scholarly journals On one generalization of the Hermite quadrature formula

2021 ◽  
Vol 57 (3) ◽  
pp. 319-329
Author(s):  
Y. A. Rouba ◽  
K. A. Smatrytski ◽  
Y. V. Dirvuk
Keyword(s):  
Numerical Analysis ◽  
Approximation Theory ◽  
Complex Variable ◽  
Jacobi Polynomials ◽  
Rational Functions ◽  
Quadrature Formulas ◽  
New Approach ◽  
Rational Chebyshev ◽  
Rational Type ◽  
Made In

In this paper we propose a new approach to the construction of quadrature formulas of interpolation rational type on an interval. In the introduction, a brief analysis of the results on the topic of the research is carried out. Most attention is paid to the works of mathematicians of the Belarusian school on approximation theory – Gauss, Lobatto, and Radau quadrature formulas with nodes at the zeros of the rational Chebyshev – Markov fractions. Rational fractions on the segment, generalizing the classical orthogonal Jacobi polynomials with one weight, are defined, and some of their properties are described. One of the main results of this paper consists in constructing quadrature formulas with nodes at zeros of the introduced rational fractions, calculating their coefficients in an explicit form, and estimating the remainder. This result is preceded by some auxiliary statements describing the properties of special rational functions. Classical methods of mathematical analysis, approximation theory, and the theory of functions of a complex variable are used for proof. In the conclusion a numerical analysis of the efficiency of the constructed quadrature formulas is carried out. Meanwhile, the choice of the parameters on which the nodes of the quadrature formulas depend is made in several standard ways. The obtained results can be applied for further research of rational quadrature formulas, as well as in numerical analysis.

scholarly journals On one interpolating rational process of Fejer – Hermite

2020 ◽  
Vol 56 (3) ◽  
pp. 263-274
Author(s):  
Ya. A. Rouba ◽  
K. A. Smatrytski ◽  
Ya. V. Dirvuk
Keyword(s):  
Numerical Analysis ◽  
Continuous Function ◽  
Approximation Theory ◽  
Complex Variable ◽  
Arbitrary Function ◽  
Rational Functions ◽  
New Approach ◽  
Definition Of ◽  
Rational Process ◽  
Made In

In this paper, a new approach to the definition of the interpolating rational process of Fejer – Hermite with first-type Chebyshev – Markov nodes on a segment is studied and some of its approximating properties are described. In the introduction a brief analysis of the results on the topic of the research is carried out. Herein, the methods of the construction of interpolating processes, in particular, Fejer – Hermite processes, in the polynomial and rational approximation are analysed. A new method to determine the interpolating rational Fejer – Hermite process is proposed. One of the main results of this paper is the proof of the uniform convergence of this process for an arbitrary function, which is continuous on the segment, under some restrictions for the poles of approximating functions. This result is preceded by some auxiliary statements describing the properties of special rational functions. The classic methods of mathematical analysis, approximation theory, and theory of functions of a complex variable are used to prove the results of the work. Moreover, we present the numerical analysis of the effectiveness of the application of the constructed interpolating Fejer – Hermite process for the approximation of a continuous function with singularities. The choice of parameters, on which the nodes of interpolation depend, is made in several standard ways. The obtained results can be applied to further study the approximating properties of interpolating processes.

scholarly journals An Analytic Expression for the Inverse Involute

2019 ◽  
Vol 2019 ◽  
pp. 1-9
Author(s):  
Alberto López Rosado ◽  
Federico Prieto Muñoz ◽  
Roberto Alvarez Fernández
Keyword(s):  
Approximation Theory ◽  
Jacobi Polynomials ◽  
Rational Functions ◽  
Rational Approximations ◽  
Efficient Computation ◽  
Remez Algorithm ◽  
Tangent Function ◽  
Analytic Expression ◽  
Involute Curve ◽  
Theoretical Foundations

This article introduces new types of rational approximations of the inverse involute function, widely used in gear engineering, allowing the processing of this function with a very low error. This approximated function is appropriate for engineering applications, with a much reduced number of operations than previous formulae in the existing literature, and a very efficient computation. The proposed expressions avoid the use of iterative methods. The theoretical foundations of the approximation theory of rational functions, the Chebyshev and Jacobi polynomials that allow these approximations to be obtained, are presented in this work, and an adaptation of the Remez algorithm is also provided, which gets a null error at the origin. This way, approximations in ranges or degrees different from those presented here can be obtained. A rational approximation of the direct involute function is computed, which avoids the computation of the tangent function. Finally, the direct polar equation of the circle involute curve is approximated with some application examples.

scholarly journals A dominated convergence theorem for Eisenstein series

2021 ◽  
Author(s):  
Johann Franke
Keyword(s):  
Fourier Series ◽  
Dirichlet Series ◽  
Eisenstein Series ◽  
Modular Forms ◽  
Convergence Theorem ◽  
Rational Functions ◽  
New Approach ◽  
Series Representations ◽  
Dominated Convergence ◽  
Generalized Dirichlet

AbstractBased on the new approach to modular forms presented in [6] that uses rational functions, we prove a dominated convergence theorem for certain modular forms in the Eisenstein space. It states that certain rearrangements of the Fourier series will converge very fast near the cusp $$\tau = 0$$ τ = 0 . As an application, we consider L-functions associated to products of Eisenstein series and present natural generalized Dirichlet series representations that converge in an expanded half plane.

A Cognitive Approach to the Mechanism of Intelligence

2011 ◽  
pp. 18-35
Author(s):  
Yi X. Zhong
Keyword(s):  
Intelligent Systems ◽  
Simulation Approach ◽  
Cognitive Approach ◽  
New Approach ◽  
The Core ◽  
Behavior Simulation ◽  
Structural Simulation ◽  
Functional Simulation ◽  
Made In

An attempt was made in the article to propose a new approach to the intelligence research, namely the cognitive approach that tries to explore in depth the core mechanism of intelligence formation of intelligent systems from the cognitive viewpoint. It is discovered, as result, that the mechanism of intelligence formation in general case is implemented by a sequence of transformations conversing the information to knowledge and further to intelligence (i.e., the intelligent strategy, the embodiment of intelligence in a narrower sense). It is also discovered that the three major approaches to AI that exist, the structural simulation approach, the functional simulation approach, and the behavior simulation approach, can all be harmoniously unified within the framework of the cognitive approach. These two discoveries, as well as the related background, will be reported here in the article.

scholarly journals Zeros of the Exceptional Laguerre and Jacobi Polynomials

2012 ◽  
Vol 2012 ◽  
pp. 1-27 ◽  
Author(s):  
Choon-Lin Ho ◽  
Ryu Sasaki
Keyword(s):  
Numerical Analysis ◽  
Orthogonal Polynomials ◽  
Jacobi Polynomials ◽  
Mathematical Physics ◽  
Positive Definite ◽  
Classical Orthogonal Polynomials ◽  
Complete Sets

An interesting discovery in the last two years in the field of mathematical physics has been the exceptional Xℓ Laguerre and Jacobi polynomials. Unlike the well-known classical orthogonal polynomials which start with constant terms, these new polynomials have the lowest degree ℓ=1,2,…, and yet they form complete sets with respect to some positive-definite measure. In this paper, we study one important aspect of these new polynomials, namely, the behaviors of their zeros as some parameters of the Hamiltonians change. Most results are of heuristic character derived by numerical analysis.

Cross-Wedge Rolling of Driving Shaft from Titanium Alloy Ti6Al4V

2016 ◽  
Vol 687 ◽  
pp. 125-132 ◽  
Author(s):  
Zbigniew Pater ◽  
Tomasz Bulzak ◽  
Janusz Tomczak
Keyword(s):  
Titanium Alloy ◽  
Numerical Analysis ◽  
Process Parameters ◽  
Axial Symmetry ◽  
Cross Wedge Rolling ◽  
University Of Technology ◽  
The Subject ◽  
Titanium Alloy Ti6al4v ◽  
Rolling Technology ◽  
Made In

This paper deals with the issue of the helicopter SW4 rear gear driving shaft forming. It was assumed that this shaft will be made from titanium alloy Ti6Al4V and it will be formed by means of cross-wedge rolling technology (CWR). It was also assumed that rolling will be realized in double configuration, which will guarantee axial symmetry of forming forces. The conception of tools guaranteeing the CWR process realization and numerical analysis results verifying the assumed CWR process parameters of the subject shaft were presented. Tests of shaft rolling in laboratory conditions at Lublin University of Technology were made, in the result of which the possibility of forming by means of CWR of a driving shaft, manufactured from titanium alloy Ti6Al4V, of the helicopter SW4 rear gear was verified.

Stress in an elastic hump: the effects of glacier flow over elastic bedrock

1975 ◽  
Vol 344 (1637) ◽  
pp. 157-173 ◽  
Keyword(s):  
Plane Stress ◽  
Complex Variable ◽  
Analytic Solution ◽  
Rational Functions ◽  
Fortran Program ◽  
Stress Fields ◽  
Infinite Domain ◽  
Principal Directions ◽  
Glacier Flow ◽  
Principal Strains

Stress fields in an isotropic elastic semi-infinite domain with a humped contour subjected to applied tractions are obtained for plane strain or plane stress conditions. Domains mapped conformally onto the half-plane by a class of rational functions are considered and a complex variable method is used to determine an analytic solution. A general Fortran program has been constructed to determine stresses, principal directions, principal strains, and maximum shear tractions at specified points.

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