{"title":"Atiyah–Bott–Lefschetz Formula on Manifolds with Periodic Ends","authors":"V.E. Nazaikinskii, A.Yu. Savin","doi":"10.1134/S1061920826020184","DOIUrl":"10.1134/S1061920826020184","url":null,"abstract":"<p> The classical Atiyah–Bott formula (1967) expresses the Lefschetz number of a geometric endomorphism of an elliptic complex of (pseudo)differential operators on a closed smooth manifold as the sum of contributions of fixed points of the corresponding diffeomorphism, assuming that all fixed points are nondegenerate. These contributions explicitly depend only on the endomorphism itself but not on the operators forming the complex. In 1999, one of the authors, together with B.-W. Schulze, B. Yu. Sternin, and V. E. Shatalov, generalized this formula to the case of manifolds with conical singularities (or, equivalently, with cylindrical ends), where the contributions of interior fixed points are supplemented by those of fixed cylindrical ends (which already depend on the operators of the complex themselves). In recent decades, a number of papers have appeared in the literature devoted to elliptic theory on manifolds with periodic ends. The present paper gives a Lefschetz formula on such manifolds; in the special case of manifolds with cylindrical ends, it strengthens previously obtained results. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"33 2","pages":"440 - 444"},"PeriodicalIF":1.9,"publicationDate":"2026-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148458754","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}
{"title":"Three Identical One-Dimensional Quantum Particles with Point Interaction as a Solvable Model: III. Positive Essential Spectrum","authors":"M.A. Lyalinov","doi":"10.1134/S1061920826600236","DOIUrl":"10.1134/S1061920826600236","url":null,"abstract":"<p> This is the third part of our work dealing with spectral analysis for the Hamiltonian of three identical one-dimensional quantum particles. It deals with positive energies and the corresponding generalized eigenfunctions. The asymptotic behavior of the generalized eigenfunctions is studied and interpreted in terms of the diffraction theory. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"33 2","pages":"392 - 406"},"PeriodicalIF":1.9,"publicationDate":"2026-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148458758","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}
{"title":"Exploring Algebraic and Probabilistic Properties for Degenerate Bivariate Appell Polynomials","authors":"M. Riyasat, S. Khan","doi":"10.1134/S1061920824600867","DOIUrl":"10.1134/S1061920824600867","url":null,"abstract":"<p> The purpose of this paper is to carry out a comprehensive analysis of the properties of degenerate bivariate Appell polynomials by making extensive use of algebraic and probabilistic techniques. The determinant expressions for the degenerate cosine-Appell and sine-Appell polynomials and for their conjugate forms are established by employing several recursive formulas. Further, we focus on exploring probabilistic properties, in which there associated a family of degenerate bivariate Appell polynomials to every random variable <span>(chi)</span> with some exponential moments <span>(e^{chi u})</span>, which provides some powerful tools for a deeper understanding of the behavior and properties of degenerate bivariate Appell polynomials. Examples are framed which introduces a random variable whose mass function is given in terms of degenerate bivariate Bernoulli polynomials. Some expressions for the expectation <span>(E[chi])</span> of the random variable associated with the degenerate bivariate Bernoulli polynomials are derived. It is demonstrated that the probabilistic approach offers interesting new results and developments for these polynomials. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"33 2","pages":"417 - 430"},"PeriodicalIF":1.9,"publicationDate":"2026-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148458755","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}
A. Piatnitski, V. Sloushch, T. Suslina, E. Zhizhina
{"title":"On the Homogenization of a Parabolic Equation with a Nonlocal Convolution Type Operator","authors":"A. Piatnitski, V. Sloushch, T. Suslina, E. Zhizhina","doi":"10.1134/S1061920826020147","DOIUrl":"10.1134/S1061920826020147","url":null,"abstract":"<p> In <span>(L_2(mathbb{R}^d))</span>, we consider a self-adjoint bounded operator <span>({mathbb A}_varepsilon(t))</span>, <span>(tge 0)</span>, <span>(varepsilon >0)</span>, of the form </p><p> It is assumed that <span>(a(cdot)in L_1(mathbb R^d))</span> is a nonnegative function such that <span>(int_{mathbb{R}^d} | mathbf{x} |^3 a(mathbf{x}),dmathbf{x}<infty)</span> and <span>(a(-mathbf{x}) = a(mathbf{x}))</span>; the function <span>(mu(mathbf{x},mathbf{y},t))</span> is measurable and periodic in all variables, moreover, <span>(mu(mathbf{x},mathbf{y},t) = mu(mathbf{y},mathbf{x},t))</span> and <span>(0< mu_- leqslant mu(mathbf{x},mathbf{y},t) leqslant mu_+< infty)</span>. Let <span>(u_varepsilon(mathbf{x},t))</span>, <span>(mathbf{x} in {mathbb R}^d)</span>, <span>(t ge 0)</span>, be the solution of the Cauchy problem </p><p> where <span>(varphi in L_2({mathbb R}^d))</span>. We show that, for a chosen <span>(t>0)</span>, the solution <span>(u_varepsilon(cdot,t))</span> converges in <span>(L_2({mathbb R}^d))</span>, as <span>(varepsilon to 0)</span>, to the solution <span>(u_0(cdot,t))</span> of the homogenized problem </p><p> where <span>({mathbb A}^0= - operatorname{div} g^0 nabla)</span> and <span>(g^0)</span> is a positive definite effective matrix. The following error estimate holds: </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"33 2","pages":"407 - 416"},"PeriodicalIF":1.9,"publicationDate":"2026-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148458752","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}
{"title":"Properties of an Operator Describing Acoustic Black Holes","authors":"N.V. Ivanov, A.I. Shafarevich","doi":"10.1134/S1061920826020081","DOIUrl":"10.1134/S1061920826020081","url":null,"abstract":"<p> The properties of the operator describing flexural waves are described. This operator is defined by a fourth-order differential expression with a singular point. A Hilbert space is constructed, and the domain of the operator and the Gelfand triple are found. For a model operator corresponding to a quadratic profile, the spectrum and corresponding (distributional) eigenfunctions are explicitly calculated. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"33 2","pages":"302 - 316"},"PeriodicalIF":1.9,"publicationDate":"2026-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148458762","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}
{"title":"Canonical Form of the Inertia Tensor for Rigid Bodies on the Lobachevskii Plane and in a Pseudo-Euclidean Space","authors":"E.A. Kudryavtseva, A.Yu. Shubert","doi":"10.1134/S1061920826020123","DOIUrl":"10.1134/S1061920826020123","url":null,"abstract":"<p> This paper studies the properties of the inertia tensor for tops in 3-dimensional (pseudo-) Euclidean space and for plates on the Lobachevskii plane. A canonical form is described to which one can reduce the (pseudo-)Euclidean inner product and the inertia operator of any top by changing the basis. All tops whose inertia operators are nondiagonalizable (every top of this kind lies in a plane tangent to the isotropic cone) are described. All realizable triples <span>((J_1,J_2,J_3))</span> of principal moments of inertia are described in terms of triangle inequalities and their pseudo-Euclidean analogs. A criterion for the realizability of such a triple by a plate on the Lobachevskii plane is obtained. All realizable pairs of triples <span>((J_1,J_2,J_3))</span> of principal moments of inertia and coordinates of the center of mass <span>((c_1,c_2,c_3))</span> of a top in the principal axes of inertia are described in both the diagonalizable and nondiagonalizable cases. In all cases, it is shown that a body consisting of no more than six points can be taken as the realizing top. Eigenvalues of all degenerate inertia operators are described. As an application, realizable examples of integrable tops in 3-dimensional pseudo-Euclidean space (the Euler and Lagrange tops) are described. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"33 2","pages":"357 - 391"},"PeriodicalIF":1.9,"publicationDate":"2026-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148458977","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}
{"title":"Asymptotics for a Class of Singular Integrals of Quotients with Highly Degenerate Denominators","authors":"A.A. Elokhin","doi":"10.1134/S1061920825601417","DOIUrl":"10.1134/S1061920825601417","url":null,"abstract":"<p> In a rigorous study of stochastic models for the wave turbulence theory and Peierls’ kinetic theory for the thermal conductivity in solids, analysis of integrals of the form <span>(int_{ mathcal{M} }frac{Fomega_ mathcal{M} }{Omega^2 + nu^2Gamma^2})</span> and <span>(int_{ mathcal{M} }frac{Fcos(nu^{-1}Omega)omega_ mathcal{M} }{Omega^2 + nu^2Gamma^2})</span> plays a crucial role, where <span>(nu>0)</span> is a small parameter, <span>( mathcal{M} )</span> is a closed Riemannian manifold with volume form <span>(omega_ mathcal{M} )</span>, and the functions <span>(Gamma > 0)</span>, <span>(F)</span>, <span>(Omega)</span> are sufficiently smooth. We investigate the asymptotic behavior of the integrals in the limit as <span>(nurightarrow 0)</span>. This paper continues the studies S. B. Kuksin, J. Math. Phys. Anal. Geom., 2018, vol. 14, no. 4, pp. 510тАУ518 and A. V. Dymov, Theoret. and Math. Phys., 2023, vol. 214, no. 2, pp. 153тАУ169, in which the authors considered similar integrals for the case in which <span>( mathcal{M} =mathbb{R}^d)</span> and the function <span>(Omega)</span> is Morse. We significantly weaken the latter assumption, which played an important role in the aforementioned papers. This makes the results thus obtained applicable to the problem of rigorous justification of Peierls’ kinetic theory. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"33 2","pages":"288 - 301"},"PeriodicalIF":1.9,"publicationDate":"2026-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148458759","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}
{"title":"Homogenization of Attractors to the Chafee–Infante Equations in a Perforated Domain with Locally Periodic Obstacles: Subcritical and Supercritical Cases","authors":"G.A. Chechkin, A.T. Karipova, A.M. Toleubay","doi":"10.1134/S1061920826600352","DOIUrl":"10.1134/S1061920826600352","url":null,"abstract":"<p> In this paper, we investigate the Chafee–Infante equation in a locally periodic perforated domain with a small parameter <span>(varepsilon>0)</span> characterizing the diameter of the cavities and the distance between them. The asymptotic behavior of the problem depends on the scaling of the boundary effects, characterized by the parameter <span>(varepsilon^theta)</span> in Fourier condition on the boundary of the cavities, leading to two distinct regimes. In contrast to the critical case (<span>(theta=1)</span>), in which the condition at the boundary of the cavities gives a potential term and new coefficients of the elliptic operator to the homogenized equation, in the subcritical and supercritical cases the situation changes radically. </p><p> The subcritical case (<span>( theta > 1)</span>): The influence of boundary conditions on perforations does not contribute to the potential term but still changes the coefficients of the homogenized equation. We prove that both trajectory and global attractors of the original problem converge to those of the homogenized problem. </p><p> The supercritical case (<span>(theta < 1)</span>): In this case, the solutions degenerate as the small parameter tends to zero. It has been proven that the attractors also tend to zero in the corresponding topology. </p><p> In the subcritical case, the homogenized limit is rigorously derived via auxiliary cell problems and asymptotic expansions. Appropriate functional spaces with weak topology are introduced, and the existence of trajectory attractors is established. In the supercritical case, we use integral estimates to prove the degeneration of solutions and attractors. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"33 2","pages":"276 - 287"},"PeriodicalIF":1.9,"publicationDate":"2026-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148458979","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}
{"title":"Phase-Field System as a Regularization of the Free-Boundary Problem for an Incompressible Immiscible Two-Component Flow","authors":"N.A. Burov, R.K. Gaydukov","doi":"10.1134/S1061920826600297","DOIUrl":"10.1134/S1061920826600297","url":null,"abstract":"<p> The spreading of a droplet of incompressible fluid over a substrate under the action of capillary forces and gravity is studied within a phase-field framework. An asymptotic analysis shows that the solution of the phase-field system (the regularized problem) converges to the solution of the corresponding limit problem when a regularization parameter tends to zero. The theoretical results are illustrated by numerical simulations for hydrophobic and hydrophilic substrates. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"33 2","pages":"263 - 275"},"PeriodicalIF":1.9,"publicationDate":"2026-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148458981","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}
{"title":"Norm Continuous Representations of Locally Compact Groups in a Hilbert Space","authors":"A.I. Shtern","doi":"10.1134/S1061920826020160","DOIUrl":"10.1134/S1061920826020160","url":null,"abstract":"<p> The structure of norm continuous representations of some locally compact groups in a Hilbert spaces is refined. </p>","PeriodicalId":763,"journal":{"name":"Russian Journal of Mathematical Physics","volume":"33 2","pages":"431 - 433"},"PeriodicalIF":1.9,"publicationDate":"2026-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"148459037","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}