Статьи журнала - Владикавказский математический журнал
Все статьи: 1021
On Narrow Operators on K¨othe–Bochner Spaces
Статья научная
In this paper we investigate narrow linear operators in the setting of K¨othe–Bochner spaces of vector-valued measurable functions. We prove that for an atomless finite measure space (A,-, μ), a K¨othe F-space E on (A,-, μ) with the absolutely continuous norm, a Banach space X and an F-space Y every bounded linear operator T : E(X) → Y from a K¨othe–Bochner space E(X) to Y is narrow, whenever the density of the range space Y is strictly less than the density of E(B) for every B ∈ - with μ(B) > 0. Our main result is based on methods and technique of the set-theoretic topology, the measure theory and the theory of F-spaces. In this connection, we mention the work of Popov, Semenov and Vatsek [1], where sufficient conditions on a rearrangement invariant space E were established for the orthogonal projection onto the closed linear span of the Rademacher system to be a hereditarily narrow operator. Moreover, in [1] the authors posed a number of open problems turned out to be important for the further development of the theory of narrow operators.
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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 Proofs of the Existence of Non-Composite Polyhedra with Rhombic Vertices
Статья научная
Previously, the author considered closed convex polyhedra in E3 with symmetrical rhombic vertices and regular faces, the so-called RR-polyhedra, in two cases: 1) regular faces of such polyhedra are of the same type, 2) regular faces are of different types, but the polyhedra are composite. From the author’s previously published works, it follows that in both cases the list of RR-polyhedra is complete. In the first case, there were twenty-four of them, and in the second, fifty-four. A vertex V of a polyhedron is called rhombic if the vertex star Star(V ) consists of n equal and identically located rhombi (not squares) having a common vertex V . If the vertex V lies on the rotation axis of order n of the star Star(V ), then V is called symmetric. A symmetric rhombic vertex V is called obtuse-angled if the rhombi of the star Star(V ) at the vertex V meet at their obtuse angles. The star Star(V ) is then called symmetric rhombic. The faces of an RR-polyhedron can be divided into two non-empty disjoint sets: the set of equal symmetric rhombic stars that do not have common edges, and the set of regular faces. An example of an RR-polyhedron is an elongated rhombododecahedron. If an RR-polyhedron has regular faces of one type, then such an RR-polyhedron is classified as the first type. Some regular-faced polyhedra with conditional edges, such as the extended rhombododecahedron and the doubly complemented cuboctahedron, are special cases of RR-polyhedrons. In this case, the rhombic faces of regular-faced polyhedra with conditional edges, unlike the rhombi of RR-polyhedra, are composed only of regular triangles. In this paper, constructive proofs are given for the existence of four non-composite RR-polyhedra. For two of them the author previously found the angles of the rhombi that form symmetrical faceted stars of the vertices. In the case of the 25-sided polyhedron, the rhombic vertex has an acute-angled faceted star, i. e. the rhombi in it meet at their acute angles. And in the case of a 31-sided polyhedron the faceted star is obtuse. The existence of two other non-composite RR-polyhedra is proven on the basis of the regular dodecahedron and the cube.
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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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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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