Majorization Research Articles

The Monte Carlo Method for Solving Large Systems of Linear Ordinary Differential Equations

The Monte Carlo method to solve the Cauchy problem for large systems of linear differential equations is proposed in this paper. Firstly, a quick overview of previously obtained results from applying the approach towards the Fredholm-type integral equations is made. In the main part of the paper, the method is applied towards a linear ODE system that is transformed into an equivalent system of the Volterra-type integral equations, which makes it possible to remove the limitations due to the conditions of convergence of the majorant series. The following key theorems are stated. Theorem 1 provides the necessary compliance conditions that should be imposed upon the transition propability and initial distribution densities that initiate the corresponding Markov chain, for which equality between the mathematical expectation of the estimate and the functional of interest would hold. Theorem 2 formulates the equation that governs the estimate’s variance. Theorem 3 states the Markov chain parameters that minimize the variance of the estimate of the functional. Proofs are given for all three theorems. In the practical part of this paper, the proposed method is used to solve a linear ODE system that describes a closed queueing system of ten conventional machines and seven conventional service persons. The solutions are obtained for systems with both constant and time-dependent matrices of coefficients, where the machine breakdown intensity is time dependent. In addition, the solutions obtained by the Monte Carlo and Runge–Kutta methods are compared. The results are presented in the corresponding tables.

Generalization of majorization theorem-II

Generalization of majorization theorem-II

Estimates for Weighted K-Functionals Using the Least Concave Majorant of Weighted Moduli of Continuity

ABSTRACTWe estimate the K-functional using the least concave majorant of modulus ωφ, where φ is a weight function on interval [a,b].

Majorization and the number of bipartite graphs for given vertex degrees

The emph{bipartite realisation problem} asks for a pair of non-negative‎, ‎non-increasing integer lists $a:=(a_1,ldots,a_n)$ and $b:=(b_1,ldots,b_{n'})$ if there is a labeled bipartite graph $G(U,V,E)$ (no loops or multiple edges) such that each vertex $u_i in U$ has degree $a_i$ and each vertex $v_i in V$ degree $b_i.$ The Gale-Ryser theorem provides characterisations for the existence of a `realisation' $G(U,V,E)$ that are strongly related to the concept of emph{majorisation}‎. ‎We prove a generalisation; list pair $(a,b)$ has more realisations than $(a',b),$ if $a'$ majorises $a.$ Furthermore‎, ‎we give explicitly list pairs which possess the largest number of realisations under all $(a,b)$ with fixed $n$‎, ‎$n'$ and $m:=sum_{i=1}^n a_i.$ We introduce the notion~emph{minconvex list pairs} for them‎. ‎If $n$ and $n'$ divide $m,$ minconvex list pairs turn in the special case of two constant lists $a=(frac{m}{n},ldots,frac{m}{n})$ and $b=(frac{m}{n'},ldots,frac{m}{n'}).$‎

Интерполяционная задача Павлова - Коревара - Диксона с мажорантой из класса сходимости

Интерполяционная задача Павлова - Коревара - Диксона с мажорантой из класса сходимости

Refinements of the majorization theorems via Fink identity and related results

Refinements of the majorization theorems via Fink identity and related results

Vector majorization and Schur-concavity of some sums generated by the Jensen and Jensen-Mercer functionals

Vector majorization and Schur-concavity of some sums generated by the Jensen and Jensen-Mercer functionals

Jessen's functional and majorization

Jessen's functional and majorization

Schur–Horn theorems in II∞-factors

Fil: Massey, Pedro Gustavo. Universidad Nacional de la Plata. Facultad de Ciencias Exactas. Departamento de Matematicas; Argentina

Majorization for certain classes of analytic functions defined by a new operator

En el presente articulo, investigamos las propiedades de mayorizacion para ciertas clases de funciones analiticas multivalentes definidas por un nuevo operador. Ademas, resaltamos algunas consecuencias -nuevas y conocidas- de nuestro resultado princresultado.

CHARACTERISTICS OF CLASS COLLABORATION NETWORKS IN LARGE JAVA SOFTWARE PROJECTS

Understanding software structural complexity and evolution plays an important role in controlling the software development and maintenance process. Recent studies have shown that the theory behind complex networks, especially the theory of scale-free networks, can be a useful approach to the analysis of concrete software systems. In this paper, class collaboration networks associated with five large Java software systems (JDK, Ant, Tomcat, Lucene and JavaCC) are analyzed in order to determine whether they belong to the class of scale-free networks, and examine their small-world characteristics. For each analyzed network, we detected (approximately) scale-free and (ultra) small-world properties. The results indicate that general conclusions from scale-free network theory can be applied to Java software systems in order to understand their structural complexity and model software evolution at the structural (class collaboration) level. Moreover, we examine class collaboration network evolution of Ant, in order to check the preferential attachment hypothesis of the Barabasi-Albert model. For several major Ant network transitions we con-clude that preferential attachment can successfully model Ant evolution at the class collaboration level. Finally, we discuss the implications of our results on software engineering, in several aspects: identification of important clas-ses/interfaces, software testing strategy, and efficient communication among software entities.http://dx.doi.org/10.5755/j01.itc.40.1.192

A THEOREM OF PHRAGMÉN-LINDELÖF TYPE FOR SUBFUNCTIONS IN A CONE

AbstractFor a subfunction u, associated with the stationary Schrödinger operator, which is dominated on the boundary by a certain function on a cone, we generalise the classical Phragmén-Lindelöf theorem by making an a-harmonic majorant of u.

LOCATING ROOTS OF A CERTAIN CLASS OF POLYNOMIALS

Abstract. We introduce a special class of real recurrent polynomials f m (m  1) of degree m+1, with positive roots s m , which are decreasing asm increases. The rst root s 1 , as well as the last one denoted by s 1 areexpressed in closed form, and enclose all s m (m > 1).This technique is also used to nd weaker than before [6] sucientconvergence conditions for some popular iterative processes convergingto solutions of equations. 1. IntroductionWe introduce a special class of recurrent polynomials f m (m1) of degreem+ 1 with real coecients.Then, we nd sucient conditions under which each polynomial f m has apositive root s m , such that s m+1 s m (m1). The rst root s 1 , as well asthe last one denoted by s 1 are expressed in simple closed form.Applications are provided. In the rst one, we show how to use s 1 and s 1 to locate any s m belonging in (s 1 ;s 1 ] (m1).In the second one, using this technique on Newton’s method (16), we showthat the famous for its simplicity and clarity Newton{Kantorovich condition(46) for solving equations can always replaced by a weaker one (49).We also show how to use our results to generate majorizing sequences ap-pearing in connection for solving abstract equations in a Banach space settingusing Newton{type methods [3], [6].2. Enclosing roots of polynomialsWe introduce the main result of this section on enclosing roots of polynomi-als.Theorem 2.1. Assume:there exist constants K0, M0, 0, L0, ‘0, and 0;

EXTENDED NEWTON’S METHOD FOR MAPPINGS ON RIEMANNIAN MANIFOLDS WITH VALUES IN A CONE

Robinson's generalized Newton's method for nonlinear functions with values in a cone is extended to mappings on Riemannian manifolds with values in a cone. When ${\cal D}f$ satisfies the $L$-average Lipschitz condition, we use the majorizing function technique to establish the semi-local quadratic convergence of the sequences generated by the extended Newton's method. As applications, we also obtain Kantorovich's type theorem, Smale's type theorem under the $\gamma$-condition and an extension of the theory of Smale's approximate zeros.

A Majorization Consequence: 11080

A Majorization Consequence: 11080

BLOOMFIELD-WATSON-KNOTT TYPE INEQUALITIES FOR EIGENVALUES

This paper is largely of expository nature. We generalize the determinantal and tracial inequalities, originating from Bloomfield-Watson and Knott, from the standpoint of majorization of eigenvalues, and observe the results as estimates of singular values of modified off-diagonal blocks of a block matrix representation in terms of the eigenvalues of the original matrix.

Logarithmic growth of theL 1-norm of the majorant of partial sums of an orthogonal series

It is proved that for anyN ×N orthogonal matrixA = {a ij} we have $$\sum\limits_{i = 1}^N {\mathop {\max }\limits_{1 \leqslant n \leqslant N} |\left| {\sum\limits_{j = 1}^n {a_{ij} } } \right|} \geqslant \frac{1}{{30}}N^{1/2} \log N.$$ A multidimensional analog of this result is also established.

Approximation of continuous functions by generalized karamata means

A sufficient condition for the order of approximation of a continuous 2π periodic function with a given majorant for the modulus of continuity by the [F, dn] means of its Fourier series to be of Jackson order is obtained. This sufficient condition is shown to be not enough for the order of approximation by partial sums of their Fourier series to be of Jackson order. The error estimate is shown to be the best possible.

Construction of a certain superharmonic majorant

Construction of a certain superharmonic majorant

Majorization and the Birthday Inequality

Majorization and the Birthday Inequality