Статьи журнала - Владикавказский математический журнал
Все статьи: 995
Inequalities for the Schwarzian derivative for subclasses of convex functions in the unit disc
Статья научная
Nehari norm of the Schwarzian derivative of an analytic function is closely related to its univalence. The famous Nehari--Kraus theorem [3, 4] and Ahlfors--Weill theorem [1] are of fundamental importance in this direction. For a non-constant meromorphic function f on D the unite disc, the Schwarzian derivative S_f of f by is holomorphic at z_0\in D if and only if f is locally univalent at z_0. The aim of this paper is to give sharp estimates of the Nehari norm for the subclasses of convex functions in the unit disc.
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Infinitely fine partitions of measures spaces
Статья научная
In any measurable space one can find a hyperfinite infinitesimal partition, that is, a hyperfinite set of disjoint inner (in the sense of nonstandard analysis) measurable subsets such that every standard measurable set is representable as a union of sets of this collection. In this paper we characterize the various properties of measures in terms of infinitesimal partitions. In particular, we characterize the non-atomicness of the measures and give a short proof of the Sobchik-Hammer theorem.
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Infinitesimals in ordered vector spaces
Статья научная
An infinitesimal approach to ordered spaces is proposed. Archimedean property and Dedekind completeness in ordered spaces are discussed from a nonstandard point of view.
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Статья научная
In this paper we consider classical solutions of an initial boundary value problem for a system of semilinear parabolic equations with absorption and nonlinear nonlocal boundary conditions. Nonlinearities in equations and boundary conditions may not satisfy the Lipschitz condition. To prove the existence of a solution we regularize the original problem. Using the Schauder–Tikhonov fixed point theorem, the existence of a local solution of regularized problem is proved. It is shown that the limit of solutions of the regularized problem is a maximal solution of the original problem. Using the properties of a maximal solution, a comparison principle is proved. In this case, no additional assumptions are made when nonlinearities in absorption do not satisfy the Lipschitz condition. Conditions are found under which solutions are positive functions. The uniqueness of the solution is established. It is shown that the trivial solution (0, 0) may not be unique.
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Inverse coefficient problem for the 2D wave equation with initial and nonlocal boundary conditions
Статья научная
In this paper, we consider direct and inverse problems for the two-dimensional wave equation. The direct problem is an initial boundary value problem for this equation with nonlocal boundary conditions. In the inverse problem, it is required to find the time-variable coefficient at the lower term of the equation. The classical solution of the direct problem is presented in the form of a biorthogonal series in eigenvalues and associated functions, and the uniqueness and stability of this solution are proven. For solution to the inverse problem, theorems of existence in local, uniqueness in global, and an estimate of conditional stability are obtained. The problems of determining the right-hand sides and variable coefficients at the lower terms from initial boundary value problems for second-order linear partial differential equations with local boundary conditions have been studied by many authors. Since the nonlinearity is convolutional, the unique solvability theorems in them are proven in a global sense. In the works, the method of separation of variables is used to find the classical solution of the direct problem in the form of a biorthogonal series in terms of eigenfunctions and associated functions. The nonlocal integral condition is used as the overdetermination condition with respect to the solution of the direct problem. The direct problem reduces to equivalent integral equations of the Fourier method. To establish integral inequalities, the generalized Gronwall-Bellman inequality is used. We obtain an a priori estimate of the solution in terms of an unknown coefficient, that are useful for studying the inverse problem.
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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 Sequence Spaces and Summing Mappings
Статья научная
This paper contributes to the theory of positive summing operators between Banach lattices by exploring the interplay between specialized sequence spaces, operator ideals, and tensor product techniques. We focus on the spaces of positive strongly p-summable sequences ℓπp(X) and positive unconditionally p-summable sequences ℓup,|ω|(X), utilizing them alongside the Banach lattice of positive weakly p-summable sequences ℓp,|ω|(X) . These tools are employed to present and characterize three central classes: positive strongly (p,q)-summing operators, positive (p,q)-summing operators, and positive Cohen (p,q)-nuclear operators. Our investigation yields new properties, including the characterization of positive (p,q)-summing operators as those which map positive unconditionally p-summable sequences into q-summable sequences, and the identification of the positive strongly (p,q)-summing class with the class of (p,q)-majorizing operators. A central achievement of this work is the unified characterization of these operator classes via tensor product continuity, a method well-established for linear operator ideals that we now extend to the context of Banach lattices. We characterize each class by the continuity of an associated tensor operator I⊗T:ℓp⊗αX→ℓq⊗βY for appropriate tensor norms α and β. This approach provides a powerful and cohesive framework that deepens the connections between summability, the order structure of Banach lattices, and tensor norms.
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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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