Статьи журнала - Владикавказский математический журнал
Все статьи: 967
Inverse problem for viscoelastic system in a vertically layered medium
Статья научная
In this paper, we consider a three-dimensional system of first-order viscoelasticity equations written with respect to displacement and stress tensor. This system contains convolution integrals of relaxation kernels with the solution of the direct problem. The direct problem is an initial-boundary value problem for the given system of integro-differential equations. In the inverse problem, it is required to determine the relaxation kernels if some components of the Fourier transform with respect to the variables x1 and x2 of the solution of the direct problem on the lateral boundaries of the region under consideration are given. At the beginning, the method of reduction to integral equations and the subsequent application of the method of successive approximations are used to study the properties of the solution of the direct problem. To ensure a continuous solution, conditions for smoothness and consistency of initial and boundary data at the corner points of the domain are obtained. To solve the inverse problem by the method of characteristics, it is reduced to an equivalent closed system of integral equations of the Volterra type of the second kind with respect to the Fourier transform in the first two spatial variables x1, x2, for solution to direct problem and the unknowns of inverse problem. Further, to this system, written in the form of an operator equation, the method of contraction mappings in the space of continuous functions with a weighted exponential norm is applied. It is shown that with an appropriate choice of the parameter in the exponent, this operator is contractive in some ball, which is a subset of the class of continuous functions. Thus, we prove the global existence and uniqueness theorem for the solution of the stated problem.
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Invitation to Boolean valued analysis
Статья научная
This is a short invitation to the field of Boolean valued analysis. Model theory evaluates and counts truth and proof. The chase of truth not only leads us close to the truth we pursue but also enables us to nearly catch up with many other instances of truth which we were not aware nor even foresaw at the start of the rally pursuit. That is what we have learned from Boolean valued models of set theory. These models stem from the famous works by Paul Cohen on the continuum hypothesis. They belong to logic and yield a profusion of the surprising and unforeseen visualizations of the ingredients of mathematics. Many promising opportunities are open to modeling the powerful habits of reasoning and verification. Boolean valued analysis is a blending of analysis and Boolean valued models. Adaptation of the ideas of Boolean valued models to functional analysis projects among the most important directions of developing the synthetic methods of mathematics. This approach yields the new models of numbers, spaces, and types of equations. The content expands of all available theorems and algorithms. The whole methodology of mathematical research is enriched and renewed, opening up absolutely fantastic opportunities. We can now transform matrices into numbers, embed function spaces into a straight line, yet having still uncharted vast territories of new knowledge. The article advertised two books that crown our thought about and research into the field.
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Isometries of real subspaces of self-adjoint operators in Banach symmetric ideals
Статья научная
Let (CE,∥⋅∥CE) be a Banach symmetric ideal of compact operators, acting in a complex separable infinite-dimensional Hilbert space H. Let ChE={x∈CE:x=x∗} be the real Banach subspace of self-adjoint operators in (CE,∥⋅∥CE). We show that in the case when (CE,∥⋅∥CE) is a separable or perfect Banach symmetric ideal (CE≠C2) any skew-Hermitian operator H:ChE→ChE has the following form H(x)=i(xa-ax) for same a∗=a∈B(H) and for all x∈ChE. Using this description of skew-Hermitian operators, we obtain the following general form of surjective linear isometries V:ChE→ChE. Let (CE,∥⋅∥CE) be a separable or a perfect Banach symmetric ideal with not uniform norm, that is ∥p∥CE>1 for any finite dimensional projection p∈CE with dimp(H)>1, let CE≠C2, and let V:ChE→ChE be a surjective linear isometry. Then there exists unitary or anti-unitary operator u on H such that V(x)=uxu∗ or V(x)=-uxu∗ for all x∈ChE.
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Статья научная
The present paper is devoted to study of certain classes of homogeneous regular subalgebras of the algebra of all complex-valued measurable functions on the unit interval. It is known that the transcendence degree of a commutative unital regular algebra is one of the important invariants of such algebras together with Boolean algebra of its idempotents. It is also known that if (Ω,Σ,μ) is a Maharam homogeneous measure space, then two homogeneous unital regular subalgebras of S(Ω) are isomorphic if and only if their Boolean algebras of idempotents are isomorphic and transcendence degrees of these algebras coincide. Let S(0,1) be the algebra of all (classes of equivalence) measurable complex-valued functions and let AD(n)(0,1) (n∈N∪{∞}) be the algebra of all (classes of equivalence of) almost everywhere n-times approximately differentiable functions on [0,1]. We prove that AD(n)(0,1) is a regular, integrally closed, ρ-closed, c-homogeneous subalgebra in S(0,1) for all n∈N∪{∞}, where c is the continuum. Further we show that the algebras S(0,1) and AD(n)(0,1) are isomorphic for all n∈N∪{∞}. As an application of these results we obtain that the dimension of the linear space of all derivations on S(0,1) and the order of the group of all band preserving automorphisms of S(0,1) coincide and are equal to 2c. Finally, we show that the Lie algebra DerS(0,1) of all derivations on S(0,1) contains a subalgebra isomorphic to the infinite dimensional Witt algebra.
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Kantorovich's principle in action: AW*-modules and injective Banach lattices
Статья научная
Making use of Boolean valued representation it is proved that Kaplansky--Hilbert lattices and injective Banach lattices may be produced from each other by means of the convexification procedure. The relationship between the Kantorovich's heuristic principle and the Boolean value transfer principle is also discussed.
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L_p-l_q-оценки для обобщенных потенциалов Рисса с осциллирующими ядрами
Статья научная
Получены $L_p-L_q$-оценки для обобщенных потенциалов Рисса с осциллирующими ядрами и однородными характеристиками бесконечно дифференцируемыми в $\mathbb{R}^n\setminus\{0\}$. Описаны выпуклые множества $(1/p,1/q)$-плоскости, для точек которых упомянутые операторы ограничены из $L_p$ в $L_q$ и указаны области, где эти операторы не ограничены.
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Статья научная
In this paper we study the large time behaviour of the solution and compactification of support to the Cauchy problem for doubly degenerate parabolic equations with strong gradient damping. Under the suitable assumptions on the structure of the equation and data of the problem we establish new sharp bound of solutions for a large time. Moreover, when the support of initial datum is compact we prove that the support of the solution contains in the ball with radius which is independent in time variable. In the critical case of the behaviour of the damping term the support of the solution depends on time variable logarithmically for a sufficiently large time. The main tool of the proof is based on nontrivial use of cylindrical Gagliardo-Nirenberg type embeddings and recursive inequalities. The sup-norm estimates of the solution is carried out by modified version of the classical method of De-Giorgi-Ladyzhenskaya-Uraltseva-DiBenedetto. The approach of the paper is flexible enough and can be used when studying the Cauchy-Dirichlet or Cauchy-Neumann problems in domains with non compact boundaries.
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Lattice structure on bounded homomorphisms between topological lattice rings
Статья научная
Suppose X is a topological ring. It is known that there are three classes of bounded group homomorphisms on X whose topological structures make them again topological rings. First, we show that if X is a Hausdorff topological ring, then so are these classes of bounded group homomorphisms on X. Now, assume that X is a locally solid lattice ring. In this paper, our aim is to consider lattice structure on these classes of bounded group homomorphisms; more precisely, we show that, under some mild assumptions, they are locally solid lattice rings. In fact, we consider bounded order bounded homomorphisms on X. Then we show that under the assumed topology, they form locally solid lattice rings. For this reason, we need a version of the remarkable Riesz-Kantorovich formulae for order bounded operators in Riesz spaces in terms of order bounded homomorphisms on topological lattice groups.
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Linear operators on abramovich--wickstead type spaces
Статья научная
In this note, we define and investigate Abramovich--Wickstead type spaces the elements of which are the sums of continuous functions and discrete functions. We give an analytic representation of regular and order continuous regular operators on these spaces with values in a Dedekind complete vector lattice.
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Статья научная
We introduce “local grand” Lebesgue spaces Lp),θ x0,a( ), 0 < p < ∞, ⊆ Rn, where the process of “grandization” relates to a single point x0 ∈ , contrast to the case of usual known grand spaces Lp),θ( ), where “grandization” relates to all the points of . We define the space L p),θ x0,a( ) by means of the weight a(|x − x0|)εp with small exponent, a(0) = 0. Under some rather wide assumptions on the choice of the local “grandizer” a(t), we prove some properties of these spaces including their equivalence under different choices of the grandizers a(t) and show that the maximal, singular and Hardy operators preserve such a “single-point grandization” of Lebesgue spaces Lp( ), 1 < p < ∞, provided that the lower Matuszewska–Orlicz index of the function a is positive. A Sobolev-type theorem is also proved in local grand spaces under the same condition on the grandizer.
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Lp-Lq-оценки для операторов типа потенциала с осциллирующими ядрами
Статья научная
Получены Lp-Lq-оценки для обобщенных потенциалов Рисса с осциллирующими ядрами и характеристиками широкого класса, включающего произведение однородной функции, бесконечно дифференцируемой в Rn∖{0}, и функции класса Cm,γ(R˙1+). Описаны выпуклые множества (1/p,1/q)-плоскости, для точек которых упомянутые операторы ограничены из Lp в Lq, и указаны области, где эти операторы не ограничены. В некоторых случаях доказана точность полученных оценок. В частности, получены необходимые и достаточные условия ограниченности исследуемых операторов в Lp. В настоящее время имеется ряд работ по Lp-Lq-оценкам для операторов свертки с осциллирующими ядрами, в частности, для операторов Бохнера - Рисса и акустических потенциалов, возникающих в различных задачах анализа и математической физики. В этих работах рассматриваются ядра, содержащие только радиальную характеристику b(r), которая стабилизируется на бесконечности как гёльдеровская функция. Благодаря этому свойству получение оценок для указанных операторов сводилось к случаю оператора с характеристикой b(r)≡1. Подобное сведение в принципе невозможно, когда ядро потенциала Рисса содержит однородную характеристику a(t′). Поэтому в работе развивается новый метод, основанный на получении специальных представлений для символов рассматриваемых операторов с последующим применением техники Фурье-мультипликаторов, вырождающихся или имеющих особенности на единичной сфере в Rn.
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Персоналии
This is a short overview of some sections of applied functional analysis, convexity, optimization, and nonstandard models.
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Mathematics and economics in the legacy of Leonid Kantorovich
Статья научная
This is a short overview of the life of Leonid Kantorovich and his contribution to the formation of the modern outlook on the interaction between mathematics and economics.
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Maximal quasi-normed extension of quasi-normed lattices
Статья научная
The purpose of this article is to extend the Abramovich's construction of a maximal normed extension of a normed lattice to quasi-Banach setting. It is proved that the maximal quasi-normed extension Xϰ of a Dedekind complete quasi-normed lattice X with the weak σ-Fatou property is a quasi-Banach lattice if and only if X is intervally complete. Moreover, Xϰ has the Fatou and the Levi property provided that X is a Dedekind complete quasi-normed space with the Fatou property. The possibility of applying this construction to the definition of a space of weakly integrable functions withrespect to a measure taking values from a quasi-Banach lattice is also discussed, since the duality based definition does not work in the quasi-Banach setting.
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Metric characteristics of classes of compact sets on Carnot groups with sub-Lorentzian structure
Статья научная
We consider classes of mappings of Carnot groups that are intrinsically Lipschitz and defined on compact subsets, and describe the metric characteristics of their images under the condition that a~sub-Lorentzian structure is introduced on the image. This structure is a sub-Riemannian generalization of Minkowski geometry. One of its features is the unlimitedness of the balls constructed with respect to the~intrinsic distance. In sub-Lorentzian geometry, the study of spacelike surfaces whose intersections with such balls are limited, is of independent interest. If the mapping is defined on an open set, then the formulation of space-likeness criterion reduces to considering the connectivity component of the intersection containing the center of the ball and analyzing the properties of the sub-Riemannian differential matrix. If the domain of definition of the mapping is not an open set, then the question arises what conditions can be set on the mapping that guarantee the boundedness of the intersection of the image of a compact set with a sub-Lorentzian ball. In this article, this problem is resolved: we consider that part of the intersection that can be parameterized by the connectivity component of the~intersection of the image of the sub-Riemannian differential and the ball. In addition, using such local parameterizations, a set function is introduced, which is constructed similarly to Hausdorff measure. We show that this set function is also a measure. As an application, the sub-Lorentzian area formula is established.
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Minimum dominating Randic energy of a graph
Статья научная
In this paper, we introduce the minimum dominating Randic energy of a graph and computed the minimum dominating Randic energy of graph. Also, obtained upper and lower bounds for the minimum dominating Randic energy of a graph.
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Mixed Problem for Even-Order Differential Equations with an Involution
Статья научная
In this manuscript we consider a mixed problem for even-order differential equations with an involution. In order to study this problem we use the corresponding differential operator with an involution, acting in the space of square integrable on a finite interval functions. Applying the method of similar operators, we transform this operator to the operator representable as orthogonal direct sum of a finite rank operator and an operators of rank 1. Moreover, it has exactly the same spectral properties as the original operator. Theorem on similarity is a basis for the construction of a group of operators, whose generator is the even-order differential operator with an involution. Using the previously obtained asymptotic formulas for the eigenvalues, we establish the main result dealing with the asymptotic representation for this group of operators. The group of operators allows us to introduce the notion of a weak solution for the corresponding mixed problem for the even-order differential operator with an involution and also to justify the Fourier method. In addition, using the representation of a group of operators, we obtain a explicit formula for a weak solution of the mixed problem and estimates for this group.
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Mollifications of contact mappings of Engel group
Статья научная
The contact mappings belonging to the metric Sobolev classes are studied on an Engel group with a left-invariant sub-Riemannian metric. In the Euclidean space one of the main methods to handle non-smooth mappings is the mollification, i.e., the convolution with a smooth kernel. An extra difficulty arising with contact mappings of Carnot groups is that the mollification of a contact mapping is usually not contact. Nevertheless, in the case considered it is possible to estimate the magnitude of deviation of contactness sufficiently to obtain useful results. We obtain estimates on convergence (or sometimes divergence) of the components of the differential of the mollified mapping to the corresponding components of the Pansu differential of the contact mapping. As an application to the quasiconformal analysis, we present alternative proofs of the convergence of mollified horizontal exterior forms and the commutativity of the pull-back of the exterior form by the Pansu differential with the exterior differential in the weak sense. These results in turn allow us to obtain such basic properties of mappings with bounded distortion as Holder continuity, differentiability almost everywhere in the sense of Pansu, Luzin N-property.
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New numerical method for solving nonlinear stochastic integral equations
Статья научная
The purpose of this paper is to propose the Chebyshev cardinal functions for solving Volterra stochastic integral equations. The method is based on expanding the required approximate solution as the element of Chebyshev cardinal functions. Though the way, a new operational matrix of integration is derived for the mentioned basis functions. More precisely, the unknown solution is expanded in terms of the Chebyshev cardinal functions including undetermined coefficients. By substituting the mentioned expansion in the original problem, the operational matrix reducing the stochastic integral equation to system of algebraic equations. The convergence and error analysis of the etablished method are investigated in Sobolev space. The method is numerically evaluated by solving test problems caught from the literature by which the computational efficiency of the method is demonstrated. From the computational point of view, the solution obtained by this method is in excellent agreement with those obtained by other works and it is efficient to use for different problems.
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Non-uniqueness of certain Hahn - Banach extensions
Статья научная
Let f be a continuous linear functional defined on a subspace M of a normed space X. If X is real or complex, there are results that characterize uniqueness of continuous extensions F of f to X for every subspace M and those that apply just to M. If X is defined over a non-Archimedean valued field K and the norm also satisfies the strong triangle inequality, the Hahn--Banach theorem holds for all subspaces M of X if and only if K is spherically complete and it is well-known that Hahn--Banach extensions are never unique in this context. We give a different proof of non-uniqueness here that is interesting for its own sake and may point a direction in which further investigation would be fruitful.
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