scholarly journals Appendix E: Definition of a Few Special Functions

2006 ◽  
pp. 241-242
Keyword(s):  
Special Functions ◽  
Definition Of

scholarly journals The Laplace equation in the exterior of the Hankel contour and novel identities for hypergeometric functions

2013 ◽  
Vol 469 (2157) ◽  
pp. 20130081 ◽  
Author(s):  
A. S. Fokas ◽  
M. L. Glasser
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Laplace Equation ◽  
Special Functions ◽  
Boundary Value ◽  
Zeta Functions ◽  
Neumann Boundary ◽  
Neumann Boundary Value Problem ◽  
Wide Range ◽  
Definition Of

By using conformal mappings, it is possible to express the solution of certain boundary-value problems for the Laplace equation in terms of a single integral involving the given boundary data. We show that such explicit formulae can be used to obtain novel identity for special functions. A convenient tool for deriving this type of identity is the so-called global relation , which has appeared recently in a wide range of boundary-value problems. As a concrete application, we analyse the Neumann boundary-value problem for the Laplace equation in the exterior of the Hankel contour, which appears in the definition of both the gamma and the Riemann zeta functions. By using the explicit solution of this problem, we derive a number of novel identities involving the hypergeometric function. Also, we point out an interesting connection between the solution of the above Neumann boundary-value problem for a particular set of Neumann data and the Riemann hypothesis.

scholarly journals Les Allemagnes et la division Est-Ouest : Une ambivalence politique

2005 ◽  
Vol 16 (3) ◽  
pp. 547-559
Author(s):  
Paul Létourneau
Keyword(s):  
International Organizations ◽  
Special Functions ◽  
States Of Affairs ◽  
Concrete Case ◽  
Long Run ◽  
Official Policy ◽  
The World ◽  
Definition Of

This article is about the role of international bureaucracies in the determination of the general policies of international organizations. In this paper it is argued that in general international organizations' Secretariats generally do wield, considerable power over the definition of the institutions' strategies, i.e. those activities, priorities and projects which taken together make up the program of the institution for a given period. Indeed, the international bureaucrats exercise tremendous control over the content of the program. This is so because international organizations have special functions in the world System. They must see to it that, certain states of affairs prevail in the world over the long run. It is, therefore, no surprise that the programs' content be more or less shielded from conjonctural fluctuations. The article then proceeds to test these hypotheses on a concrete case: the analysis of the processus through which Unesco's program goes before becoming the official policy of the organization.

scholarly journals Investigation of the k-Analogue of Gauss Hypergeometric Functions Constructed by the Hadamard Product

Symmetry ◽  
2021 ◽  
Vol 13 (4) ◽  
pp. 714
Author(s):  
Mohamed Abdalla ◽  
Muajebah Hidan
Keyword(s):  
Hypergeometric Functions ◽  
Hadamard Product ◽  
Special Function ◽  
Function Theory ◽  
Special Functions ◽  
Integral Transforms ◽  
Integral Representations ◽  
Convergence Properties ◽  
Pochhammer Symbol ◽  
Definition Of

Traditionally, the special function theory has many applications in various areas of mathematical physics, economics, statistics, engineering, and many other branches of science. Inspired by certain recent extensions of the k-analogue of gamma, the Pochhammer symbol, and hypergeometric functions, this work is devoted to the study of the k-analogue of Gauss hypergeometric functions by the Hadamard product. We give a definition of the Hadamard product of k-Gauss hypergeometric functions (HPkGHF) associated with the fourth numerator and two denominator parameters. In addition, convergence properties are derived from this function. We also discuss interesting properties such as derivative formulae, integral representations, and integral transforms including beta transform and Laplace transform. Furthermore, we investigate some contiguous function relations and differential equations connecting the HPkGHF. The current results are more general than previous ones. Moreover, the proposed results are useful in the theory of k-special functions where the hypergeometric function naturally occurs.

THE JONES POLYNOMIAL AND THE INTERSECTION NUMBERS OF TWISTED CYCLES ASSOCIATED WITH A SELBERG TYPE INTEGRAL

2011 ◽  
Vol 20 (03) ◽  
pp. 469-496 ◽  
Author(s):  
KATSUHISA MIMACHI
Keyword(s):  
Special Functions ◽  
Jones Polynomial ◽  
Intersection Number ◽  
Homology Theory ◽  
Intersection Numbers ◽  
Type Integral ◽  
Jones Polynomials ◽  
Twisted Homology ◽  
Hypergeometric Type ◽  
Definition Of

We give a new definition of the Jones polynomial by means of the intersection number of loaded (or twisted) cycles associated with a Selberg type integral. Our definition is naturally formulated in the framework of the twisted homology theory, which is developd by Aomoto to study the special functions of hypergeometric type. The naturality of the definition leads to evaluate the Jones polynomials in several cases: well-known results in the case of two-bridge link, a formula for (3, s)-torus and that for the Prezel with 3 parameters. Our definition is motivated by the work of Bigelow.

scholarly journals ON GENERALIZATION OF SPECIAL FUNCTIONS RELATED TO WEYL GROUPS

2016 ◽  
Vol 56 (6) ◽  
pp. 440-447
Author(s):  
Lenka Háková ◽  
Agnieszka Tereszkiewicz
Keyword(s):  
Weyl Group ◽  
Lie Algebras ◽  
Special Functions ◽  
Irreducible Representations ◽  
Weyl Groups ◽  
One Dimensional ◽  
Complex Functions ◽  
Group Orbit ◽  
Rank 2 ◽  
Definition Of

Weyl group orbit functions are defined in the context of Weyl groups of simple Lie algebras. They are multivariable complex functions possessing remarkable properties such as (anti)invariance with respect to the corresponding Weyl group, continuous and discrete orthogonality. A crucial tool in their definition are so-called sign homomorphisms, which coincide with one-dimensional irreducible representations. In this work we generalize the definition of orbit functions using characters of irreducible representations of higher dimensions. We describe their properties and give examples for Weyl groups of rank 2 and 3.

scholarly journals Fractional Hermite-Hadamard Integral Inequalities for a New Class of Convex Functions

Symmetry ◽  
2020 ◽  
Vol 12 (9) ◽  
pp. 1485
Author(s):  
Pshtiwan Othman Mohammed ◽  
Thabet Abdeljawad ◽  
Shengda Zeng ◽  
Artion Kashuri
Keyword(s):  
Convex Functions ◽  
Fractional Integral ◽  
Special Functions ◽  
Applied Mathematics ◽  
Strong Relationship ◽  
Integral Inequalities ◽  
Mathematical Methods ◽  
Fractional Integral Operators ◽  
New Class ◽  
Definition Of

Fractional integral inequality plays a significant role in pure and applied mathematics fields. It aims to develop and extend various mathematical methods. Therefore, nowadays we need to seek accurate fractional integral inequalities in obtaining the existence and uniqueness of the fractional methods. Besides, the convexity theory plays a concrete role in the field of fractional integral inequalities due to the behavior of its definition and properties. There is also a strong relationship between convexity and symmetric theories. So, whichever one we work on, we can then apply it to the other one due to the strong correlation produced between them, specifically in the last few decades. First, we recall the definition of φ-Riemann–Liouville fractional integral operators and the recently defined class of convex functions, namely the σ˘-convex functions. Based on these, we will obtain few integral inequalities of Hermite–Hadamard’s type for a σ˘-convex function with respect to an increasing function involving the φ-Riemann–Liouville fractional integral operator. We can conclude that all derived inequalities in our study generalize numerous well-known inequalities involving both classical and Riemann–Liouville fractional integral inequalities. Finally, application to certain special functions are pointed out.

scholarly journals Analytic Study of Complex Fractional Tsallis’ Entropy with Applications in CNNs

Entropy ◽  
2018 ◽  
Vol 20 (10) ◽  
pp. 722 ◽  
Author(s):  
Rabha Ibrahim ◽  
Maslina Darus
Keyword(s):  
Neural Networks ◽  
Lower Bounds ◽  
Special Functions ◽  
Tsallis Entropy ◽  
Analytic Study ◽  
Complex Domain ◽  
Upper And Lower Bounds ◽  
Probability Functions ◽  
Fractional Entropy ◽  
Definition Of

In this paper, we study Tsallis’ fractional entropy (TFE) in a complex domain by applying the definition of the complex probability functions. We study the upper and lower bounds of TFE based on some special functions. Moreover, applications in complex neural networks (CNNs) are illustrated to recognize the accuracy of CNNs.

Introductory paper

1966 ◽  
Vol 24 ◽  
pp. 3-5
Author(s):  
W. W. Morgan
Keyword(s):  
Line Intensity ◽  
Classification Scheme ◽  
Two Dimensional ◽  
Spectral Classification ◽  
Dimensional Array ◽  
Rr Lyrae ◽  
Metallic Line ◽  
Introductory Paper ◽  
Parameter Varying ◽  
Definition Of

1. The definition of “normal” stars in spectral classification changes with time; at the time of the publication of theYerkes Spectral Atlasthe term “normal” was applied to stars whose spectra could be fitted smoothly into a two-dimensional array. Thus, at that time, weak-lined spectra (RR Lyrae and HD 140283) would have been considered peculiar. At the present time we would tend to classify such spectra as “normal”—in a more complicated classification scheme which would have a parameter varying with metallic-line intensity within a specific spectral subdivision.

2.2. Working Group 2: Conceptual Definitions of Reference Coordinate Systems for Earth Dynamics

1975 ◽  
Vol 26 ◽  
pp. 21-26
Keyword(s):  
Coordinate System ◽  
Working Group ◽  
General Character ◽  
Coordinate Systems ◽  
Reference Coordinate System ◽  
Group 2 ◽  
Definition Of ◽  
Earth Dynamics

An ideal definition of a reference coordinate system should meet the following general requirements:1. It should be as conceptually simple as possible, so its philosophy is well understood by the users.2. It should imply as few physical assumptions as possible. Wherever they are necessary, such assumptions should be of a very general character and, in particular, they should not be dependent upon astronomical and geophysical detailed theories.3. It should suggest a materialization that is dynamically stable and is accessible to observations with the required accuracy.

Symbiotic Stars at Optical, Infrared and Radio Wavelengths

1979 ◽  
Vol 46 ◽  
pp. 125-149 ◽  
Author(s):  
David A. Allen
Keyword(s):  
Temperature Regime ◽  
Type Star ◽  
Stellar Systems ◽  
Symbiotic Stars ◽  
Lower Excitation ◽  
Definition Of ◽  
Original Type

No paper of this nature should begin without a definition of symbiotic stars. It was Paul Merrill who, borrowing on his botanical background, coined the termsymbioticto describe apparently single stellar systems which combine the TiO absorption of M giants (temperature regime ≲ 3500 K) with He II emission (temperature regime ≳ 100,000 K). He and Milton Humason had in 1932 first drawn attention to three such stars: AX Per, CI Cyg and RW Hya. At the conclusion of the Mount Wilson Ha emission survey nearly a dozen had been identified, and Z And had become their type star. The numbers slowly grew, as much because the definition widened to include lower-excitation specimens as because new examples of the original type were found. In 1970 Wackerling listed 30; this was the last compendium of symbiotic stars published.

Sign in / Sign up

Export Citation Format

Share Document