Siberian Mathematical Journal最新文献

{"title":"A Stability Estimate for a Solution to an Inverse Problem for a Nonlinear Hyperbolic Equation","authors":"V. G. Romanov","doi":"10.1134/s0037446624030108","DOIUrl":"https://doi.org/10.1134/s0037446624030108","url":null,"abstract":"<p>We consider a hyperbolic equation with variable leading part and nonlinearity in the lower-order term.\u0000The coefficients of the equation are smooth functions\u0000constant beyond some compact domain in the three-dimensional space.\u0000A plane wave with direction <span>( ell )</span> falls to the heterogeneity from the exterior of this domain.\u0000A solution to the corresponding Cauchy problem for the original equation is measured at boundary points of the domain for\u0000a time interval including the moment of arrival of the wave at these points.\u0000The unit vector <span>( ell )</span> is assumed to be a parameter of the problem and\u0000can run through all possible values sequentially.\u0000We study the inverse problem of determining the coefficient of the nonlinearity on using this\u0000information about solutions. We describe the structure of a solution to the direct problem and\u0000demonstrate that the inverse problem reduces to an integral geometry problem.\u0000The latter problem consists of constructing the desired function on using given integrals\u0000of the product of this function and a weight function.\u0000The integrals are taken along the geodesic lines of the Riemannian metric\u0000associated with the leading part of the differential equation. We analyze this new problem\u0000and find some stability estimate for its solution, which yields\u0000a stability estimate for solutions to the inverse problem.</p>","PeriodicalId":49533,"journal":{"name":"Siberian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141169486","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Trace and Integrable Commutators of the Measurable Operators Affiliated to a Semifinite von Neumann Algebra","authors":"A. M. Bikchentaev","doi":"10.1134/s0037446624030030","DOIUrl":"https://doi.org/10.1134/s0037446624030030","url":null,"abstract":"<p>Assume that <span>( tau )</span> is a faithful normal semifinite trace\u0000on a von Neumann algebra <span>( {mathcal{M}} )</span>, <span>( I )</span> is the unit of <span>( mathcal{M} )</span>,\u0000<span>( S({mathcal{M}},tau) )</span> is the <span>( * )</span>-algebra of <span>( tau )</span>-measurable operators,\u0000and <span>( L_{1}({mathcal{M}},tau) )</span> is the Banach space of <span>( tau )</span>-integrable operators.\u0000We present a new proof of the following generalization\u0000of Putnam’s theorem (1951):\u0000No positive self-commutator\u0000<span>( [A^{*},A] )</span>\u0000with\u0000<span>( Ain S({mathcal{M}},tau) )</span>\u0000is invertible in <span>( {mathcal{M}} )</span>.\u0000If <span>( tau )</span>\u0000is infinite\u0000then no positive self-commutator\u0000<span>( [A^{*},A] )</span>\u0000with\u0000<span>( Ain S({mathcal{M}},tau) )</span>\u0000can be of the form\u0000<span>( lambda I+K )</span>,\u0000where <span>( lambda )</span>\u0000is a nonzero complex number and <span>( K )</span>\u0000is a <span>( tau )</span>-compact operator.\u0000Given\u0000<span>( A,Bin S({mathcal{M}},tau) )</span>\u0000with\u0000<span>( [A,B]in L_{1}({mathcal{M}},tau) )</span>\u0000we seek for the conditions that\u0000<span>( tau([A,B])=0 )</span>.\u0000If\u0000<span>( Xin S({mathcal{M}},tau) )</span>\u0000and\u0000<span>( Y=Y^{3}in{mathcal{M}} )</span>\u0000with\u0000<span>( [X,Y]in L_{1}({mathcal{M}},tau) )</span>\u0000then\u0000<span>( tau([X,Y])=0 )</span>.\u0000If\u0000<span>( A^{2}=Ain S({mathcal{M}},tau) )</span>\u0000and\u0000<span>( [A^{*},A]in L_{1}({mathcal{M}},tau) )</span>\u0000then\u0000<span>( tau([A^{*},A])=0 )</span>.\u0000If a partial isometry <span>( U )</span>\u0000lies in <span>( {mathcal{M}} )</span>\u0000and\u0000<span>( U^{n}=0 )</span>\u0000for some integer\u0000<span>( ngeq 2 )</span>\u0000then <span>( U^{n-1} )</span>\u0000is a commutator\u0000and\u0000<span>( U^{n-1}in L_{1}({mathcal{M}},tau) )</span>\u0000implies that\u0000<span>( tau(U^{n-1})=0 )</span>.</p>","PeriodicalId":49533,"journal":{"name":"Siberian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141169556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Upper Bounds for Volumes of Generalized Hyperbolic Polyhedra and Hyperbolic Links","authors":"A. Yu. Vesnin, A. A. Egorov","doi":"10.1134/s0037446624030042","DOIUrl":"https://doi.org/10.1134/s0037446624030042","url":null,"abstract":"<p>Call a polyhedron in a three-dimensional hyperbolic space\u0000generalized if finite, ideal, and truncated vertices are admitted.\u0000By Belletti’s theorem of 2021 the exact upper bound for the volumes\u0000of generalized hyperbolic polyhedra with the same one-dimensional skeleton <span>( Gamma )</span>\u0000equals the volume of an ideal right-angled hyperbolic polyhedron\u0000whose one-dimensional skeleton is the medial graph for <span>( Gamma )</span>.\u0000We give the upper bounds for the volume of\u0000an arbitrary generalized hyperbolic polyhedron\u0000such that the bounds depend linearly on\u0000the number of edges. Moreover, we show that the bounds can be improved\u0000if the polyhedron has triangular faces and trivalent vertices.\u0000As application we obtain some new upper bounds for the volume\u0000of the complement of the hyperbolic link with more than eight twists in a diagram.</p>","PeriodicalId":49533,"journal":{"name":"Siberian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141169705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Periodic Trajectories of Nonlinear Circular Gene Network Models","authors":"L. S. Minushkina","doi":"10.1134/s0037446624030212","DOIUrl":"https://doi.org/10.1134/s0037446624030212","url":null,"abstract":"<p>The article addresses the qualitative analysis of the two dynamical systems simulating circular gene network functioning.\u0000The equations of a three-dimensional\u0000dynamical system contain some monotonically decreasing smooth functions that describe negative feedback.\u0000A six-dimensional dynamical system consists of three equations with monotonically decreasing smooth functions\u0000and three equations with monotonically increasing smooth functions that characterize negative and positive feedbacks.\u0000In both models the process of degradation is described by smooth nonlinear functions.\u0000We construct invariants domains in order to localize cycles for both systems,\u0000show that each of the two systems has a unique stationary point\u0000in the invariant domain, and find the conditions for this point to be hyperbolic.\u0000The main result is the proof of existence of a cycle in the invariant subdomain\u0000from which the trajectories cannot pass to other subdomains obtained by discretization of the phase portrait.\u0000The cycles of three- and six-dimensional systems bound the\u0000two-dimensional invariant surfaces including the trajectories of the systems.</p>","PeriodicalId":49533,"journal":{"name":"Siberian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141169484","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Diffusion Instability Domains for Systems of Parabolic Equations","authors":"S. V. Revina","doi":"10.1134/s0037446624020216","DOIUrl":"https://doi.org/10.1134/s0037446624020216","url":null,"abstract":"<p>We consider a system of two reaction-diffusion equations in\u0000a bounded domain of the <span>( m )</span>-dimensional space\u0000with Neumann boundary conditions\u0000on the boundary for which the reaction terms <span>( f(u,v) )</span> and <span>( g(u,v) )</span>\u0000depend on two parameters <span>( a )</span> and <span>( b )</span>.\u0000Assume that the system has a spatially homogeneous solution <span>( (u_{0},v_{0}) )</span>,\u0000with <span>( f_{u}(u_{0},v_{0})&gt;0 )</span> and <span>( -g_{v}(u_{0},v_{0})=F(operatorname{Det}(operatorname{J})) )</span>,\u0000where <span>( operatorname{J} )</span> is the Jacobian\u0000of the corresponding linearized system in the diffusionless approximation and <span>( F )</span>\u0000is a smooth monotonically increasing function.\u0000We propose some method for the analytical description of the domain\u0000of necessary and sufficient conditions of\u0000Turing instability on the plane of system parameters\u0000for a fixed diffusion coefficient <span>( d )</span>.\u0000Also, we show that the domain\u0000of necessary conditions of Turing instability on\u0000the plane <span>( (operatorname{Det}(operatorname{J}),f_{u}) )</span> is bounded by the zero-trace curve,\u0000the discriminant curve, and the locus of points <span>( operatorname{Det(operatorname{J})}=0 )</span>.\u0000Explicit expressions are found for the curves of\u0000sufficient conditions and we prove that the discriminant curve is\u0000the envelope of the family of these curves.\u0000It is shown that one of the boundaries of the Turing instability domain\u0000consists of the fragments of the curves of sufficient conditions\u0000and is expressed in terms of the function <span>( F )</span> and the eigenvalues\u0000of the Laplace operator in the domain under consideration.\u0000We find the points of intersection of the curves of sufficient conditions\u0000and show that their abscissas do not depend on\u0000the form of <span>( F )</span> and are expressed in terms of\u0000the diffusion coefficient and the eigenvalues of the Laplace operator.\u0000In the special case\u0000<span>( F(operatorname{Det}(operatorname{J}))=operatorname{Det}(operatorname{J}) )</span>.\u0000For this case,\u0000the range of wave numbers at which Turing instability occurs is indicated.\u0000We obtain some partition of the semiaxis <span>( d&gt;1 )</span> into half-intervals\u0000each of which corresponds to its own minimum critical wave number.\u0000The points of intersection of the curves of sufficient conditions lie\u0000on straight lines independent of the diffusion coefficient <span>( d )</span>.\u0000By way of applications of the statements proven,\u0000we consider the Schnakenberg system and the Brusselator equations.</p>","PeriodicalId":49533,"journal":{"name":"Siberian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140297453","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Riordan Arrays and Difference Equations of Subdiagonal Lattice Paths","authors":"S. Chandragiri","doi":"10.1134/s0037446624020149","DOIUrl":"https://doi.org/10.1134/s0037446624020149","url":null,"abstract":"<p>We study lattice paths by combinatorial methods on the positive lattice. We give some identity that produces the functional equations\u0000and generating functions to counting the lattice paths on or below the main diagonal.\u0000Also, we consider the subdiagonal lattice paths in relation to lower triangular arrays.\u0000This presents a Riordan array in conjunction with the columns of the matrix of the coefficients of\u0000certain formal power series by implying an infinite lower triangular matrix <span>( F=(f_{x,y})_{x,ygeqslant 0} )</span>.\u0000We derive new combinatorial interpretations in terms of restricted lattice paths for some Riordan arrays.</p>","PeriodicalId":49533,"journal":{"name":"Siberian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140297520","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"The Scale-Dependent Deformation Model of a Layered Rectangle","authors":"A. O. Vatulyan, S. A. Nesterov","doi":"10.1134/s0037446624020198","DOIUrl":"https://doi.org/10.1134/s0037446624020198","url":null,"abstract":"<p>We consider the problem of\u0000deformation of a layered rectangle whose lower side is rigidly clamped, a\u0000distributed normal load acts on the upper side, and the lateral sides are in conditions of sliding\u0000termination. One-parameter gradient elasticity theory is used to account for the\u0000scale effects. The boundary conditions on the lateral faces allow us to use\u0000separation of variables. The displacements and mechanical loads are\u0000expanded in Fourier series. To find the harmonics of\u0000displacements, we have a system of two fourth order differential equations.\u0000We seek a solution to the system of differential equations\u0000by using the elastic potential of\u0000displacements and find the unknown integration constants by\u0000satisfying the boundary and transmission conditions\u0000for the harmonics of displacements. Considering some particular examples,\u0000we calculate the horizontal and vertical distribution of\u0000displacements as well as the couple and total stresses of a layered rectangle.\u0000We exhibit the difference between the distributions of\u0000displacements and stresses which are found on using the solutions to the\u0000problem in the classical and gradient formulations.\u0000Also, we show that the total stresses have a small\u0000jump on the transmission line due to the fact that, in accord with the\u0000gradient elasticity theory, not the total stresses, but the\u0000components of the load vectors should be continuous on the\u0000transmission line.\u0000Furthermore, we reveal\u0000a significant influence of the increase of the scale parameter on the\u0000changes of the values of displacements and total and\u0000couple stresses.</p>","PeriodicalId":49533,"journal":{"name":"Siberian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299772","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"On the $ K $ -functionals of Absolutely Calderón Elements of the Banach Pair  $ (l_{1},c_{0}) $","authors":"V. I. Dmitriev, E. V. Zhuravleva, O. Yu. Mikhailova, I. N. Burilich","doi":"10.1134/s0037446624020046","DOIUrl":"https://doi.org/10.1134/s0037446624020046","url":null,"abstract":"<p>We characterize the absolutely Calderón elements\u0000of the canonical pair <span>( (l_{1},c_{0}) )</span> of sequence spaces in terms of the Peetre\u0000<span>( K )</span>-functional. This result has been known to the first\u0000author since rather long ago but the proof appears here.\u0000Also, we formulate a few unsolved problems.</p>","PeriodicalId":49533,"journal":{"name":"Siberian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299774","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Geodesics and Shortest Arcs of Some Sub-Riemannian Metrics on the Lie Groups $ operatorname{SU}(1,1)times �� $ and $ operatorname{SO}_{0}(2,1)times �� $ with Three-Dimensional Generating Distributions","authors":"I. A. Zubareva","doi":"10.1134/s003744662402006x","DOIUrl":"https://doi.org/10.1134/s003744662402006x","url":null,"abstract":"<p>We find geodesics, shortest arcs, cut loci, and first conjugate loci for some left-invariant sub-Riemannian metrics on the Lie groups\u0000<span>( operatorname{SU}(1,1)times 𝕉 )</span> and <span>( operatorname{SO}_{0}(2,1)times 𝕉 )</span>.</p>","PeriodicalId":49533,"journal":{"name":"Siberian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140299705","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Optimal Recovery of a Family of Operators from Inaccurate Measurements on a Compact Set","authors":"E. O. Sivkova","doi":"10.1134/s0037446624020228","DOIUrl":"https://doi.org/10.1134/s0037446624020228","url":null,"abstract":"<p>Given a one-parameter family of continuous linear operators\u0000<span>( T(t):L_{2}(𝕉^{d})to L_{2}(𝕉^{d}) )</span>, with\u0000<span>( 0leq t&lt;infty )</span>, we consider the optimal\u0000recovery of the values of\u0000<span>( T(tau) )</span> on the whole space by approximate information\u0000on the values of\u0000<span>( T(t) )</span>, where <span>( t )</span> runs over a compact set\u0000<span>( Ksubset 𝕉_{+} )</span> and <span>( taunotin K )</span>.\u0000We find a family of optimal methods for recovering the\u0000values of <span>( T(tau) )</span>.\u0000Each of these methods uses approximate measurements\u0000at no more than two points in <span>( K )</span> and\u0000depends linearly on these measurements.\u0000As a corollary, we provide some families of optimal methods\u0000for recovering the solution of the heat equation\u0000at a given moment of time from\u0000inaccurate measurements on other time intervals and for\u0000solving the Dirichlet problem for\u0000a half-space on a hyperplane by inaccurate\u0000measurements on other hyperplanes.\u0000The optimal recovery of the values of\u0000<span>( T(tau) )</span> from the indicated\u0000information reduces to finding the value of\u0000an extremal problem for the maximum with\u0000continuum many inequality-type constraints, i.e.,\u0000to finding the exact upper bound of the\u0000maximized functional under these constraints.\u0000This rather complicated task reduces\u0000to the infinite-dimensional problem of linear\u0000programming on the vector space of all\u0000finite real measures on the <span>( sigma )</span>-algebra of\u0000Lebesgue measurable sets in <span>( 𝕉^{d} )</span>.\u0000This problem can be solved by some generalization of\u0000the Karush–Kuhn–Tucker theorem,\u0000and its significance coincides with the significance\u0000of the original problem.</p>","PeriodicalId":49533,"journal":{"name":"Siberian Mathematical Journal","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140297410","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}