Pub Date : 2020-04-13DOI: 10.1007/978-3-030-61346-4_13
H. Feichtinger, F. Nicola, S. I. Trapasso
{"title":"On Exceptional Times for Pointwise Convergence of Integral Kernels in Feynman–Trotter Path Integrals","authors":"H. Feichtinger, F. Nicola, S. I. Trapasso","doi":"10.1007/978-3-030-61346-4_13","DOIUrl":"https://doi.org/10.1007/978-3-030-61346-4_13","url":null,"abstract":"","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84769703","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The Lohe tensor model is a first-order tensor-valued continuous-time model for the aggregation of tensors with the same rank and size. It reduces to well-known aggregation models such as the Kuramoto model, the Lohe sphere model and the Lohe matrix model as special cases for low-rank tensors. We present a sufficient and necessary framework for the solution splitting property(SSP) and analyze two possible asymptotic states(completely aggregate state and bi-polar state) which can emerge from a set of initial data. Moreover, we provide a sufficient framework leading to the aforementioned two asymptotic states in terms of initial data and system parameters.
{"title":"Complete aggregation of the Lohe tensor model with the same free flow","authors":"Seung‐Yeal Ha, Hansol Park","doi":"10.1063/5.0007292","DOIUrl":"https://doi.org/10.1063/5.0007292","url":null,"abstract":"The Lohe tensor model is a first-order tensor-valued continuous-time model for the aggregation of tensors with the same rank and size. It reduces to well-known aggregation models such as the Kuramoto model, the Lohe sphere model and the Lohe matrix model as special cases for low-rank tensors. We present a sufficient and necessary framework for the solution splitting property(SSP) and analyze two possible asymptotic states(completely aggregate state and bi-polar state) which can emerge from a set of initial data. Moreover, we provide a sufficient framework leading to the aforementioned two asymptotic states in terms of initial data and system parameters.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87650900","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The geometry and topology of cosmological spacetimes and vector bundles thereon are discussed. Global hyperbolicty and factorization properties that are normally assumed in bulk in the literature are derived from a minimal set of assumptions using recent progress in pure mathematics.
{"title":"Global hyperbolicity and factorization in cosmological models","authors":"Zh. Avetisyan","doi":"10.1063/5.0038970","DOIUrl":"https://doi.org/10.1063/5.0038970","url":null,"abstract":"The geometry and topology of cosmological spacetimes and vector bundles thereon are discussed. Global hyperbolicty and factorization properties that are normally assumed in bulk in the literature are derived from a minimal set of assumptions using recent progress in pure mathematics.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77708686","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-03DOI: 10.1007/978-3-030-63253-3_6
A. Duyunova, V. Lychagin, S. Tychkov
{"title":"Differential Invariants for Flows of Fluids and Gases","authors":"A. Duyunova, V. Lychagin, S. Tychkov","doi":"10.1007/978-3-030-63253-3_6","DOIUrl":"https://doi.org/10.1007/978-3-030-63253-3_6","url":null,"abstract":"","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"77336687","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We consider a quasilinear system of hyperbolic equations that describes plane one-dimensional non-relativistic oscillations of electrons in a cold plasma with allowance for electron-ion collisions. Accounting for collisions leads to the appearance of a term analogous to dry friction in a mechanical system, leading to a decrease in the total energy. We obtain a criterion for the existence of a global in time smooth solution to the Cauchy problem. It allows to accurately separate the initial data into two classes: one corresponds to a globally in time smooth solutions, and the other leads to a finite-time blowup. The influence of electron collision frequency $ nu $ on the solution is investigated. It is shown that there is a threshold value, after exceeding which the regime of damped oscillations is replaced by the regime of monotonic damping. The set of initial data corresponding to a globally in time smooth solution of the Cauchy problem expands with increasing $ nu $, however, at an arbitrarily large value there are smooth initial data for which the solution forms a singularity in a finite time, and this time tends to zero as $ nu $ tends to infinity. The character of the emerging singularities is illustrated by numerical examples.
我们考虑了一个准线性双曲方程系统，它描述了冷等离子体中允许电子-离子碰撞的平面一维非相对论性电子振荡。考虑碰撞导致出现一个类似于机械系统中干摩擦的术语，导致总能量的减少。得到了柯西问题的全局时间光滑解存在的一个判据。它允许将初始数据精确地分为两类:一类对应于全局时间平滑解，另一类导致有限时间爆炸。研究了电子碰撞频率对解的影响。结果表明，存在一个阈值，超过该阈值后，阻尼振荡的状态就被单调阻尼的状态所取代。柯西问题的全局时间光滑解对应的初始数据集随着$ nu $的增加而扩展，然而，在任意大的值下，存在其解在有限时间内形成奇点的光滑初始数据，并且该时间随着$ nu $趋于无穷而趋于零。通过数值算例说明了出现奇点的性质。
{"title":"Exact thresholds in the dynamics of cold plasma with electron-ion collisions","authors":"O. Rozanova, E. Chizhonkov, Maria I. Delova","doi":"10.1063/5.0033619","DOIUrl":"https://doi.org/10.1063/5.0033619","url":null,"abstract":"We consider a quasilinear system of hyperbolic equations that describes plane one-dimensional non-relativistic oscillations of electrons in a cold plasma with allowance for electron-ion collisions. Accounting for collisions leads to the appearance of a term analogous to dry friction in a mechanical system, leading to a decrease in the total energy. We obtain a criterion for the existence of a global in time smooth solution to the Cauchy problem. It allows to accurately separate the initial data into two classes: one corresponds to a globally in time smooth solutions, and the other leads to a finite-time blowup. The influence of electron collision frequency $ nu $ on the solution is investigated. It is shown that there is a threshold value, after exceeding which the regime of damped oscillations is replaced by the regime of monotonic damping. The set of initial data corresponding to a globally in time smooth solution of the Cauchy problem expands with increasing $ nu $, however, at an arbitrarily large value there are smooth initial data for which the solution forms a singularity in a finite time, and this time tends to zero as $ nu $ tends to infinity. The character of the emerging singularities is illustrated by numerical examples.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"88589058","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Motivated by the notion of Lagrangian multiforms, which provide a Lagrangian formulation of integrability, and by results of the authors on the role of covariant Hamiltonian formalism for integrable field theories, we propose the notion of Hamiltonian multiforms for integrable $1+1$-dimensional field theories. They provide the Hamiltonian counterpart of Lagrangian multiforms and encapsulate in a single object an arbitrary number of flows within an integrable hierarchy. For a given hierarchy, taking a Lagrangian multiform as starting point, we provide a systematic construction of a Hamiltonian multiform based on a generalisation of techniques of covariant Hamiltonian field theory. This also produces two other important objects: a symplectic multiform and the related multi-time Poisson bracket. They reduce to a multisymplectic form and the related covariant Poisson bracket if we restrict our attention to a single flow in the hierarchy. Our framework offers an alternative approach to define and derive conservation laws for a hierarchy. We illustrate our results on three examples: the potential Korteweg-de Vries hierarchy, the sine-Gordon hierarchy (in light cone coordinates) and the Ablowitz-Kaup-Newell-Segur hierarchy.
基于可积性的拉格朗日多重形式的概念，以及作者关于协变哈密顿形式在可积场论中的作用的结果，我们提出了可积1+1维场论的哈密顿多重形式的概念。它们提供了拉格朗日多形式的哈密顿形式，并将可积层次结构中的任意数量的流封装在单个对象中。对于给定的层次，以拉格朗日多重形式为出发点，基于协变哈密顿场论技术的推广，给出了哈密顿多重形式的系统构造。这也产生了另外两个重要的对象:辛多重形式和相关的多时间泊松括号。如果我们将注意力限制在层次结构中的单个流上，它们将简化为多辛形式和相关的协变泊松括号。我们的框架提供了另一种方法来定义和推导层次结构的守恒定律。我们用三个例子来说明我们的结果:潜在的Korteweg-de Vries层次结构，sin - gordon层次结构(光锥坐标)和ablowitz - kap - newwell - segur层次结构。
{"title":"Hamiltonian multiform description of an integrable hierarchy","authors":"V. Caudrelier, Matteo Stoppato","doi":"10.1063/5.0012153","DOIUrl":"https://doi.org/10.1063/5.0012153","url":null,"abstract":"Motivated by the notion of Lagrangian multiforms, which provide a Lagrangian formulation of integrability, and by results of the authors on the role of covariant Hamiltonian formalism for integrable field theories, we propose the notion of Hamiltonian multiforms for integrable $1+1$-dimensional field theories. They provide the Hamiltonian counterpart of Lagrangian multiforms and encapsulate in a single object an arbitrary number of flows within an integrable hierarchy. For a given hierarchy, taking a Lagrangian multiform as starting point, we provide a systematic construction of a Hamiltonian multiform based on a generalisation of techniques of covariant Hamiltonian field theory. This also produces two other important objects: a symplectic multiform and the related multi-time Poisson bracket. They reduce to a multisymplectic form and the related covariant Poisson bracket if we restrict our attention to a single flow in the hierarchy. Our framework offers an alternative approach to define and derive conservation laws for a hierarchy. We illustrate our results on three examples: the potential Korteweg-de Vries hierarchy, the sine-Gordon hierarchy (in light cone coordinates) and the Ablowitz-Kaup-Newell-Segur hierarchy.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87519973","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
A notion of local algebras is introduced in the theory of causal fermion systems. Their properties are studied in the example of the regularized Dirac sea vacuum in Minkowski space. The commutation relations are worked out, and the differences to the canonical commutation relations are discussed. It is shown that the spacetime point operators associated to a Cauchy surface satisfy a time slice axiom. It is proven that the algebra generated by operators in an open set is irreducible as a consequence of Hegerfeldt's theorem. The light cone structure is recovered by analyzing expectation values of the operators in the algebra in the limit when the regularization is removed. It is shown that every spacetime point operator commutes with the algebras localized away from its null cone, up to small corrections involving the regularization length.
{"title":"Local algebras for causal fermion systems in Minkowski space","authors":"F. Finster, Marco Oppio","doi":"10.1063/5.0011371","DOIUrl":"https://doi.org/10.1063/5.0011371","url":null,"abstract":"A notion of local algebras is introduced in the theory of causal fermion systems. Their properties are studied in the example of the regularized Dirac sea vacuum in Minkowski space. The commutation relations are worked out, and the differences to the canonical commutation relations are discussed. It is shown that the spacetime point operators associated to a Cauchy surface satisfy a time slice axiom. It is proven that the algebra generated by operators in an open set is irreducible as a consequence of Hegerfeldt's theorem. The light cone structure is recovered by analyzing expectation values of the operators in the algebra in the limit when the regularization is removed. It is shown that every spacetime point operator commutes with the algebras localized away from its null cone, up to small corrections involving the regularization length.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79440841","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2020-04-01DOI: 10.14311/ap.2020.60.0098
Amlan K. Halder, A. Paliathanasis, P. Leach
We study the similarity solutions and we determine the conservation laws of the various forms of beam equation, such as, Euler-Bernoulli, Rayleigh and Timoshenko-Prescott. The travelling-wave reduction leads to solvable fourth-order odes for all the forms. In addition, the reduction based on the scaling symmetry for the Euler-Bernoulli form leads to certain odes for which there exists zero symmetries. Therefore, we conduct the singularity analysis to ascertain the integrability. We study two reduced odes of order second and third. The reduced second-order ode is a perturbed form of Painleve-Ince equation, which is integrable and the third-order ode falls into the category of equations studied by Chazy, Bureau and Cosgrove. Moreover, we derived the symmetries and its corresponding reductions and conservation laws for the forced form of the above mentioned beam forms. The Lie Algebra is mentioned explicitly for all the cases.
{"title":"SIMILARITY SOLUTIONS AND CONSERVATION LAWS FOR THE BEAM EQUATIONS: A COMPLETE STUDY","authors":"Amlan K. Halder, A. Paliathanasis, P. Leach","doi":"10.14311/ap.2020.60.0098","DOIUrl":"https://doi.org/10.14311/ap.2020.60.0098","url":null,"abstract":"We study the similarity solutions and we determine the conservation laws of the various forms of beam equation, such as, Euler-Bernoulli, Rayleigh and Timoshenko-Prescott. The travelling-wave reduction leads to solvable fourth-order odes for all the forms. In addition, the reduction based on the scaling symmetry for the Euler-Bernoulli form leads to certain odes for which there exists zero symmetries. Therefore, we conduct the singularity analysis to ascertain the integrability. We study two reduced odes of order second and third. The reduced second-order ode is a perturbed form of Painleve-Ince equation, which is integrable and the third-order ode falls into the category of equations studied by Chazy, Bureau and Cosgrove. Moreover, we derived the symmetries and its corresponding reductions and conservation laws for the forced form of the above mentioned beam forms. The Lie Algebra is mentioned explicitly for all the cases.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"89244862","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this work, we present the analytical approach to the evaluation of the conditional measure Wiener path integral. We consider the time-dependent model parameters. We find the differential equation for the variable, determining the behavior of the harmonic as well the an-harmonic parts of the oscillator. We present the an-harmonic part of the result in the form of the operator function.
{"title":"Time-dependent propagator for an-harmonic oscillator with quartic term in potential","authors":"J. Boháčik, P. Prešnajder, P. August'in","doi":"10.1063/5.0018545","DOIUrl":"https://doi.org/10.1063/5.0018545","url":null,"abstract":"In this work, we present the analytical approach to the evaluation of the conditional measure Wiener path integral. We consider the time-dependent model parameters. We find the differential equation for the variable, determining the behavior of the harmonic as well the an-harmonic parts of the oscillator. We present the an-harmonic part of the result in the form of the operator function.","PeriodicalId":8469,"journal":{"name":"arXiv: Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2020-03-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"79690121","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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