Статьи журнала - Владикавказский математический журнал
Все статьи: 930
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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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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Nonlinear viscosity algorithm with perturbation for non-expansive multi-valued mappings
Статья научная
The viscosity iterative algorithms for finding a common element of the set of fixed points for nonlinear operators and the set of solutions of variational inequality problems have been investigated by many authors. The viscosity technique allow us to apply this method to convex optimization, linear programming and monoton inclusions. In this paper, based on viscosity technique with perturbation, we introduce a new nonlinear viscosity algorithm for finding an element of the set of fixed points of nonexpansive multi-valued mappings in a Hilbert spaces. Furthermore, strong convergence theorems of this algorithm were established under suitable assumptions imposed on parameters. Our results can be viewed as a generalization and improvement of various existing results in the current literature. Moreover, some numerical examples that show the efficiency and implementation of our algorithm are presented.
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Nonstandard models and optimization
Статья научная
This is an overview of a few possibilities that are open by model theory in optimization. Most attention is paid to the impact of infinitesimal analysis and Boolean valued models to convexity, Pareto optimality, and hyperapproximation.
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Note on surjective polynomial operators
Статья научная
A linear Markov chain is a discrete time stochastic process whose transitions depend only on the current state of the process. A nonlinear Markov chain is a discrete time stochastic process whose transitions may depend on both the current state and the current distribution of the process. These processes arise naturally in the study of the limit behavior of a large number of weakly interacting Markov processes. The nonlinear Markov processes were introduced by McKean and have been extensively studied in the context of nonlinear Chapman-Kolmogorov equations as well as nonlinear Fokker-Planck equations. The nonlinear Markov chain over a finite state space can be identified by a continuous mapping (a nonlinear Markov operator) defined on a set of all probability distributions (which is a simplex) of the finite state space and by a family of transition matrices depending on occupation probability distributions of states. Particularly, a linear Markov operator is a linear operator associated with a square stochastic matrix. It is well-known that a linear Markov operator is a surjection of the simplex if and only if it is a bijection. The similar problem was open for a nonlinear Markov operator associated with a stochastic hyper-matrix. We solve it in this paper. Namely, we show that a nonlinear Markov operator associated with a stochastic hyper-matrix is a surjection of the simplex if and only if it is a permutation of the Lotka-Volterra operator.
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On Borel's extension theorem for general Beurling classes of ultradifferentiable functions
Статья научная
We obtain necessary and sufficient conditions under which general Beurling class of ultradifferentiable functions admits a version of Borel's extension theorem.
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