Let $G=(V,E)$ be a graph. A subset $D$ of $V$ is called common neighbourhood dominating set (CN-do\-mi\-nating set) if for every $v\in V-D$ there exists a vertex $u\in D$ such that $uv\in E(G)$ and $|\Gamma(u,v)|\geq1$, where $|\Gamma(u,v)|$ is the number of common neighbourhood between the vertices $u$ and $v$. The minimum cardinality of such CN-dominating set denoted by $\gamma_{cn}(G)$ and is called common neighbourhood domination number (CN-edge domination) of $G$. In this paper we introduce the concept of common neighbourhood edge domination (CN-edge domination) and common neighbourhood edge domatic number (CN-edge domatic number) in a graph, exact values for some standard graphs, bounds and some interesting results are established.

Бесплатно

Calculus of tangents and beyond

Kusraev Anatoly G., Kutateladze Semen S.

Статья научная

Optimization is the choice of what is most preferable. Geometry and local analysis of nonsmooth objects are needed for variational analysis which embraces optimization. These involved admissible directions and tangents as the limiting positions of the former. The calculus of tangents is one of the main techniques of optimization. Calculus reduces forecast to numbers, which is scalarization in modern parlance. Spontaneous solutions are often labile and rarely optimal. Thus, optimization as well as calculus of tangents deals with inequality, scalarization and stability. The purpose of this article is to give an overview of the modern approach to this range of questions based on non-standard models of set theory. A model of a mathematical theory is usually called nonstandard if the membership within the model has interpretation different from that of set theory. In the recent decades much research is done into the nonstandard methods located at the junctions of analysis and logic. This area requires the study of some new opportunities of modeling that open broad vistas for consideration and solution of various theoretical and applied problems.

Бесплатно

Casey Type Theorem and Laguerre Transformations

Kostin A.V.

Статья научная

The article explores the connections between Casey's theorems and their generalizations on the Euclidean and pseudo-Euclidean planes. Along with Casey type theorems about circles and "tangent distances'' between them, Laguerre transformations that preserve such distances are considered. Using non-Euclidean geometry, some connections between such transformations are described. In Casey's theorem, which is one of the generalizations of Ptolemy's theorem on an inscribed quadrilateral, four circles are considered that are tangent to one circle on the Euclidean plane. Instead of the lengths of the sides and diagonals, Casey's theorem takes the lengths of the common tangents of the corresponding pairs of circles. This theorem can be easily generalized to a larger number of circles. In addition, this theorem has various analogs and generalizations in spaces of constant curvature. On the pseudo-Euclidean plane, one can also consider analogs of Casey's theorem and its generalizations. Theorems of this type on the pseudo-Euclidean plane are a direct consequence of the corresponding Euclidean theorems. In this paper, a correspondence is constructed between configurations of circles on the Euclidean plane and configurations of circles of imaginary radius on the pseudo-Euclidean plane. In this case, the relationship from Euclidean geometry corresponds to the same relationship in pseudo-Euclidean geometry. Laguerre transformations on the Euclidean plane affect oriented lines. In this case, the family of straight lines enveloping the circle, under the influence of Laguerre transformations, passes into a similar family. If a straight line belongs to two such families, then under Laguerre transformations the length of the straight line segment between the points of contact with the circles is preserved. Using isotropic projection, Laguerre transformations on Euclidean and pseudo-Euclidean planes can be considered as transformations induced by the movements of three-dimensional pseudo-Euclidean space. To describe the properties of one-parameter subgroups of the Laguerre group on the Euclidean and pseudo-Euclidean planes, the Lobachevsky and de Sitter geometries are used.

Бесплатно

Characterizations of finite dimensional Archimedean vector lattices

Polat Faruk, Toumi Mohamed Ali

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.

Бесплатно

Chernoff Approximations of the Solution of Linear ODE with Variable Coefficients

Remizov I.D.

Статья научная

The Chernoff approximation method is a powerful and flexible tool of functional analysis, which allows in many cases to express exp(tL) in terms of variable coefficients of a linear differential operator L. In this paper, we prove a theorem that allows us to apply this method to find the resolvent of L. Our theorem states that the Laplace transforms of Chernoff approximations of a C0-semigroup converge to the resolvent of the generator of this semigroup. We demonstrate the proposed method on a second-order differential operator with variable coefficients. As a consequence, we obtain a new representation of the solution of a~nonhomogeneous linear ordinary differential equation of the second order in terms of functions that are coefficients of this equation, playing the role of parameters of the problem. For the Chernoff function, based on the shift operator, we give an estimate for the rate of convergence of approximations to the solution.

Бесплатно

Classification of Dynamical Systems Near a Cosymmetric Equilibrium

Kurakin L.G., Kurdoglyan A.V.

Статья научная

A local classification is developed in a neighborhood of a cosymmetric equilibrium for differential equations with invertible cosymmetry and a vector parameter, under the assumption that the kernel of the linearization matrix at the cosymmetric equilibrium is two-dimensional and that the entire stability spectrum, except for the double zero eigenvalue, is stable. Equations with such properties are of codimension one among even-dimensional systems with a cosymmetric equilibrium. In all cases, such a system admits a straightenable family of non-cosymmetric equilibria near the cosymmetric one. The classification is based on the following properties: the type of the cosymmetric equilibrium (node, focus, saddle); the relative position of the cosymmetric equilibrium and the family (including the case where the cosymmetric equilibrium belongs to the family); the number of boundary equilibria of the family separating its stable and unstable regions (⩽3); the number of intersections of each separatrix of the cosymmetric saddle equilibrium with the family (⩽3). Each property is determined by polynomial conditions, and the classification therefore reduces to identifying sets of conditions with a non-empty intersection. The defining polynomial conditions and corresponding phase portraits are presented for each identified class. The existence of each nonempty class is established by a scalable example for non-obvious cases, while the emptiness of the remaining classes is established separately. This work continues the studies of L. G. Kurakin and V. I. Yudovich [1, 2], where analogous results were obtained in the neighborhood of a non-cosymmetric equilibrium.

Бесплатно

Color energy of some cluster graphs

Dsouza Sabitha, Girija Kulambi Parameshwarappa, Gowtham Halgar Jagadeesh, Bhat Pradeep Ganapati

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.

Бесплатно

Conditions for the Limit Summability of Solutions of Nonlinear Elliptic Equations with Degenerate Coercivity and L1-Data

Kovalevsky A.A.

Статья научная

We study entropy and weak solutions of the Dirichlet problem for a class of second-order nonlinear elliptic equations with degenerate coercivity and right-hand side f in L1( ), where is a bounded open set in Rn (n > 2). The growth condition on the coefficients of the equations admits any their growth with respect to the unknown function itself. Estimates for the distribution function of an entropy solution and its gradient are obtained using a function ˜ f : [0,+∞) → R generated by the function f. Applying these estimates, we establish integral conditions on the function ˜ f which guarantee the belonging of entropy solutions and their gradients to limit Lebesgue spaces. As a consequence, we obtain conditions for the belonging of entropy solutions to a limit Sobolev space W1,r 0 ( ) and, as a particular case, to the space W1,1 0 ( ). In addition, we establish conditions for the existence of weak solutions of the considered problem belonging to the space W1,r 0 ( ). The obtained results generalize the known ones for equations whose coefficients satisfy the usual coercivity condition.

Бесплатно

Conformal Ricci soliton in an indefinite trans-Sasakian manifold

Girish Babu Shivanna, Reddy Polaepalli Siva Kota, Somashekhara Ganganna

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.

Бесплатно

Cyclically compact operators in banach spaces

Kusraev A.G.

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.

Бесплатно

Cергей Михайлович Никольский (к столетию со дня рождения)

Вакулов Б.Г., Карапетянц Н.К., Коробейник Ю.Ф., Кусраев А.Г., Самко С.Г.

Статья

Бесплатно

Cлабая непрерывность оператора суперпозиции в пространствах последовательностей

Алехно Егор Александрович

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.

Бесплатно

Derivations on Banach *-ideals in von Neumann algebras

Ber Aleksey Feliksovich, Chilin Vladimir Ivanovich, Sukochev Fedor Anatolevich

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.

Бесплатно

$Determination of a coefficient and kernel in a D-dimensional fractional integro-differential equation$

Determination of a coefficient and kernel in a D-dimensional fractional integro-differential equation

Rahmonov A.A.

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.

Бесплатно

Dynamics of quadratic Volterra-type stochastic operators corresponding to strange tournaments

Ganikhodzhaev R.N., Kurganov K.A., Tadzhieva M.A., Haydarov F.H.

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.

Бесплатно

Erratum to: "Infinitesimals in ordered vector spaces"

Emelyanov E. Yu.

Другой

In this note, Theorem 1 in the article which is cited in the title is corrected.

Бесплатно

Every lateral band is the kernel of an orthogonally additive operator

Pliev Marat A.

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?

Бесплатно

Existence and uniqueness theorems for a differential equation with a discontinuous right-hand side

Magomed-kasumov Magomedrasul G.

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.

Бесплатно

Existence of a Local Renormalized Solution of an Elliptic Equation with Variable Exponents in Rn

Kozhevnikova L.M.

Статья научная

The article is devoted to the study of second-order quasilinear elliptic equations with variable nonlinearity exponents and a locally integrable right-hand side in the space Rn. The author adapts the concept of a locally renormalized solution for equations with variable growth exponents, generalizing the results of M. F. Bidaut-V´eron and L. V´eron obtained for equations with constant exponents. The work establishes conditions on the structure of the quasilinear elliptic operator with variable growth that are sufficient for the correct definition of a locally renormalized solution. The author derives a priori local estimates characterizing the regularity of the solution and, based on these, proves the existence of a locally renormalized solution in the space Rn without additional restrictions on its growth at infinity. Furthermore, the work demonstrates that for a non-negative right-hand side, the solution is also nonnegative almost everywhere. The research employs methods of functional analysis, including the theory of Lebesgue and Sobolev spaces with variable exponents. The proofs are based on compactness and monotonicity techniques, as well as the use of special test functions. The results of the work are significant for the theory of nonlinear elliptic equations and can be applied to further studies of degenerate equations and problems with measure-valued data. The study contributes to the development of analytical methods for equations with variable nonlinearity exponents and expands the applicability of the concept of locally renormalized solutions.

Бесплатно

Existence of classical solutions for a class of the Khokhlov-Zabolotskaya-Kuznetsov type equations

Bouakaz A., Bouhmila F., Georgiev S.G., Kheloufi A., Khoufache S.

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.

Бесплатно

1
2
3
4
5
6
7
...
В конец

Журнал