scholarly journals Viscovatov-Like Algorithm of Thiele–Newton’s Blending Expansion for a Bivariate Function

Mathematics ◽  
2019 ◽  
Vol 7 (8) ◽  
pp. 696 ◽  
Author(s):  
Shengfeng Li ◽  
Yi Dong
Keyword(s):  
Numerical Experiment ◽  
Continued Fraction ◽  
Polynomial Expansion ◽  
Suggested Algorithm ◽  
Bivariate Function

In this paper, Thiele–Newton’s blending expansion of a bivariate function is firstly suggested by means of combining Thiele’s continued fraction in one variable with Taylor’s polynomial expansion in another variable. Then, the Viscovatov-like algorithm is given for the computations of the coefficients of this rational expansion. Finally, a numerical experiment is presented to illustrate the practicability of the suggested algorithm. Henceforth, the Viscovatov-like algorithm has been considered as the imperative generalization to find out the coefficients of Thiele–Newton’s blending expansion of a bivariate function.

The Solvability of Polynomial Pell’s Equation

2020 ◽  
Vol 25 (2) ◽  
pp. 125-132
Author(s):  
Bal Bahadur Tamang ◽  
Ajay Singh
Keyword(s):  
Trivial Solution ◽  
Continued Fraction ◽  
Laurent Series ◽  
Quadratic Polynomial ◽  
Continued Fraction Expansion ◽  
Pell’S Equation ◽  
Integer Polynomials

This article attempts to describe the continued fraction expansion of ÖD viewed as a Laurent series x-1. As the behavior of the continued fraction expansion of ÖD is related to the solvability of the polynomial Pell’s equation p2-Dq2=1  where D=f2+2g  is monic quadratic polynomial with deg g<deg f  and the solutions p, q  must be integer polynomials. It gives a non-trivial solution if and only if the continued fraction expansion of ÖD  is periodic.

Mathematical modeling of Couette flow in anomalous thermoviscous liquid

2011 ◽  
Vol 8 (1) ◽  
pp. 143-152
Author(s):  
S.F. Khizbullina
Keyword(s):  
Mathematical Modeling ◽  
Angular Velocity ◽  
Numerical Experiment ◽  
Temperature Dependence ◽  
Steady Flow ◽  
Couette Flow ◽  
Outer Cylinder ◽  
Flow Regimes ◽  
Constant Angular Velocity ◽  
Coaxial Cylinders

The steady flow of anomalous thermoviscous liquid between the coaxial cylinders is considered. The inner cylinder rotates at a constant angular velocity while the outer cylinder is at rest. On the basis of numerical experiment various flow regimes depending on the parameter of viscosity temperature dependence are found.

Microwave Field Distribution Uniformity Coefficient in the Grain Mass

2020 ◽  
Vol 67 (2) ◽  
pp. 87-92
Author(s):  
Dmitriy A. Budnikov
Keyword(s):  
Electromagnetic Field ◽  
Numerical Experiment ◽  
Microwave Field ◽  
Research Purpose ◽  
Coefficient Of Determination ◽  
Material Research ◽  
Dense Layer ◽  
Uniformity Coefficient ◽  
Average Value ◽  
Processed Material

The article considers the microwave electromagnetic fields as one of the options for improving the thermal drying of grain. Their application is limited by the high unevenness of the field propagation in the layer of the processed material. (Research purpose) The research purpose is in justifying the uniformity of distribution of microwave field in the layer of the processed grain. (Materials and methods) The article presents the scheme of computer models of microwave processing zones and waveguides, properties of materials for conducting a numerical experiment. (Results and discussion) A numerical experiment was performed to determine the uniformity coefficient of propagation of the microwave field in a layer of grain material. The article presents the dependencies. (Conclusions) It was found that the results of modeling the distribution of the electromagnetic field in the zone of microwave convective influence of the installation containing two sources of microwave power for processing the grain layer indicate a high level of its unevenness in the volume of the product pipeline. To assess the uniformity of the distribution of the electromagnetic field in the working area of a laboratory installation, there used a coefficient that is the ratio of the average value of the intensity in the zone of microwave convective action to its average value of the wave strength passing through the output of the waveguide. The values of the uniformity coefficient in the considered implementation options are in the range of 0.1757-0.4946 for a dense layer of wheat. To ensure a sufficient level of uniformity of the electromagnetic wave distribution in the volume of the microwave convective zone, the uniformity coefficient must be higher than 0.37. The article presents the dependence of the uniformity coefficient of the electromagnetic field on the humidity of the processed material by a third-degree polynomial with a coefficient of determination higher than 0.98.

Analysis of the pore structure of adsorbents and its characterization by a numerical method

1984 ◽  
Vol 49 (12) ◽  
pp. 2721-2738 ◽  
Author(s):  
Ondřej Kadlec ◽  
Jerzy Choma ◽  
Helena Jankowska ◽  
Andrzej Swiatkowski
Keyword(s):  
Distribution Function ◽  
Pore Structure ◽  
Numerical Method ◽  
Active Carbon ◽  
Numerical Evaluation ◽  
Suggested Algorithm

This paper describes the algorithm of numerical evaluation of the parameters of the pore structure of adsorbents ( the micro, mezo and macropores). The structure of individual types of pores is described with the equation proposed by one of the present authors and giving the total distribution function of the pores with respect to their radii. The reliability of the suggested algorithm was verified in a number of calculations using a specially developed program. The results of the analysis and characterization of three different specimens of active carbon are shown as an example.

scholarly journals Ramanujan’s function k(τ)=r(τ)r 2(2τ) and its modularity

2020 ◽  
Vol 18 (1) ◽  
pp. 1727-1741
Author(s):  
Yoonjin Lee ◽  
Yoon Kyung Park
Keyword(s):  
Continued Fraction ◽  
Quadratic Field ◽  
Singular Values ◽  
Modular Function ◽  
Imaginary Quadratic Field ◽  
Modular Equations ◽  
Positive Rational Number ◽  
Congruence Relations ◽  
Symmetry Relations ◽  
Ray Class Field

Abstract We study the modularity of Ramanujan’s function k ( τ ) = r ( τ ) r 2 ( 2 τ ) k(\tau )=r(\tau ){r}^{2}(2\tau ) , where r ( τ ) r(\tau ) is the Rogers-Ramanujan continued fraction. We first find the modular equation of k ( τ ) k(\tau ) of “an” level, and we obtain some symmetry relations and some congruence relations which are satisfied by the modular equations; these relations are quite useful for reduction of the computation cost for finding the modular equations. We also show that for some τ \tau in an imaginary quadratic field, the value k ( τ ) k(\tau ) generates the ray class field over an imaginary quadratic field modulo 10; this is because the function k is a generator of the field of the modular function on Γ 1 ( 10 ) {{\mathrm{\Gamma}}}_{1}(10) . Furthermore, we suggest a rather optimal way of evaluating the singular values of k ( τ ) k(\tau ) using the modular equations in the following two ways: one is that if j ( τ ) j(\tau ) is the elliptic modular function, then one can explicitly evaluate the value k ( τ ) k(\tau ) , and the other is that once the value k ( τ ) k(\tau ) is given, we can obtain the value k ( r τ ) k(r\tau ) for any positive rational number r immediately.

scholarly journals A Lochs-Type Approach via Entropy in Comparing the Efficiency of Different Continued Fraction Algorithms

Mathematics ◽  
2021 ◽  
Vol 9 (3) ◽  
pp. 255
Author(s):  
Dan Lascu ◽  
Gabriela Ileana Sebe
Keyword(s):  
Dynamical Systems ◽  
Continued Fractions ◽  
Continued Fraction ◽  
Unit Interval ◽  
Probability Measures ◽  
Absolutely Continuous ◽  
Continued Fraction Expansions ◽  
Absolutely Continuous Invariant

We investigate the efficiency of several types of continued fraction expansions of a number in the unit interval using a generalization of Lochs theorem from 1964. Thus, we aim to compare the efficiency by describing the rate at which the digits of one number-theoretic expansion determine those of another. We study Chan’s continued fractions, θ-expansions, N-continued fractions, and Rényi-type continued fractions. A central role in fulfilling our goal is played by the entropy of the absolutely continuous invariant probability measures of the associated dynamical systems.

scholarly journals Coefficients of a Comprehensive Subclass of Meromorphic Bi-Univalent Functions Associated with the Faber Polynomial Expansion

Axioms ◽  
2021 ◽  
Vol 10 (1) ◽  
pp. 27
Author(s):  
Hari Mohan Srivastava ◽  
Ahmad Motamednezhad ◽  
Safa Salehian
Keyword(s):  
Unit Disk ◽  
Univalent Functions ◽  
Expansion Method ◽  
Upper Bounds ◽  
Polynomial Expansion ◽  
Open Unit ◽  
Open Unit Disk ◽  
Faber Polynomial ◽  
The Subject ◽  
Polynomial Expansion Method

In this paper, we introduce a new comprehensive subclass ΣB(λ,μ,β) of meromorphic bi-univalent functions in the open unit disk U. We also find the upper bounds for the initial Taylor-Maclaurin coefficients |b0|, |b1| and |b2| for functions in this comprehensive subclass. Moreover, we obtain estimates for the general coefficients |bn|(n≧1) for functions in the subclass ΣB(λ,μ,β) by making use of the Faber polynomial expansion method. The results presented in this paper would generalize and improve several recent works on the subject.

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