http://www.mdpi.com/journal/axioms Latest open access articles published in Axioms at http://www.mdpi.com/journal/axioms http://www.mdpi.com/2075-1680/2/2/224 We introduce and study Clarkson, Dol’nikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Sch¨affer constants in order to establish quantitative versions of Hahn-Banach separability theorem and to characterise the isometric extendability of H¨older-Lipschitz mappings. Abstract results are further applied to the spaces and pairs from the wide classes IG and IG+ and non-commutative Lp-spaces. The intimate relation between the subspaces and quotients of the IG-spaces on one side and various types of anisotropic Besov, Lizorkin-Triebel and Sobolev spaces of functions on open subsets of an Euclidean space defined in terms of differences, local polynomial approximations, wavelet decompositions and other means (as well as the duals and the lp-sums of all these spaces) on the other side, allows us to present the algorithm of extending the main results of the article to the latter spaces and pairs. Special attention is paid to the matter of sharpness. Our approach is quasi-Euclidean in its nature because it relies on the extrapolation of properties of Hilbert spaces and the study of 1-complemented subspaces of the spaces under consideration. Axioms 2013-05-23 2 2 Article 10.3390/axioms2020224 224 270 2075-1680 2013-05-23 doi: 10.3390/axioms2020224 Sergey Ajiev http://www.mdpi.com/2075-1680/2/2/208 In this paper we present a new fuzzy reasoning method in which the Choquet integral is used as aggregation function. In this manner, we can take into account the interaction among the rules of the system. For this reason, we consider several fuzzy measures, since it is a key point on the subsequent success of the Choquet integral, and we apply the new method with the same fuzzy measure for all the classes. However, the relationship among the set of rules of each class can be different and therefore the best fuzzy measure can change depending on the class. Consequently, we propose a learning method by means of a genetic algorithm in which the most suitable fuzzy measure for each class is computed. From the obtained results it is shown that our new proposal allows the performance of the classical fuzzy reasoning methods of the winning rule and additive combination to be enhanced whenever the fuzzy measure is appropriate for the tackled problem. Axioms 2013-04-23 2 2 Article 10.3390/axioms2020208 208 223 2075-1680 2013-04-23 doi: 10.3390/axioms2020208 Edurne Barrenechea Humberto Bustince Javier Fernandez Daniel Paternain José Sanz http://www.mdpi.com/2075-1680/2/2/182 This paper adds to the literature on the information content of different spreads for real activity by explicitly taking into account the time scale relationship between a variety of monetary and financial indicators (real interest rate, term and credit spreads) and output growth. By means of wavelet-based exploratory data analysis we obtain richer results relative to the aggregate analysis by identifying the dominant scales of variation in the data and the scales and location at which structural breaks have occurred. Moreover, using the “double residuals” regression analysis on a scale-by-scale basis, we find that changes in the spread in several markets have different information content for output at different time frames. This is consistent with the idea that allowing for different time scales of variation in the data can provide a fruitful understanding of the complex dynamics of economic relationships between variables with non-stationary or transient components, certainly richer than those obtained using standard time domain methods. Axioms 2013-04-23 2 2 Article 10.3390/axioms2020182 182 207 2075-1680 2013-04-23 doi: 10.3390/axioms2020182 Marco Gallegati James Ramsey Willi Semmler http://www.mdpi.com/2075-1680/2/2/142 This paper describes and tests a wavelet-based implicit numerical method for solving partial differential equations. Intended for problems with localized small-scale interactions, the method exploits the form of the wavelet decomposition to divide the implicit system created by the time-discretization into multiple smaller systems that can be solved sequentially. Included is a test on a basic non-linear problem, with both the results of the test, and the time required to calculate them, compared with control results based on a single system with fine resolution. The method is then tested on a non-trivial problem, its computational time and accuracy checked against control results. In both tests, it was found that the method requires less computational expense than the control. Furthermore, the method showed convergence towards the fine resolution control results. Axioms 2013-04-23 2 2 Article 10.3390/axioms2020142 142 181 2075-1680 2013-04-23 doi: 10.3390/axioms2020142 Donald McLaren Lucy Campbell Rémi Vaillancourt http://www.mdpi.com/2075-1680/2/2/122 Boundary functions for wavelets on a finite interval are often constructed as linear combinations of boundary-crossing scaling functions. An alternative approach is based on linear algebra techniques for truncating the infinite matrix of the DiscreteWavelet Transform to a finite one. In this article we show how an algorithm of Madych for scalar wavelets can be generalized to multiwavelets, given an extra assumption. We then develop a new algorithm that does not require this additional condition. Finally, we apply results from a previous paper to resolve the non-uniqueness of the algorithm by imposing regularity conditions (including approximation orders) on the boundary functions. Axioms 2013-04-17 2 2 Article 10.3390/axioms2020122 122 141 2075-1680 2013-04-17 doi: 10.3390/axioms2020122 Ahmet Altürk Fritz Keinert http://www.mdpi.com/2075-1680/2/2/100 We use the biorthogonal multiwavelets related by differentiation constructed in previous work to construct compactly supported biorthogonal multiwavelet bases for the space of vector fields on the upper half plane R2 + such that the reconstruction wavelets are divergence-free and have vanishing normal components on the boundary of R2 +. Such wavelets are suitable to study the Navier–Stokes equations on a half plane when imposing a Navier boundary condition. Axioms 2013-04-11 2 2 Article 10.3390/axioms2020100 100 121 2075-1680 2013-04-11 doi: 10.3390/axioms2020100 Joseph Lakey Phan Nguyen http://www.mdpi.com/2075-1680/2/2/85 We find Euler integral formulas, summation and reduction formulas for q-analogues of Srivastava’s three triple hypergeometric functions. The proofs use q-analogues of Picard’s integral formula for the first Appell function, a summation formula for the first Appell function based on the Bayley–Daum formula, and a general triple series reduction formula of Karlsson. Many of the formulas are purely formal, since it is difficult to find convergence regions for these functions of several complex variables. We use the Ward q-addition to describe the known convergence regions of q-Appell and q-Lauricella functions. Axioms 2013-04-11 2 2 Article 10.3390/axioms2020085 85 99 2075-1680 2013-04-11 doi: 10.3390/axioms2020085 Thomas Ernst http://www.mdpi.com/2075-1680/2/2/67 The mollification obtained by truncating the expansion in wavelets is studied, where the wavelets are so chosen that noise is reduced and the Gibbs phenomenon does not occur. The estimations of the error of approximation of the mollification are given for the case when the fractional derivative of a function is calculated. Noting that the estimations are applicable even when the orthogonality of the wavelets is not satisfied, we study mollifications using unorthogonalized wavelets, as well as those using orthogonal wavelets. Axioms 2013-03-25 2 2 Article 10.3390/axioms2020067 67 84 2075-1680 2013-03-25 doi: 10.3390/axioms2020067 Tohru Morita Ken-ichi Sato http://www.mdpi.com/2075-1680/2/1/58 In this paper, we give a pedagogical introduction to several beautiful formulas discovered by Ramanujan. Using these results, we evaluate a Ramanujan-type integral formula. The result can be expressed in terms of the Golden Ratio. Axioms 2013-03-20 2 1 Article 10.3390/axioms2010058 58 66 2075-1680 2013-03-20 doi: 10.3390/axioms2010058 Hei-Chi Chan http://www.mdpi.com/2075-1680/2/1/44 Wavelets are explored as a data smoothing (or de-noising) option for solution monitoring data in nuclear safeguards. In wavelet-smoothed data, the Gibbs phenomenon can obscure important data features that may be of interest. This paper compares wavelet smoothing to piecewise linear smoothing and local kernel smoothing, and illustrates that the Haar wavelet basis is effective for reducing the Gibbs phenomenon. Axioms 2013-03-20 2 1 Article 10.3390/axioms2010044 44 57 2075-1680 2013-03-20 doi: 10.3390/axioms2010044 Tom Burr Claire Longo http://www.mdpi.com/2075-1680/2/1/20 Recently, the authors have established a large class of modular relations involving the Rogers-Ramanujan type functions J(q) and K(q) of order ten. In this paper, we establish further modular relations connecting these two functions with Rogers-Ramanujan functions, Göllnitz-Gordon functions and cubic functions, which are analogues to the Ramanujan’s forty identities for Rogers-Ramanujan functions. Furthermore, we give partition theoretic interpretations of some of our modular relations. Axioms 2013-02-18 2 1 Article 10.3390/axioms2010020 20 43 2075-1680 2013-02-18 doi: 10.3390/axioms2010020 Chandrashekar Adiga Nasser Bulkhali http://www.mdpi.com/2075-1680/2/1/10 In this paper, we define the generating functions for the generalized q-Stirling numbers of the second kind. By applying Mellin transform to these functions, we construct interpolation functions of these numbers at negative integers. We also derive some identities and relations related to q-Bernoulli numbers and polynomials and q-Stirling numbers of the second kind. Axioms 2013-02-08 2 1 Article 10.3390/axioms2010010 10 19 2075-1680 2013-02-08 doi: 10.3390/axioms2010010 Hacer Ozden Ismail Cangul Yilmaz Simsek http://www.mdpi.com/2075-1680/2/1/1 Let K be a finite extension of Q and let S = {ν} denote the collection of K normalized absolute values on K. Let V+K denote the additive group of adeles over K and let K ≥0   c : V + → R denote the content map defined as c({aν }) = Q K   ν ∈S ν (aν ) for {aν } ∈ V+K A classical result of J. W. S. Cassels states that there is a constant c > 0 depending only on the field K  with the following property: if {aν } ∈ V+K with c({aν })  > c, then there exists a non-zero element b  ∈ K for which ν (b) ≤ ν (aν ), ∀ν  ∈ S. Let cK be the greatest lower bound of the set of all c that satisfy this property. In the case that K is a real quadratic extension there is a known upper bound for cK due to S. Lang. The purpose of this paper is to construct a new upper bound for cK in the case that K has class number one. We compare our new bound with Lang’s bound for various real quadratic extensions and find that our new bound is better than Lang’s in many instances. Axioms 2012-12-28 2 1 Article 10.3390/axioms2010001 1 9 2075-1680 2012-12-28 doi: 10.3390/axioms2010001 Robert Underwood http://www.mdpi.com/2075-1680/1/3/395 The aim of this paper is to construct generating functions, related to nonnegative real parameters, for q-Eulerian type polynomials and numbers (or q-Apostol type Frobenius–Euler polynomials and numbers). We derive some identities for these polynomials and numbers based on the generating functions and functional equations. We also give multiplication formula for the generalized Apostol type Frobenius–Euler polynomials. Axioms 2012-12-07 1 3 Article 10.3390/axioms1030395 395 403 2075-1680 2012-12-07 doi: 10.3390/axioms1030395 Yilmaz Simsek http://www.mdpi.com/2075-1680/1/3/384 This research work presents original properties of the equilibrium critical (ideal) points sets for an important class of generalized dynamical systems. The existence and significant results regarding such points are specified. Strong connections with the Vector Optimization by the Efficiency and the Potential Theory together with its applications following Choquet’s boundaries are provided. Axioms 2012-12-06 1 3 Article 10.3390/axioms1030384 384 394 2075-1680 2012-12-06 doi: 10.3390/axioms1030384 Vasile Postolică http://www.mdpi.com/2075-1680/1/3/372 It is well known that the number of 5-core partitions of 5kn + 5k − 1 is a multiple of 5k. In [1] a statistic called a crank was developed to sort the 5-core partitions of 5n + 4 and 25n + 24 into 5 and 25 classes of equal size, respectively. In this paper we will develop the cranks that can be used to sort the 5-core partitions of 5kn + 5k − 1 into 5k classes of equal size. Axioms 2012-12-03 1 3 Article 10.3390/axioms1030372 372 383 2075-1680 2012-12-03 doi: 10.3390/axioms1030372 Louis Kolitsch http://www.mdpi.com/2075-1680/1/3/365 By applying a classical method, already employed by Cauchy, to a terminating curious summation by one of the authors, a new curious bilateral q-series identity is derived. We also apply the same method to a quadratic summation by Gessel and Stanton, and to a cubic summation by Gasper, respectively, to derive a bilateral quadratic and a bilateral cubic summation formula. Axioms 2012-10-31 1 3 Article 10.3390/axioms1030365 365 371 2075-1680 2012-10-31 doi: 10.3390/axioms1030365 Frédéric Jouhet Michael J. Schlosser http://www.mdpi.com/2075-1680/1/3/324 We introduce the Frobenius–Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius–Schur theorem including that for semisimple quasi-Hopf algebras, weak Hopf C*-algebras and association schemes. Our framework also clarifies a mechanism of how the “twisted” theory arises from the ordinary case. As a demonstration, we establish twisted versions of the Frobenius–Schur theorem for various algebraic objects. We also give several applications to the quantum SL2. Axioms 2012-10-23 1 3 Article 10.3390/axioms1030324 324 364 2075-1680 2012-10-23 doi: 10.3390/axioms1030324 Kenichi Shimizu http://www.mdpi.com/2075-1680/1/3/291 We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid—the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest an explicit connection to new higher isomorphisms arising from incidence geometries, which are solutions of the Zamolodchikov tetrahedron equation. This paper is expository in style and is meant as a companion to Higher Dimensional Algebra VII: Groupoidification and an exploration of structures arising in the work in progress, Higher Dimensional Algebra VIII: The Hecke Bicategory, which introduces the Hecke bicategory in detail. Axioms 2012-10-09 1 3 Communication 10.3390/axioms1030291 291 323 2075-1680 2012-10-09 doi: 10.3390/axioms1030291 Alexander E. Hoffnung http://www.mdpi.com/2075-1680/1/3/259 Motivated by the orthogonality relations for irreducible characters of a finite group, we evaluate the sum of a finite group of linear characters of a Hopf algebra, at all grouplike and skew-primitive elements. We then discuss results for products of skew-primitive elements. Examples include groups, (quantum groups over) Lie algebras, the small quantum groups of Lusztig, and their variations (by Andruskiewitsch and Schneider). Axioms 2012-10-05 1 3 Article 10.3390/axioms1030259 259 290 2075-1680 2012-10-05 doi: 10.3390/axioms1030259 Apoorva Khare http://www.mdpi.com/2075-1680/1/3/238 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. Axioms 2012-10-05 1 3 Article 10.3390/axioms1030238 238 258 2075-1680 2012-10-05 doi: 10.3390/axioms1030238 H. M. Srivastava Rakesh K. Parmar Purnima Chopra http://www.mdpi.com/2075-1680/1/2/226 Since the advent of Drinfel’d’s double construction, Hopf algebraic structures have been a centrepiece for many developments in the theory and analysis of integrable quantum systems. An integrable anyonic pairing Hamiltonian will be shown to admit Hopf algebra symmetries for particular values of its coupling parameters. While the integrable structure of the model relates to the well-known six-vertex solution of the Yang–Baxter equation, the Hopf algebra symmetries are not in terms of the quantum algebra Uq(sl(2)). Rather, they are associated with the Drinfel’d doubles of dihedral group algebras D(Dn). Axioms 2012-09-20 1 2 Article 10.3390/axioms1020226 226 237 2075-1680 2012-09-20 doi: 10.3390/axioms1020226 Jon Links http://www.mdpi.com/2075-1680/1/2/201 The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed. Principal U(1)-bundles over quantum real weighted projective spaces are constructed. As the spaces in question fall into two separate classes, the negative or odd class that generalises quantum real projective planes and the positive or even class that generalises the quantum disc, so do the constructed principal bundles. In the negative case the principal bundle is proven to be non-trivial and associated projective modules are described. In the positive case the principal bundles turn out to be trivial, and so all the associated modules are free. It is also shown that the circle (co)actions on the quantum Seifert manifold that define quantum real weighted projective spaces are almost free. Axioms 2012-09-17 1 2 Article 10.3390/axioms1020201 201 225 2075-1680 2012-09-17 doi: 10.3390/axioms1020201 Tomasz Brzeziński Simon A. Fairfax http://www.mdpi.com/2075-1680/1/2/186 Using the most elementary methods and considerations, the solution of the star-triangle condition (a2+b2-c2)/2ab = ((a’)^2+(b’)^2-(c’))^2/2a’b’ is shown to be a necessary condition for the extension of the operator coalgebra of the six-vertex model to a bialgebra. A portion of the bialgebra acts as a spectrum-generating algebra for the algebraic Bethe ansatz, with which higher-dimensional representations of the bialgebra can be constructed. The star-triangle relation is proved to be necessary for the commutativity of the transfer matrices T(a, b, c) and T(a’, b’, c’). Axioms 2012-08-27 1 2 Article 10.3390/axioms1020186 186 200 2075-1680 2012-08-27 doi: 10.3390/axioms1020186 Jeffrey R. Schmidt http://www.mdpi.com/2075-1680/1/2/173 We study the duality between corings and ring extensions. We construct a new category with a self-dual functor acting on it, which extends that duality. This construction can be seen as the non-commutative case of another duality extension: the duality between finite dimensional algebras and coalgebra. Both these duality extensions have some similarities with the Pontryagin-van Kampen duality theorem. Axioms 2012-08-10 1 2 Article 10.3390/axioms1020173 173 185 2075-1680 2012-08-10 doi: 10.3390/axioms1020173 Florin F. Nichita Bartosz Zielinski http://www.mdpi.com/2075-1680/1/2/155 In computer science the Myhill–Nerode Theorem states that a set L of words in a finite alphabet is accepted by a finite automaton if and only if the equivalence relation ∼L, defined as x ∼L y if and only if xz ∈ L exactly when yz ∈ L, ∀z, has finite index. The Myhill–Nerode Theorem can be generalized to an algebraic setting giving rise to a collection of bialgebras which we call Myhill–Nerode bialgebras. In this paper we investigate the quasitriangular structure of Myhill–Nerode bialgebras. Axioms 2012-07-24 1 2 Article 10.3390/axioms1020155 155 172 2075-1680 2012-07-24 doi: 10.3390/axioms1020155 Robert G. Underwood http://www.mdpi.com/2075-1680/1/2/149 Let NSymm be the Hopf algebra of non-commutative symmetric functions (in an infinity of indeterminates): . It is shown that an associative algebra A with a Hasse-Schmidt derivation ) on it is exactly the same as an NSymm module algebra. The primitives of NSymm act as ordinary derivations. There are many formulas for the generators in terms of the primitives (and vice-versa). This leads to formulas for the higher derivations in a Hasse-Schmidt derivation in terms of ordinary derivations, such as the known formulas of Heerema and Mirzavaziri (and also formulas for ordinary derivations in terms of the elements of a Hasse-Schmidt derivation). These formulas are over the rationals; no such formulas are possible over the integers. Many more formulas are derivable. Axioms 2012-07-16 1 2 Communication 10.3390/axioms1020149 149 154 2075-1680 2012-07-16 doi: 10.3390/axioms1020149 Michiel Hazewinkel http://www.mdpi.com/2075-1680/1/2/111 Quivers (directed graphs), species (a generalization of quivers) and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this paper, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field). Namely, we show that the category of K -species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K -species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver. Axioms 2012-07-04 1 2 Article 10.3390/axioms1020111 111 148 2075-1680 2012-07-04 doi: 10.3390/axioms1020111 Joel Lemay http://www.mdpi.com/2075-1680/1/2/99 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. Axioms 2012-07-04 1 2 Article 10.3390/axioms1020099 99 110 2075-1680 2012-07-04 doi: 10.3390/axioms1020099 Emil Saucan Meir Katchalski http://www.mdpi.com/2075-1680/1/1/74 A research proposal on the algebraic structure, the representations and the possible applications of paraparticle algebras is structured in three modules: The first part stems from an attempt to classify the inequivalent gradings and braided group structures present in the various parastatistical algebraic models. The second part of the proposal aims at refining and utilizing a previously published methodology for the study of the Fock-like representations of the parabosonic algebra, in such a way that it can also be directly applied to the other parastatistics algebras. Finally, in the third part, a couple of Hamiltonians is proposed, suitable for modeling the radiation matter interaction via a parastatistical algebraic model. Axioms 2012-06-15 1 1 Communication 10.3390/axioms1010074 74 98 2075-1680 2012-06-15 doi: 10.3390/axioms1010074 Konstantinos Kanakoglou http://www.mdpi.com/2075-1680/1/1/38 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. Axioms 2012-06-15 1 1 Article 10.3390/axioms1010038 38 73 2075-1680 2012-06-15 doi: 10.3390/axioms1010038 Kevin H. Knuth John Skilling 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. Axioms 2012-04-26 1 1 Communication 10.3390/axioms1010033 33 37 2075-1680 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. Axioms 2012-04-20 1 1 Article 10.3390/axioms1010021 21 32 2075-1680 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. Axioms 2012-04-20 1 1 Article 10.3390/axioms1010009 9 20 2075-1680 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. Axioms 2012-03-21 1 1 Communication 10.3390/axioms1010004 4 8 2075-1680 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. [...] Axioms 2011-09-01 1 1 Editorial 10.3390/axioms1010001 1 3 2075-1680 2011-09-01 doi: 10.3390/axioms1010001 Angel Garrido