Theoretical and Mathematical Physics
Journal Prestige (SJR): 0.409 Citation Impact (citeScore): 1 Number of Followers: 8 Hybrid journal (It can contain Open Access articles) ISSN (Print) 15739333  ISSN (Online) 00405779 Published by SpringerVerlag [2468 journals] 
 Multidimensional Zaremba problem for the $$p(\,\cdot\,)$$ Laplace
equation. A Boyarsky–Meyers estimate
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Abstract: We prove the higher integrability of the gradient of solutions of the Zaremba problem in a bounded strongly Lipschitz domain for an inhomogeneous \(p(\,\cdot\,)\) Laplace equation with a variable exponent \(p\) having a logarithmic continuity modulus.
PubDate: 20240101

 Recent progress in the theory of functions of several complex variables
and complex geometry
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Abstract: We give a survey on recent progress on converses of \(L^2\) existence theorem and \(L^2\) extension theorem which are two main parts in \(L^2\) theory, and their applications in getting criteria of Griffiths positivity and characterizations of Nakano positivity of (singular) Hermitian metrics of holomorphic vector bundles, as well as the strong openness property and stability property of multiplier submodule sheaves associated to singular Nakano semipositive Hermitian metrics on holomorphic vector bundles.
PubDate: 20240101

 Existence of an entropic solution of a nonlinear elliptic problem in an
unbounded domain
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Abstract: We consider a secondorder quasilinear elliptic equation with an integrable righthand 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.
PubDate: 20240101

 On Dirichlet problem

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Abstract: During almost two centuries after the Gauss’ formulation of the Dirichlet problem for Laplace equation, many famous mathematicians devoted their studies to this subject and to its various generalizations. Many interesting and important results have been obtained, which become already classical ones. Our paper is an extended presentation of the author’s talk on the international conference dedicate to the century of V. S. Vladimirov birthday. Its main content is the review of the results in that direction, including the proves of new statements and discussion of unsolved problems. Our goal is to convince readers that, in this “principal” problem of mathematical physics, we know far from everything even about the case of linear equation. There are many interesting and important unsolved problems in that direction.
PubDate: 20240101

 Geometry and quasiclassical quantization of magnetic monopoles

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Abstract: We present the basic physical and mathematical ideas (P. Curie, Darboux, Poincaré, Dirac) that led to the concept of magnetic charge, the general construction of magnetic Laplacians for magnetic monopoles on Riemannian manifolds, and the results of Kordyukov and the author on the quasiclassical approximation for eigensections of these operators.
PubDate: 20240101

 On the combination of Lebesgue and Riemann integrals in theory of
convolution equations
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Abstract: Using the example of scalar and vector Wiener–Hopf equations, we consider two methods for combining the options for the Riemann integral and Lebesgue functional spaces in problems of studying and solving integral convolution equations. The method of nonlinear factorization equations and the kernel averaging method are used. A generalization of the direct Riemann integrability is introduced and applied.
PubDate: 20240101

 A characterization of Gibbs semigroups

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Abstract: We propose a new characterization of Gibbs semigroups, which is an extension of a similar characterization for compact semigroups.
PubDate: 20240101

 Arnold Lagrangian singularity in the asymptotics of the solution of a

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Abstract: A model Helmholtz equation with a localized righthand 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.
PubDate: 20240101

 On qualitative properties of the solution of a boundary value problem for
a system of nonlinear integral equations
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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.
PubDate: 20240101

 Ternary $$Z_3$$ symmetric algebra and generalized quantum oscillators

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Abstract: We present a generalized version of a quantum oscillator described by means of a ternary Heisenberg algebra. The model leads to a sixthorder Hamiltonian whose energy levels can be discretized using the Bohr–Sommerfeld quantization procedure. We note the similarity with the \(Z_3\) extended version of Dirac’s equation applied to quark color dynamics, which also leads to sixthorder field equations. The paper also contains a comprehensive guide to \(Z_3\) graded structures, including ternary algebras, which form a mathematical basis for the proposed generalization. The symmetry properties of the model are also discussed.
PubDate: 20240101

 Some new methods for studying boundary value problems for general partial
differential equations
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Abstract: We consider methods for studying boundary value problems for linear partial differential equations in a domain regardless of the type of the equation. We propose several methods for studying boundary value problems, typically based on the Green’s formula. Our previous publications were devoted to these methods, and we present these results in a summarized form in this paper.
PubDate: 20240101

 On the quarkyonic phase in the holographic approach

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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.
PubDate: 20231201

 The structure of quantum corrections and exact results in supersymmetric
theories from the higher covariant derivative regularization
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Abstract: We review some recent results of the studies of quantum corrections in supersymmetric theories derived using Slavnov’s higher covariant derivative regularization. In particular, we demonstrate that the \(\beta\) function of \(\mathcal{N}=1\) supersymmetric theories is related to the anomalous dimensions of matter superfields by the NSVZ relation if the theory is regularized by higher covariant derivatives and the renormalization group functions are defined in terms of the bare couplings, because the corresponding loop corrections are given by integrals of double total derivatives in the momentum space. For the standard renormalizationgroup functions, we show that an allloop NSVZ renormalization scheme is given by the HD \(\,+\,\) MSL renormalization prescription when the higher covariant derivative regularization is supplemented by minimal subtractions of logarithms. Applications of these results to the precise calculations in various supersymmetric theories are briefly described.
PubDate: 20231201

 Nonexplicit versions of integrable equations

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Abstract: We consider some generalizations of a \((2+1)\) dimensional Davey–Stewartsontype equation. In particular, we propose a dynamical system that does not admit an explicit formulation in terms of differential equations, but needs an additional independent variable.
PubDate: 20231201

 Comments on a 4derivative scalar theory in 4 dimensions

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Abstract: We review and elaborate on some aspects of the classically scaleinvariant renormalizable \(4\) derivative scalar theory \(L=\phi\,\partial^4\phi+g(\partial\phi)^4\) . 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 Poincaréinvariant scattering amplitudes by assuming that only massless oscillating (nongrowing) modes appear as external states. In such shiftsymmetric 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 oneloop massless scattering amplitude.
PubDate: 20231201

 A new solvable twomatrix model and the BKP tau function

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Abstract: We present exactly solvable modifications of the twomatrix ZinnJustin–Zuber model and write it as a tau function. The grand partition function of these matrix integrals is written as the fermion expectation value. The perturbation theory series is written explicitly in terms of a series in strict partitions. The related string equations are presented.
PubDate: 20231201

 Quantum corrections to the effective potential in nonrenormalizable
theories
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Abstract: For the effective potential in the leading logarithmic approximation, we construct a renormalization group equation that holds for arbitrary scalar field theories, including nonrenormalizable ones, in four dimensions. This equation reduces to the usual renormalization group equation with a oneloop betafunction in the renormalizable case. The solution of this equation sums up the leading logarithmic contributions in the field in all orders of the perturbation theory. This is a nonlinear secondorder partial differential equation in general, but it can be reduced to an ordinary one in some cases. In specific examples, we propose a numerical solution of this equation and construct the effective potential in the leading logarithmic approximation. We consider two examples as an illustration: a powerlaw potential and a cosmological potential of the \(\tan^2\phi\) type. The obtained equation in physically interesting cases opens up the possibility of studying the properties of the effective potential, the presence of additional minima, spontaneous symmetry breaking, stability of the ground state, etc.
PubDate: 20231201

 Cluster variables for affine Lie–Poisson systems

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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 clusteralgebra 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 \(({n_1\times n_2})\) 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 infinitedimensional algebra.
PubDate: 20231201

 Novel integrability in string theory from automorphic symmetries

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Abstract: We develop a technique based on the boost automorphism for finding new lattice integrable models with various dimensions of local Hilbert spaces. We initiate the method by implementing it in twodimensional models and resolve a classification problem, which not only confirms the known vertex model solution space but also extends to the new \(\mathfrak{sl}_2\) deformed sector. A generalization of the approach to integrable string backgrounds is provided and allows finding new integrable deformations and associated \(R\) matrices. The new integrable solutions appear to be of a nondifference or pseudodifference form admitting \(AdS_2\) and \(AdS_3\) \(S\) matrices as special cases (embeddings), which also includes a map of the doubledeformed sigma model \(R\) matrix. The corresponding braiding and conjugation operators of the novel models are derived. We also demonstrate implications of the obtained freefermion analogue for \(AdS\) deformations.
PubDate: 20231201

 A new stability equation for the Abelian Higgs–Kibble model with a
dimension6 derivative operator
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Abstract: We show that the dynamics of the scalar Higgs field in the Abelian Higgs–Kibble model supplemented with a dimension6 derivative operator can be constrained at the quantum level by a certain stability equation. It holds in the Landau gauge and is derived within the recently proposed extended field formalism, where the physical scalar is described by a gaugeinvariant field variable. Physical implications of the stability equation are discussed.
PubDate: 20231201
