynamic behavior is given by the heat equation and depending on the value of <span>(theta )</span>, the equation is supplemented with different boundary conditions. Setting <span>(alpha = 1)</span> we find the results known in Baldasso et al. (J Stat Phys 167(5):1112–1142, 2017) and Bernardin et al. (Markov Processes Relat. Fields 25:217–274, 2017) for the symmetric simple exclusion process.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-05-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4350497","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 4
A Riemann Hilbert Approach to the Study of the Generating Function Associated to the Pearcey Process 用黎曼希尔伯特方法研究与皮尔斯过程相关的生成函数
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-04-25 DOI: 10.1007/s11040-023-09455-8
Thomas Chouteau
{"title":"A Riemann Hilbert Approach to the Study of the Generating Function Associated to the Pearcey Process","authors":"Thomas Chouteau","doi":"10.1007/s11040-023-09455-8","DOIUrl":"10.1007/s11040-023-09455-8","url":null,"abstract":"<div><p>Using Riemann–Hilbert methods, we establish a Tracy–Widom like formula for the generating function of the occupancy numbers of the Pearcey process. This formula is linked to a coupled vector differential equation of order three. We also obtain a non linear coupled heat equation. Combining these two equations we obtain a PDE for the logarithm of the the generating function of the Pearcey process.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-04-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4961850","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 1
Gibbs Measures for HC-Model with a Cuountable Set of Spin Values on a Cayley Tree Cayley树上具有可计数自旋值集的hc模型的Gibbs测度
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-03-28 DOI: 10.1007/s11040-023-09453-w
R. M. Khakimov, M. T. Makhammadaliev, U. A. Rozikov
{"title":"Gibbs Measures for HC-Model with a Cuountable Set of Spin Values on a Cayley Tree","authors":"R. M. Khakimov,&nbsp;M. T. Makhammadaliev,&nbsp;U. A. Rozikov","doi":"10.1007/s11040-023-09453-w","DOIUrl":"10.1007/s11040-023-09453-w","url":null,"abstract":"<div><p>In this paper, we study the HC-model with a countable set <span>(mathbb Z)</span> of spin values on a Cayley tree of order <span>(kge 2)</span>. This model is defined by a countable set of parameters (that is, the activity function <span>(lambda _i&gt;0)</span>, <span>(iin mathbb Z)</span>). A functional equation is obtained that provides the consistency condition for finite-dimensional Gibbs distributions. Analyzing this equation, the following results are obtained:</p><ul>\u0000 <li>\u0000 <p>Let <span>(Lambda =sum _ilambda _i)</span>. For <span>(Lambda =+infty )</span> there is no translation-invariant Gibbs measure (TIGM) and no two-periodic Gibbs measure (TPGM);</p>\u0000 </li>\u0000 <li>\u0000 <p>For <span>(Lambda &lt;+infty )</span>, the uniqueness of TIGM is proved;</p>\u0000 </li>\u0000 <li>\u0000 <p>Let <span>(Lambda _textrm{cr}(k)=frac{k^k}{(k-1)^{k+1}})</span>. If <span>(0&lt;Lambda le Lambda _textrm{cr})</span>, then there is exactly one TPGM that is TIGM;</p>\u0000 </li>\u0000 <li>\u0000 <p>For <span>(Lambda &gt;Lambda _textrm{cr})</span>, there are exactly three TPGMs, one of which is TIGM.</p>\u0000 </li>\u0000 </ul></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 2","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s11040-023-09453-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"5090994","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
A Comparison of Two Quantum Distances 两个量子距离的比较
IF 1 3区 数学
Mathematical Physics, Analysis and Geometry Pub Date : 2023-03-13 DOI: 10.1007/s11040-023-09451-y
Jens Kaad, David Kyed
{"title":"A Comparison of Two Quantum Distances","authors":"Jens Kaad,&nbsp;David Kyed","doi":"10.1007/s11040-023-09451-y","DOIUrl":"10.1007/s11040-023-09451-y","url":null,"abstract":"<div><p>We show that Rieffel’s quantum Gromov–Hausdorff distance between two compact quantum metric spaces is not equivalent to the ordinary Gromov–Hausdorff distance applied to the associated state spaces.</p></div>","PeriodicalId":694,"journal":{"name":"Mathematical Physics, Analysis and Geometry","volume":"26 1","pages":""},"PeriodicalIF":1.0,"publicationDate":"2023-03-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"4549296","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 2
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