scholarly journals The two-dimensional g-fraction with independent variables for double power series

2012 ◽  
Vol 164 (12) ◽  
pp. 1520-1539 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Two Dimensional ◽  
Independent Variables ◽  
Double Power Series

scholarly journals Some properties of approximants for branched continued fractions of the special form with positive and alternating-sign partial numerators

2018 ◽  
Vol 10 (1) ◽  
pp. 3-13 ◽  
Author(s):  
T.M. Antonova ◽  
M.V. Dmytryshyn ◽  
S.M. Vozna
Keyword(s):  
Continued Fractions ◽  
Special Form ◽  
Sufficient Conditions ◽  
Two Dimensional ◽  
Rational Approximants ◽  
Independent Variables ◽  
Function Of Two Variables ◽  
Necessary And Sufficient ◽  
Double Power Series ◽  
Two Independent Variables

The paper deals with research of convergence for one of the generalizations of continued fractions -- branched continued fractions of the special form with two branches. Such branched continued fractions, similarly as the two-dimensional continued fractions and the branched continued fractions with two independent variables are connected with the problem of  the correspondence between a formal double power series and a sequence of the rational approximants of a function of two variables. Unlike continued fractions, approximants of which are constructed unambiguously, there are many ways to construct approximants of branched continued fractions of the general and the special form. The paper examines the ordinary approximants and one of the structures of figured approximants of the studied branched continued fractions, which is connected with the problem of correspondence. We consider some properties of approximants of such fractions, whose partial numerators are positive and alternating-sign  and partial denominators are equal to one. Some necessary and sufficient conditions for figured convergence are established. It is proved that under these conditions from the convergence of the sequence of figured approximants it follows the convergence of the sequence of ordinary approximants  to the same limit.

A Difference Method for Plane Problems in Magnetoelastodynamics

1972 ◽  
Vol 39 (3) ◽  
pp. 689-695 ◽  
Author(s):  
W. W. Recker
Keyword(s):  
Partial Differential Equations ◽  
Differential Equations ◽  
Numerical Method ◽  
Difference Method ◽  
Necessary Condition ◽  
Two Dimensional ◽  
Symmetric Hyperbolic System ◽  
Von Neumann ◽  
First Order ◽  
Independent Variables

The two-dimensional equations of magnetoelastodynamics are considered as a symmetric hyperbolic system of linear first-order partial-differential equations in three independent variables. The characteristic properties of the system are determined and a numerical method for obtaining the solution to mixed initial and boundary-value problems in plane magnetoelastodynamics is presented. Results on the von Neumann necessary condition are presented. Application of the method to a problem which has a known solution provides further numerical evidence of the convergence and stability of the method.

Multidimensional regular C-fraction with independent variables corresponding to formal multiple power series

2019 ◽  
Vol 150 (4) ◽  
pp. 1853-1870 ◽  
Author(s):  
R. I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Numerical Experiments ◽  
The Other ◽  
Independent Variables ◽  
Classical Algorithm ◽  
Multiple Power ◽  
The One ◽  
The Given ◽  
Multiple Power Series

AbstractIn the paper the correspondence between a formal multiple power series and a special type of branched continued fractions, the so-called ‘multidimensional regular C-fractions with independent variables’ is analysed providing with an algorithm based upon the classical algorithm and that enables us to compute from the coefficients of the given formal multiple power series, the coefficients of the corresponding multidimensional regular C-fraction with independent variables. A few numerical experiments show, on the one hand, the efficiency of the proposed algorithm and, on the other, the power and feasibility of the method in order to numerically approximate certain multivariable functions from their formal multiple power series.

scholarly journals A NOTE ON TWO DIMENSIONAL INTEGRAL INEQUALITY

1990 ◽  
Vol 21 (3) ◽  
pp. 211-213
Author(s):  
B. G. PACHPATTE
Keyword(s):  
Integral Inequality ◽  
Present Note ◽  
Two Dimensional ◽  
Partial Derivatives ◽  
Independent Variables ◽  
Dimensional Integral ◽  
Two Independent Variables

In the present note we establish a new integral inequality involving a function of two independent variables and its partial derivatives.

scholarly journals On the convergence of multidimensional regular C-fractions with independent variables

2018 ◽  
Vol 26 (1) ◽  
pp. 18 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Open Disk ◽  
Efficient Tool ◽  
Independent Variables ◽  
Multidimensional Generalization ◽  
Multiple Power ◽  
Multiple Power Series ◽  
Branched Continued Fraction

In this paper, we investigate the convergence of multidimensional regular С-fractions with independent variables, which are a multidimensional generalization of regular С-fractions. These branched continued fractions are an efficient tool for the approximation of multivariable functions, which are represented by formal multiple power series. We have shown that the intersection of the interior of the parabola and the open disk is the domain of convergence of a multidimensional regular С-fraction with independent variables. And, in addition, we have shown that the interior of the parabola is the domain of convergence of a branched continued fraction, which is reciprocal to the multidimensional regular С-fraction with independent variables.

scholarly journals On the convergence criterion for branched continued fractions with independent variables

2018 ◽  
Vol 9 (2) ◽  
pp. 120-127 ◽  
Author(s):  
R.I. Dmytryshyn
Keyword(s):  
Power Series ◽  
Continued Fractions ◽  
Special Form ◽  
Absolute Convergence ◽  
Convergence Criterion ◽  
Complex Numbers ◽  
Independent Variables ◽  
Nonnegative Coefficients ◽  
Multiple Power ◽  
Multiple Power Series

In this paper, we consider the problem of convergence of an important type of multidimensional generalization of continued fractions, the branched continued fractions with independent variables. These fractions are an efficient apparatus for the approximation of multivariable functions, which are represented by multiple power series. We have established the effective criterion of absolute convergence of branched continued fractions of the special form in the case when the partial numerators are complex numbers and partial denominators are equal to one. This result is a multidimensional analog of the Worpitzky's criterion for continued fractions. We have investigated the polycircular domain of uniform convergence for multidimensional C-fractions with independent variables in the case of nonnegative coefficients of this fraction.

Chromatic Solutions

1982 ◽  
Vol 34 (3) ◽  
pp. 741-758 ◽  
Author(s):  
W. T. Tutte
Keyword(s):  
Power Series ◽  
Formal Power Series ◽  
Chromatic Polynomial ◽  
Basic Theory ◽  
Root Vertex ◽  
Independent Variables ◽  
Image Position ◽  
Formal Power ◽  
Two Independent Variables

Early in the Seventies I sought the number of rooted λ-coloured triangulations of the sphere with 2p faces. In these triangulations double joins, but not loops, were permitted. The investigation soon took the form of a discussion of a certain formal power series l(y, z, λ) in two independent variables y and z.The basic theory of l is set out in [1]. There l is defined as the coefficient of x2 in a more complicated power series g(x, y, z, λ). But the definition is equivalent to the following formula.1Here T denotes a general rooted triangulation. n(T) is the valency of its root-vertex, and 2p(T) is the number of its faces. P(T, λ) is the chromatic polynomial of the graph of T.

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