Статьи журнала - Владикавказский математический журнал
Все статьи: 883
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.
Бесплатно
Персоналии
This is a short overview of some sections of applied functional analysis, convexity, optimization, and nonstandard models.
Бесплатно
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.
Бесплатно
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.
Бесплатно
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.
Бесплатно
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.
Бесплатно
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.
Бесплатно
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.
Бесплатно
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.
Бесплатно
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.
Бесплатно
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.
Бесплатно
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.
Бесплатно
Статья научная
We present an elementary proof of the (known) fact that a CD_0(K)-space is a Banach lattice and is lattice isometrically isomorphic to a particular C(\widetilde{K}) for some compact space \widetilde{K}.
Бесплатно
On Janowski type harmonic functions associated with the Wright hypergeometric functions
Статья научная
In our present study we consider Janowski type harmonic functions class introduced and studied by Dziok, whose members are given by h(z)=z+∑∞n=2hnzn and g(z)=∑∞n=1gnzn, such that STH(F,G)={f=h+g¯∈H:DHf(z)f(z) ≺ 1+Fz1+Gz;(-G ≤ F
Бесплатно
On Poletsky-type modulus inequalities for some classes of mappings
Статья научная
It is well-known that the theory of mappings with bounded distortion was laid by Yu. G. Reshetnyak in 60-th of the last century [1]. In papers [2, 3], there was introduced the two-index scale of mappings with weighted bounded (q,p)-distortion. This scale of mappings includes, in particular, mappings with bounded distortion mentioned above (under q=p=n and the trivial weight function). In paper [4], for the two-index scale of mappings with weighted bounded (q,p)-distortion, the Poletsky-type modulus inequality was proved under minimal regularity; many examples of mappings were given to which the results of [4] can be applied. In this paper we show how to apply results of [4] to one such class. Another goal of this paper is to exhibit a new class of mappings in which Poletsky-type modulus inequalities is valid. To this end, for n=2, we extend the validity of the assertions in [4] to the limiting exponents of summability: 1
Бесплатно
On Riesz spaces with b-property and b-weakly compact operators
Статья научная
An operator Т: E→ X between a Banach lattice E and a Banach space X is called b-weakly compact if T(B) is relatively weakly compact for each b-bounded set B in E. We characterize b-weakly compact operators among o-weakly compact operators. We show summing operators are b-weakly compact and discuss relation between Dunford--Pettis and b-weakly compact operators. We give necessary conditions for b-weakly compact operators to be compact and give characterizations of KB-spaces in terms of b-weakly compact operators defined on them.
Бесплатно
On a decomposition equality in modular group rings
Статья научная
Let G be an abelian group such that A\leq G with p-component A_p and B\leq G, and let R be a commutative ring with 1 of prime characteristic p with nil-radical N(R). It is proved that if A_p\not\subseteq B_p or N(R)\not= 0, then S(RG)=S(RA)(1+I_p(RG; B)) \iff G=AB and G_p=A_pB_p. In particular, if A_p\not= 1 or N(R)\not= 0, then S(RG)=S(RA)\times (1+I_p(RG; B)) \iff G=A\times B. So, the question concerning the validity of this formula is completely exhausted. The main statement encompasses both the results of this type established by the author in (Hokkaido Math. J., 2000) and (Miskolc Math. Notes, 2005). We also point out and eliminate in a concrete situation an error in the proof of a statement due to T. Zh. Mollov on a decomposition formula in commutative modular group rings (Proceedings of the Plovdiv University-Math., 1973).
Бесплатно
Статья научная
On a new combination of orthogonal polynomials sequences
Статья научная
In this paper, we are interested in the following inverse problem. We assume that {Pn}n≥0 is a monic orthogonal polynomials sequence with respect to a quasi-definite linear functional u and we analyze the existence of a sequence of orthogonal polynomials {Qn}n≥0 such that we have a following decomposition Qn(x)+rnQn-1(x)=Pn(x)+snPn-1(x)+tnPn-2(x)+vnPn-3(x), n≥0, when vnrn≠0, for every n≥4. Moreover, we show that the orthogonality of the sequence {Qn}n≥0 can be also characterized by the existence of sequences depending on the parameters rn, sn, tn, vn and the recurrence coefficients which remain constants. Furthermore, we show that the relation between the corresponding linear functionals is k(x-c)u=(x3+ax2+bx+d)v, where c,a,b,d∈C and k∈C∖{0}. We also study some subcases in which the parameters rn, sn, tn and vn can be computed more easily. We end by giving an illustration for a special example of the above type relation.
Бесплатно
Статья научная
In this article the problems of the unique classical solvability and the construction of the solution of a nonlinear boundary value problem for a fifth order partial integro-differential equations with degenerate kernel are studied. Dirichlet boundary conditions are specified with respect to the spatial variable. So, the Fourier series method, based on the separation of variables is used. A countable system of the second order ordinary integro-differential equations with degenerate kernel is obtained. The method of degenerate kernel is applied to this countable system of ordinary integro-differential equations. A system of countable systems of algebraic equations is derived. Then the countable system of nonlinear Fredholm integral equations is obtained. Iteration process of solving this integral equation is constructed. Sufficient coefficient conditions of the unique solvability of the countable system of nonlinear integral equations are established for the regular values of parameter. In proof of unique solvability of the obtained countable system of nonlinear integral equations the method of successive approximations in combination with the contraction mapping method is used. In the proof of the convergence of Fourier series the Cauchy-Schwarz and Bessel inequalities are applied. The smoothness of solution of the boundary value problem is also proved.
Бесплатно
On a question on Banach - Stone theorem
Краткое сообщение
We present a very simple and elementary proof of the main theorem of [l]. This also gives an answer to a conjecture in [1].
Бесплатно
