Special Issue "Axioms: Feature Papers"

A special issue of Axioms (ISSN 2075-1680).

Deadline for manuscript submissions: closed (1 June 2012).

Special Issue Editor

Editor-in-Chief
Prof. Dr. Angel Garrido Website E-Mail
Department of Fundamental Mathematics, Faculty of Sciences, UNED, Paseo Senda del Rey No. 9, 28040 Madrid, Spain
Phone: +34 91 3987237
Fax: +34 91 3987237
Interests: Mathematical Analysis; Measure Theory; Fuzzy Measures, in particular symmetry and entropy; Graph Theory; Discrete Mathematics; Automata Theory; Mathematical Education; Heuristics; Automata Theory; Artificial Intelligence

Published Papers (7 papers)

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Open AccessArticle
A Class of Extended Fractional Derivative Operators and Associated Generating Relations Involving Hypergeometric Functions
Axioms 2012, 1(3), 238-258; https://doi.org/10.3390/axioms1030238 - 05 Oct 2012
Cited by 34
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)
Open AccessArticle
Fat Triangulations, Curvature and Quasiconformal Mappings
Axioms 2012, 1(2), 99-110; https://doi.org/10.3390/axioms1020099 - 04 Jul 2012
Cited by 1
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)
Open AccessArticle
Foundations of Inference
Axioms 2012, 1(1), 38-73; https://doi.org/10.3390/axioms1010038 - 15 Jun 2012
Cited by 31
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)
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Open AccessCommunication
Introduction to the Yang-Baxter Equation with Open Problems
Axioms 2012, 1(1), 33-37; https://doi.org/10.3390/axioms1010033 - 26 Apr 2012
Cited by 5
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)
Open AccessArticle
Axiomatic of Fuzzy Complex Numbers
Axioms 2012, 1(1), 21-32; https://doi.org/10.3390/axioms1010021 - 20 Apr 2012
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)
Open AccessArticle
Discrete Integrals and Axiomatically Defined Functionals
Axioms 2012, 1(1), 9-20; https://doi.org/10.3390/axioms1010009 - 20 Apr 2012
Cited by 6
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)
Open AccessCommunication
An Itô Formula for an Accretive Operator
Axioms 2012, 1(1), 4-8; https://doi.org/10.3390/axioms1010004 - 21 Mar 2012
Cited by 4
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)
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