Ural Mathematical Journal (UMJ)
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ON PLURIHARMONICITY CRITERIA OF HARMONIC AND -HARMONIC FUNCTIONS IN THE ARBITRARY BALL OF
No full textThis paper introduces the concept of -harmonic function in an arbitrary ball of and proves some criteria for pluriharmonicity of harmonic functions in this ball. It is devoted to the study of the connection between the invariant Laplacian and the specificity of the domain and for this purpose we will try to define the invariant Laplacian for arbitrary ball of $. Moreover, we will give the criteria for pluriharmonicity in terms of -harmonicity with respect to two different balls. The main goal of this work is expanding the properties of pluriharmonic functions furthermore and study their connection with harmonic and -subharmonic functions
TWO METHODS OF DESCRIBING 2-LOCAL DERIVATIONS AND AUTOMORPHISMS
Get PDFIn the present paper, we investigate 2-local linear operators on vector spaces. Sufficient conditions are obtained for the linearity of a 2-local linear operator on a finite-dimensional vector space. To do this, families of matrices of a certain type are selected and it is proved that every 2-local linear operator generated by these families is a linear operator. Based on these results we prove that each 2-local derivation of a finite-dimensional null-filiform Zinbiel algebra is a derivation. Also, we develop a method of construction of 2-local linear operators which are not linear operators. To this end, we select matrices of a certain type and using these matrices we construct a 2-local linear operator. If these matrices are distinct, then the 2-local linear operator constructed using these matrices is not a linear operator. Applying this method we prove that each finite-dimensional filiform Zinbiel algebra has a 2-local derivation that is not a derivation. We also prove that each finite-dimensional naturally graded quasi-filiform Leibniz algebras of type I has a 2-local automorphism that is not an automorphism
IMPROVED BRANCH-AND-PRICE ALGORITHM FOR THE EFFICIENT 2-TERMINAL RELIABILITY PROBLEM
Get PDFThe Efficient 2-Terminal Reliability Problem is a nonlinear optimization problem aimed at designing a minimal-cost network within the reliability guaranties. Recent research has provided Branch-and-Price (BnP) solution based on probability relaxation, the Dantzig–Wolfe decomposition, followed by the column generation technique, and Branch-and-Bound scheme. Unfortunately, the performance of this algorithm deteriorates in cases of high-density graphs and stringent unreliability thresholds. By extending our recent approach, we introduce an improved BnP algorithm supplemented with novel valid inequalities, more efficient nonlinear integer pricing problem solver, primal heuristics, and branching strategies. Evaluation results on benchmarking instances demonstrate significant performance advantage of the proposed method
TOPOLOGIES ON THE FUNCTION SPACE WITH VALUES IN A TOPOLOGICAL GROUP
Get PDFLet denote the set of all functions from to . When is a topological space, various topologies can be defined on . In this paper, we study these topologies within the framework of function spaces. To characterize different topologies and their properties, we employ generalized open sets in the topological space . This approach also applies to the set of all continuous functions from to , denoted by , particularly when is a topological group. In investigating various topologies on both and , the concept of limit points plays a crucial role. The notion of a topological ideal provides a useful tool for defining limit points in such spaces. Thus, we utilize topological ideals to study the properties and consequences for function spaces and topological groups
FIXED POINTS IN THE CONSTRUCTION OF A MINIMAX SOLUTION FOR A CLASS OF BOUNDARY VALUE PROBLEMS FOR HAMILTON–JACOBI EQUATIONS
Get PDFThis paper deals with analytical and numerical methods for constructing a minimax (generalized) solution to the Dirichlet problem for the Hamilton–Jacobi equation. The case of a closed planar nonconvex boundary set is considered, where the boundary points have a smoothness defect in the coordinate functions with respect to third-order derivatives. These points belong to the pseudo-vertices of the boundary set. Pseudo-vertices generate branches of a singular set, which are one-dimensional manifolds where the smoothness of the minimax solution breaks down. To construct a branch of a singular set, it is necessary to find markers, i.e., numerical characteristics of the corresponding pseudo-vertex. The markers (left and right ones) establish a link between the characteristics of the Hamilton–Jacobi equation and the geometry of the boundary set. For the markers, a relation with the structure of the equation at a fixed point is obtained. An iterative procedure for calculating a solution based on Newton's method is proposed. The convergence of the procedure to the pseudo-vertex marker is proved. An example of constructing a minimax solution is given, demonstrating the effectiveness of the developed approaches for solving nonsmooth boundary value problems
GENERALIZED ASYMPTOTIC NOTATIONS VIA FILTERS: A NEW FRAMEWORK FOR ALGORITHM ANALYSIS
Get PDFThis paper introduces a generalization of asymptotic notation using filters, a topological structure. We present key properties of this generalization, including reflexivity, symmetry, and transitivity, supported by illustrative examples. Our research demonstrates that classical asymptotic notations imply their filter-generalized counterparts, but we provide examples showing the converse is not universally true. We also propose a characterization of traditional asymptotic notations using filters, offering a new perspective on these fundamental concepts. Furthermore, we establish relationships between bounded or vanishing sequences and filter-based asymptotic notations, enabling the determination of properties central to this study. This generalization provides a more nuanced framework for analyzing algorithmic complexity, potentially capturing behaviors overlooked by classical notations and opening new avenues for theoretical computer science research
ASYMPTOTIC BEHAVIOR OF REACHABLE SETS WITH -BOUNDED CONTROLS
Get PDFThe paper studies the reachable sets of control systems over a fixed time interval, subject to control constraints defined as a ball in the space for . The dependence of reachable sets on the parameter is investigated. For affine-control nonlinear systems, it is established that these sets are continuous in the Hausdorff metric for all , including and . In the case of linear systems, estimates for the Hausdorff distance between the sets are derived, and their asymptotic behavior as and is analyzed. For , the reachable set, up to closure, coincides with the reachable set of the system with impulse control under a constraint on the magnitude of the impulse. The case corresponds to geometric (instantaneous) constraints on the control
ON AN INITIAL BOUNDARY–VALUE PROBLEM FOR A DEGENERATE EQUATION OF HIGH EVEN ORDER
Get PDFIn this paper, we formulate and study an initial boundary-value problem of the type of the third boundary condition for a degenerate partial differential equation of high even order in a rectangle. Using the Fouriers method, based on separation of variables, a spectral problem for an ordinary differential equation is obtained. Using the Green's function method, the latter problem is equivalently reduced to the Fredholm integral equation of the second kind with a symmetric kernel, which implies the existence of eigenvalues and a system of eigenfunctions of the spectral problem. Using the found integral equation and Mercer's theorem, the uniform convergence of certain bilinear series depending on the eigenfunctions is proved. The order of the Fourier coefficients has been established. The solution to the considered problem has been written as a sum of the Fourier series over the system of eigenfunctions of the spectral problem. The uniqueness of the solution to the problem was proved using the method of energy integrals. An estimate for solution of the problem was obtained, which implies its continuous dependence on the given functions
INTEGRAL ANALOGUE OF TURÁN-TYPE INEQUALITIES CONCERNING THE POLAR DERIVATIVE OF A POLYNOMIAL
Get PDFIf is a polynomial of degree with all its zeros in and any real , Aziz proved the integral inequality [1] In this article, we establish a refined extension of the above integral inequality by using the polar derivative instead of the ordinary derivative consisting of the leading coefficient and the constant term of the polynomial. Besides, our result also yields other intriguing inequalities as special cases
ON A GROUP EXTENSION INVOLVING THE SPORADIC JANKO GROUP
Get PDFUsing the electronic Atlas of Wilson [21], the group J_2 has an absolutely irreducible module of dimension 6 over F_4. Therefor a split extension group of the form 4^6:J_2:= \bar{G} exists. In this paper we study this group, where we determine its conjugacy classes and character table using the coset analysis technique together with Clifford-Fischer Theory. We determined the inertia factor groups of \bar{G} by analysing the maximal subgroups of J_2 and maximal of the maximal subgroups of J_2 together with other various information. It turns out that the character table of \bar{G} is a 53 x 53 real valued matrix, while the Fischer matrices are all integer valued matrices with sizes ranging from 1 to 8