http://www.mdpi.com/journal/axioms Latest open access articles published in Axioms at http://www.mdpi.com/journal/axioms http://www.mdpi.com/2075-1680/1/1/33 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. http://www.mdpi.com/2075-1680/1/1/33 Thu, 26 Apr 2012 00:00:00 CEST Axioms 2012-04-26 1 1 Communication 33 37 2075-1680 Introduction to the Yang-Baxter Equation with Open Problems 2012-04-26 doi: 10.3390/axioms1010033 Florin Nichita http://www.mdpi.com/2075-1680/1/1/21 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. http://www.mdpi.com/2075-1680/1/1/21 Fri, 20 Apr 2012 00:00:00 CEST Axioms 2012-04-20 1 1 Article 21 32 2075-1680 Axiomatic of Fuzzy Complex Numbers 2012-04-20 doi: 10.3390/axioms1010021 Angel Garrido http://www.mdpi.com/2075-1680/1/1/9 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. http://www.mdpi.com/2075-1680/1/1/9 Fri, 20 Apr 2012 00:00:00 CEST Axioms 2012-04-20 1 1 Article 9 20 2075-1680 Discrete Integrals and Axiomatically Defined Functionals 2012-04-20 doi: 10.3390/axioms1010009 Erich Peter Klement Radko Mesiar http://www.mdpi.com/2075-1680/1/1/4 We give an Itô formula associated to a non-linear semi-group associated to a m-accretive operator. http://www.mdpi.com/2075-1680/1/1/4 Wed, 21 Mar 2012 00:00:00 CET Axioms 2012-03-21 1 1 Communication 4 8 2075-1680 An Itô Formula for an Accretive Operator 2012-03-21 doi: 10.3390/axioms1010004 Rémi Léandre http://www.mdpi.com/2075-1680/1/1/1 It is my great pleasure to welcome you to Axioms: Mathematical Logic and Mathematical Physics, a new open access journal, which is dedicated to the foundations (structure and axiomatic basis, in particular) of mathematical and physical theories, not only on crisp or strictly classical sense, but also on fuzzy and generalized sense. This includes the more innovative current scientific trends, devoted to discover and solving new, defying problems. Our new journal does not try to be the same as those journals already dedicated to this field. Below we highlight what makes Axioms: Mathematical Logic and Mathematical Physics different. [...] http://www.mdpi.com/2075-1680/1/1/1 Thu, 01 Sep 2011 00:00:00 CEST Axioms 2011-09-01 1 1 Editorial 1 3 2075-1680 Another Journal on Mathematical Logic and Mathematical Physics? 2011-09-01 doi: 10.3390/axioms1010001 Angel Garrido