Статьи журнала - Владикавказский математический журнал
Все статьи: 930
Characterizations of finite dimensional Archimedean vector lattices
Статья научная
In this paper, we give some necessary and sufficient conditions for an Archimedean vector lattice A to be of finite dimension. In this context, we give three characterizations. The first one contains the relation between the vector lattice A to be of finite dimension and its universal completion Au. The second one shows that the vector lattice A is of finite dimension if and only if one of the following two equivalent conditions holds : (a) every maximal modular algebra ideal in Au is relatively uniformly complete or (b) Orth(A,Au)=Z(A,Au) where Orth(A,Au) and Z(A,Au) denote the vector lattice of all orthomorphisms from A to Au and the sublattice consisting of orthomorphisms π with |π(x)|≤λ|x| (x∈A) for some 0≤λ∈R, respectively. It is well-known that any universally complete vector lattice A is of the form C∞(X) for some Hausdorff extremally disconnected compact topological space X. The point x∈X is called σ- isolated if the intersection of every sequence of neighborhoods of x is a neighborhood of x. The last characterization of finite dimensional Archimedean vector lattices is the following. Let A be a vector lattice and let Au(=C∞(X)) be its universal completion. Then A is of finite dimension if and only if each element of X is σ-isolated. Bresar in \cite{4} raised a question to find new examples of zero product determined algebras. Finally, as an application, we give a positive answer to this question.
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Color energy of some cluster graphs
Статья научная
Let G be a simple connected graph. The energy of a graph G is defined as sum of the absolute eigenvalues of an adjacency matrix of the graph G. It represents a proper generalization of a formula valid for the total π-electron energy of a conjugated hydrocarbon as calculated by the Huckel molecular orbital (HMO) method in quantum chemistry. A coloring of a graph G is a coloring of its vertices such that no two adjacent vertices share the same color. The minimum number of colors needed for the coloring of a graph G is called the chromatic number of G and is denoted by χ(G). The color energy of a graph G is defined as the sum of absolute values of the color eigenvalues of G. The graphs with large number of edges are referred as cluster graphs. Cluster graphs are graphs obtained from complete graphs by deleting few edges according to some criteria. It can be obtained on deleting some edges incident on a vertex, deletion of independent edges/triangles/cliques/path P3 etc. Bipartite cluster graphs are obtained by deleting few edges from complete bipartite graphs according to some rule. In this paper, the color energy of cluster graphs and bipartite cluster graphs are studied.
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Conformal Ricci soliton in an indefinite trans-Sasakian manifold
Статья научная
Conformal Ricci solitons are self similar solutions of the conformal Ricci flow equation. A new class of n-dimensional almost contact manifold namely trans-Sasakian manifold was introduced by Oubina in 1985 and further study about the local structures of trans-Sasakian manifolds was carried by several authors. As a natural generalization of both Sasakian and Kenmotsu manifolds, the notion of trans-Sasakian manifolds, which are closely related to the locally conformal Kahler manifolds introduced by Oubina. This paper deals with the study of conformal Ricci solitons within the framework of indefinite trans-Sasakian manifold. Further, we investigate the certain curvature tensor on indefinite trans-Sasakian manifold. Also, we have proved some important results.
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Cyclically compact operators in banach spaces
Статья научная
The Boolean-valued interpretation of compactness gives rise to the new notions of cyclically compact sets and operators which deserves an independent study. A part of the corresponding theory is presented in this work. General form of cyclically compact operators in Kaplansky--Hilbert module as well as a variant of Fredholm Alternative for cyclically compact operators are also given.
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Cлабая непрерывность оператора суперпозиции в пространствах последовательностей
Статья научная
Изучаются условия слабой непрерывности оператора суперпозиции, действующего в некотором пространстве последовательностей. Даны условия, при которых слабая непрерывность оператора суперпозиции равносильна его аффинности. В то же самое время, в пространстве сходящихся к нулю последовательностей любая ограниченная непрерывная функция порождает слабо непрерывный оператор суперпозиции. Приведены примеры, показывающие существенность предположения об ограниченности. Показывается, что в произвольном бесконечномерном пространстве последовательностей всегда существует оператор суперпозиции, являющийся слабо непрерывным и не представимый в виде суммы аффинного оператора и оператора обладающего конечномерной областью значений.
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Derivations on Banach *-ideals in von Neumann algebras
Статья научная
It is known that any derivation δ:M→M on the von Neumann algebra M is an inner, i.e. δ(x):=δa(x)=[a,x]=ax-xa, x∈M, for some a∈M. If H is a separable infinite-dimensional complex Hilbert space and K(H) is a C∗-subalgebra of compact operators in C∗-algebra B(H) of all bounded linear operators acting in H, then any derivation δ:K(H)→K(H) is a spatial derivation, i.e. there exists an operator a∈B(H) such that δ(x)=[x,a] for all x∈K(H). In addition, it has recently been established by Ber A. F., Chilin V. I., Levitina G. B. and Sukochev F. A. (JMAA, 2013) that any derivation δ:E→E on Banach symmetric ideal of compact operators E⊆K(H) is a spatial derivation. We show that the same result is also true for an arbitrary Banach ∗-ideal in every von Neumann algebra M. More precisely: If M is an arbitrary von Neumann algebra, E be a Banach ∗-ideal in M and δ:E→E is a derivation on E, then there exists an element a∈M such that δ(x)=[x,a] for all x∈E, i.e. δ is a spatial derivation.
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Статья научная
This paper is devoted to obtaining a unique solution to an inverse problem for a multid-mensional time-fractional integro-differential equation. In the case of additional data, we consider an inverse problem. The unknown coefficient and kernel are uniquely determined by the additional data. By using the fixed point theorem in suitable Sobolev spaces, the global in time existence and uniqueness results of this inverse problem are obtained. The weak solvability of a nonlinear inverse boundary value problem for a $d$-dimensional fractional diffusion-wave equation with natural initial conditions was studied in the work. First, the existence and uniqueness of the direct problem were investigated. The considered problem was reduced to an auxiliary inverse boundary value problem in a certain sense and its equivalence to the original problem was shown. Then, the local existence and uniqueness theorem for the auxiliary problem is proved using the Fourier method and contraction mappings principle. Further, based on the equivalency of these problems, the global existence and uniqueness theorem for the weak solution of the original inverse coefficient problem was established for any value of time.
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Dynamics of quadratic Volterra-type stochastic operators corresponding to strange tournaments
Статья научная
By studying the dynamics of these operators on the simplex, focusing on the presence of an interior fixed point, we investigate the conditions under which the operators exhibit nonergodic behavior. Through rigorous analysis and numerical simulations, we demonstrate that certain parameter regimes lead to nonergodicity, characterized by the convergence of initial distributions to a limited subset of the simplex. Our findings shed light on the intricate dynamics of quadratic stochastic operators with interior fixed points and provide insights into the emergence of nonergodic behavior in complex dynamical systems. Also, the nonergodicity of quadratic stochastic operators of Volterra type with an interior fixed point defined in a simplex introduces additional complexity to the already intricate dynamics of such systems. In this context, the presence of an interior fixed point within the simplex further complicates the exploration of the state space and convergence properties of the operator. In this paper, we give sufficiency and necessary conditions for the existence of strange tournaments. Also, we prove the nonergodicity of quadratic stochastic operators of Volterra type with an interior fixed point, defined in a simplex.
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Erratum to: "Infinitesimals in ordered vector spaces"
Другой
In this note, Theorem 1 in the article which is cited in the title is corrected.
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Every lateral band is the kernel of an orthogonally additive operator
Статья научная
In this paper we continue a study of relationships between the lateral partial order ⊑ in a vector lattice (the relation x⊑y means that x is a fragment of y) and the theory of orthogonally additive operators on vector lattices. It was shown in [1] that the concepts of lateral ideal and lateral band play the same important role in the theory of orthogonally additive operators as ideals and bands play in the theory for linear operators in vector lattices. We show that, for a vector lattice E and a lateral band G of E, there exists a vector lattice F and a positive, disjointness preserving orthogonally additive operator T:E→F such that kerT=G. As a consequence, we partially resolve the following open problem suggested in [1]: Are there a vector lattice E and a lateral ideal in E which is not equal to the kernel of any positive orthogonally additive operator T:E→F for any vector lattice F?
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Existence and uniqueness theorems for a differential equation with a discontinuous right-hand side
Статья научная
We consider new conditions for existence and uniqueness of a Caratheodory solution for an initial value problem with a discontinuous right-hand side. The method used here is based on: 1) the representation of the solution as a Fourier series in a system of functions orthogonal in Sobolev sense and generated by a classical orthogonal system; 2) the use of a specially constructed operator A acting in l2, the fixed point of which are the coefficients of the Fourier series of the solution. Under conditions given here the operator A is contractive. This property can be employed to construct robust, fast and easy to implement spectral numerical methods of solving an initial value problem with discontinuous right-hand side. Relationship of new conditions with classical ones (Caratheodory conditions with Lipschitz condition) is also studied. Namely, we show that if in classical conditions we replace L1 by L2, then they become equivalent to the conditions given in this article.
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Existence of classical solutions for a class of the Khokhlov-Zabolotskaya-Kuznetsov type equations
Статья научная
In medical sciences, during medical exploration and diagnosis of tissues or in medical imaging, we often use mathematical models to answer questions related to these examinations. Among these models, the nonlinear partial differential equation of the Khokhlov-Zabolotskaya-Kuznetsov type (abbreviated as the KZK equation) is of proven interest in ultrasound acoustics problems. This mathematical model describes the nonlinear propagation of a sound pulse of finite amplitude in a thermo-viscous medium. The equation is obtained by combining the conservation of mass equation, the conservation of momentum equation and the equations of state. It should be noted that for this equation little mathematical analysis is reserved. This equation takes into account three combined effects: the diffraction of the wave, the absorption of energy and the nonlinearity of the medium in which the wave propagates. KZK-type equation introduced in this paper is a modified version of the KZK model known in acoustics. We study a class of the Khokhlov-Zabolotskaya-Kuznetsov type equations for the existence of global classical solutions. We give conditions under which the considered equations have at least one and at least two classical solutions. To prove our main results, we propose a new approach based on recent theoretical results.
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Existence of global classical solutions for the Saint-Venant equations
Статья научная
Nowadays, investigations of the existence of global classical solutions for non linear evolution equations is a topic of active mathematical research. In this article, we are concerned with a classical system of shallow water equations which describes long surface waves in a fluid of variable depth. This system was proposed in 1871 by Adhemar Jean-Claude Barre de Saint-Venant. Namely, we investigate an initial value problem for the one dimensional Saint-Venant equations. We are especially interested in question of what sufficient conditions the initial data and the topography of the bottom must verify in order that the considered system has global classical solutions. In order to prove our main results we use a new topological approach based on the fixed point abstract theory of the sum of two operators in Banach spaces. This basic and new idea yields global existence theorems for many of the interesting equations of mathematical physics.
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Existence of solutions for a class of impulsive Burgers equation
Статья научная
We study a class of impulsive Burgers equations. A new topological approach is applied to prove the existence of at least one and at least two nonnegative classical solutions. The arguments are based on recent theoretical results. Here we focus our attention on a class of Burgers equations and we investigate it for the existence of classical solutions. The Burgers equation can be used for modeling both traveling and standing nonlinear plane waves. The simplest model equation can describe the second-order nonlinear effects connected with the propagation of high-amplitude (finite-amplitude waves) plane waves and, in addition, the dissipative effects in real fluids. There are several approximate solutions to the Burgers equation. These solutions are always fixed to areas before and after the shock formation. For an area where the shock wave is forming no approximate solution has yet been found. Therefore, it is therefore necessary to solve the Burgers equation numerically in this area.
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Existence results for a Dirichlet boundary value problem involving the p(x)-Laplacian operator
Статья научная
The aim of this paper is to establish the existence of weak solutions, in W1,p(x)0(Ω), for a Dirichlet boundary value problem involving the p(x)-Laplacian operator. Our technical approach is based on the Berkovits topological degree theory for a class of demicontinuous operators of generalized (S+) type. We also use as a necessary tool the properties of variable Lebesgue and Sobolev spaces, and specially properties of p(x)-Laplacian operator. In order to use this theory, we will transform our problem into an abstract Hammerstein equation of the form v+S∘Tv=0 in the reflexive Banach space W-1,p′(x)(Ω) which is the dual space of W1,p(x)0(Ω). Note also that the problem can be seen as a nonlinear eigenvalue problem of the formAu=λu, where Au:=-Div(|∇u|p(x)-2∇u)-f(x,u). When this problem admits a non-zero weak solution u, λ is an eigenvalue of it and u is an associated eigenfunction.
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Статья научная
In this paper, we provide sufficient conditions for the existence of solutions of initial value problem, for perturbed partial functional hyperbolic differential equations of fractional order involving Caputo fractional derivative with state-dependent delay by reducing the research to the search of the existence and the uniqueness of fixed points of appropriate operators. Our main result for this problem is based on a nonlinear alternative fixed point theorem for the sum of a completely continuous operator and a contraction one in Banach spaces due to Burton and Kirk and a fractional version of Gronwall's inequality. We should observe the structure of the space and the properties of the operators to obtain existence results. To our knowledge, there are very few papers devoted to fractional differential equations with finite and/or infinite constant delay on bounded domains. Many other questions and issues can be investigated regarding the existence in the space of weighted continuous functions, the uniqueness, the structure of the solutions set and also whether or not the condition satisfied by the operators are optimal. This paper can be considered as a contribution in this setting case. Examples are given to illustrate this work.
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Existence theorem for a fractal Sturm-Liouville problem
Статья научная
In this article, using a new calculus defined on fractal subsets of the set of real numbers, a Sturm-Lioville type problem is discussed, namely the fractal Sturm-Liouville problem. The existence and uniqueness theorem has been proved for such equations. In this context, the historical development of the subject is discussed in the introduction. In Section 2, the basic concepts of Fα-calculus defined on fractal subsets of real numbers are given, i.e., Fα-continuity, Fα-derivative and fractal integral definitions are given and some theorems to be used in the article are given. In Section 3, the existence and uniqueness of the solutions for the fractal Sturm-Liouville problem are obtained by using the successive approximations method. Thus, the well-known existence and uniqueness problem for Sturm-Liouville equations in ordinary calculus is handled on the fractal calculus axis, and the existing results are generalized.
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Exponential stability for a swelling porous-heat system with thermodiffusion effects and delay
Статья научная
In the present work, we consider a one-dimensional swelling porous-heat system with single time-delay in a bounded domain under Dirichlet-Neumann boundary conditions subject to thermodiffusion effects and frictional damping to control the delay term. The coupling gives new contributions to the theory associated with asymptotic behaviors of swelling porous-heat. At first, we state and prove the well-posedness of the solution of the system by the semigroup approach using Lumer-Philips theorem under suitable assumption on the weight of the delay. Then, we show that the considered dissipation in which we depended on are strong enough to guarantee an exponential decay result by using the energy method that consists to construct an appropriate Lyapunov functional based on the multiplier technique, this result is obtained without the equal-speed requirement. Our result is new and an extension of many other works in this area.
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Extension of an almost f-algebra multiplication
Статья научная
It is proved that an almost f-algebra multiplication and a d-algebra multiplication defined on a majorizing vector sublattice of a Dedekind complete vector lattice can be extended to the whole vector lattice by using purely algebraic and order theoretical means.
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