Ural Mathematical Journal (UMJ)
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ON DOUBLE SIGNAL NUMBER OF A GRAPH
Get PDFA set of vertices in a connected graph is called a signal set if every vertex not in lies on a signal path between two vertices from . A set is called a double signal set of if if for each pair of vertices there exist such that . The double signal number of is the minimum cardinality of a double signal set. Any double signal set of cardinality is called -set of . In this paper we introduce and initiate some properties on double signal number of a graph. We have also given relation between geodetic number, signal number and double signal number for some classes of graphs
APPROXIMATION OF POSITIONAL IMPULSE CONTROLS FOR DIFFERENTIAL INCLUSIONS
Get PDFNonlinear control systems presented as differential inclusions with positional impulse controls are investigated. By such a control we mean some abstract operator with the Dirac function concentrated at each time. Such a control ("running impulse"), as a generalized function, has no meaning and is formalized as a sequence of correcting impulse actions on the system corresponding to a directed set of partitions of the control interval. The system responds to such control by discontinuous trajectories, which form a network of so-called "Euler's broken lines." If, as a result of each such correction, the phase point of the object under study is on some given manifold (hypersurface), then a slip-type effect is introduced into the motion of the system, and then the network of "Euler's broken lines" is called an impulse-sliding mode. The paper deals with the problem of approximating impulse-sliding modes using sequences of continuous delta-like functions. The research is based on Yosida's approximation of set-valued mappings and some well-known facts for ordinary differential equations with impulses
ON SOME VERTEX-TRANSITIVE DISTANCE-REGULAR ANTIPODAL COVERS OF COMPLETE GRAPHS
Get PDFIn the present paper, we classify abelian antipodal distance-regular graphs of diameter 3 with the following property: has a transitive group of automorphisms that induces a primitive almost simple permutation group on the set of its antipodal classes. There are several infinite families of (arc-transitive) examples in the case when the permutation rank of equals 2 moreover, all such graphs are now known. Here we focus on the case .Under this condition the socle of turns out to be either a sporadic simple group, or an alternating group, or a simple group of exceptional Lie type, or a classical simple group. Earlier, it was shown that the family of non-bipartite graphs with the property such that and the socle of is a sporadic or an alternating group is finite and limited to a small number of potential examples. The present paper is aimed to study the case of classical simple socle for . We follow a classification scheme that is based on a reduction to minimal quotients of that inherit the property . For each given group with simple classical socle of degree , we determine potential minimal quotients of , applying some previously developed techniques for bounding their spectrum and parameters in combination with the classification of primitive rank 3 groups of the corresponding type and associated rank 3 graphs. This allows us to essentially restrict the sets of feasible parameters of in the case of classical socle for under condition \(|{\Sigma}|\le 2500.\
ON LOCAL IRREGULARITY OF THE VERTEX COLORING OF THE CORONA PRODUCT OF A TREE GRAPH
Get PDFLet be a graph with a vertex set and an edge set . The graph is said to be with a local irregular vertex coloring if there is a function called a local irregularity vertex coloring with the properties: (i) as a vertex irregular -labeling and for every where and (ii) . The chromatic number of the local irregularity vertex coloring of denoted by , is the minimum cardinality of the largest label over all such local irregularity vertex colorings. In this paper, we study a local irregular vertex coloring of when is a family of tree graphs, centipede , double star graph , Weed graph , and graph .
INDUCED DECOMPOSITION OF INFINITE SQUARE GRIDS AND INFINITE HEXAGONAL GRIDS
Get PDFThe induced decomposition of infinite square grids and hexagonal grids are described here. We use the multi-level distance edge labeling as an effective technique in the decomposition of square grids. If the edges are adjacent, then their color difference is at least 2 and if they are separated by exactly a single edge, then their colors must be distinct. Only non-negative integers are used for labeling. The proposed partitioning technique per the edge labels to get the induced decomposition of the ladder graph is the square grid and the hexagonal grid
EVOLUTION OF A MULTISCALE SINGULARITY OF THE SOLUTION OF THE BURGERS EQUATION IN THE 4-DIMENSIONAL SPACE–TIME
Get PDFThe solution of the Cauchy problem for the vector Burgers equation with a small parameter of dissipation in the -dimensional space-time is studied: With the help of the Cole–Hopf transform the exact solution and its leading asymptotic approximation, depending on six space-time scales, near a singular point are found. A formula for the growth of partial derivatives of the components of the vector field on the time interval from the initial moment to the singular point, called the formula of the gradient catastrophe, is established: The asymptotics of the solution far from the singular point, involving a multistep reconstruction of the space-time scales, is also obtained: u_{\nu} (\mathbf{x}, t, \varepsilon) \approx - 2 \left( \frac{t}{\nu + 1} \right)^{1/2\nu} \tanh \left[ \frac{x_{\nu}}{\varepsilon} \left( \frac{t}{\nu + 1} \right)^{1/2\nu} \right]\!, \quad \frac{t}{\varepsilon^{\nu /(\nu + 1)} } \to +\infty. $
A CHARACTERIZATION OF DERIVATIONS AND AUTOMORPHISMS ON SOME SIMPLE ALGEBRAS
Get PDFIn the present paper, we study simple algebras, which do not belong to the well-known classes of algebras (associative algebras, alternative algebras, Lie algebras, Jordan algebras, etc.). The simple finite-dimensional algebras over a field of characteristic 0 without finite basis of identities, constructed by Kislitsin, are such algebras. In the present paper, we consider two such algebras: the simple seven-dimensional anticommutative algebra and the seven-dimensional central simple commutative algebra . We prove that every local derivation of these algebras and is a derivation, and every 2-local derivation of these algebras and is also a derivation. We also prove that every local automorphism of these algebras and is an automorphism, and every 2-local automorphism of these algebras and is also an automorphism
OUTPUT CONTROLLABILITY OF DELAYED CONTROL SYSTEMS IN A LONG TIME HORIZON
Get PDFIn this paper, we consider the output controllability of finite-dimensional control systems governed by a distributed delayed control. For systems with ordinary controls, this problem was investigated earlier. Nevertheless, in many practical and technical problems the control acts with some delay. We give the necessary and sufficient condition for the output controllability. The main goal of our control is to govern the output of the system to some position on a subspace in a given instant, and then keep this output fixed for the remaining times. This property is called the long-time output controllability. For this, sufficient conditions are given. The introduced notions are applied for the investigation of averaged controllability of systems with delayed controls. The general approach for that is to approximate the system by the ordinary ones. Some examples are considered
FIXED POINT THEOREM FOR MULTIVALUED NON-SELF MAPPINGS SATISFYING JS-CONTRACTION WITH AN APPLICATION
Get PDFIn this paper, we present some fixed point results for multivalued non-self mappings. We generalize the fixed point theorem due to Altun and Minak [2] by using Jleli and Sameti [9] -contraction. To validate the results proved here, we provide an appropriate application of our main result
MONOPOLISTIC COMPETITION MODEL WITH ENTRANCE FEE
Get PDFWe study the monopolistic competition model with producer-retailer-consumers two-level interaction. The industry is organized according to the Dixit–Stiglitz model. The retailer is the only monopolist. A quadratic utility function represents consumer preferences. We consider the case of the retailer's leadership; namely, we study two types of behavior: with and without the free entry condition. Earlier, we obtained the result: to increase social welfare and/or consumer surplus, the government needs to subsidize (not tax!) retailers. In the presented paper, we develop these results for the situation when the producer imposes an entrance fee for retailers