Статьи журнала - Владикавказский математический журнал
Все статьи: 1009
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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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].
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On b-weakly demicompact operators on Banach lattices
Статья научная
Aqzzouz and Elbour proved that an operator T on a Banach lattice E is b-weakly compact if and only if ∥Txn∥→0 as n→∞ for each b-order bounded weakly sequence {xn} in E+. In this present paper, we introduce and study new concept of operators that we call b-weakly demicompact, use it to generalize known classes of operators which defined by b-weakly compact operators. An operator T on a Banach lattice E is said to be b-weakly demicompact if for every b-order bounded sequence {xn} in E+ such that xn→0 in σ(E,E′) and ∥xn-Txn∥→0 as n→∞, we have ∥xn∥→0 as n→∞. As consequence, we obtain a characterization of KB-spaces in terms of b-weakly demicompact operators. After that, we investigate the relationships between b-weakly demicompact operators and some other classes of operators on Banach lattices espaciallly their relationships with demi Dunford-Pettis operators and order weakly demicompact operators.
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On collectively-Levi sets of operators
Статья научная
The Levi operators are operator abstractions of the Levy property of Banach lattices. Such operators have been studied recently by several authors. The present paper deals with the collective properties of the Levi operators of several kinds: σ-Levi operators; quasi c-σ-Levi operators; and quasi σ-Levi operators. A notion of collectively σ-Levi set generalizes the notion of a single σ-Levi operator to the families of operators. Working with families of sequences of elements of a vector lattice requires the notion of the collective order convergence. This notion that is introduced and studied in the present paper may have its own interest and further possible applications. Various relations of the collectively quasi σ-Levi sets to the collectively compact sets are investigated. The domination problem for the collectively quasi σ-Levi sets is studied. In this study a special notion of a set of operators dominated by another set of operators is used.
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On combined nonstandard methods in functional analysis
Статья научная
The main nonstandard tool-kits are known as infinitesimal analysis (Robinson's nonstandard analysis) and Boolean-valued analysis. Sharp distinctions between these two versions of nonstandard analysis in content and technique notwithstanding, many ways are open to their simultaneous application. One of the simplest approaches consists in successive application of different nonstandard methods. It is demonstrated that combining is often useful in settling the problems of functional analysis which stem mainly from the theory of vector lattices.
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On conservation laws in affine toda systems
Статья научная
With the help of certain matrix decomposition and projectors of special forms we show that non-Abelian Toda systems associated with loop groups possess infinite sets of conserved quantities following from essentially different conservation laws.
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On extreme extension of positive operators
Статья научная
Given vector lattices E, F and a positive operator S from a majorzing subspace D of E to F, denote by E(S) the collection of all positive extensions of S to all of E. This note aims to describe the collection of extreme points of the convex set E(T∘S). It is proved, in particular, that E(T∘S) and T∘E(S) coincide and every extreme point of E(T∘S) is an extreme point of T∘E(S), whenever T:F→G is a Maharam operator between Dedekind complete vector lattices. The proofs of the main results are based on the three ingredients: a characterization of extreme points of subdifferentials, abstract disintegration in Kantorovich spaces, and an intrinsic characterization of subdifferentials.
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On finite homogeneous metric spaces
Статья научная
This survey is devoted to recently obtained results on finite homogeneous metric spaces. The main subject of discussion is the classification of regular and semiregular polytopes in Euclidean spaces by whether or not their vertex sets have the normal homogeneity property or the Clifford - Wolf homogeneity property. Every finite homogeneous metric subspace of an Euclidean space represents the vertex set of a compact convex polytope with the isometry group that is transitive on the set of vertices, moreover, all these vertices lie on some sphere. Consequently, the study of such subsets is closely related to the theory of convex polytopes in Euclidean spaces. The normal generalized homogeneity and the Clifford - Wolf homogeneity describe more stronger properties than the homogeneity. Therefore, it is natural to first check the presence of these properties for the vertex sets of regular and semiregular polytopes. In addition to the classification results, the paper contains a description of the main tools for the study of the relevant objects.
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On generalization of Fourier and Hartley transforms for some quotient class of sequences
Статья научная
In this paper we consider a class of distributions and generate two spaces of Boehmians for certain class of integral operators. We derive a convolution theorem and generate two spaces of Boehmians. The integral operator under concern is well-defined, linear and one-to-one in the class of Boehmians. An inverse problem is also discussed in some details.
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On generalized Kenmotsu manifolds as hypersurfaces of Vaisman-Gray manifolds
Статья научная
In this paper, we conclude that the hypersurfaces of Vaisman-Gray manifolds have generalized Kenmotsu structures under some conditions for the Lee form, Kirichenko's tensors and the second fundamental form of the immersion of the hypersurface into the manifold of Vaisman-Gray class. Moreover, the components of the second fundamental form are determined when the foregoing hypersurfaces have generalized Kenmotsu structures or any special kind of it or Kenmotsu structures, such that some of these components are vanish. Also, some components of Lee form and some components of some Kirichenko's tensors in the Vaisman-Gray class are equal to zero. On the other hand, the minimality of totally umbilical, totally geodesic hypersurfaces of Vaisman-Gray manifolds with generalized Kenmotsu structures are investigated. In addition, we deduced that the hypersurface of Vaisman-Gray manifold that have generalized Kenmotsu structure is totally geodesic if and only if it is totally umbilical and some components of Lee form are constants.
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On ideal of compact operators in real factors
Статья научная
In the present paper the real ideals of relatively compact operators of W*-algebras are considered. A description (up to isomorphism) of real two-sided ideal of relatively compact operators of the complex W*-factors is given.
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On meromorphic function with maximal deficiency sum and it's difference operators
Статья научная
The paper deals with characteristic funtion and deficiency of a meromorphic function. We mainly focused on the relation between the characteristic function of a product of difference operators with the characteristic function of a meromorphic function with maximal deficiency sum. The concept of maximal deficiency sum of a meromorphic function is employed as an effective tool for our research. In the same context, the notion of a difference polynomial of a difference operator is discussed. The paper contains the details analysis and discussion of some asymptotic behaviour of the product of difference operators, such as limr→∞T(r,∏qi=1Δηif)T(r,f), limr→∞N(r,0;∏qi=1Δηif)T(r,∏qi=1Δηif), lim r→∞N(r,∞;∏qi=1Δηif)+N(r,0;∏qi=1Δηif)T(r,∏qi=1Δηif) etc. and same resolution and discussion also developed for the difference polynomial of difference operators. Several innovative idea to establish some inequalities on the zeros and poles for ∏qi=1Δηif and L(Δηf) are also introduced. We broadly elaborate our results with many remarks and corollaries, and give two excellent examples for proper justification of our results. The results on product and polynomial of difference operators of our article improved and generalised the results of Z. Wu.
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On minimax theorems for sets closed in measure
Статья научная
This article is devoted to the Ky Fan minimax theorem for convex sets closed in measure in L^1. In general, these sets do not carry any formal compactness properties for any reasonable topology.
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On neveu decomposition and ergodic type theorems for semi-finite von Neumann algebras
Статья научная
Some ergodic type theorems for automorphisms of semi-finite von Neumann algebras are considered. Neveu decomposition is employed in order to prove stochastical convergence. This work is a generalization of authors results from [5] to the case of semi-finite von Neumann algebras.
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On non-commutative ergodic type theorems for free finitely generated semigroups
Статья научная
In the paper the authors generalize Bufetov's Ergodic Type Theorems to the case of the actions of free finitely generated semigroups on von Neumann algebras.
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On normal subgroups of the group representation of the Cayley tree
Статья научная
Gibbs measure plays an important role in statistical mechanics. On a Cayley tree, for describing periodic Gibbs measures for models in statistical mechanics we need subgroups of the group representation of the Cayley tree. A normal subgroup of the group representation of the Cayley tree keeps the invariance property which is a significant tool in finding Gibbs measures. By this occasion, a full description of normal subgroups of the group representation of the Cayley tree is a significant problem in Gibbs measure theory. For instance, in [1, 2] a full description of normal subgroups of indices four, six, eight, and ten for the group representation of a Cayley tree is given. The present paper is a generalization of these papers, i. e., in this paper, for any odd prime number p, we give a characterization of the normal subgroups of indices 2n, n∈{p,2p} and 2i,i∈N, of the group representation of the Cayley tree.
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