Pub Date : 2024-02-01DOI: 10.1134/s004057792402003x
Abstract
Random Hamiltonian flows in an infinite-dimensional phase space is represented by random unitary groups in a Hilbert space. For this, the phase space is equipped with a measure that is invariant under a group of symplectomorphisms. The obtained representation of random flows allows applying the Chernoff averaging technique to random processes with values in the group of nonlinear operators. The properties of random unitary groups and the limit distribution for their compositions are described.
{"title":"Unitary representation of walks along random vector fields and the Kolmogorov–Fokker–Planck equation in a Hilbert space","authors":"","doi":"10.1134/s004057792402003x","DOIUrl":"https://doi.org/10.1134/s004057792402003x","url":null,"abstract":"<span> <h3>Abstract</h3> <p> Random Hamiltonian flows in an infinite-dimensional phase space is represented by random unitary groups in a Hilbert space. For this, the phase space is equipped with a measure that is invariant under a group of symplectomorphisms. The obtained representation of random flows allows applying the Chernoff averaging technique to random processes with values in the group of nonlinear operators. The properties of random unitary groups and the limit distribution for their compositions are described. </p> </span>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139987945","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 : 2024-01-01DOI: 10.1134/s0040577924010100
Abstract
For a system of nonlinear integral equations on the semiaxis, we study a boundary value problem whose matrix kernel has unit spectral radius. This boundary value problem has applications in various areas of physics and biology. In particular, such problems arise in the dynamical theory of (p)-adic strings for the scalar field of tachyons, in the mathematical theory of spread of epidemic diseases, in the kinetic theory of gases, and in the theory of radiative transfer. The questions of the existence, absence, and uniqueness of a nontrivial solution of this boundary value problem are discussed. In particular, it is proved that a boundary value problem with a zero boundary conditions at infinity has only a trivial solution in the class of nonnegative and bounded functions. It is also proved that if at least one of the values at infinity is positive, then this problem has a convex nontrivial nonnegative bounded and continuous solution. At the end of this paper, examples of the matrix kernel and nonlinearity are provided that satisfy all the conditions of the proved theorems.
{"title":"On qualitative properties of the solution of a boundary value problem for a system of nonlinear integral equations","authors":"","doi":"10.1134/s0040577924010100","DOIUrl":"https://doi.org/10.1134/s0040577924010100","url":null,"abstract":"<span> <h3>Abstract</h3> <p> For a system of nonlinear integral equations on the semiaxis, we study a boundary value problem whose matrix kernel has unit spectral radius. This boundary value problem has applications in various areas of physics and biology. In particular, such problems arise in the dynamical theory of <span> <span>(p)</span> </span>-adic strings for the scalar field of tachyons, in the mathematical theory of spread of epidemic diseases, in the kinetic theory of gases, and in the theory of radiative transfer. The questions of the existence, absence, and uniqueness of a nontrivial solution of this boundary value problem are discussed. In particular, it is proved that a boundary value problem with a zero boundary conditions at infinity has only a trivial solution in the class of nonnegative and bounded functions. It is also proved that if at least one of the values at infinity is positive, then this problem has a convex nontrivial nonnegative bounded and continuous solution. At the end of this paper, examples of the matrix kernel and nonlinearity are provided that satisfy all the conditions of the proved theorems. </p> </span>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139662021","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 : 2024-01-01DOI: 10.1134/s0040577924010082
Abstract
We consider a second-order quasilinear elliptic equation with an integrable right-hand side. We formulate constraints on the structure of the equation in terms of a generalized (N)-function. We prove the existence of an entropic solution of the Dirichlet problem in nonreflexive Musielak–Orlicz–Sobolev spaces in an arbitrary unbounded strictly Lipschitz domain.
{"title":"Existence of an entropic solution of a nonlinear elliptic problem in an unbounded domain","authors":"","doi":"10.1134/s0040577924010082","DOIUrl":"https://doi.org/10.1134/s0040577924010082","url":null,"abstract":"<span> <h3>Abstract</h3> <p> We consider a second-order quasilinear elliptic equation with an integrable right-hand side. We formulate constraints on the structure of the equation in terms of a generalized <span> <span>(N)</span> </span>-function. We prove the existence of an entropic solution of the Dirichlet problem in nonreflexive Musielak–Orlicz–Sobolev spaces in an arbitrary unbounded strictly Lipschitz domain. </p> </span>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139662033","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 : 2024-01-01DOI: 10.1134/s0040577924010021
Abstract
A model Helmholtz equation with a localized right-hand side is considered. When writing asymptotics of a solution satisfying the limit absorption principle, a Lagrangian surface naturally appears that has a logarithmic singularity at one point. Because of this singularity, the solution is localized not only in a neighborhood of the projection of the Lagrangian surface onto the coordinate space but also in a neighborhood of a certain ray “escaping” from the Lagrangian surface and going into the region forbidden in the classical approximation.
{"title":"Arnold Lagrangian singularity in the asymptotics of the solution of a model two-dimensional Helmholtz equation with a localized right-hand side","authors":"","doi":"10.1134/s0040577924010021","DOIUrl":"https://doi.org/10.1134/s0040577924010021","url":null,"abstract":"<span> <h3>Abstract</h3> <p> A model Helmholtz equation with a localized right-hand side is considered. When writing asymptotics of a solution satisfying the limit absorption principle, a Lagrangian surface naturally appears that has a logarithmic singularity at one point. Because of this singularity, the solution is localized not only in a neighborhood of the projection of the Lagrangian surface onto the coordinate space but also in a neighborhood of a certain ray “escaping” from the Lagrangian surface and going into the region forbidden in the classical approximation. </p> </span>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2024-01-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139661921","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 : 2023-12-22DOI: 10.1134/s0040577923120024
I. Ya. Aref’eva
Abstract
We study the problem of the existence of the quarkyonic phase in quantum chromodynamics. This phase can exists under certain conditions in quantum chromodynamics along with the phase of free quarks and the confinement phase. As is known, the confinement phase is characterized by the presence of a linear potential between quarks, and the quarks are confined to one hadron (meson or baryon). A linear potential between quarks also exists in the quarkyonic phase; however, it is not so strong to confine quarks inside one hadron. The characteristics of the quarkyonic phase, as well as the confinement phase, can be calculated in quantum chromodynamics only in a nonperturbative framework. We interpret the previously obtained results of Wilson loop calculations in the holographic approach in terms of a phase transition to the quarkyonic phase.
{"title":"On the quarkyonic phase in the holographic approach","authors":"I. Ya. Aref’eva","doi":"10.1134/s0040577923120024","DOIUrl":"https://doi.org/10.1134/s0040577923120024","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the problem of the existence of the quarkyonic phase in quantum chromodynamics. This phase can exists under certain conditions in quantum chromodynamics along with the phase of free quarks and the confinement phase. As is known, the confinement phase is characterized by the presence of a linear potential between quarks, and the quarks are confined to one hadron (meson or baryon). A linear potential between quarks also exists in the quarkyonic phase; however, it is not so strong to confine quarks inside one hadron. The characteristics of the quarkyonic phase, as well as the confinement phase, can be calculated in quantum chromodynamics only in a nonperturbative framework. We interpret the previously obtained results of Wilson loop calculations in the holographic approach in terms of a phase transition to the quarkyonic phase. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139024064","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 : 2023-12-22DOI: 10.1134/s0040577923120140
L. O. Chekhov
Abstract
We show that having any planar (cyclic or acyclic) directed network on a disc with the only condition that all (n_1+m) sources are separated from all (n_2+m) sinks, we can construct a cluster-algebra realization of elements of an affine Lie–Poisson algebra (R(lambda,mu){stackrel{1}{T}}(lambda){stackrel{2}{T}}(mu) ={stackrel{2}{T}}(mu){stackrel{1}{T}}(lambda)R(lambda,mu)) with (({n_1times n_2}))-matrices (T(lambda)). Upon satisfaction of some invertibility conditions, we can construct a realization of a quantum loop algebra. Having the quantum loop algebra, we can also construct a realization of the twisted Yangian algebra or of the quantum reflection equation. For each such a planar network, we can therefore construct a symplectic leaf of the corresponding infinite-dimensional algebra.
Abstract 我们证明，在一个圆盘上有任何平面（循环或非循环）有向网络，唯一的条件是所有（n_1+m）源与所有（n_2+m）汇分离、我们可以构建一个由仿射李-泊松代数（R(lambda、mu){stackrel{1}{T}}(lambda){stackrel{2}{T}}(mu) ={stackrel{2}{T}}(mu){stackrel{1}{T}}(lambda)R(lambda,mu)) with (({n_1times n_2}))-matrices (T(lambda)).在满足一些可逆性条件后，我们就可以构造一个量子环代数的实现。有了量子环代数，我们还可以构造一个扭曲扬基代数或量子反射方程的实现。因此，对于每一个这样的平面网络，我们都可以构造出相应无穷维代数的交点叶。
{"title":"Cluster variables for affine Lie–Poisson systems","authors":"L. O. Chekhov","doi":"10.1134/s0040577923120140","DOIUrl":"https://doi.org/10.1134/s0040577923120140","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We show that having any planar (cyclic or acyclic) directed network on a disc with the only condition that all <span>(n_1+m)</span> sources are separated from all <span>(n_2+m)</span> sinks, we can construct a cluster-algebra realization of elements of an affine Lie–Poisson algebra <span>(R(lambda,mu){stackrel{1}{T}}(lambda){stackrel{2}{T}}(mu) ={stackrel{2}{T}}(mu){stackrel{1}{T}}(lambda)R(lambda,mu))</span> with <span>(({n_1times n_2}))</span>-matrices <span>(T(lambda))</span>. Upon satisfaction of some invertibility conditions, we can construct a realization of a quantum loop algebra. Having the quantum loop algebra, we can also construct a realization of the twisted Yangian algebra or of the quantum reflection equation. For each such a planar network, we can therefore construct a symplectic leaf of the corresponding infinite-dimensional algebra. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139030404","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 : 2023-12-22DOI: 10.1134/s0040577923120152
A. V. Chukhnova
Abstract
We study the simultaneous interaction of a neutrino with matter and the electromagnetic field in the two-flavor model. We show that (T)-invariance violating terms can appear in the probabilities of not only spin-flip transitions but also flavor transitions between states with the same helicity in the case of the interaction via charged currents.
{"title":"Violation of the $$T$$ invariance in the probabilities of spin–flavor transitions of neutrino characterized by a real mixing matrix","authors":"A. V. Chukhnova","doi":"10.1134/s0040577923120152","DOIUrl":"https://doi.org/10.1134/s0040577923120152","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We study the simultaneous interaction of a neutrino with matter and the electromagnetic field in the two-flavor model. We show that <span>(T)</span>-invariance violating terms can appear in the probabilities of not only spin-flip transitions but also flavor transitions between states with the same helicity in the case of the interaction via charged currents. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139030428","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 : 2023-12-22DOI: 10.1134/s0040577923120115
A. V. Razumov
Abstract
We give a detailed derivation of the commutation relations for the Poincaré–Birkhoff–Witt generators of the quantum superalgebra (mathrm U_q(mathfrak{gl}_{M|N})).
{"title":"On Poincaré–Birkhoff–Witt basis of the quantum general linear superalgebra","authors":"A. V. Razumov","doi":"10.1134/s0040577923120115","DOIUrl":"https://doi.org/10.1134/s0040577923120115","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We give a detailed derivation of the commutation relations for the Poincaré–Birkhoff–Witt generators of the quantum superalgebra <span>(mathrm U_q(mathfrak{gl}_{M|N}))</span>. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139030455","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 : 2023-12-22DOI: 10.1134/s0040577923120085
G. Kulkarni, N. A. Slavnov
Abstract
We consider an (XYZ) spin chain within the framework of the generalized algebraic Bethe ansatz. We calculate the actions of monodromy matrix elements on Bethe vectors as linear combinations of new Bethe vectors. We also compute the multiple action of the gauge-transformed monodromy matrix elements on the pre-Bethe vector and express the results in terms of the partition function of the (8)-vertex model.
{"title":"Action of the monodromy matrix elements in the generalized algebraic Bethe ansatz","authors":"G. Kulkarni, N. A. Slavnov","doi":"10.1134/s0040577923120085","DOIUrl":"https://doi.org/10.1134/s0040577923120085","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We consider an <span>(XYZ)</span> spin chain within the framework of the generalized algebraic Bethe ansatz. We calculate the actions of monodromy matrix elements on Bethe vectors as linear combinations of new Bethe vectors. We also compute the multiple action of the gauge-transformed monodromy matrix elements on the pre-Bethe vector and express the results in terms of the partition function of the <span>(8)</span>-vertex model. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139030506","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 : 2023-12-22DOI: 10.1134/s0040577923120139
A. A. Tseytlin
Abstract
We review and elaborate on some aspects of the classically scale-invariant renormalizable (4)-derivative scalar theory (L=phi,partial^4phi+g(partialphi)^4). Similar models appear, e.g., in the context of conformal supergravity or in the description of the crystalline phase of membranes. Considering this theory in Minkowski signature, we suggest how to define Poincaré-invariant scattering amplitudes by assuming that only massless oscillating (nongrowing) modes appear as external states. In such shift-symmetric interacting theory, there are no IR divergences despite the presence of (1/q^4) internal propagators. We discuss how nonunitarity of this theory manifests itself at the level of the one-loop massless scattering amplitude.
{"title":"Comments on a 4-derivative scalar theory in 4 dimensions","authors":"A. A. Tseytlin","doi":"10.1134/s0040577923120139","DOIUrl":"https://doi.org/10.1134/s0040577923120139","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p> We review and elaborate on some aspects of the classically scale-invariant renormalizable <span>(4)</span>-derivative scalar theory <span>(L=phi,partial^4phi+g(partialphi)^4)</span>. Similar models appear, e.g., in the context of conformal supergravity or in the description of the crystalline phase of membranes. Considering this theory in Minkowski signature, we suggest how to define Poincaré-invariant scattering amplitudes by assuming that only massless oscillating (nongrowing) modes appear as external states. In such shift-symmetric interacting theory, there are no IR divergences despite the presence of <span>(1/q^4)</span> internal propagators. We discuss how nonunitarity of this theory manifests itself at the level of the one-loop massless scattering amplitude. </p>","PeriodicalId":797,"journal":{"name":"Theoretical and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":1.0,"publicationDate":"2023-12-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139024065","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}
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