Research

238 KiB

Open AccessArticle

A Class of Extended Fractional Derivative Operators and Associated Generating Relations Involving Hypergeometric Functions

by H. M. Srivastava, Rakesh K. Parmar and Purnima Chopra

Axioms 2012, 1(3), 238-258; https://doi.org/10.3390/axioms1030238 - 05 Oct 2012

Cited by 69 | Viewed by 7626

Abstract

Recently, an extended operator of fractional derivative related to a generalized Beta function was used in order to obtain some generating relations involving the extended hypergeometric functions [1]. The main object of this paper is to present a further generalization of the extended [...] Read more.

Recently, an extended operator of fractional derivative related to a generalized Beta function was used in order to obtain some generating relations involving the extended hypergeometric functions [1]. The main object of this paper is to present a further generalization of the extended fractional derivative operator and apply the generalized extended fractional derivative operator to derive linear and bilinear generating relations for the generalized extended Gauss, Appell and Lauricella hypergeometric functions in one, two and more variables. Some other properties and relationships involving the Mellin transforms and the generalized extended fractional derivative operator are also given. Full article

(This article belongs to the Special Issue Axioms: Feature Papers)

212 KiB

Open AccessArticle

Fat Triangulations, Curvature and Quasiconformal Mappings

by Emil Saucan and Meir Katchalski

Axioms 2012, 1(2), 99-110; https://doi.org/10.3390/axioms1020099 - 04 Jul 2012

Cited by 1 | Viewed by 5458

Abstract

We investigate the interplay between the existence of fat triangulations, P L approximations of Lipschitz–Killing curvatures and the existence of quasiconformal mappings. In particular we prove that if there exists a quasiconformal mapping between two P L or smooth n-manifolds, then their [...] Read more.

We investigate the interplay between the existence of fat triangulations, P L approximations of Lipschitz–Killing curvatures and the existence of quasiconformal mappings. In particular we prove that if there exists a quasiconformal mapping between two P L or smooth n-manifolds, then their Lipschitz–Killing curvatures are bilipschitz equivalent. An extension to the case of almost Riemannian manifolds, of a previous existence result of quasimeromorphic mappings on manifolds due to the first author is also given. Full article

(This article belongs to the Special Issue Axioms: Feature Papers)

347 KiB

Open AccessArticle

Foundations of Inference

by Kevin H. Knuth and John Skilling

Axioms 2012, 1(1), 38-73; https://doi.org/10.3390/axioms1010038 - 15 Jun 2012

Cited by 39 | Viewed by 12694

Abstract

We present a simple and clear foundation for finite inference that unites and significantly extends the approaches of Kolmogorov and Cox. Our approach is based on quantifying lattices of logical statements in a way that satisfies general lattice symmetries. With other applications such [...] Read more.

We present a simple and clear foundation for finite inference that unites and significantly extends the approaches of Kolmogorov and Cox. Our approach is based on quantifying lattices of logical statements in a way that satisfies general lattice symmetries. With other applications such as measure theory in mind, our derivations assume minimal symmetries, relying on neither negation nor continuity nor differentiability. Each relevant symmetry corresponds to an axiom of quantification, and these axioms are used to derive a unique set of quantifying rules that form the familiar probability calculus. We also derive a unique quantification of divergence, entropy and information. Full article

(This article belongs to the Special Issue Axioms: Feature Papers)

► Show Figures

Graphical abstract

201 KiB

Open AccessCommunication

Introduction to the Yang-Baxter Equation with Open Problems

by Florin Nichita

Axioms 2012, 1(1), 33-37; https://doi.org/10.3390/axioms1010033 - 26 Apr 2012

Cited by 18 | Viewed by 8394

Abstract

The Yang-Baxter equation first appeared in theoretical physics, in a paper by the Nobel laureate C. N. Yang, and in statistical mechanics, in R. J. Baxter’s work. Later, it turned out that this equation plays a crucial role in: quantum groups, knot theory, [...] Read more.

The Yang-Baxter equation first appeared in theoretical physics, in a paper by the Nobel laureate C. N. Yang, and in statistical mechanics, in R. J. Baxter’s work. Later, it turned out that this equation plays a crucial role in: quantum groups, knot theory, braided categories, analysis of integrable systems, quantum mechanics, non-commutative descent theory, quantum computing, non-commutative geometry, etc. Many scientists have found solutions for the Yang-Baxter equation, obtaining qualitative results (using the axioms of various algebraic structures) or quantitative results (usually using computer calculations). However, the full classification of its solutions remains an open problem. In this paper, we present the (set-theoretical) Yang-Baxter equation, we sketch the proof of a new theorem, we state some problems, and discuss about directions for future research. Full article

(This article belongs to the Special Issue Axioms: Feature Papers)

211 KiB

Open AccessArticle

Axiomatic of Fuzzy Complex Numbers

by Angel Garrido

Axioms 2012, 1(1), 21-32; https://doi.org/10.3390/axioms1010021 - 20 Apr 2012

Cited by 1 | Viewed by 6563

Abstract

Fuzzy numbers are fuzzy subsets of the set of real numbers satisfying some additional conditions. Fuzzy numbers allow us to model very difficult uncertainties in a very easy way. Arithmetic operations on fuzzy numbers have also been developed, and are based mainly on [...] Read more.

Fuzzy numbers are fuzzy subsets of the set of real numbers satisfying some additional conditions. Fuzzy numbers allow us to model very difficult uncertainties in a very easy way. Arithmetic operations on fuzzy numbers have also been developed, and are based mainly on the crucial Extension Principle. When operating with fuzzy numbers, the results of our calculations strongly depend on the shape of the membership functions of these numbers. Logically, less regular membership functions may lead to very complicated calculi. Moreover, fuzzy numbers with a simpler shape of membership functions often have more intuitive and more natural interpretations. But not only must we apply the concept and the use of fuzzy sets, and its particular case of fuzzy number, but also the new and interesting mathematical construct designed by Fuzzy Complex Numbers, which is much more than a correlate of Complex Numbers in Mathematical Analysis. The selected perspective attempts here that of advancing through axiomatic descriptions. Full article

(This article belongs to the Special Issue Axioms: Feature Papers)

204 KiB

Open AccessArticle

Discrete Integrals and Axiomatically Defined Functionals

by Erich Peter Klement and Radko Mesiar

Axioms 2012, 1(1), 9-20; https://doi.org/10.3390/axioms1010009 - 20 Apr 2012

Cited by 18 | Viewed by 5321

Abstract

Several discrete universal integrals on finite universes are discussed from an axiomatic point of view. We start from the first attempt due to B. Riemann and cover also most recent approaches based on level dependent capacities. Our survey includes, among others, the Choquet [...] Read more.

Several discrete universal integrals on finite universes are discussed from an axiomatic point of view. We start from the first attempt due to B. Riemann and cover also most recent approaches based on level dependent capacities. Our survey includes, among others, the Choquet and the Sugeno integral and general copula-based integrals. Full article

(This article belongs to the Special Issue Axioms: Feature Papers)

125 KiB

Open AccessCommunication

An Itô Formula for an Accretive Operator

by Rémi Léandre

Axioms 2012, 1(1), 4-8; https://doi.org/10.3390/axioms1010004 - 21 Mar 2012

Cited by 5 | Viewed by 4927

Abstract

We give an Itô formula associated to a non-linear semi-group associated to a m-accretive operator. Full article

(This article belongs to the Special Issue Axioms: Feature Papers)

Journal Menu

Journal Browser

Axioms: Feature Papers

Share This Special Issue

Special Issue Editor

Published Papers (7 papers)

Research

Further Information

Guidelines

MDPI Initiatives

Follow MDPI