Статьи журнала - Владикавказский математический журнал
Все статьи: 995
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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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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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}.
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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
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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
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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.
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On Self-Similar Solutions of a Multi-Phase Stefan Problem in a Moving Ray
Статья научная
We study self-similar solutions of a multi-phase Stefan problem for a heat equation on the moving ray x > √t with Dirichlet or Neumann boundary conditions at the boundary x = √t. In the case of Dirichlet condition we prove that an algebraic system for determination of the free boundaries is gradient one and the corresponding potential is an explicitly written strictly convex and coercive function. Therefore, there exists a unique minimum point of the potential, which determines free boundaries and provides the solution. In the case of Neumann condition solutions with different numbers (called types) of phase transitions appear. For each fixed type the system for determination of the free boundaries is again gradient with a strictly convex potential. This allows to find precise conditions for existence and uniqueness of a solution. In the last section we study Stefan–Dirichlet problem on the half-line x > 0 with infinitely many phase transitions. Using again a variational approach, we find sufficient conditions of existence and uniqueness of a solution to the problem under consideration.
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Статья научная
We consider some multiplicative interpolation inequalities between the H¨older space and the Lebesgue space. Multiplicative interpolation inequalities of the Gagliardo–Nirenberg type are used in the investigations of partial differential equations. Several such inequalities involving the H¨older norm (seminorm) were already proved and applied. In the present paper we generalise previous results to the anisotropic “parabolic” case with another simple proof due to idea of Olga Ladyzhenskaya. The manuscript also contains an application of such Gagliardo–Nirenberg type inequality with the H¨older norm. Some integral estimate and this inequality give a priori estimate of the solution to quasilinear parabolic problem in the smooth H¨older classes. Moreover, using this a priori estimate, we establish the existence of solution of the quasilinear parabolic problem. In order to prove multiplicative inequality of the Gagliardo–Nirenberg type with the H¨older norm we use an equivalent normalization of the higher order H¨older spaces over higher order finite differences. The key technical tool is the representation of a function u(x, t) at an arbitrary fixed point (x, t) over a higher order finite difference at this point and the corresponding additional sum of values at neighboring points. After that we integrate with respect to the neighboring points over the balls Br((x, t)) of small radius r. Estimating the finite difference over the corresponding H¨older seminorm, we obtain an additive inequality with the parameter r, involving the H¨older and integral norms. Optimizing this inequality over r we get the multiplicative estimate of the Gagliardo–Nirenberg type with the H¨older norm and the Lebesgue norm.
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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).
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On a new class of meromorphic functions associated with Mittag-Leffler function
Статья научная
The Mittag-Leffler function arises naturally in solving differential and integral equations of fractional order and especially in the study of fractional generalization of kinetic equation, random walks, Levy flights, super-diffusive transport and in the study of complex systems. In the present investigation, the authors define a new class Mτ,κς,ϱ(ϑ,℘) of meromorphic functions defined in the punctured unit disk Δ∗:={z∈C:0
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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.
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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.
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