Статьи журнала - Владикавказский математический журнал
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Generalization of Eberlein's and sine's ergodic theorems to lr-nets
Статья научная
The notion of LR-nets provides an appropriate setting for study of various ergodic theorems in Banach spaces. In the present paper, we prove Theorems 2.1, 3.1 which extend Eberlein's and Sine's ergodic theorems to LR-nets. Together with Theorem 1.1, these two theorems form the necessary background for further investigation of strongly convergent LR-nets. Theorem 2.1 is due to F. Rabiger, and was announced without a proof in [1].
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Generalization of the Ostrowski inequalities on time scales
Статья научная
The idea of time scales calculus’ theory was initiated and introduced by Hilger (1988) in his PhD thesis order to unify discret and continuous analysis and to expend the discrete and continous theories to cases ``in between''. Since then, mathematical research in this field has exceeded more than 1000 publications and a lot of applications in the fields of science, i.e., operations research, economics, physics, engineering, statistics, finance and biology. Ostrowski proved an inequality to estimate the absolute deviation of a differentiable function from its integral mean. This result was obtained by applying the Montgomery identity. In the present paper we derive a generalization of the Montgomery identity to the various time scale versions such as discrete case, continuous case and the case of quantum calculus, by obtaining this generalization of Montgomery identity we would prove our results about the generalization of the Ostrowski inequalities (without weighted case) to the several time scales such as discrete case, continuous case and the case of quantum calculus and recapture the several published results of different authors of various papers and thus unify corresponding discrete version and continuous version. Similarly we would also derive our results about the generalization of the Ostrowski inequalities (weighted case) to the different time scales such as discrete case and continuous case and recapture the different published results of several authors of various papers and thus unify corresponding discrete version and continuous version. Moreover, we would use our obtained results (without weighted case) to the case of quantum calculus.
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Статья научная
The so called grand spaces nowadays are one of the main objects in the theory of function spaces. Grand Lebesgue spaces were introduced by T. Iwaniec and C. Sbordone in the case of sets Ω with finite measure |Ω|
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H-операторы в идеальных пространствах со смешанной квазинормой E (\ Omega)
Статья научная
Цель настоящей работы - с единой точки зрения рассмотреть поведение нелинейных операторов типа суперпозиции, интегральных операторов Гаммерштейна и Урысона в общих квазинормированных идеальных пространствах. Некоторые из нижеприведенных результатов анонсированы нами ранее в статье [1].
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![H\"Older type inequalities for orthosymmetric bilinear operators](/file/thumb/14318219/holder-type-inequalities-for-orthosymmetric-bilinear-operators.png "H\\"Older type inequalities for orthosymmetric bilinear operators")
H\"Older type inequalities for orthosymmetric bilinear operators
Статья научная
An interplay between squares of vector lattice and homogeneous functional calculus is considered and H\"older type inequalities for orthosymmetric bilinear operators are obtained.
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Hankel determinant of third kind for certain subclass of multivalent analytic functions
Статья научная
The objective of this paper is to obtain an upper bound (not sharp) to the third order Hankel determinant for certain subclass of multivalent (p-valent) analytic functions, defined in the open unit disc E. Using the Toeplitz determinants, we may estimate the Hankel determinant of third kind for the normalized multivalent analytic functions belongng to this subclass. But, using the technique adopted by Zaprawa [1], i.e., grouping the suitable terms in order to apply Lemmas due to Hayami [2], Livingston [3] and Pommerenke [4], we observe that, the bound estimated by the method adopted by Zaprawa is more refined than using upon applying the Toeplitz determinants.
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Hardy type inequalities in classical and grand Lebesgue spaces lp), 0
Статья научная
In 2020 Rovshan A. Bandaliev et al. proved the boundedness of Hardy operator for monotone functions in grand Lebesgue spaces Lp)(0,1), 0
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Homogeneous functions of regular linear and bilinear operators
Статья научная
Using envelope representations explicit formulae for computing \widehat{\varphi}(T_1,...,T_N) for any finite sequence of regular linear or bilinear operators T_1,...,T_N on vector lattices are derived.
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Increasing unions of Stein spaces with singularities
Статья научная
We show that if X is a Stein space and, if Ω⊂X is exhaustable by a sequence Ω1⊂Ω2⊂…⊂Ωn⊂… of open Stein subsets of X, then Ω is Stein. This generalizes a well-known result of Behnke and Stein which is obtained for X=Cn and solves the union problem, one of the most classical questions in Complex Analytic Geometry. When X has dimension 2, we prove that the same result follows if we assume only that Ω⊂⊂X is a domain of holomorphy in a Stein normal space. It is known, however, that if X is an arbitrary complex space which is exhaustable by an increasing sequence of open Stein subsets X1⊂X2⊂⋯⊂Xn⊂…, it does not follow in general that X is holomorphically-convex or holomorphically-separate (even if X has no singularities). One can even obtain 2-dimensional complex manifolds on which all holomorphic functions are constant.
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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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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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