scholarly journals The relationship between algebraic equations and $(n,m)$-forms, their degrees and recurrent fractions

2019 ◽  
Vol 11 (1) ◽  
pp. 96-106
Author(s):  
I.I. Lishchynsky
Keyword(s):  
Characteristic Equation ◽  
Rational Approximation ◽  
Rational Numbers ◽  
Rational Approximations ◽  
Algebraic Equations ◽  
Triangular Matrices ◽  
Pell’S Equation ◽  
Recurrence Equations ◽  
The Relationship

Algebraic and recursion equations are widely used in different areas of mathematics, so various objects and methods of research that are associated with them are very important. In this article we investigate the relationship between $(n,m)$-forms with generalized Diophantine Pell's equation, algebraic equations of $n$ degree and recurrent fractions. The properties of the $(n,m^n+1)$-forms and their characteristic equation are considered. The author applied parafunctions of triangular matrices to the study of algebraic equations and corresponding recurrence equations. The form of adjacent roots of the annihilating polynomial of arbitrary $(n,m)$-forms over the field of rational numbers are explored. The following question is very important for some applied problems: Is a given form the largest by module among its adjacent roots? If it is so, then there is a periodic recurrence fraction of $n$-order that is equal to this $(n,m)$-form, and its $m$th rational shortening will be its rational approximation. The author has identified the class $(n,m)$-forms with the largest module among their adjacent roots and showed how to find periodic recurrence fractions of $n$-order and rational approximations for them.

scholarly journals CARACTERIZAÇÃO HIDRÁULICA DO MICROASPERSOR DAN SPRINKLERS GRUPO MODULAR

Irriga ◽  
2001 ◽  
Vol 6 (1) ◽  
pp. 13-18 ◽  
Author(s):  
Marcio Antonio Vilas Boas ◽  
Eurides Kuster Macedo Júnior ◽  
Silvio Cesar Sampaio ◽  
Melânia Inês Valiati
Keyword(s):  
Production Process ◽  
Power Function ◽  
Characteristic Equation ◽  
State University ◽  
The West ◽  
Potential Equation ◽  
Group 2 ◽  
The Relationship

CARACTERIZAÇÃO HIDRÁULICA DO MICROASPERSOR DAN SPRINKLERS GRUPO MODULAR                                     Márcio Antônio Vilas BoasEurides Kuster Macedo JuniorSilvio César SampaioMelânia Inês ValiatiUNIOESTE - Universidade Estadual do Oeste do ParanáCEP: 85814-110 - Cascavel – PR - Brasil - Cx. Postal   711Fone: (045) 225 -2100  (R-249) - Fax : (045) [email protected]  1 RESUMO  Este  trabalho  teve  como  objetivo  avaliar as características hidráulicas  do  microaspersor DAN SPRINKLERS do grupo modular de fabricação da DAN SPRINKLERS - ISRAEL, de uso recente no Oeste do Paraná. Os ensaios foram realizados no Laboratório de Hidráulica do Departamento de Engenharia da Universidade Estadual do Oeste do Paraná – UNIOESTE. Na  avaliação dos microaspersores  estudou-se, a variação decorrente do processo de fabricação e a determinação da equação característica da relação vazão–pressão. Os microaspersores do Grupo modular com diâmetros de bocais 0,94; 1,16;1,41;1,92 e 2,34 mm,  foram submetidos às pressões de 100, 150, 200, 250, 300 e 350 kPa. As equações características determinadas indicaram que o microaspersor testado não é auto-compensante , tolerante a sensibilidade de variações de pressões e que a equação potencial se ajusta bem aos dados. Os coeficientes de variação de fabricação obtidos foram menores que 5%, classificando-se, de acordo com a Norma ISO, como de categoria A. UNITERMOS: Microaspersão, coeficiente de variação, modelo potencial.  VILAS BÔAS, M. A., MACEDO JUNIOR, E. K. HYDRAULIC CHARACTERIZATION OF MICROSPRINKLER DAN SPRINKLER - MODULATE GROUP   2 ABSTRACT This work had as objective to evaluate the characteristics hydraulic of the microsprinklers of the group to modulate of production of DAN SPRINKLERS - ISRAEL, of recent use in the West of Paraná. The tests was accomplished in the Laboratory of Hydraulics of the Department of Engineering of the State University of the West of Paraná - UNIOESTE. In the evaluation of the microasprinklers it was studied such characteristics as, the variation due to the production process and the determination of the characteristic equation of the relationship vazão-pressure. The microsprinklers of the Group to modulate with diameters of nozzle 0,94; 1,16;1,41;1,92 and 2,34 mm, the pressures were submitted 100, 150, 200, 250, 300 and 350 kPa. The certain characteristic equations showed that the tested microsprinklers is not solemnity-compensante and that the potential equation was fit well to the data. The obtained coefficients of production variation were everybody below 5% being able to not this way to classify them in agreement with for ISO  category as A. KEYWORDS: Microsprinkler, coefficient variation, power function.

Solving Systems of Linear Algebraic Equations with the Use of an Selective Regularization

2021 ◽  
pp. 11-15
Author(s):  
Vladimir N. Lutay
Keyword(s):  
Diagonal Element ◽  
Decomposition Process ◽  
Algebraic Equations ◽  
Original Matrix ◽  
Triangular Matrices ◽  
First Case ◽  
Linear Algebraic Equations ◽  
The Matrix ◽  
Gauss Method ◽  
Scalar Products

The solution of systems of linear algebraic equations, which matrices can be poorly conditioned or singular is considered. As a solution method, the original matrix is decomposed into triangular components by Gauss or Chole-sky with an additional operation, which consists in increasing the small or zero diagonal terms of triangular matrices during the decomposition process. In the first case, the scalar products calculated during decomposition are divided into two positive numbers such that the first is greater than the second, and their sum is equal to the original one. In further operations, the first number replaces the scalar product, as a result of which the value of the diagonal term increases, and the second number is stored and used after the decomposition process is completed to correct the result of calculations. This operation increases the diagonal elements of triangular matrices and prevents the appearance of very small numbers in the Gauss method and a negative root expression in the Cholesky method. If the matrix is singular, then the calculated diagonal element is zero, and an arbitrary positive number is added to it. This allows you to complete the decomposition process and calculate the pseudo-inverse matrix using the Greville method. The results of computational experiments are presented.

On Padé and Best Rational Approximation

1983 ◽  
Vol 26 (1) ◽  
pp. 50-57 ◽  
Author(s):  
Peter B. Borwein
Keyword(s):  
Rational Approximation ◽  
Analytic Functions ◽  
Unit Disc ◽  
Padé Approximants ◽  
Main Diagonal ◽  
Rational Approximations ◽  
Pade Approximants ◽  
Best Rational Approximation

AbstractIt is reasonable to expect that, under suitable conditions, Padé approximants should provide nearly optimal rational approximations to analytic functions in the unit disc. This is shown to be the case for ez in the sense that main diagonal Padé approximants are shown to converge as expeditiously as best uniform approximants. Some more general but less precise related results are discussed.

Adaptive Fourier decompositions and rational approximations, part I: Theory

2014 ◽  
Vol 12 (05) ◽  
pp. 1461008 ◽  
Author(s):  
Tao Qian
Keyword(s):  
Rational Approximation ◽  
One Dimension ◽  
Rational Approximations ◽  
Different Types ◽  
Best Rational Approximation

In this paper, we will give a survey on adaptive Fourier decompositions (AFDs) in one- and multi-dimensions. Theoretical formulations of three different types of AFDs in one-dimension, viz., Core AFD, Cyclic AFD in conjunction with best rational approximation and Unwending AFD are provided.

Simultaneous diophantine approximations and Hermite's method

1980 ◽  
Vol 21 (3) ◽  
pp. 463-470 ◽  
Author(s):  
Alain Durand
Keyword(s):  
Exponential Function ◽  
Integer Point ◽  
Rational Numbers ◽  
Rational Points ◽  
Rational Approximations ◽  
Diophantine Approximations ◽  
Simultaneous Diophantine Approximations ◽  
Image Position

In this paper we generalize a result of Mahler on rational approximations of the exponential function at rational points by proving the following theorem: let n ε N* and αl, …, αn be distinct non-zero rational numbers; there exists a constant c = c(n, αl, …, αn) ≥ 0 such thatfor every non-zero integer point (qo, ql, …, qn)and q = max {|ql|, … |qn|, 3}.

Expressions for rational approximations to square roots of integers using Pell's equation

2019 ◽  
Vol 103 (556) ◽  
pp. 101-110
Author(s):  
Ken Surendran ◽  
Desarazu Krishna Babu
Keyword(s):  
Positive Integer ◽  
Continued Fractions ◽  
Integer Solution ◽  
Rational Approximations ◽  
Positive Integer Solution ◽  
Pell’S Equation ◽  
Square Roots ◽  
Integer Solutions

There are recursive expressions (see [1]) for sequentially generating the integer solutions to Pell's equation:p2 −Dq2 = 1, whereDis any positive non-square integer. With known positive integer solutionp1 andq1 we can compute, using these recursive expressions,pnandqnfor alln> 1. See Table in [2] for a list of smallest integer, orfundamental, solutionsp1 andq1 forD≤ 128. These (pn,qn) pairs also formrational approximationstothat, as noted in [3, Chapter 3], match with convergents (Cn=pn/qn) of the Regular Continued Fractions (RCF, continued fractions with the numerator of all fractions equal to 1) for.

scholarly journals Automatic rational approximation and linearization of nonlinear eigenvalue problems

2021 ◽  
Author(s):  
Pieter Lietaert ◽  
Karl Meerbergen ◽  
Javier Pérez ◽  
Bart Vandereycken
Keyword(s):  
Rational Approximation ◽  
Large Scale ◽  
Eigenvalue Problems ◽  
Space Representation ◽  
Rational Approximations ◽  
Krylov Methods ◽  
Nonlinear Eigenvalue Problems ◽  
State Space Representation ◽  
Rational Polynomial ◽  
Nonlinear Eigenvalue

Abstract We present a method for solving nonlinear eigenvalue problems (NEPs) using rational approximation. The method uses the Antoulas–Anderson algorithm (AAA) of Nakatsukasa, Sète and Trefethen to approximate the NEP via a rational eigenvalue problem. A set-valued variant of the AAA algorithm is also presented for building low-degree rational approximations of NEPs with a large number of nonlinear functions. The rational approximation is embedded in the state-space representation of a rational polynomial by Su and Bai. This procedure perfectly fits the framework of the compact rational Krylov methods (CORK and TS-CORK), allowing solve large-scale NEPs to be efficiently solved. One advantage of our method, compared to related techniques such as NLEIGS and infinite Arnoldi, is that it automatically selects the poles and zeros of the rational approximations. Numerical examples show that the presented framework is competitive with NLEIGS and usually produces smaller linearizations with the same accuracy but with less effort for the user.

scholarly journals On 'Best' Rational Approximations to $\pi$ and $\pi+e$

2020 ◽  
Author(s):  
Bertrand Teguia Tabuguia
Keyword(s):  
Power Series ◽  
Unit Circle ◽  
Infinite Series ◽  
Rational Number ◽  
Upper Bound ◽  
Continued Fraction ◽  
Series Representation ◽  
Rational Numbers ◽  
Similar Process ◽  
Rational Approximations

Through the half-unit circle area computation using the integration of the corresponding curve power series representation, we deduce a slow converging positive infinite series to $\pi$. However, by studying the remainder of that series we establish sufficiently close inequalities with equivalent lower and upper bound terms allowing us to estimate, more precisely, how the series approaches $\pi$. We use the obtained inequalities to compute up to four-digit denominator, what are in this sense, the best rational numbers that can replace $\pi$. It turns out that the well-known convergents of the continued fraction of $\pi$, $22/7$ and $355/113$ called, respectively, Yuel\"{u} and Mil\"{u} in China are the only ones found. Thus we apply a similar process to find rational estimations to $\pi+e$ where $e$ is taken as the power series of the exponential function evaluated at $1$. For rational numbers with denominators less than $2000$, the convergent $920/157$ of the continued fraction of $\pi+e$ turns out to be the only rational number of this type.

scholarly journals Energy product in a crack-like defect model under loading of mode II type

2019 ◽  
pp. 48-58
Author(s):  
V V Glagolev ◽  
L V Glagolev ◽  
A A Markin
Keyword(s):  
Characteristic Equation ◽  
Linear Size ◽  
Mode Ii ◽  
Crack Model ◽  
Interaction Layer ◽  
Real Roots ◽  
Energy Product ◽  
The Face ◽  
Simplified Solution ◽  
The Relationship

The loading of a crack-like defect in mode II is considered. In contrast to the classical representation of a crack in the form of a mathematical cut, the proposed model defines a crack in the form of a physical cut with a characteristic linear size. The mental continuation of a physical cut in a solid forms an interaction layer. It is significant that the stress-strain state of the layer does not introduce a singularity to the crack model. The product of the increment of the specific free energy in the face square element of the layer by the linear size determines its energy product. The object of the study is a double-cantilever sample, and the subject of study is the energy product in the face element of the interaction layer. The external load of the cantilevers leads to their horizontal antisymmetric displacements, which form uniform shear deformations in the interaction layer. From the equilibrium conditions of the cantilevers in the variation form, taking into account the hypothesis of axial deformation homogeneity and their reduction, a system of differential equations is obtained, which relates the stress state in the layer and the cantilevers. The solution of the characteristic equation of the system is investigated for various ratios of layer thickness and cantilevers. It is shown that when the relationship is less than a certain value, depending on the Poisson's ratio, real roots take place. In the framework of the real roots of the characteristic equation, an analytical solution of the problem is obtained. Subject to the neglect of compression cantilevers found a simplified solution. The deformations in the layer are determined taking into account the compression of the consoles and without it. The analysis of the dependence of the energy product on the relationship of the thickness of the layer and cantilevers. It is shown that with a thickness ratio of 10-6 or less, the energy product practically does not change its value. Accounting for the compression of cantilevers gives a difference in the values of the energy product of the order of 20 % in relation to the simplified solution of the problem.

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