Axioms
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Latest open access articles published in Axioms at http://www.mdpi.com/journal/axioms<![CDATA[Axioms, Vol. 6, Pages 13: No Uncountable Polish Group Can be a Right-Angled Artin Group]]>
http://www.mdpi.com/2075-1680/6/2/13
We prove that if G is a Polish group and A a group admitting a system of generators whose associated length function satisfies: (i) if 0 &lt; k &lt; ω , then l g ( x ) ≤ l g ( x k ) ; (ii) if l g ( y ) &lt; k &lt; ω and x k = y , then x = e , then there exists a subgroup G * of G of size b (the bounding number) such that G * is not embeddable in A. In particular, we prove that the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes analogous results for free and free abelian uncountable groups.Axioms2017-05-1162Article10.3390/axioms6020013132075-16802017-05-11doi: 10.3390/axioms6020013Gianluca PaoliniSaharon Shelah<![CDATA[Axioms, Vol. 6, Pages 12: Toward Measuring Network Aesthetics Based on Symmetry]]>
http://www.mdpi.com/2075-1680/6/2/12
In this exploratory paper, we discuss quantitative graph-theoretical measures of network aesthetics. Related work in this area has typically focused on geometrical features (e.g., line crossings or edge bendiness) of drawings or visual representations of graphs which purportedly affect an observer’s perception. Here we take a very different approach, abandoning reliance on geometrical properties, and apply information-theoretic measures to abstract graphs and networks directly (rather than to their visual representaions) as a means of capturing classical appreciation of structural symmetry. Examples are used solely to motivate the approach to measurement, and to elucidate our symmetry-based mathematical theory of network aesthetics.Axioms2017-05-0662Article10.3390/axioms6020012122075-16802017-05-06doi: 10.3390/axioms6020012Zengqiang ChenMatthias DehmerFrank Emmert-StreibAbbe MowshowitzYongtang Shi<![CDATA[Axioms, Vol. 6, Pages 10: Orientation Asymmetric Surface Model for Membranes: Finsler Geometry Modeling]]>
http://www.mdpi.com/2075-1680/6/2/10
We study triangulated surface models with nontrivial surface metrices for membranes. The surface model is defined by a mapping r from a two-dimensional parameter space M to the three-dimensional Euclidean space R 3 . The metric variable g a b , which is always fixed to the Euclidean metric δ a b , can be extended to a more general non-Euclidean metric on M in the continuous model. The problem we focus on in this paper is whether such an extension is well defined or not in the discrete model. We find that a discrete surface model with a nontrivial metric becomes well defined if it is treated in the context of Finsler geometry (FG) modeling, where triangle edge length in M depends on the direction. It is also shown that the discrete FG model is orientation asymmetric on invertible surfaces in general, and for this reason, the FG model has a potential advantage for describing real physical membranes, which are expected to have some asymmetries for orientation-changing transformations.Axioms2017-04-2562Article10.3390/axioms6020010102075-16802017-04-25doi: 10.3390/axioms6020010Evgenii ProutorovHiroshi Koibuchi<![CDATA[Axioms, Vol. 6, Pages 11: Multivariate Extended Gamma Distribution]]>
http://www.mdpi.com/2075-1680/6/2/11
In this paper, I consider multivariate analogues of the extended gamma density, which will provide multivariate extensions to Tsallis statistics and superstatistics. By making use of the pathway parameter β , multivariate generalized gamma density can be obtained from the model considered here. Some of its special cases and limiting cases are also mentioned. Conditional density, best predictor function, regression theory, etc., connected with this model are also introduced.Axioms2017-04-2462Article10.3390/axioms6020011112075-16802017-04-24doi: 10.3390/axioms6020011Dhannya Joseph<![CDATA[Axioms, Vol. 6, Pages 9: Euclidean Algorithm for Extension of Symmetric Laurent Polynomial Matrix and Its Application in Construction of Multiband Symmetric Perfect Reconstruction Filter Bank]]>
http://www.mdpi.com/2075-1680/6/2/9
For a given pair of s-dimensional real Laurent polynomials ( a → ( z ) , b → ( z ) ) , which has a certain type of symmetry and satisfies the dual condition b → ( z ) T a → ( z ) = 1 , an s × s Laurent polynomial matrix A ( z ) (together with its inverse A - 1 ( z ) ) is called a symmetric Laurent polynomial matrix extension of the dual pair ( a → ( z ) , b → ( z ) ) if A ( z ) has similar symmetry, the inverse A - 1 ( Z ) also is a Laurent polynomial matrix, the first column of A ( z ) is a → ( z ) and the first row of A - 1 ( z ) is ( b → ( z ) ) T . In this paper, we introduce the Euclidean symmetric division and the symmetric elementary matrices in the Laurent polynomial ring and reveal their relation. Based on the Euclidean symmetric division algorithm in the Laurent polynomial ring, we develop a novel and effective algorithm for symmetric Laurent polynomial matrix extension. We also apply the algorithm in the construction of multi-band symmetric perfect reconstruction filter banks.Axioms2017-04-2062Article10.3390/axioms602000992075-16802017-04-20doi: 10.3390/axioms6020009Jianzhong Wang<![CDATA[Axioms, Vol. 6, Pages 8: Expansion of the Kullback-Leibler Divergence, and a New Class of Information Metrics]]>
http://www.mdpi.com/2075-1680/6/2/8
Inferring and comparing complex, multivariable probability density functions is fundamental to problems in several fields, including probabilistic learning, network theory, and data analysis. Classification and prediction are the two faces of this class of problem. This study takes an approach that simplifies many aspects of these problems by presenting a structured, series expansion of the Kullback-Leibler divergence—a function central to information theory—and devise a distance metric based on this divergence. Using the Möbius inversion duality between multivariable entropies and multivariable interaction information, we express the divergence as an additive series in the number of interacting variables, which provides a restricted and simplified set of distributions to use as approximation and with which to model data. Truncations of this series yield approximations based on the number of interacting variables. The first few terms of the expansion-truncation are illustrated and shown to lead naturally to familiar approximations, including the well-known Kirkwood superposition approximation. Truncation can also induce a simple relation between the multi-information and the interaction information. A measure of distance between distributions, based on Kullback-Leibler divergence, is then described and shown to be a true metric if properly restricted. The expansion is shown to generate a hierarchy of metrics and connects this work to information geometry formalisms. An example of the application of these metrics to a graph comparison problem is given that shows that the formalism can be applied to a wide range of network problems and provides a general approach for systematic approximations in numbers of interactions or connections, as well as a related quantitative metric.Axioms2017-04-0162Article10.3390/axioms602000882075-16802017-04-01doi: 10.3390/axioms6020008David GalasGregory DeweyJames Kunert-GrafNikita Sakhanenko<![CDATA[Axioms, Vol. 6, Pages 7: Fourier Series for Singular Measures]]>
http://www.mdpi.com/2075-1680/6/2/7
Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure μ on [ 0 , 1 ) , every f ∈ L 2 ( μ ) possesses a Fourier series of the form f ( x ) = ∑ n = 0 ∞ c n e 2 π i n x . We show that the coefficients c n can be computed in terms of the quantities f ^ ( n ) = ∫ 0 1 f ( x ) e − 2 π i n x d μ ( x ) . We also demonstrate a Shannon-type sampling theorem for functions that are in a sense μ -bandlimited.Axioms2017-03-2862Article10.3390/axioms602000772075-16802017-03-28doi: 10.3390/axioms6020007John HerrEric Weber<![CDATA[Axioms, Vol. 6, Pages 6: Norm Retrieval and Phase Retrieval by Projections]]>
http://www.mdpi.com/2075-1680/6/1/6
We make a detailed study of norm retrieval. We give several classification theorems for norm retrieval and give a large number of examples to go with the theory. One consequence is a new result about Parseval frames: If a Parseval frame is divided into two subsets with spans W 1 , W 2 and W 1 ∩ W 2 = { 0 } , then W 1 ⊥ W 2 .Axioms2017-03-0461Article10.3390/axioms601000662075-16802017-03-04doi: 10.3390/axioms6010006Peter CasazzaDorsa GhoreishiShani JoseJanet Tremain<![CDATA[Axioms, Vol. 6, Pages 5: Kullback-Leibler Divergence and Mutual Information of Experiments in the Fuzzy Case]]>
http://www.mdpi.com/2075-1680/6/1/5
The main aim of this contribution is to define the notions of Kullback-Leibler divergence and conditional mutual information in fuzzy probability spaces and to derive the basic properties of the suggested measures. In particular, chain rules for mutual information of fuzzy partitions and for Kullback-Leibler divergence with respect to fuzzy P-measures are established. In addition, a convexity of Kullback-Leibler divergence and mutual information with respect to fuzzy P-measures is studied.Axioms2017-03-0361Article10.3390/axioms601000552075-16802017-03-03doi: 10.3390/axioms6010005Dagmar Markechová<![CDATA[Axioms, Vol. 6, Pages 4: Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefﬁcients]]>
http://www.mdpi.com/2075-1680/6/1/4
We propose a construction of a Hermite cubic spline-wavelet basis on the interval and hypercube. The basis is adapted to homogeneous Dirichlet boundary conditions. The wavelets are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Therefore, the wavelets have eight vanishing moments, and the matrices arising from discretization of differential equations with coefﬁcients that are piecewise polynomials of degree at most four on uniform grids are sparse. Numerical examples demonstrate the efﬁciency of an adaptive wavelet method with the constructed wavelet basis for solving the one-dimensional elliptic equation and the two-dimensional Black–Scholes equation with a quadratic volatility.Axioms2017-02-2261Article10.3390/axioms601000442075-16802017-02-22doi: 10.3390/axioms6010004Dana ČernáVáclav Finĕk<![CDATA[Axioms, Vol. 6, Pages 3: Discrete Frames on Finite Dimensional Left Quaternion Hilbert Spaces]]>
http://www.mdpi.com/2075-1680/6/1/3
An introductory theory of frames on finite dimensional left quaternion Hilbert spaces is demonstrated along the lines of their complex counterpart.Axioms2017-02-2161Article10.3390/axioms601000332075-16802017-02-21doi: 10.3390/axioms6010003M. KhokulanK. ThirulogasantharS. Srisatkunarajah<![CDATA[Axioms, Vol. 6, Pages 2: Acknowledgement to Reviewers of Axioms in 2016]]>
http://www.mdpi.com/2075-1680/6/1/2
n/aAxioms2017-01-1161Editorial10.3390/axioms601000222075-16802017-01-11doi: 10.3390/axioms6010002 Axioms Editorial Office<![CDATA[Axioms, Vol. 6, Pages 1: Cuntz Semigroups of Compact-Type Hopf C*-Algebras]]>
http://www.mdpi.com/2075-1680/6/1/1
The classical Cuntz semigroup has an important role in the study of C*-algebras, being one of the main invariants used to classify recalcitrant C*-algebras up to isomorphism. We consider C*-algebras that have Hopf algebra structure, and find additional structure in their Cuntz semigroups. We show that in many cases, isomorphisms of Cuntz semigroups that respect this additional structure can be lifted to Hopf algebra (bi)isomorphisms, up to a possible flip of the co-product. This shows that the Cuntz semigroup provides an interesting invariant of C*-algebraic quantum groups.Axioms2017-01-0461Article10.3390/axioms601000112075-16802017-01-04doi: 10.3390/axioms6010001Dan Kučerovský<![CDATA[Axioms, Vol. 5, Pages 29: Operational Solution of Non-Integer Ordinary and Evolution-Type Partial Differential Equations]]>
http://www.mdpi.com/2075-1680/5/4/29
A method for the solution of linear differential equations (DE) of non-integer order and of partial differential equations (PDE) by means of inverse differential operators is proposed. The solutions of non-integer order ordinary differential equations are obtained with recourse to the integral transforms and the exponent operators. The generalized forms of Laguerre and Hermite orthogonal polynomials as members of more general Appèl polynomial family are used to find the solutions. Operational definitions of these polynomials are used in the context of the operational approach. Special functions are employed to write solutions of DE in convolution form. Some linear partial differential equations (PDE) are also explored by the operational method. The Schrödinger and the Black–Scholes-like evolution equations and solved with the help of the operational technique. Examples of the solution of DE of non-integer order and of PDE are considered with various initial functions, such as polynomial, exponential, and their combinations.Axioms2016-12-1354Article10.3390/axioms5040029292075-16802016-12-13doi: 10.3390/axioms5040029Konstantin ZhukovskyHari Srivastava<![CDATA[Axioms, Vol. 5, Pages 28: Operational Approach and Solutions of Hyperbolic Heat Conduction Equations]]>
http://www.mdpi.com/2075-1680/5/4/28
We studied physical problems related to heat transport and the corresponding differential equations, which describe a wider range of physical processes. The operational method was employed to construct particular solutions for them. Inverse differential operators and operational exponent as well as operational definitions and operational rules for generalized orthogonal polynomials were used together with integral transforms and special functions. Examples of an electric charge in a constant electric field passing under a potential barrier and of heat diffusion were compared and explored in two dimensions. Non-Fourier heat propagation models were studied and compared with each other and with Fourier heat transfer. Exact analytical solutions for the hyperbolic heat equation and for its extensions were explored. The exact analytical solution for the Guyer-Krumhansl type heat equation was derived. Using the latter, the heat surge propagation and relaxation was studied for the Guyer-Krumhansl heat transport model, for the Cattaneo and for the Fourier models. The comparison between them was drawn. Space-time propagation of a power–exponential function and of a periodic signal, obeying the Fourier law, the hyperbolic heat equation and its extended Guyer-Krumhansl form were studied by the operational technique. The role of various terms in the equations was explored and their influence on the solutions demonstrated. The accordance of the solutions with maximum principle is discussed. The application of our theoretical study for heat propagation in thin films is considered. The examples of the relaxation of the initial laser flash, the wide heat spot, and the harmonic function are considered and solved analytically.Axioms2016-12-1254Article10.3390/axioms5040028282075-16802016-12-12doi: 10.3390/axioms5040028Konstantin Zhukovsky<![CDATA[Axioms, Vol. 5, Pages 27: Discrete Geometry—From Theory to Applications: A Case Study]]>
http://www.mdpi.com/2075-1680/5/4/27
n/aAxioms2016-12-0954Editorial10.3390/axioms5040027272075-16802016-12-09doi: 10.3390/axioms5040027David GuEmil Saucan<![CDATA[Axioms, Vol. 5, Pages 26: Forman-Ricci Flow for Change Detection in Large Dynamic Data Sets]]>
http://www.mdpi.com/2075-1680/5/4/26
We present a viable geometric solution for the detection of dynamic effects in complex networks. Building on Forman’s discretization of the classical notion of Ricci curvature, we introduce a novel geometric method to characterize different types of real-world networks with an emphasis on peer-to-peer networks. We study the classical Ricci-flow in a network-theoretic setting and introduce an analytic tool for characterizing dynamic effects. The formalism suggests a computational method for change detection and the identification of fast evolving network regions and yields insights into topological properties and the structure of the underlying data.Axioms2016-11-1054Article10.3390/axioms5040026262075-16802016-11-10doi: 10.3390/axioms5040026Melanie WeberJürgen JostEmil Saucan<![CDATA[Axioms, Vol. 5, Pages 25: Quantum Quasigroups and the Quantum Yang–Baxter Equation]]>
http://www.mdpi.com/2075-1680/5/4/25
Quantum quasigroups are algebraic structures providing a general self-dual framework for the nonassociative extension of Hopf algebra techniques. They also have one-sided analogues, which are not self-dual. The paper presents a survey of recent work on these structures, showing how they furnish various solutions to the quantum Yang–Baxter equation.Axioms2016-11-0954Article10.3390/axioms5040025252075-16802016-11-09doi: 10.3390/axioms5040025Jonathan Smith<![CDATA[Axioms, Vol. 5, Pages 23: The Universe in Leśniewski’s Mereology: Some Comments on Sobociński’s Reflections]]>
http://www.mdpi.com/2075-1680/5/3/23
Stanisław Leśniewski’s mereology was originally conceived as a theory of foundations of mathematics and it is also for this reason that it has philosophical connotations. The ‘philosophical significance’ of mereology was upheld by Bolesław Sobociński who expressed the view in his correspondence with J.M. Bocheński. As he wrote to Bocheński in 1948: “[...] it is interesting that, being such a simple deductive theory, mereology may prove a number of very general theses reminiscent of metaphysical ontology”. The theses which Sobociński had in mind were related to the mereological notion of “the Universe”. Sobociński listed them in the letter adding his philosophical commentary but he did not give proofs for them and did not specify precisely the theory lying behind them. This is what we want to supply in the first part of our paper. We indicate some connections between the notion of the universe and other specific mereological notions. Motivated by Sobociński’s informal suggestions showing his preference for mereology over the axiomatic set theory in application to philosophy we propose to consider Sobociński’s formalism in a new frame which is the ZFM theory—an extension of Zermelo-Fraenkel set theory by mereological axioms, developed by A. Pietruszczak. In this systematic part we investigate reasons of ’philosophical hopes’ mentioned by Sobociński, pinned on the mereological concept of “the Universe”.Axioms2016-09-0653Article10.3390/axioms5030023232075-16802016-09-06doi: 10.3390/axioms5030023Marcin ŁyczakMarek PorwolikKordula Świętorzecka<![CDATA[Axioms, Vol. 5, Pages 24: On the q-Laplace Transform and Related Special Functions]]>
http://www.mdpi.com/2075-1680/5/3/24
Motivated by statistical mechanics contexts, we study the properties of the q-Laplace transform, which is an extension of the well-known Laplace transform. In many circumstances, the kernel function to evaluate certain integral forms has been studied. In this article, we establish relationships between q-exponential and other well-known functional forms, such as Mittag–Leffler functions, hypergeometric and H-function, by means of the kernel function of the integral. Traditionally, we have been applying the Laplace transform method to solve differential equations and boundary value problems. Here, we propose an alternative, the q-Laplace transform method, to solve differential equations, such as as the fractional space-time diffusion equation, the generalized kinetic equation and the time fractional heat equation.Axioms2016-09-0653Article10.3390/axioms5030024242075-16802016-09-06doi: 10.3390/axioms5030024Shanoja NaikHans Haubold<![CDATA[Axioms, Vol. 5, Pages 22: A Method for Ordering of LR-Type Fuzzy Numbers: An Important Decision Criteria]]>
http://www.mdpi.com/2075-1680/5/3/22
Methods for ordering fuzzy numbers play an important role as decision criteria, with applications in areas such as optimization and data mining, among others. Although there are several proposals for ordering methods in the fuzzy literature, many of them are difficult to apply and present some problems with ranking computation. For that reason, this work proposes an ordering method for fuzzy numbers based on a simple application of a polynomial function. We study some properties of our new method, comparing our results with those generated by other methods previously discussed in literature.Axioms2016-08-3153Review10.3390/axioms5030022222075-16802016-08-31doi: 10.3390/axioms5030022José González CamposRonald Manríquez Peñafiel<![CDATA[Axioms, Vol. 5, Pages 21: Is Kazimierz Ajdukiewicz’s Concept of a Real Definition Still Important?]]>
http://www.mdpi.com/2075-1680/5/3/21
The concept of a real definition worked out by Kazimierz Ajdukiewicz is still important in the theory of definition and can be developed by applying Hilary Putnam’s theory of reference of natural kind terms and Karl Popper’s fallibilism. On the one hand, the definiendum of a real definition refers to a natural kind of things and, on the other hand, the definiens of such a definition expresses actual, empirical, fallible knowledge which can be revised and changed.Axioms2016-08-1753Article10.3390/axioms5030021212075-16802016-08-17doi: 10.3390/axioms5030021Robert Kublikowski<![CDATA[Axioms, Vol. 5, Pages 20: Approach of Complexity in Nature: Entropic Nonuniqueness]]>
http://www.mdpi.com/2075-1680/5/3/20
Boltzmann introduced in the 1870s a logarithmic measure for the connection between the thermodynamical entropy and the probabilities of the microscopic configurations of the system. His celebrated entropic functional for classical systems was then extended by Gibbs to the entire phase space of a many-body system and by von Neumann in order to cover quantum systems, as well. Finally, it was used by Shannon within the theory of information. The simplest expression of this functional corresponds to a discrete set of W microscopic possibilities and is given by S B G = − k ∑ i = 1 W p i ln p i (k is a positive universal constant; BG stands for Boltzmann–Gibbs). This relation enables the construction of BGstatistical mechanics, which, together with the Maxwell equations and classical, quantum and relativistic mechanics, constitutes one of the pillars of contemporary physics. The BG theory has provided uncountable important applications in physics, chemistry, computational sciences, economics, biology, networks and others. As argued in the textbooks, its application in physical systems is legitimate whenever the hypothesis of ergodicity is satisfied, i.e., when ensemble and time averages coincide. However, what can we do when ergodicity and similar simple hypotheses are violated, which indeed happens in very many natural, artificial and social complex systems. The possibility of generalizing BG statistical mechanics through a family of non-additive entropies was advanced in 1988, namely S q = k 1 − ∑ i = 1 W p i q q − 1 , which recovers the additive S B G entropy in the q→ 1 limit. The index q is to be determined from mechanical first principles, corresponding to complexity universality classes. Along three decades, this idea intensively evolved world-wide (see the Bibliography in http://tsallis.cat.cbpf.br/biblio.htm) and led to a plethora of predictions, verifications and applications in physical systems and elsewhere. As expected, whenever a paradigm shift is explored, some controversy naturally emerged, as well, in the community. The present status of the general picture is here described, starting from its dynamical and thermodynamical foundations and ending with its most recent physical applications.Axioms2016-08-1253Review10.3390/axioms5030020202075-16802016-08-12doi: 10.3390/axioms5030020Constantino Tsallis<![CDATA[Axioms, Vol. 5, Pages 19: A Logical Analysis of Existential Dependence and Some Other Ontological Concepts—A Comment to Some Ideas of Eugenia Ginsberg-Blaustein]]>
http://www.mdpi.com/2075-1680/5/3/19
This paper deals with several problems concerning notion of existential dependence and ontological notions of existence, necessity and fusion. Following some ideas of Eugenia Ginsberg-Blaustein, the notions are treated in reference to objects, in relation to the concepts of state of affairs and subject of state of affairs. It provides an axiomatic characterization of these concepts within the framework of a multi-modal propositional logic and then presents a semantic analysis of these concepts. The semantics are a slight modification to the standard relational semantics for normal modal propositional logic.Axioms2016-07-1553Article10.3390/axioms5030019192075-16802016-07-15doi: 10.3390/axioms5030019Marek Magdziak<![CDATA[Axioms, Vol. 5, Pages 18: Potential Infinity, Abstraction Principles and Arithmetic (Leśniewski Style)]]>
http://www.mdpi.com/2075-1680/5/2/18
This paper starts with an explanation of how the logicist research program can be approached within the framework of Leśniewski’s systems. One nice feature of the system is that Hume’s Principle is derivable in it from an explicit definition of natural numbers. I generalize this result to show that all predicative abstraction principles corresponding to second-level relations, which are provably equivalence relations, are provable. However, the system fails, despite being much neater than the construction of Principia Mathematica (PM). One of the key reasons is that, just as in the case of the system of PM, without the assumption that infinitely many objects exist, (renderings of) most of the standard axioms of Peano Arithmetic are not derivable in the system. I prove that introducing modal quantifiers meant to capture the intuitions behind potential infinity results in the (renderings of) axioms of Peano Arithmetic (PA) being valid in all relational models (i.e. Kripke-style models, to be defined later on) of the extended language. The second, historical part of the paper contains a user-friendly description of Leśniewski’s own arithmetic and a brief investigation into its properties.Axioms2016-06-1552Article10.3390/axioms5020018182075-16802016-06-15doi: 10.3390/axioms5020018Rafal Urbaniak<![CDATA[Axioms, Vol. 5, Pages 17: An Overview of the Fuzzy Axiomatic Systems and Characterizations Proposed at Ghent University]]>
http://www.mdpi.com/2075-1680/5/2/17
During the past 40 years of fuzzy research at the Fuzziness and Uncertainty Modeling research unit of Ghent University several axiomatic systems and characterizations have been introduced. In this paper we highlight some of them. The main purpose of this paper consists of an invitation to continue research on these first attempts to axiomatize important concepts and systems in fuzzy set theory. Currently, these attempts are spread over many journals; with this paper they are now collected in a neat overview. In the literature, many axiom systems have been introduced, but as far as we know the axiomatic system of Huntington concerning a Boolean algebra has been the only one where the axioms have been proven independent. Another line of further research could be with respect to the simplification of these systems, in discovering redundancies between the axioms.Axioms2016-06-0752Article10.3390/axioms5020017172075-16802016-06-07doi: 10.3390/axioms5020017Etienne KerreLynn D´eerBart Van Gasse<![CDATA[Axioms, Vol. 5, Pages 15: On the Mutual Definability of the Notions of Entailment, Rejection, and Inconsistency]]>
http://www.mdpi.com/2075-1680/5/2/15
In this paper, two axiomatic theories T− and T′ are constructed, which are dual to Tarski’s theory T+ (1930) of deductive systems based on classical propositional calculus. While in Tarski’s theory T+ the primitive notion is the classical consequence function (entailment) Cn+, in the dual theory T− it is replaced by the notion of Słupecki’s rejection consequence Cn− and in the dual theory T′ it is replaced by the notion of the family Incons of inconsistent sets. The author has proved that the theories T+, T−, and T′ are equivalent.Axioms2016-06-0752Article10.3390/axioms5020015152075-16802016-06-07doi: 10.3390/axioms5020015Urszula Wybraniec-Skardowska<![CDATA[Axioms, Vol. 5, Pages 16: Contribution of Warsaw Logicians to Computational Logic]]>
http://www.mdpi.com/2075-1680/5/2/16
The newly emerging branch of research of Computer Science received encouragement from the successors of the Warsaw mathematical school: Kuratowski, Mazur, Mostowski, Grzegorczyk, and Rasiowa. Rasiowa realized very early that the spectrum of computer programs should be incorporated into the realm of mathematical logic in order to make a rigorous treatment of program correctness. This gave rise to the concept of algorithmic logic developed since the 1970s by Rasiowa, Salwicki, Mirkowska, and their followers. Together with Pratt’s dynamic logic, algorithmic logic evolved into a mainstream branch of research: logic of programs. In the late 1980s, Warsaw logicians Tiuryn and Urzyczyn categorized various logics of programs, depending on the class of programs involved. Quite unexpectedly, they discovered that some persistent open questions about the expressive power of logics are equivalent to famous open problems in complexity theory. This, along with parallel discoveries by Harel, Immerman and Vardi, contributed to the creation of an important area of theoretical computer science: descriptive complexity. By that time, the modal μ-calculus was recognized as a sort of a universal logic of programs. The mid 1990s saw a landmark result by Walukiewicz, who showed completeness of a natural axiomatization for the μ-calculus proposed by Kozen. The difficult proof of this result, based on automata theory, opened a path to further investigations. Later, Bojanczyk opened a new chapter by introducing an unboundedness quantifier, which allowed for expressing some quantitative properties of programs. Yet another topic, linking the past with the future, is the subject of automata founded in the Fraenkel-Mostowski set theory. The studies on intuitionism found their continuation in the studies of Curry-Howard isomorphism. ukasiewicz’s landmark idea of many-valued logic found its continuation in various approaches to incompleteness and uncertainty.Axioms2016-06-0352Article10.3390/axioms5020016162075-16802016-06-03doi: 10.3390/axioms5020016Damian Niwiński<![CDATA[Axioms, Vol. 5, Pages 14: An Axiomatic Account of Question Evocation: The Propositional Case]]>
http://www.mdpi.com/2075-1680/5/2/14
An axiomatic system for question evocation in Classical Propositional Logic is proposed. Soundness and completeness of the system are proven.Axioms2016-05-2652Article10.3390/axioms5020014142075-16802016-05-26doi: 10.3390/axioms5020014Andrzej Wiśniewski<![CDATA[Axioms, Vol. 5, Pages 13: Fundamental Results for Pseudo-Differential Operators of Type 1, 1]]>
http://www.mdpi.com/2075-1680/5/2/13
This paper develops some deeper consequences of an extended definition, proposed previously by the author, of pseudo-differential operators that are of type 1 , 1 in Hörmander’s sense. Thus, it contributes to the long-standing problem of creating a systematic theory of such operators. It is shown that type 1 , 1 -operators are defined and continuous on the full space of temperate distributions, if they fulfil Hörmander’s twisted diagonal condition, or more generally if they belong to the self-adjoint subclass; and that they are always defined on the temperate smooth functions. As a main tool the paradifferential decomposition is derived for type 1 , 1 -operators, and to confirm a natural hypothesis the symmetric term is shown to cause the domain restrictions; whereas the other terms are shown to define nice type 1 , 1 -operators fulfilling the twisted diagonal condition. The decomposition is analysed in the type 1 , 1 -context by combining the Spectral Support Rule and the factorisation inequality, which gives pointwise estimates of pseudo-differential operators in terms of maximal functions.Axioms2016-05-1952Article10.3390/axioms5020013132075-16802016-05-19doi: 10.3390/axioms5020013Jon Johnsen<![CDATA[Axioms, Vol. 5, Pages 12: Infinite-dimensional Lie Algebras, Representations, Hermitian Duality and the Operators of Stochastic Calculus]]>
http://www.mdpi.com/2075-1680/5/2/12
We study densely defined unbounded operators acting between different Hilbert spaces. For these, we introduce a notion of symmetric (closable) pairs of operators. The purpose of our paper is to give applications to selected themes at the cross road of operator commutation relations and stochastic calculus. We study a family of representations of the canonical commutation relations (CCR)-algebra (an infinite number of degrees of freedom), which we call admissible. The family of admissible representations includes the Fock-vacuum representation. We show that, to every admissible representation, there is an associated Gaussian stochastic calculus, and we point out that the case of the Fock-vacuum CCR-representation in a natural way yields the operators of Malliavin calculus. We thus get the operators of Malliavin’s calculus of variation from a more algebraic approach than is common. We further obtain explicit and natural formulas, and rules, for the operators of stochastic calculus. Our approach makes use of a notion of symmetric (closable) pairs of operators. The Fock-vacuum representation yields a maximal symmetric pair. This duality viewpoint has the further advantage that issues with unbounded operators and dense domains can be resolved much easier than what is possible with alternative tools. With the use of CCR representation theory, we also obtain, as a byproduct, a number of new results in multi-variable operator theory which we feel are of independent interest.Axioms2016-05-1752Article10.3390/axioms5020012122075-16802016-05-17doi: 10.3390/axioms5020012Palle JorgensenFeng Tian<![CDATA[Axioms, Vol. 5, Pages 11: An Overview of Topological Groups: Yesterday, Today, Tomorrow]]>
http://www.mdpi.com/2075-1680/5/2/11
n/aAxioms2016-05-0552Editorial10.3390/axioms5020011112075-16802016-05-05doi: 10.3390/axioms5020011Sidney Morris<![CDATA[Axioms, Vol. 5, Pages 10: Applications of Skew Models Using Generalized Logistic Distribution]]>
http://www.mdpi.com/2075-1680/5/2/10
We use the skew distribution generation procedure proposed by Azzalini [Scand. J. Stat., 1985, 12, 171–178] to create three new probability distribution functions. These models make use of normal, student-t and generalized logistic distribution, see Rathie and Swamee [Technical Research Report No. 07/2006. Department of Statistics, University of Brasilia: Brasilia, Brazil, 2006]. Expressions for the moments about origin are derived. Graphical illustrations are also provided. The distributions derived in this paper can be seen as generalizations of the distributions given by Nadarajah and Kotz [Acta Appl. Math., 2006, 91, 1–37]. Applications with unimodal and bimodal data are given to illustrate the applicability of the results derived in this paper. The applications include the analysis of the following data sets: (a) spending on public education in various countries in 2003; (b) total expenditure on health in 2009 in various countries and (c) waiting time between eruptions of the Old Faithful Geyser in the Yellow Stone National Park, Wyoming, USA. We compare the fit of the distributions introduced in this paper with the distributions given by Nadarajah and Kotz [Acta Appl. Math., 2006, 91, 1–37]. The results show that our distributions, in general, fit better the data sets. The general R codes for fitting the distributions introduced in this paper are given in Appendix A.Axioms2016-04-1552Article10.3390/axioms5020010102075-16802016-04-15doi: 10.3390/axioms5020010Pushpa RathiePaulo SilvaGabriela Olinto<![CDATA[Axioms, Vol. 5, Pages 9: The Lvov-Warsaw School and Its Future]]>
http://www.mdpi.com/2075-1680/5/2/9
The Lvov-Warsaw School (L-WS) was the most important movement in the history of Polish philosophy, and certainly prominent in the general history of philosophy, and 20th century logics and mathematics in particular.[...]Axioms2016-04-1152Editorial10.3390/axioms502000992075-16802016-04-11doi: 10.3390/axioms5020009Angel GarridoPiedad Yuste<![CDATA[Axioms, Vol. 5, Pages 8: Summary of Data Farming]]>
http://www.mdpi.com/2075-1680/5/1/8
Data Farming is a process that has been developed to support decision-makers by answering questions that are not currently addressed. Data farming uses an inter-disciplinary approach that includes modeling and simulation, high performance computing, and statistical analysis to examine questions of interest with a large number of alternatives. Data farming allows for the examination of uncertain events with numerous possible outcomes and provides the capability of executing enough experiments so that both overall and unexpected results may be captured and examined for insights. Harnessing the power of data farming to apply it to our questions is essential to providing support not currently available to decision-makers. This support is critically needed in answering questions inherent in the scenarios we expect to confront in the future as the challenges our forces face become more complex and uncertain. This article was created on the basis of work conducted by Task Group MSG-088 “Data Farming in Support of NATO”, which is being applied in MSG-124 “Developing Actionable Data Farming Decision Support for NATO” of the Science and Technology Organization, North Atlantic Treaty Organization (STO NATO).Axioms2016-03-0151Article10.3390/axioms501000882075-16802016-03-01doi: 10.3390/axioms5010008Gary HorneKlaus-Peter Schwierz<![CDATA[Axioms, Vol. 5, Pages 7: Tactical Size Unit as Distribution in a Data Farming Environment]]>
http://www.mdpi.com/2075-1680/5/1/7
In agent based models, the agents are usually platforms (individual soldiers, tanks, helicopters, etc.), not military units. In the Sandis software, the agents can be platoon size units. As there are about 30 soldiers in a platoon, there is a need for strength distribution in simulations. The contribution of this paper is a conceptual model of the platoon level agent, the needed mathematical models and concepts, and references earlier studies of how simulations have been conducted in a data farming environment with platoon/squad size unit agents with strength distribution.Axioms2016-02-2251Article10.3390/axioms501000772075-16802016-02-22doi: 10.3390/axioms5010007Esa LappiBernt Åkesson<![CDATA[Axioms, Vol. 5, Pages 6: Entropy Production Rate of a One-Dimensional Alpha-Fractional Diffusion Process]]>
http://www.mdpi.com/2075-1680/5/1/6
In this paper, the one-dimensional α-fractional diffusion equation is revisited. This equation is a particular case of the time- and space-fractional diffusion equation with the quotient of the orders of the time- and space-fractional derivatives equal to one-half. First, some integral representations of its fundamental solution including the Mellin-Barnes integral representation are derived. Then a series representation and asymptotics of the fundamental solution are discussed. The fundamental solution is interpreted as a probability density function and its entropy in the Shannon sense is calculated. The entropy production rate of the stochastic process governed by the α-fractional diffusion equation is shown to be equal to one of the conventional diffusion equation.Axioms2016-02-0551Article10.3390/axioms501000662075-16802016-02-05doi: 10.3390/axioms5010006Yuri Luchko<![CDATA[Axioms, Vol. 5, Pages 5: Modular Nuclearity: A Generally Covariant Perspective]]>
http://www.mdpi.com/2075-1680/5/1/5
A quantum field theory in its algebraic description may admit many irregular states. So far, selection criteria to distinguish physically reasonable states have been restricted to free fields (Hadamard condition) or to flat spacetimes (e.g., Buchholz-Wichmann nuclearity). We propose instead to use a modular ℓp -condition, which is an extension of a strengthened modular nuclearity condition to generally covariant theories. The modular nuclearity condition was previously introduced in Minkowski space, where it played an important role in constructive two dimensional algebraic QFT’s. We show that our generally covariant extension of this condition makes sense for a vast range of theories, and that it behaves well under causal propagation and taking mixtures. In addition we show that our modular ℓp -condition holds for every quasi-free Hadamard state of a free scalar quantum field (regardless of mass or scalar curvature coupling). However, our condition is not equivalent to the Hadamard condition.Axioms2016-01-2951Article10.3390/axioms501000552075-16802016-01-29doi: 10.3390/axioms5010005Gandalf LechnerKo Sanders<![CDATA[Axioms, Vol. 5, Pages 4: Data Farming Process and Initial Network Analysis Capabilities]]>
http://www.mdpi.com/2075-1680/5/1/4
Data Farming, network applications and approaches to integrate network analysis and processes to the data farming paradigm are presented as approaches to address complex system questions. Data Farming is a quantified approach that examines questions in large possibility spaces using modeling and simulation. It evaluates whole landscapes of outcomes to draw insights from outcome distributions and outliers. Social network analysis and graph theory are widely used techniques for the evaluation of social systems. Incorporation of these techniques into the data farming process provides analysts examining complex systems with a powerful new suite of tools for more fully exploring and understanding the effect of interactions in complex systems. The integration of network analysis with data farming techniques provides modelers with the capability to gain insight into the effect of network attributes, whether the network is explicitly defined or emergent, on the breadth of the model outcome space and the effect of model inputs on the resultant network statistics.Axioms2016-01-2751Article10.3390/axioms501000442075-16802016-01-27doi: 10.3390/axioms5010004Gary HorneTheodore Meyer<![CDATA[Axioms, Vol. 5, Pages 3: Acknowledgement to Reviewers of Axioms in 2015]]>
http://www.mdpi.com/2075-1680/5/1/3
The editors of Axioms would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2015. [...]Axioms2016-01-2551Editorial10.3390/axioms501000332075-16802016-01-25doi: 10.3390/axioms5010003 Axioms Editorial Office<![CDATA[Axioms, Vol. 5, Pages 2: Non-Abelian Pseudocompact Groups]]>
http://www.mdpi.com/2075-1680/5/1/2
Here are three recently-established theorems from the literature. (A) (2006) Every non-metrizable compact abelian group K has 2|K| -many proper dense pseudocompact subgroups. (B) (2003) Every non-metrizable compact abelian group K admits 22|K| -many strictly finer pseudocompact topological group refinements. (C) (2007) Every non-metrizable pseudocompact abelian group has a proper dense pseudocompact subgroup and a strictly finer pseudocompact topological group refinement. (Theorems (A), (B) and (C) become false if the non-metrizable hypothesis is omitted.) With a detailed view toward the relevant literature, the present authors ask: What happens to (A), (B), (C) and to similar known facts about pseudocompact abelian groups if the abelian hypothesis is omitted? Are the resulting statements true, false, true under certain natural additional hypotheses, etc.? Several new results responding in part to these questions are given, and several specific additional questions are posed.Axioms2016-01-1251Article10.3390/axioms501000222075-16802016-01-12doi: 10.3390/axioms5010002W. ComfortDieter Remus<![CDATA[Axioms, Vol. 5, Pages 1: On some Integral Representations of Certain G-Functions]]>
http://www.mdpi.com/2075-1680/5/1/1
This is a brief exposition of some statistical techniques utilized to obtain several useful integral equations involving G-functions.Axioms2015-12-3151Article10.3390/axioms501000112075-16802015-12-31doi: 10.3390/axioms5010001Seemon Thomas<![CDATA[Axioms, Vol. 4, Pages 530-553: An Overview of the Pathway Idea and Its Applications in Statistical and Physical Sciences]]>
http://www.mdpi.com/2075-1680/4/4/530
Pathway idea is a switching mechanism by which one can go from one functional form to another, and to yet another. It is shown that through a parameter α, called the pathway parameter, one can connect generalized type-1 beta family of densities, generalized type-2 beta family of densities, and generalized gamma family of densities, in the scalar as well as the matrix cases, also in the real and complex domains. It is shown that when the model is applied to physical situations then the current hot topics of Tsallis statistics and superstatistics in statistical mechanics become special cases of the pathway model, and the model is capable of capturing many stable situations as well as the unstable or chaotic neighborhoods of the stable situations and transitional stages. The pathway model is shown to be connected to generalized information measures or entropies, power law, likelihood ratio criterion or λ - criterion in multivariate statistical analysis, generalized Dirichlet densities, fractional calculus, Mittag-Leffler stochastic process, Krätzel integral in applied analysis, and many other topics in different disciplines. The pathway model enables one to extend the current results on quadratic and bilinear forms, when the samples come from Gaussian populations, to wider classes of populations.Axioms2015-12-1944Review10.3390/axioms40405305305532075-16802015-12-19doi: 10.3390/axioms4040530Nicy SebastianSeema S. NairDhannya P. Joseph<![CDATA[Axioms, Vol. 4, Pages 518-529: On Limiting Behavior of Contaminant Transport Models in Coupled Surface and Groundwater Flows]]>
http://www.mdpi.com/2075-1680/4/4/518
There has been a surge of work on models for coupling surface-water with groundwater flows which is at its core the Stokes-Darcy problem. The resulting (Stokes-Darcy) fluid velocity is important because the flow transports contaminants. The analysis of models including the transport of contaminants has, however, focused on a quasi-static Stokes-Darcy model. Herein we consider the fully evolutionary system including contaminant transport and analyze its quasi-static limits.Axioms2015-11-0644Article10.3390/axioms40405185185292075-16802015-11-06doi: 10.3390/axioms4040518Vincent ErvinMichaela KubackiWilliam LaytonMarina MoraitiZhiyong SiCatalin Trenchea<![CDATA[Axioms, Vol. 4, Pages 492-517: Free Boolean Topological Groups]]>
http://www.mdpi.com/2075-1680/4/4/492
Known and new results on free Boolean topological groups are collected. An account of the properties that these groups share with free or free Abelian topological groups and properties specific to free Boolean groups is given. Special emphasis is placed on the application of set-theoretic methods to the study of Boolean topological groups.Axioms2015-11-0344Article10.3390/axioms40404924925172075-16802015-11-03doi: 10.3390/axioms4040492Ol’ga Sipacheva<![CDATA[Axioms, Vol. 4, Pages 459-491: Characterized Subgroups of Topological Abelian Groups]]>
http://www.mdpi.com/2075-1680/4/4/459
A subgroup H of a topological abelian group X is said to be characterized by a sequence v = (vn) of characters of X if H = {x ∈ X : vn(x) → 0 in T}. We study the basic properties of characterized subgroups in the general setting, extending results known in the compact case. For a better description, we isolate various types of characterized subgroups. Moreover, we introduce the relevant class of auto-characterized groups (namely, the groups that are characterized subgroups of themselves by means of a sequence of non-null characters); in the case of locally compact abelian groups, these are proven to be exactly the non-compact ones. As a by-product of our results, we find a complete description of the characterized subgroups of discrete abelian groups.Axioms2015-10-1644Article10.3390/axioms40404594594912075-16802015-10-16doi: 10.3390/axioms4040459Dikran DikranjanAnna Giordano BrunoDaniele Impieri<![CDATA[Axioms, Vol. 4, Pages 436-458: Locally Quasi-Convex Compatible Topologies on a Topological Group]]>
http://www.mdpi.com/2075-1680/4/4/436
For a locally quasi-convex topological abelian group (G,τ), we study the poset \(\mathscr{C}(G,τ)\) of all locally quasi-convex topologies on (G) that are compatible with (τ) (i.e., have the same dual as (G,τ) ordered by inclusion. Obviously, this poset has always a bottom element, namely the weak topology σ(G,\(\widehat{G})\) . Whether it has also a top element is an open question. We study both quantitative aspects of this poset (its size) and its qualitative aspects, e.g., its chains and anti-chains. Since we are mostly interested in estimates ``from below'', our strategy consists of finding appropriate subgroups (H) of (G) that are easier to handle and show that \(\mathscr{C} (H)\) and \(\mathscr{C} (G/H)\) are large and embed, as a poset, in \(\mathscr{C}(G,τ)\). Important special results are: (i) if \(K\) is a compact subgroup of a locally quasi-convex group \(G\), then \(\mathscr{C}(G)\) and \(\mathscr{C}(G/K)\) are quasi-isomorphic (3.15); (ii) if (D) is a discrete abelian group of infinite rank, then \(\mathscr{C}(D)\) is quasi-isomorphic to the poset \(\mathfrak{F}_D\) of filters on D (4.5). Combining both results, we prove that for an LCA (locally compact abelian) group \(G \) with an open subgroup of infinite co-rank (this class includes, among others, all non-σ-compact LCA groups), the poset \( \mathscr{C} (G) \) is as big as the underlying topological structure of (G,τ) (and set theory) allows. For a metrizable connected compact group \(X\), the group of null sequences \(G=c_0(X)\) with the topology of uniform convergence is studied. We prove that \(\mathscr{C}(G)\) is quasi-isomorphic to \(\mathscr{P}(\mathbb{R})\) (6.9).Axioms2015-10-1344Article10.3390/axioms40404364364582075-16802015-10-13doi: 10.3390/axioms4040436Lydia AußenhoferDikran DikranjanElena Martín-Peinador<![CDATA[Axioms, Vol. 4, Pages 423-435: Yang–Baxter Equations, Computational Methods and Applications]]>
http://www.mdpi.com/2075-1680/4/4/423
Computational methods are an important tool for solving the Yang–Baxter equations (in small dimensions), for classifying (unifying) structures and for solving related problems. This paper is an account of some of the latest developments on the Yang–Baxter equation, its set-theoretical version and its applications. We construct new set-theoretical solutions for the Yang–Baxter equation. Unification theories and other results are proposed or proven.Axioms2015-10-0944Article10.3390/axioms40404234234352075-16802015-10-09doi: 10.3390/axioms4040423Florin Nichita<![CDATA[Axioms, Vol. 4, Pages 412-422: Some Aspects of Extended Kinetic Equation]]>
http://www.mdpi.com/2075-1680/4/3/412
Motivated by the pathway model of Mathai introduced in 2005 [Linear Algebra and Its Applications, 396, 317–328] we extend the standard kinetic equations. Connection of the extended kinetic equation with fractional calculus operator is established. The solution of the general form of the fractional kinetic equation is obtained through Laplace transform. The results for the standard kinetic equation are obtained as the limiting case.Axioms2015-09-1843Article10.3390/axioms40304124124222075-16802015-09-18doi: 10.3390/axioms4030412Dilip Kumar<![CDATA[Axioms, Vol. 4, Pages 400-411: POVMs and the Two Theorems of Naimark and Sz.-Nagy]]>
http://www.mdpi.com/2075-1680/4/3/400
In 1940 Naimark showed that if a set of quantum observables are positive semi-definite and sum to the identity then, on a larger space, they have a joint resolution as commuting projectors. In 1955 Sz.-Nagy showed that any set of observables could be so resolved, with the resolution respecting all linear sums. Crucially, both resolutions return the correct Born probabilities for the original observables. Here, an alternative proof of the Sz.-Nagy result is given using elementary inner product spaces. A version of the resolution is then shown to respect all products of observables on the base space. Practical and theoretical consequences are indicated. For example, quantum statistical inference problems that involve any algebraic functionals can now be studied using classical statistical methods over commuting observables. The estimation of quantum states is a problem of this type. Further, as theoretical objects, classical and quantum systems are now distinguished by only more or less degrees of freedom.Axioms2015-09-0143Article10.3390/axioms40304004004112075-16802015-09-01doi: 10.3390/axioms4030400James MalleyAnthony Fletcher<![CDATA[Axioms, Vol. 4, Pages 385-399: Limiting Approach to Generalized Gamma Bessel Model via Fractional Calculus and Its Applications in Various Disciplines]]>
http://www.mdpi.com/2075-1680/4/3/385
The essentials of fractional calculus according to different approaches that can be useful for our applications in the theory of probability and stochastic processes are established. In addition to this, from this fractional integral, one can list out almost all of the extended densities for the pathway parameter q &lt; 1 and q → 1. Here, we bring out the idea of thicker- or thinner-tailed models associated with a gamma-type distribution as a limiting case of the pathway operator. Applications of this extended gamma model in statistical mechanics, input-output models, solar spectral irradiance modeling, etc., are established.Axioms2015-08-2643Article10.3390/axioms40303853853992075-16802015-08-26doi: 10.3390/axioms4030385Nicy Sebastian<![CDATA[Axioms, Vol. 4, Pages 365-384: An Overview of Generalized Gamma Mittag–Leffler Model and Its Applications]]>
http://www.mdpi.com/2075-1680/4/3/365
Recently, probability models with thicker or thinner tails have gained more importance among statisticians and physicists because of their vast applications in random walks, Lévi flights, financial modeling, etc. In this connection, we introduce here a new family of generalized probability distributions associated with the Mittag–Leffler function. This family gives an extension to the generalized gamma family, opens up a vast area of potential applications and establishes connections to the topics of fractional calculus, nonextensive statistical mechanics, Tsallis statistics, superstatistics, the Mittag–Leffler stochastic process, the Lévi process and time series. Apart from examining the properties, the matrix-variate analogue and the connection to fractional calculus are also explained. By using the pathway model of Mathai, the model is further generalized. Connections to Mittag–Leffler distributions and corresponding autoregressive processes are also discussed.Axioms2015-08-2643Article10.3390/axioms40303653653842075-16802015-08-26doi: 10.3390/axioms4030365Seema Nair<![CDATA[Axioms, Vol. 4, Pages 345-364: Almost Periodic Solutions of Nonlinear Volterra Difference Equations with Unbounded Delay]]>
http://www.mdpi.com/2075-1680/4/3/345
In order to obtain the conditions for the existence of periodic and almost periodic solutions of Volterra difference equations, \( x(n+1)=f(n,x(n))+\sum_{s=-\infty}^{n}F(n,s, {x(n+s)},x(n)) \), we consider certain stability properties, which are referred to as (K, \( \rho \))-weakly uniformly-asymptotic stability and (K, \( \rho \))-uniformly asymptotic stability. Moreover, we discuss the relationship between the \( \rho \)-separation condition and the uniformly-asymptotic stability property in the \( \rho \) sense.Axioms2015-08-2443Article10.3390/axioms40303453453642075-16802015-08-24doi: 10.3390/axioms4030345Yoshihiro HamayaTomomi ItokazuKaori Saito<![CDATA[Axioms, Vol. 4, Pages 321-344: On the Fractional Poisson Process and the Discretized Stable Subordinator]]>
http://www.mdpi.com/2075-1680/4/3/321
We consider the renewal counting number process N = N(t) as a forward march over the non-negative integers with independent identically distributed waiting times. We embed the values of the counting numbers N in a “pseudo-spatial” non-negative half-line x ≥ 0 and observe that for physical time likewise we have t ≥ 0. Thus we apply the Laplace transform with respect to both variables x and t. Applying then a modification of the Montroll-Weiss-Cox formalism of continuous time random walk we obtain the essential characteristics of a renewal process in the transform domain and, if we are lucky, also in the physical domain. The process t = t(N) of accumulation of waiting times is inverse to the counting number process, in honour of the Danish mathematician and telecommunication engineer A.K. Erlang we call it the Erlang process. It yields the probability of exactly n renewal events in the interval (0; t]. We apply our Laplace-Laplace formalism to the fractional Poisson process whose waiting times are of Mittag-Leffler type and to a renewal process whose waiting times are of Wright type. The process of Mittag-Leffler type includes as a limiting case the classical Poisson process, the process of Wright type represents the discretized stable subordinator and a re-scaled version of it was used in our method of parametric subordination of time-space fractional diffusion processes. Properly rescaling the counting number process N(t) and the Erlang process t(N) yields as diffusion limits the inverse stable and the stable subordinator, respectively.Axioms2015-08-0443Article10.3390/axioms40303213213442075-16802015-08-04doi: 10.3390/axioms4030321Rudolf GorenfloFrancesco Mainardi<![CDATA[Axioms, Vol. 4, Pages 313-320: Fixed Points of Local Actions of Lie Groups on Real and Complex 2-Manifolds]]>
http://www.mdpi.com/2075-1680/4/3/313
I discuss old and new results on fixed points of local actions by Lie groups G on real and complex 2-manifolds, and zero sets of Lie algebras of vector fields. Results of E. Lima, J. Plante and C. Bonatti are reviewed.Axioms2015-07-2743Article10.3390/axioms40303133133202075-16802015-07-27doi: 10.3390/axioms4030313Morris Hirsch<![CDATA[Axioms, Vol. 4, Pages 294-312: Pro-Lie Groups: A Survey with Open Problems]]>
http://www.mdpi.com/2075-1680/4/3/294
A topological group is called a pro-Lie group if it is isomorphic to a closed subgroup of a product of finite-dimensional real Lie groups. This class of groups is closed under the formation of arbitrary products and closed subgroups and forms a complete category. It includes each finite-dimensional Lie group, each locally-compact group that has a compact quotient group modulo its identity component and, thus, in particular, each compact and each connected locally-compact group; it also includes all locally-compact Abelian groups. This paper provides an overview of the structure theory and the Lie theory of pro-Lie groups, including results more recent than those in the authors’ reference book on pro-Lie groups. Significantly, it also includes a review of the recent insight that weakly-complete unital algebras provide a natural habitat for both pro-Lie algebras and pro-Lie groups, indeed for the exponential function that links the two. (A topological vector space is weakly complete if it is isomorphic to a power RX of an arbitrary set of copies of R. This class of real vector spaces is at the basis of the Lie theory of pro-Lie groups.) The article also lists 12 open questions connected to pro-Lie groups.Axioms2015-07-2443Article10.3390/axioms40302942943122075-16802015-07-24doi: 10.3390/axioms4030294Karl HofmannSidney Morris<![CDATA[Axioms, Vol. 4, Pages 275-293: Heat Kernel Embeddings, Differential Geometry and Graph Structure]]>
http://www.mdpi.com/2075-1680/4/3/275
In this paper, we investigate the heat kernel embedding as a route to graph representation. The heat kernel of the graph encapsulates information concerning the distribution of path lengths and, hence, node affinities on the graph; and is found by exponentiating the Laplacian eigen-system over time. A Young–Householder decomposition is performed on the heat kernel to obtain the matrix of the embedded coordinates for the nodes of the graph. With the embeddings at hand, we establish a graph characterization based on differential geometry by computing sets of curvatures associated with the graph edges and triangular faces. A sectional curvature computed from the difference between geodesic and Euclidean distances between nodes is associated with the edges of the graph. Furthermore, we use the Gauss–Bonnet theorem to compute the Gaussian curvatures associated with triangular faces of the graph.Axioms2015-07-2143Article10.3390/axioms40302752752932075-16802015-07-21doi: 10.3390/axioms4030275Hewayda ElGhawalbyEdwin Hancock<![CDATA[Axioms, Vol. 4, Pages 268-274: Closed-Form Representations of the Density Function and Integer Moments of the Sample Correlation Coefficient]]>
http://www.mdpi.com/2075-1680/4/3/268
This paper provides a simplified representation of the exact density function of R, the sample correlation coefficient. The odd and even moments of R are also obtained in closed forms. Being expressed in terms of generalized hypergeometric functions, the resulting representations are readily computable. Some numerical examples corroborate the validity of the results derived herein.Axioms2015-07-2043Article10.3390/axioms40302682682742075-16802015-07-20doi: 10.3390/axioms4030268Serge Provost<![CDATA[Axioms, Vol. 4, Pages 254-267: Lindelöf Σ-Spaces and R-Factorizable Paratopological Groups]]>
http://www.mdpi.com/2075-1680/4/3/254
We prove that if a paratopological group G is a continuous image of an arbitrary product of regular Lindelöf Σ-spaces, then it is R-factorizable and has countable cellularity. If in addition, G is regular, then it is totally w-narrow and satisfies celw(G) ≤ w, and the Hewitt–Nachbin completion of G is again an R-factorizable paratopological group.Axioms2015-07-1043Article10.3390/axioms40302542542672075-16802015-07-10doi: 10.3390/axioms4030254Mikhail Tkachenko<![CDATA[Axioms, Vol. 4, Pages 235-253: On Elliptic and Hyperbolic Modular Functions and the Corresponding Gudermann Peeta Functions]]>
http://www.mdpi.com/2075-1680/4/3/235
In this article, we move back almost 200 years to Christoph Gudermann, the great expert on elliptic functions, who successfully put the twelve Jacobi functions in a didactic setting. We prove the second hyperbolic series expansions for elliptic functions again, and express generalizations of many of Gudermann’s formulas in Carlson’s modern notation. The transformations between squares of elliptic functions can be expressed as general Möbius transformations, and a conjecture of twelve formulas, extending a Gudermannian formula, is presented. In the second part of the paper, we consider the corresponding formulas for hyperbolic modular functions, and show that these Möbius transformations can be used to prove integral formulas for the inverses of hyperbolic modular functions, which are in fact Schwarz-Christoffel transformations. Finally, we present the simplest formulas for the Gudermann Peeta functions, variations of the Jacobi theta functions. 2010 Mathematics Subject Classification: Primary 33E05; Secondary 33D15.Axioms2015-07-0843Article10.3390/axioms40302352352532075-16802015-07-08doi: 10.3390/axioms4030235Thomas Ernst<![CDATA[Axioms, Vol. 4, Pages 213-234: Scientific Endeavors of A.M. Mathai: An Appraisal on the Occasion of his Eightieth Birthday, 28 April 2015]]>
http://www.mdpi.com/2075-1680/4/3/213
A.M. Mathai is Emeritus Professor of Mathematics and Statistics at McGill University, Canada. He is currently the Director of the Centre for Mathematical and Statistical Sciences India. His research contributions cover a wide spectrum of topics in mathematics, statistics, physics, astronomy, and biology. He is a Fellow of the Institute of Mathematical Statistics, National Academy of Sciences of India, and a member of the International Statistical Institute. He is a founder of the Canadian Journal of Statistics and the Statistical Society of Canada. He was instrumental in the implementation of the United Nations Basic Space Science Initiative (1991–2012). This paper highlights research results of A.M. Mathai in the period of time from 1962 to 2015. He published over 300 research papers and over 25 books.Axioms2015-07-0343Editorial10.3390/axioms40302132132342075-16802015-07-03doi: 10.3390/axioms4030213Hans HauboldArak Mathai<![CDATA[Axioms, Vol. 4, Pages 194-212: On T-Characterized Subgroups of Compact Abelian Groups]]>
http://www.mdpi.com/2075-1680/4/2/194
A sequence \(\{ u_n \}_{n\in \omega}\) in abstract additively-written Abelian group \(G\) is called a \(T\)-sequence if there is a Hausdorff group topology on \(G\) relative to which \(\lim_n u_n =0\). We say that a subgroup \(H\) of an infinite compact Abelian group \(X\) is \(T\)-characterized if there is a \(T\)-sequence \(\mathbf{u} =\{ u_n \}\) in the dual group of \(X\), such that \(H=\{ x\in X: \; (u_n, x)\to 1 \}\). We show that a closed subgroup \(H\) of \(X\) is \(T\)-characterized if and only if \(H\) is a \(G_\delta\)-subgroup of \(X\) and the annihilator of \(H\) admits a Hausdorff minimally almost periodic group topology. All closed subgroups of an infinite compact Abelian group \(X\) are \(T\)-characterized if and only if \(X\) is metrizable and connected. We prove that every compact Abelian group \(X\) of infinite exponent has a \(T\)-characterized subgroup, which is not an \(F_{\sigma}\)-subgroup of \(X\), that gives a negative answer to Problem 3.3 in Dikranjan and Gabriyelyan (Topol. Appl. 2013, 160, 2427–2442).Axioms2015-06-1942Article10.3390/axioms40201941942122075-16802015-06-19doi: 10.3390/axioms4020194Saak Gabriyelyan<![CDATA[Axioms, Vol. 4, Pages 177-193: Generalized Yang–Baxter Operators for Dieudonné Modules]]>
http://www.mdpi.com/2075-1680/4/2/177
An enrichment of a category of Dieudonné modules is made by considering Yang–Baxter conditions, and these are used to obtain ring and coring operations on the corresponding Hopf algebras. Some examples of these induced structures are discussed, including those relating to the Morava K-theory of Eilenberg–MacLane spaces.Axioms2015-05-0842Article10.3390/axioms40201771771932075-16802015-05-08doi: 10.3390/axioms4020177Rui Saramago<![CDATA[Axioms, Vol. 4, Pages 156-176: Diffeomorphism Spline]]>
http://www.mdpi.com/2075-1680/4/2/156
Conventional splines offer powerful means for modeling surfaces and volumes in three-dimensional Euclidean space. A one-dimensional quaternion spline has been applied for animation purpose, where the splines are defined to model a one-dimensional submanifold in the three-dimensional Lie group. Given two surfaces, all of the diffeomorphisms between them form an infinite dimensional manifold, the so-called diffeomorphism space. In this work, we propose a novel scheme to model finite dimensional submanifolds in the diffeomorphism space by generalizing conventional splines. According to quasiconformal geometry theorem, each diffeomorphism determines a Beltrami differential on the source surface. Inversely, the diffeomorphism is determined by its Beltrami differential with normalization conditions. Therefore, the diffeomorphism space has one-to-one correspondence to the space of a special differential form. The convex combination of Beltrami differentials is still a Beltrami differential. Therefore, the conventional spline scheme can be generalized to the Beltrami differential space and, consequently, to the diffeomorphism space. Our experiments demonstrate the efficiency and efficacy of diffeomorphism splines. The diffeomorphism spline has many potential applications, such as surface registration, tracking and animation.Axioms2015-04-1042Article10.3390/axioms40201561561762075-16802015-04-10doi: 10.3390/axioms4020156Wei ZengMuhammad RazibAbdur Shahid<![CDATA[Axioms, Vol. 4, Pages 134-155: Convergence Aspects for Generalizations of q-Hypergeometric Functions]]>
http://www.mdpi.com/2075-1680/4/2/134
In an earlier paper, we found transformation and summation formulas for 43 q-hypergeometric functions of 2n variables. The aim of the present article is to find convergence regions and a few conjectures of convergence regions for these functions based on a vector version of the Nova q-addition. These convergence regions are given in a purely formal way, extending the results of Karlsson (1976). The Γq-function and the q-binomial coefficients, which are used in the proofs, are adjusted accordingly. Furthermore, limits and special cases for the new functions, e.g., q-Lauricella functions and q-Horn functions, are pointed out.Axioms2015-04-0842Article10.3390/axioms40201341341552075-16802015-04-08doi: 10.3390/axioms4020134Thomas Ernst<![CDATA[Axioms, Vol. 4, Pages 120-133: Computational Solutions of Distributed Order Reaction-Diffusion Systems Associated with Riemann-Liouville Derivatives]]>
http://www.mdpi.com/2075-1680/4/2/120
This article is in continuation of the authors research attempts to derive computational solutions of an unified reaction-diffusion equation of distributed order associated with Caputo derivatives as the time-derivative and Riesz-Feller derivative as space derivative. This article presents computational solutions of distributed order fractional reaction-diffusion equations associated with Riemann-Liouville derivatives of fractional orders as the time-derivatives and Riesz-Feller fractional derivatives as the space derivatives. The method followed in deriving the solution is that of joint Laplace and Fourier transforms. The solution is derived in a closed and computational form in terms of the familiar Mittag-Leffler function. It provides an elegant extension of results available in the literature. The results obtained are presented in the form of two theorems. Some results associated specifically with fractional Riesz derivatives are also derived as special cases of the most general result. It will be seen that in case of distributed order fractional reaction-diffusion, the solution comes in a compact and closed form in terms of a generalization of the Kampé de Fériet hypergeometric series in two variables. The convergence of the double series occurring in the solution is also given.Axioms2015-04-0242Article10.3390/axioms40201201201332075-16802015-04-02doi: 10.3390/axioms4020120Ram SaxenaArak MathaiHans Haubold<![CDATA[Axioms, Vol. 4, Pages 102-119: A Model for the Universe that Begins to Resemble a Quantum Computer]]>
http://www.mdpi.com/2075-1680/4/1/102
This article presents a sequential growth model for the Universe that acts like a quantum computer. The basic constituents of the model are a special type of causal set (causet) called a c-causet. A c-causet is defined to be a causet that has a unique labeling. We characterize c-causets as those causets that form a multipartite graph or equivalently those causets whose elements are comparable whenever their heights are different. We show that a c-causet has precisely two c-causet offspring. It follows that there are 2n c-causets of cardinality n + 1. This enables us to classify c-causets of cardinality n + 1 in terms of n-bits. We then quantize the model by introducing a quantum sequential growth process. This is accomplished by replacing the n-bits by n-qubits and defining transition amplitudes for the growth transitions. We mainly consider two types of processes, called stationary and completely stationary. We show that for stationary processes, the probability operators are tensor products of positive rank-one qubit operators. Moreover, the converse of this result holds. Simplifications occur for completely stationary processes. We close with examples of precluded events.Axioms2015-03-0941Article10.3390/axioms40101021021192075-16802015-03-09doi: 10.3390/axioms4010102Stan Gudder<![CDATA[Axioms, Vol. 4, Pages 84-101: Open and Dense Topological Transitivity of Extensions by Non-Compact Fiber of Hyperbolic Systems: A Review]]>
http://www.mdpi.com/2075-1680/4/1/84
Currently, there is great renewed interest in proving the topological transitivity of various classes of continuous dynamical systems. Even though this is one of the most basic dynamical properties that can be investigated, the tools used by various authors are quite diverse and are strongly related to the class of dynamical systems under consideration. The goal of this review article is to present the state of the art for the class of Hölder extensions of hyperbolic systems with non-compact connected Lie group fiber. The hyperbolic systems we consider are mostly discrete time. In particular, we address the stability and genericity of topological transitivity in large classes of such transformations. The paper lists several open problems and conjectures and tries to place this topic of research in the general context of hyperbolic and topological dynamics.Axioms2015-02-0441Review10.3390/axioms4010084841012075-16802015-02-04doi: 10.3390/axioms4010084Viorel NiticaAndrei Török<![CDATA[Axioms, Vol. 4, Pages 71-83: Boas’ Formula and Sampling Theorem]]>
http://www.mdpi.com/2075-1680/4/1/71
In 1937, Boas gave a smart proof for an extension of the Bernstein theorem for trigonometric series. It is the purpose of the present note (i) to point out that a formula which Boas used in the proof is related with the Shannon sampling theorem; (ii) to present a generalized Parseval formula, which is suggested by the Boas’ formula; and (iii) to show that this provides a very smart derivation of the Shannon sampling theorem for a function which is the Fourier transform of a distribution involving the Dirac delta function. It is also shows that, by the argument giving Boas’ formula for the derivative f'(x) of a function f(x), we can derive the corresponding formula for f'''(x), by which we can obtain an upperbound of |f'''(x)+3R2f'(x)|. Discussions are given also on an extension of the Szegö theorem for trigonometric series, which Boas mentioned in the same paper.Axioms2015-01-2641Article10.3390/axioms401007171832075-16802015-01-26doi: 10.3390/axioms4010071Tohru MoritaKen-ichi Sato<![CDATA[Axioms, Vol. 4, Pages 32-70: Azumaya Monads and Comonads]]>
http://www.mdpi.com/2075-1680/4/1/32
The definition of Azumaya algebras over commutative rings \(R\) requires the tensor product of modules over \(R\) and the twist map for the tensor product of any two \(R\)-modules. Similar constructions are available in braided monoidal categories, and Azumaya algebras were defined in these settings. Here, we introduce Azumaya monads on any category \(\mathbb{A}\) by considering a monad \((F,m,e)\) on \(\mathbb{A}\) endowed with a distributive law \(\lambda: FF\to FF\) satisfying the Yang–Baxter equation (BD%please define -law). This allows to introduce an opposite monad \((F^\lambda,m\cdot \lambda,e)\) and a monad structure on \(FF^\lambda\). The quadruple \((F,m,e,\lambda)\) is called an Azumaya monad, provided that the canonical comparison functor induces an equivalence between the category \(\mathbb{A}\) and the category of \(FF^\lambda\)-modules. Properties and characterizations of these monads are studied, in particular for the case when \(F\) allows for a right adjoint functor. Dual to Azumaya monads, we define Azumaya comonads and investigate the interplay between these notions. In braided categories (V\(,\otimes,I,\tau)\), for any V-algebra \(A\), the braiding induces a BD-law \(\tau_{A,A}:A\otimes A\to A\otimes A\), and \(A\) is called left (right) Azumaya, provided the monad \(A\otimes-\) (resp. \(-\otimes A\)) is Azumaya. If \(\tau\) is a symmetry or if the category V admits equalizers and coequalizers, the notions of left and right Azumaya algebras coincide.Axioms2015-01-1941Article10.3390/axioms401003232702075-16802015-01-19doi: 10.3390/axioms4010032Bachuki MesablishviliRobert Wisbauer<![CDATA[Axioms, Vol. 4, Pages 30-31: Acknowledgement to Reviewers of Axioms in 2014]]>
http://www.mdpi.com/2075-1680/4/1/30
The editors of Axioms would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2014:[...]Axioms2015-01-0841Editorial10.3390/axioms401003030312075-16802015-01-08doi: 10.3390/axioms4010030 Axioms Editorial Office<![CDATA[Axioms, Vol. 4, Pages 1-29: Positive-Operator Valued Measure (POVM) Quantization]]>
http://www.mdpi.com/2075-1680/4/1/1
We present a general formalism for giving a measure space paired with a separable Hilbert space a quantum version based on a normalized positive operator-valued measure. The latter are built from families of density operators labeled by points of the measure space. We especially focus on various probabilistic aspects of these constructions. Simple ormore elaborate examples illustrate the procedure: circle, two-sphere, plane and half-plane. Links with Positive-Operator Valued Measure (POVM) quantum measurement and quantum statistical inference are sketched.Axioms2014-12-2541Article10.3390/axioms40100011292075-16802014-12-25doi: 10.3390/axioms4010001Jean GazeauBarbara Heller<![CDATA[Axioms, Vol. 3, Pages 369-379: A Simplified Algorithm for Inverting Higher Order Diffusion Tensors]]>
http://www.mdpi.com/2075-1680/3/4/369
In Riemannian geometry, a distance function is determined by an inner product on the tangent space. In Riemann–Finsler geometry, this distance function can be determined by a norm. This gives more freedom on the form of the so-called indicatrix or the set of unit vectors. This has some interesting applications, e.g., in medical image analysis, especially in diffusion weighted imaging (DWI). An important application of DWI is in the inference of the local architecture of the tissue, typically consisting of thin elongated structures, such as axons or muscle fibers, by measuring the constrained diffusion of water within the tissue. From high angular resolution diffusion imaging (HARDI) data, one can estimate the diffusion orientation distribution function (dODF), which indicates the relative diffusivity in all directions and can be represented by a spherical polynomial. We express this dODF as an equivalent spherical monomial (higher order tensor) to directly generalize the (second order) diffusion tensor approach. To enable efficient computation of Riemann–Finslerian quantities on diffusion weighted (DW)-images, such as the metric/norm tensor, we present a simple and efficient algorithm to invert even order spherical monomials, which extends the familiar inversion of diffusion tensors, i.e., symmetric matrices.Axioms2014-11-1434Article10.3390/axioms30403693693792075-16802014-11-14doi: 10.3390/axioms3040369Laura AstolaNeda SepasianTom HaijeAndrea FusterLuc Florack<![CDATA[Axioms, Vol. 3, Pages 360-368: The Yang-Baxter Equation, (Quantum) Computers and Unifying Theories]]>
http://www.mdpi.com/2075-1680/3/4/360
Quantum mechanics has had an important influence on building computers;nowadays, quantum mechanics principles are used for the processing and transmission ofinformation. The Yang-Baxter equation is related to the universal gates from quantumcomputing and it realizes a unification of certain non-associative structures. Unifyingstructures could be seen as structures which comprise the information contained in other(algebraic) structures. Recently, we gave the axioms of a structure which unifies associativealgebras, Lie algebras and Jordan algebras. Our paper is a review and a continuation of thatapproach. It also contains several geometric considerations.Axioms2014-11-1434Communication10.3390/axioms30403603603682075-16802014-11-14doi: 10.3390/axioms3040360Radu IordanescuFlorin NichitaIon Nichita<![CDATA[Axioms, Vol. 3, Pages 342-359: Weak n-Ary Relational Products in Allegories]]>
http://www.mdpi.com/2075-1680/3/4/342
Allegories are enriched categories generalizing a category of sets and binary relations. Accordingly, relational products in an allegory can be viewed as a generalization of Cartesian products. There are several definitions of relational products currently in the literature. Interestingly, definitions for binary products do not generalize easily to n-ary ones. In this paper, we provide a new definition of an n-ary relational product, and we examine its properties.Axioms2014-10-3034Article10.3390/axioms30403423423592075-16802014-10-30doi: 10.3390/axioms3040342Bartosz ZielińskiPaweł Maślanka<![CDATA[Axioms, Vol. 3, Pages 335-341: The Gromov–Wasserstein Distance: A Brief Overview]]>
http://www.mdpi.com/2075-1680/3/3/335
We recall the construction of the Gromov–Wasserstein distance and concentrate on quantitative aspects of the definition.Axioms2014-09-0233Article10.3390/axioms30303353353412075-16802014-09-02doi: 10.3390/axioms3030335Facundo Mémoli<![CDATA[Axioms, Vol. 3, Pages 320-334: Space-Time Fractional Reaction-Diffusion Equations Associated with a Generalized Riemann–Liouville Fractional Derivative]]>
http://www.mdpi.com/2075-1680/3/3/320
This paper deals with the investigation of the computational solutions of a unified fractional reaction-diffusion equation, which is obtained from the standard diffusion equation by replacing the time derivative of first order by the generalized Riemann–Liouville fractional derivative defined by others and the space derivative of second order by the Riesz–Feller fractional derivative and adding a function ɸ(x, t). The solution is derived by the application of the Laplace and Fourier transforms in a compact and closed form in terms of Mittag–Leffler functions. The main result obtained in this paper provides an elegant extension of the fundamental solution for the space-time fractional diffusion equation obtained by others and the result very recently given by others. At the end, extensions of the derived results, associated with a finite number of Riesz–Feller space fractional derivatives, are also investigated.Axioms2014-08-0433Article10.3390/axioms30303203203342075-16802014-08-04doi: 10.3390/axioms3030320Ram SaxenaArak MathaiHans Haubold<![CDATA[Axioms, Vol. 3, Pages 300-319: Matching the LBO Eigenspace of Non-Rigid Shapes via High Order Statistics]]>
http://www.mdpi.com/2075-1680/3/3/300
A fundamental tool in shape analysis is the virtual embedding of the Riemannian manifold describing the geometry of a shape into Euclidean space. Several methods have been proposed to embed isometric shapes into flat domains, while preserving the distances measured on the manifold. Recently, attention has been given to embedding shapes into the eigenspace of the Laplace–Beltrami operator. The Laplace–Beltrami eigenspace preserves the diffusion distance and is invariant under isometric transformations. However, Laplace–Beltrami eigenfunctions computed independently for different shapes are often incompatible with each other. Applications involving multiple shapes, such as pointwise correspondence, would greatly benefit if their respective eigenfunctions were somehow matched. Here, we introduce a statistical approach for matching eigenfunctions. We consider the values of the eigenfunctions over the manifold as the sampling of random variables and try to match their multivariate distributions. Comparing distributions is done indirectly, using high order statistics. We show that the permutation and sign ambiguities of low order eigenfunctions can be inferred by minimizing the difference of their third order moments. The sign ambiguities of antisymmetric eigenfunctions can be resolved by exploiting isometric invariant relations between the gradients of the eigenfunctions and the surface normal. We present experiments demonstrating the success of the proposed method applied to feature point correspondence.Axioms2014-07-1533Article10.3390/axioms30303003003192075-16802014-07-15doi: 10.3390/axioms3030300Alon ShternRon Kimmel<![CDATA[Axioms, Vol. 3, Pages 280-299: A Comparative Study of Several Classical, Discrete Differential and Isogeometric Methods for Solving Poisson’s Equation on the Disk]]>
http://www.mdpi.com/2075-1680/3/2/280
This paper outlines and qualitatively compares the implementations of seven different methods for solving Poisson’s equation on the disk. The methods include two classical finite elements, a cotan formula-based discrete differential geometry approach and four isogeometric constructions. The comparison reveals numerical convergence rates and, particularly for isogeometric constructions based on Catmull–Clark elements, the need to carefully choose quadrature formulas. The seven methods include two that are new to isogeometric analysis. Both new methods yield O(h3) convergence in the L2 norm, also when points are included where n 6≠ 4 pieces meet. One construction is based on a polar, singular parameterization; the other is a G1 tensor-product construction.Axioms2014-06-1132Article10.3390/axioms30202802802992075-16802014-06-11doi: 10.3390/axioms3020280Thien NguyenKeçstutis KarčiauskasJörg Peters<![CDATA[Axioms, Vol. 3, Pages 260-279: Modalities for an Allegorical Conceptual Data Model]]>
http://www.mdpi.com/2075-1680/3/2/260
Allegories are enriched categories generalizing a category of sets and binary relations. In this paper, we extend a new, recently-introduced conceptual data model based on allegories by adding support for modal operators and developing a modal interpretation of the model in any allegory satisfying certain additional (but natural) axioms. The possibility of using different allegories allows us to transparently use alternative logical frameworks, such as fuzzy relations. Mathematically, our work demonstrates how to enrich with modal operators and to give a many world semantics to an abstract algebraic logic framework. We also give some examples of applications of the modal extension.Axioms2014-05-3032Article10.3390/axioms30202602602792075-16802014-05-30doi: 10.3390/axioms3020260Bartosz ZielińskiPaweł MaślankaŚcibor Sobieski<![CDATA[Axioms, Vol. 3, Pages 244-259: Classical Probability and Quantum Outcomes]]>
http://www.mdpi.com/2075-1680/3/2/244
There is a contact problem between classical probability and quantum outcomes. Thus, a standard result from classical probability on the existence of joint distributions ultimately implies that all quantum observables must commute. An essential task here is a closer identification of this conflict based on deriving commutativity from the weakest possible assumptions, and showing that stronger assumptions in some of the existing no-go proofs are unnecessary. An example of an unnecessary assumption in such proofs is an entangled system involving nonlocal observables. Another example involves the Kochen-Specker hidden variable model, features of which are also not needed to derive commutativity. A diagram is provided by which user-selected projectors can be easily assembled into many new, graphical no-go proofs.Axioms2014-05-2632Article10.3390/axioms30202442442592075-16802014-05-26doi: 10.3390/axioms3020244James Malley<![CDATA[Axioms, Vol. 3, Pages 222-243: Conformal-Based Surface Morphing and Multi-Scale Representation]]>
http://www.mdpi.com/2075-1680/3/2/222
This paper presents two algorithms, based on conformal geometry, for the multi-scale representations of geometric shapes and surface morphing. A multi-scale surface representation aims to describe a 3D shape at different levels of geometric detail, which allows analyzing or editing surfaces at the global or local scales effectively. Surface morphing refers to the process of interpolating between two geometric shapes, which has been widely applied to estimate or analyze deformations in computer graphics, computer vision and medical imaging. In this work, we propose two geometric models for surface morphing and multi-scale representation for 3D surfaces. The basic idea is to represent a 3D surface by its mean curvature function, H, and conformal factor function λ, which uniquely determine the geometry of the surface according to Riemann surface theory. Once we have the (λ, H) parameterization of the surface, post-processing of the surface can be done directly on the conformal parameter domain. In particular, the problem of multi-scale representations of shapes can be reduced to the signal filtering on the λ and H parameters. On the other hand, the surface morphing problem can be transformed to an interpolation process of two sets of (λ, H) parameters. We test the proposed algorithms on 3D human face data and MRI-derived brain surfaces. Experimental results show that our proposed methods can effectively obtain multi-scale surface representations and give natural surface morphing results.Axioms2014-05-2032Article10.3390/axioms30202222222432075-16802014-05-20doi: 10.3390/axioms3020222Ka LamChengfeng WenLok Lui<![CDATA[Axioms, Vol. 3, Pages 202-221: Characteristic Number: Theory and Its Application to Shape Analysis]]>
http://www.mdpi.com/2075-1680/3/2/202
Geometric invariants are important for shape recognition and matching. Existing invariants in projective geometry are typically defined on the limited number (e.g., five for the classical cross-ratio) of collinear planar points and also lack the ability to characterize the curve or surface underlying the given points. In this paper, we present a projective invariant named after the characteristic number of planar algebraic curves. The characteristic number in this work reveals an intrinsic property of an algebraic hypersurface or curve, which relies no more on the existence of the surface or curve as its planar version. The new definition also generalizes the cross-ratio by relaxing the collinearity and number of points for the cross-ratio. We employ the characteristic number to construct more informative shape descriptors that improve the performance of shape recognition, especially when severe affine and perspective deformations occur. In addition to the application to shape recognition, we incorporate the geometric constraints on facial feature points derived from the characteristic number into facial feature matching. The experiments show the improvements on accuracy and robustness to pose and view changes over the method with the collinearity and cross-ratio constraints.Axioms2014-05-1532Article10.3390/axioms30202022022212075-16802014-05-15doi: 10.3390/axioms3020202Xin FanZhongxuan LuoJielin ZhangXinchen ZhouQi JiaDaiyun Luo<![CDATA[Axioms, Vol. 3, Pages 177-201: Deterministic Greedy Routing with Guaranteed Delivery in 3D Wireless Sensor Networks]]>
http://www.mdpi.com/2075-1680/3/2/177
With both computational complexity and storage space bounded by a small constant, greedy routing is recognized as an appealing approach to support scalable routing in wireless sensor networks. However, significant challenges have been encountered in extending greedy routing from 2D to 3D space. In this research, we develop decentralized solutions to achieve greedy routing in 3D sensor networks. Our proposed approach is based on a unit tetrahedron cell (UTC) mesh structure. We propose a distributed algorithm to realize volumetric harmonic mapping (VHM) of the UTC mesh under spherical boundary condition. It is a one-to-one map that yields virtual coordinates for each node in the network without or with one internal hole. Since a boundary has been mapped to a sphere, node-based greedy routing is always successful thereon. At the same time, we exploit the UTC mesh to develop a face-based greedy routing algorithm and prove its success at internal nodes. To deliver a data packet to its destination, face-based and node-based greedy routing algorithms are employed alternately at internal and boundary UTCs, respectively. For networks with multiple internal holes, a segmentation and tunnel-based routing strategy is proposed on top of VHM to support global end-to-end routing. As far as we know, this is the first work that realizes truly deterministic routing with constant-bounded storage and computation in general 3D wireless sensor networks.Axioms2014-05-1532Article10.3390/axioms30201771772012075-16802014-05-15doi: 10.3390/axioms3020177Su XiaXiaotian YinHongyi WuMiao JinXianfeng Gu<![CDATA[Axioms, Vol. 3, Pages 166-176: Joint Distributions and Quantum Nonlocal Models]]>
http://www.mdpi.com/2075-1680/3/2/166
A standard result in quantum mechanics is this: if two observables are commuting then they have a classical joint distribution in every state. A converse is demonstrated here: If a classical joint distribution for the pair agrees with standard quantum facts, then the observables must commute. This has consequences for some historical and recent quantum nonlocal models: they are analytically disallowed without the need for experiment, as they imply that all local observables must commute among themselves.Axioms2014-04-1532Communication10.3390/axioms30201661661762075-16802014-04-15doi: 10.3390/axioms3020166James MalleyAnthony Fletcher<![CDATA[Axioms, Vol. 3, Pages 153-165: Bell Length as Mutual Information in Quantum Interference]]>
http://www.mdpi.com/2075-1680/3/2/153
The necessity of a rigorously operative formulation of quantum mechanics, functional to the exigencies of quantum computing, has raised the interest again in the nature of probability and the inference in quantum mechanics. In this work, we show a relation among the probabilities of a quantum system in terms of information of non-local correlation by means of a new quantity, the Bell length.Axioms2014-04-1032Article10.3390/axioms30201531531652075-16802014-04-10doi: 10.3390/axioms3020153Ignazio LicataDavide Fiscaletti<![CDATA[Axioms, Vol. 3, Pages 140-152: Continuous Stieltjes-Wigert Limiting Behaviour of a Family of Confluent q-Chu-Vandermonde Distributions]]>
http://www.mdpi.com/2075-1680/3/2/140
From Kemp [1], we have a family of confluent q-Chu- Vandermonde distributions, consisted by three members I, II and III, interpreted as a family of q-steady-state distributions from Markov chains. In this article, we provide the moments of the distributions of this family and we establish a continuous limiting behavior for the members I and II, in the sense of pointwise convergence, by applying a q-analogue of the usual Stirling asymptotic formula for the factorial number of order n. Specifically, we initially give the q-factorial moments and the usual moments for the family of confluent q-Chu- Vandermonde distributions and then we designate as a main theorem the conditions under which the confluent q-Chu-Vandermonde distributions I and II converge to a continuous Stieltjes-Wigert distribution. For the member III we give a continuous analogue. Moreover, as applications of this study we present a modified q-Bessel distribution, a generalized q-negative Binomial distribution and a generalized over/underdispersed (O/U) distribution. Note that in this article we prove the convergence of a family of discrete distributions to a continuous distribution which is not of a Gaussian type.Axioms2014-04-1032Article10.3390/axioms30201401401522075-16802014-04-10doi: 10.3390/axioms3020140Andreas KyriakoussisMalvina Vamvakari<![CDATA[Axioms, Vol. 3, Pages 119-139: Ricci Curvature on Polyhedral Surfaces via Optimal Transportation]]>
http://www.mdpi.com/2075-1680/3/1/119
The problem of correctly defining geometric objects, such as the curvature, is a hard one in discrete geometry. In 2009, Ollivier defined a notion of curvature applicable to a wide category of measured metric spaces, in particular to graphs. He named it coarse Ricci curvature because it coincides, up to some given factor, with the classical Ricci curvature, when the space is a smooth manifold. Lin, Lu and Yau and Jost and Liu have used and extended this notion for graphs, giving estimates for the curvature and, hence, the diameter, in terms of the combinatorics. In this paper, we describe a method for computing the coarse Ricci curvature and give sharper results, in the specific, but crucial case of polyhedral surfaces.Axioms2014-03-0631Article10.3390/axioms30101191191392075-16802014-03-06doi: 10.3390/axioms3010119Benoît LoiselPascal Romon<![CDATA[Axioms, Vol. 3, Pages 109-118: Optimization Models for Reaction Networks: Information Divergence, Quadratic Programming and Kirchhoff’s Laws]]>
http://www.mdpi.com/2075-1680/3/1/109
This article presents a simple derivation of optimization models for reaction networks leading to a generalized form of the mass-action law, and compares the formal structure of Minimum Information Divergence, Quadratic Programming and Kirchhoff type network models. These optimization models are used in related articles to develop and illustrate the operation of ontology alignment algorithms and to discuss closely connected issues concerning the epistemological and statistical significance of sharp or precise hypotheses in empirical science.Axioms2014-03-0531Article10.3390/axioms30101091091182075-16802014-03-05doi: 10.3390/axioms3010109Julio SternFabio Nakano<![CDATA[Axioms, Vol. 3, Pages 84-108: Increasing Personal Value Congruence in Computerized Decision Support Using System Feedback]]>
http://www.mdpi.com/2075-1680/3/1/84
The Theory of Universals in Values (TUV), a reliable and validated conceptualization of personal values used in psychology, is used to examine the effect of system feedback delivered by a Decision Support System (DSS) on personal values. The results indicate that value-based decision-making behavior can be influenced by DSS feedback to address value congruence in decision-making. User behavior was shown to follow the outcomes expected by operant theory when feedback was supportive and to follow the outcomes of reactance theory when feedback was challenging. This result suggests that practitioners and Information System (IS) researchers should consider user values when designing computerized decision feedback to adjust a system’s design such that the potential user backlash is avoided or congruence between organizational and personal values is achieved.Axioms2014-02-2531Article10.3390/axioms3010084841082075-16802014-02-25doi: 10.3390/axioms3010084Bryan HosackDavid Paradice<![CDATA[Axioms, Vol. 3, Pages 82-83: Acknowledgement to Reviewers of Axioms in 2013]]>
http://www.mdpi.com/2075-1680/3/1/82
The editors of Axioms would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2013. [...]Axioms2014-02-2531Editorial10.3390/axioms301008282832075-16802014-02-25doi: 10.3390/axioms3010082 Axioms Editorial Office<![CDATA[Axioms, Vol. 3, Pages 70-81: Canonical Coordinates for Retino-Cortical Magnification]]>
http://www.mdpi.com/2075-1680/3/1/70
A geometric model for a biologically-inspired visual front-end is proposed, based on an isotropic, scale-invariant two-form field. The model incorporates a foveal property typical of biological visual systems, with an approximately linear decrease of resolution as a function of eccentricity, and by a physical size constant that measures the radius of the geometric foveola, the central region characterized by maximal resolving power. It admits a description in singularity-free canonical coordinates generalizing the familiar log-polar coordinates and reducing to these in the asymptotic case of negligibly-sized geometric foveola or, equivalently, at peripheral locations in the visual field. It has predictive power to the extent that quantitative geometric relationships pertaining to retino-cortical magnification along the primary visual pathway, such as receptive field size distribution and spatial arrangement in retina and striate cortex, can be deduced in a principled manner. The biological plausibility of the model is demonstrated by comparison with known facts of human vision.Axioms2014-02-2431Article10.3390/axioms301007070812075-16802014-02-24doi: 10.3390/axioms3010070Luc Florack<![CDATA[Axioms, Vol. 3, Pages 64-69: On Transcendental Numbers]]>
http://www.mdpi.com/2075-1680/3/1/64
Transcendental numbers play an important role in many areas of science. This paper contains a short survey on transcendental numbers and some relations among them. New inequalities for transcendental numbers are stated in Section 2 and proved in Section 4. Also, in relationship with these topics, we study the exponential function axioms related to the Yang-Baxter equation.Axioms2014-02-2131Communication10.3390/axioms301006464692075-16802014-02-21doi: 10.3390/axioms3010064Florin Nichita<![CDATA[Axioms, Vol. 3, Pages 50-63: A Hybrid Artificial Reputation Model Involving Interaction Trust, Witness Information and the Trust Model to Calculate the Trust Value of Service Providers]]>
http://www.mdpi.com/2075-1680/3/1/50
Agent interaction in a community, such as the online buyer-seller scenario, is often uncertain, as when an agent comes in contact with other agents they initially know nothing about each other. Currently, many reputation models are developed that help service consumers select better service providers. Reputation models also help agents to make a decision on who they should trust and transact with in the future. These reputation models are either built on interaction trust that involves direct experience as a source of information or they are built upon witness information also known as word-of-mouth that involves the reports provided by others. Neither the interaction trust nor the witness information models alone succeed in such uncertain interactions. In this paper we propose a hybrid reputation model involving both interaction trust and witness information to address the shortcomings of existing reputation models when taken separately. A sample simulation is built to setup buyer-seller services and uncertain interactions. Experiments reveal that the hybrid approach leads to better selection of trustworthy agents where consumers select more reputable service providers, eventually helping consumers obtain more gains. Furthermore, the trust model developed is used in calculating trust values of service providers.Axioms2014-02-1931Article10.3390/axioms301005050632075-16802014-02-19doi: 10.3390/axioms3010050Gurdeep RansiZiad Kobti<![CDATA[Axioms, Vol. 3, Pages 46-49: The Three Laws of Thought, Plus One: The Law of Comparisons]]>
http://www.mdpi.com/2075-1680/3/1/46
The rules of logic are nearly 2500 years old and date back to Plato and Aristotle who set down the three laws of thought: identity, non-contradiction, and excluded middle. The use of language and logic has been adequate for us to develop mathematics, prove theorems, and create scientific knowledge. However, the laws of thought are incomplete. We need to extend our logical system by adding to the very old laws of thought an essential yet poorly understood law. It is a necessary law of thought that resides in our biology even deeper than the other three laws. It is related to the rudiments of how we as living beings, and even nonliving things, respond to influences as stimuli. It helps us discriminate between being ourselves and sensing that there is something else that is not ourselves that even amoebas seem to know. It is the intrinsic ability to sense and distinguish. This fourth law is the law of comparisons. Although it has been missing from our logical deductions it underlies the other three laws of thought because without it we cannot know what is and what is not.Axioms2014-02-1031Concept Paper10.3390/axioms301004646492075-16802014-02-10doi: 10.3390/axioms3010046Thomas Saaty<![CDATA[Axioms, Vol. 3, Pages 31-45: Second-Order Risk Constraints in Decision Analysis]]>
http://www.mdpi.com/2075-1680/3/1/31
Recently, representations and methods aimed at analysing decision problems where probabilities and values (utilities) are associated with distributions over them (second-order representations) have been suggested. In this paper we present an approach to how imprecise information can be modelled by means of second-order distributions and how a risk evaluation process can be elaborated by integrating procedures for numerically imprecise probabilities and utilities. We discuss some shortcomings of the use of the principle of maximising the expected utility and of utility theory in general, and offer remedies by the introduction of supplementary decision rules based on a concept of risk constraints taking advantage of second-order distributions.Axioms2014-01-1731Article10.3390/axioms301003131452075-16802014-01-17doi: 10.3390/axioms3010031Love EkenbergMats DanielsonAron LarssonDavid Sundgren<![CDATA[Axioms, Vol. 3, Pages 10-30: Business Decision-Making Using Geospatial Data: A Research Framework and Literature Review]]>
http://www.mdpi.com/2075-1680/3/1/10
Organizations that leverage their increasing volume of geospatial data have the potential to enhance their strategic and organizational decisions. However, literature describing the best techniques to make decisions using geospatial data and the best approaches to take advantage of geospatial data’s unique visualization capabilities is limited. This paper reviews the use of geospatial visualization and its effects on decision performance, which is one of the many components of decision-making when using geospatial data. Additionally, this paper proposes a comprehensive model allowing researchers to better understand decision-making using geospatial data and provides a robust foundation for future research. Finally, this paper makes an argument for further research of information-presentation, task-characteristics, user-characteristics and their effects on decision-performance when utilizing geospatial data.Axioms2013-12-2331Review10.3390/axioms301001010302075-16802013-12-23doi: 10.3390/axioms3010010Michael ErskineDawn GreggJahangir KarimiJudy Scott<![CDATA[Axioms, Vol. 3, Pages 1-9: A Method for Negotiating Various Customer Requirements for Public Service Design]]>
http://www.mdpi.com/2075-1680/3/1/1
A method for public service design, which enables designers to realize high-value added service design by considering plural different customer groups in parallel, is proposed. In General, service designs focus on specific customers. However, because of the diversity of customer requirements, it is difficult to design a public service that addresses the requirements of all customers. To achieve higher customer satisfaction, it is imperative to summarize the requirements of various customers and design a service by considering customers belonging to different categories. In this article, we propose a method that enables highly public service development by considering groups of various customers and minimizing customer dissatisfaction by adopting a group-decision-making approach. As a consequence, improvement of effectiveness of highly public service development can be expected.Axioms2013-12-2031Communication10.3390/axioms3010001192075-16802013-12-20doi: 10.3390/axioms3010001Yoshiki ShimomuraYutaro NemotoFumiya AkasakaKoji Kimita<![CDATA[Axioms, Vol. 2, Pages 477-489: Orthogonality and Dimensionality]]>
http://www.mdpi.com/2075-1680/2/4/477
In this article, we present what we believe to be a simple way to motivate the use of Hilbert spaces in quantum mechanics. To achieve this, we study the way the notion of dimension can, at a very primitive level, be defined as the cardinality of a maximal collection of mutually orthogonal elements (which, for instance, can be seen as spatial directions). Following this idea, we develop a formalism based on two basic ingredients, namely an orthogonality relation and matroids which are a very generic algebraic structure permitting to define a notion of dimension. Having obtained what we call orthomatroids, we then show that, in high enough dimension, the basic constituants of orthomatroids (more precisely the simple and irreducible ones) are isomorphic to generalized Hilbert lattices, so that their presence is a direct consequence of an orthogonality-based characterization of dimension.Axioms2013-12-1324Article10.3390/axioms20404774774892075-16802013-12-13doi: 10.3390/axioms2040477Olivier Brunet<![CDATA[Axioms, Vol. 2, Pages 443-476: R-Matrices, Yetter-Drinfel'd Modules and Yang-Baxter Equation]]>
http://www.mdpi.com/2075-1680/2/3/443
In the first part we recall two famous sources of solutions to the Yang-Baxter equation—R-matrices and Yetter-Drinfel0d (=YD) modules—and an interpretation of the former as a particular case of the latter. We show that this result holds true in the more general case of weak R-matrices, introduced here. In the second part we continue exploring the “braided” aspects of YD module structure, exhibiting a braided system encoding all the axioms from the definition of YD modules. The functoriality and several generalizations of this construction are studied using the original machinery of YD systems. As consequences, we get a conceptual interpretation of the tensor product structures for YD modules, and a generalization of the deformation cohomology of YD modules. This homology theory is thus included into the unifying framework of braided homologies, which contains among others Hochschild, Chevalley-Eilenberg, Gerstenhaber-Schack and quandle homologies.Axioms2013-09-0523Article10.3390/axioms20304434434762075-16802013-09-05doi: 10.3390/axioms2030443Victoria Lebed