Ural Mathematical Journal (UMJ)
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ON NEW HYBRID ROOT-FINDING ALGORITHMS FOR SOLVING TRANSCENDENTAL EQUATIONS USING EXPONENTIAL AND HALLEY'S METHODS
Get PDFThe objective of this paper is to propose two new hybrid root finding algorithms for solving transcendental equations. The proposed algorithms are based on the well-known root finding methods namely the Halley's method, regula-falsi method and exponential method. We show using numerical examples that the proposed algorithms converge faster than other related methods. The first hybrid algorithm consists of regula-falsi method and exponential method (RF-EXP). In the second hybrid algorithm, we use regula falsi method and Halley's method (RF-Halley). Several numerical examples are presented to illustrate the proposed algorithms, and comparison of these algorithms with other existing methods are presented to show the efficiency and accuracy. The implementation of the proposed algorithms is presented in Microsoft Excel (MS Excel) and the mathematical software tool Maple
ON TWO-SIDED UNIDIRECTIONAL MEAN VALUE INEQUALITY IN A FRÉCHET SMOOTH SPACE
Get PDFThe paper is devoted to a new unidirectional mean value inequality for the Fréchet subdifferential of a continuous function. This mean value inequality finds an intermediate point and localizes its value both from above and from below; for this reason, the inequality is called two-sided. The inequality is considered for a continuous function defined on a Fréchet smooth space. This class of Banach spaces includes the case of a reflexive space and the case of a separable Asplund space. As some application of these inequalities, we give an upper estimate for the Fréchet subdifferential of the upper limit of continuous functions defined on a reflexive space
WEIGHTED -PSEUDO -ASYMPTOTICALLY PERIODIC SOLUTIONS FOR SOME SYSTEMS OF NONLINEAR DELAY INTEGRAL EQUATIONS WITH SUPERLINEAR PERTURBATION
Get PDFThis work is concerned with the existence of positive weighted pseudo -asymptotically periodic solution in Stepanov-like sense for some systems of nonlinear delay integral equations. In this context, we will first be interested in establishing a suitable composition theorem, and then some existing results concerning the -asymptotic periodicity in the scalar case are developed here for the vector case. We point out that, in this paper, we adopt some changes in the definitions, which, although slight, are necessary to accomplish the work
COUNTABLE COMPACTNESS MODULO AN IDEAL OF NATURAL NUMBERS
Get PDFIn this article, we introduce the idea of -compactness as a covering property through ideals of and regardless of the -convergent sequences of points. The frameworks of -compactness, compactness and sequential compactness are compared to the structure of -compact space. We began our research by looking at some fundamental characteristics, such as the nature of a subspace of an -compact space, then investigated its attributes in regular and separable space. Finally, various features resembling finite intersection property have been investigated, and a connection between -compactness and sequential -compactness has been established
CONTROL PROBLEM FOR A PARABOLIC SYSTEM WITH UNCERTAINTIES AND A NON-CONVEX GOAL
Get PDFWe consider the control problem for a parabolic system that describes the heating of a given number of rods. Control is carried out through heat sources that are located at the ends of the rods (only at one end or at both). The density functions of the internal heat sources and exact values of the temperature at the right ends of some rods are unknown, and only the segments of their change are given. The goal of choosing control is to ensure that at a fixed time moment the weighted sum of the average temperatures of the rods belongs to a non-convex terminal set for any admissible unknown functions. After a change of variables, this problem reduces to a one-dimensional differential game. Necessary and sufficient conditions for the game termination are found
ON THE PROPERTIES OF THE SET OF TRAJECTORIES OF THE NONLINEAR CONTROL SYSTEM WITH QUADRATIC INTEGRAL CONSTRAINT ON THE CONTROL FUNCTIONS
Get PDFIn this paper the control system described by a nonlinear differential equation is studied. It is assumed that the control functions have a quadratic integral constraint, more precisely, the admissible control functions are chosen from the ellipsoid of the space . Different properties of the set of trajectories are investigated. It is proved that a small perturbation of the set of control functions causes also appropriate small perturbation of the set of trajectories. It is also shown that the set of trajectories has a small change if along with the integral constraint on the control functions, a sufficiently large norm type geometric constraint on the control functions is introduced. It is established that every trajectory is robust with respect to the fast consumption of the remaining control resource, and hence every trajectory of the system can be approximated by a trajectory generated by full consumption of the total control resource
ON CAUCHY-TYPE BOUNDS FOR THE EIGENVALUES OF A SPECIAL CLASS OF MATRIX POLYNOMIALS
Get PDFLet be the set of all matrices whose entries are in the set of complex numbers. Then is called a matrix polynomial. If , then is said to be a matrix polynomial of degree . In this paper we prove some results for the bound estimates of the eigenvalues of some lacunary type of matrix polynomials
AROUND THE ERDÖS–GALLAI CRITERION
Get PDFBy an (integer) partition we mean a non-increasing sequence of non-negative integers that contains a finite number of non-zero components. A partition is said to be graphic if there exists a graph such that , where we denote by the degree partition of composed of the degrees of its vertices, taken in non-increasing order and added with zeros. In this paper, we propose to consider another criterion for a partition to be graphic, the ht-criterion, which, in essence, is a convenient and natural reformulation of the well-known Erdös–Gallai criterion for a sequence to be graphical. The ht-criterion fits well into the general study of lattices of integer partitions and is convenient for applications. The paper shows the equivalence of the Gale–Ryser criterion on the realizability of a pair of partitions by bipartite graphs, the ht-criterion and the Erdös–Gallai criterion. New proofs of the Gale–Ryser criterion and the Erdös–Gallai criterion are given. It is also proved that for any graphical partition there exists a realization that is obtained from some splitable graph in a natural way. A number of information of an overview nature is also given on the results previously obtained by the authors which are close in subject matter to those considered in this paper
APPROXIMATION OF DIFFERENTIATION OPERATORS BY BOUNDED LINEAR OPERATORS IN LEBESGUE SPACES ON THE AXIS AND RELATED PROBLEMS IN THE SPACES OF -MULTIPLIERS AND THEIR PREDUAL SPACES
Get PDFWe consider a variant of the four-parameter Stechkin problem on the best approximation of differentiation operators of order on the class of times differentiable functions in Lebesgue spaces on the real axis. We discuss the state of research in this problem and related problems in the spaces of multipliers of Lebesgue spaces and their predual spaces. We give two-sided estimates for . The paper is based on the author's talk at the S.B. Stechkin's International Workshop-Conference on Function Theory (Kyshtym, Chelyabinsk region, August 1–10, 2023)
CANONICAL APPROXIMATIONS IN IMPULSE STABILIZATION FOR A SYSTEM WITH AFTEREFFECT
Get PDFFor optimal stabilization of an autonomous linear system of differential equations with aftereffect and impulse controls, the formulation of the problem in the functional state space is used. For a system with aftereffect, approximating systems of ordinary differential equations proposed by S.N. Shimanov and J. Hale are used. A method for constructing approximations for optimal stabilizing control of an autonomous linear system with aftereffect and impulse controls is proposed. Matrix Riccati equations are used to find approximating controls