Axioms
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Latest open access articles published in Axioms at http://www.mdpi.com/journal/axioms<![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<![CDATA[Axioms, Vol. 2, Pages 437-442: Yang-Baxter Systems, Algebra Factorizations and Braided Categories]]>
http://www.mdpi.com/2075-1680/2/3/437
The Yang-Baxter equation first appeared in a paper by the Nobel laureate, C.N. Yang, and in R.J. Baxter’s work. Later, Vladimir Drinfeld, Vaughan F. R. Jones and Edward Witten were awarded Fields Medals for their work related to the Yang-Baxter equation. After a short review on this equation and the Yang-Baxter systems, we consider the problem of constructing algebra factorizations from Yang-Baxter systems. Our sketch of proof uses braided categories. Other problems are also proposed.Axioms2013-09-0323Communication10.3390/axioms20304374374422075-16802013-09-03doi: 10.3390/axioms2030437Florin Nichita<![CDATA[Axioms, Vol. 2, Pages 435-436: Special Issue: “q-Series and Related Topics in Special Functions and Analytic Number Theory”—Foreword]]>
http://www.mdpi.com/2075-1680/2/3/435
It is indeed a fairly common practice for scientific research journals and scientific research periodicals to publish special issues as well as conference proceedings. Quite frequently, these special issues are devoted exclusively to specific topics and/or are dedicated respectfully to commemorate the celebrated works of renowned research scientists. The following Special Issue: “q-Series and Related Topics in Special Functions and Analytic Number Theory” (see [1–8] below) is an outcome of the ongoing importance and popularity of such topics as Basic (or q-) Series and Basic (or q-) Polynomials. [...]Axioms2013-09-0323Editorial10.3390/axioms20304354354362075-16802013-09-03doi: 10.3390/axioms2030435Hari Srivastava<![CDATA[Axioms, Vol. 2, Pages 404-434: On Solutions of Holonomic Divided-Difference Equations on Nonuniform Lattices]]>
http://www.mdpi.com/2075-1680/2/3/404
The main aim of this paper is the development of suitable bases that enable the direct series representation of orthogonal polynomial systems on nonuniform lattices (quadratic lattices of a discrete or a q-discrete variable). We present two bases of this type, the first of which allows one to write solutions of arbitrary divided-difference equations in terms of series representations, extending results given by Sprenger for the q-case. Furthermore, it enables the representation of the Stieltjes function, which has already been used to prove the equivalence between the Pearson equation for a given linear functional and the Riccati equation for the formal Stieltjes function. If the Askey-Wilson polynomials are written in terms of this basis, however, the coefficients turn out to be not q-hypergeometric. Therefore, we present a second basis, which shares several relevant properties with the first one. This basis enables one to generate the defining representation of the Askey-Wilson polynomials directly from their divided-difference equation. For this purpose, the divided-difference equation must be rewritten in terms of suitable divided-difference operators developed in previous work by the first author.Axioms2013-07-2323Article10.3390/axioms20304044044342075-16802013-07-23doi: 10.3390/axioms2030404Mama FoupouagnigniWolfram KoepfMaurice Kenfack-NanghoSalifou Mboutngam<![CDATA[Axioms, Vol. 2, Pages 390-403: Discrete Integrals Based on Comonotonic Modularity]]>
http://www.mdpi.com/2075-1680/2/3/390
It is known that several discrete integrals, including the Choquet and Sugeno integrals, as well as some of their generalizations, are comonotonically modular functions. Based on a recent description of the class of comonotonically modular functions, we axiomatically identify more general families of discrete integrals that are comonotonically modular, including signed Choquet integrals and symmetric signed Choquet integrals, as well as natural extensions of Sugeno integrals.Axioms2013-07-2323Article10.3390/axioms20303903904032075-16802013-07-23doi: 10.3390/axioms2030390Miguel CouceiroJean-Luc Marichal<![CDATA[Axioms, Vol. 2, Pages 371-389: Nonnegative Scaling Vectors on the Interval]]>
http://www.mdpi.com/2075-1680/2/3/371
In this paper, we outline a method for constructing nonnegative scaling vectors on the interval. Scaling vectors for the interval have been constructed in [1–3]. The approach here is different in that the we start with an existing scaling vector ϕ that generates a multi-resolution analysis for L2(R) to create a scaling vector for the interval. If desired, the scaling vector can be constructed so that its components are nonnegative. Our construction uses ideas from [4,5] and we give results for scaling vectors satisfying certain support and continuity properties. These results also show that less edge functions are required to build multi-resolution analyses for L2 ([a; b]) than the methods described in [5,6].Axioms2013-07-0923Article10.3390/axioms20303713713892075-16802013-07-09doi: 10.3390/axioms2030371David RuchPatrick Van Fleet<![CDATA[Axioms, Vol. 2, Pages 345-370: Wavelet-Based Monitoring for Biosurveillance]]>
http://www.mdpi.com/2075-1680/2/3/345
Biosurveillance, focused on the early detection of disease outbreaks, relies on classical statistical control charts for detecting disease outbreaks. However, such methods are not always suitable in this context. Assumptions of normality, independence and stationarity are typically violated in syndromic data. Furthermore, outbreak signatures are typically of unknown patterns and, therefore, call for general detectors. We propose wavelet-based methods, which make less assumptions and are suitable for detecting abnormalities of unknown form. Wavelets have been widely used for data denoising and compression, but little work has been published on using them for monitoring. We discuss monitoring-based issues and illustrate them using data on military clinic visits in the USA.Axioms2013-07-0923Article10.3390/axioms20303453453702075-16802013-07-09doi: 10.3390/axioms2030345Galit Shmueli<![CDATA[Axioms, Vol. 2, Pages 311-344: Complexity L0-Penalized M-Estimation: Consistency in More Dimensions]]>
http://www.mdpi.com/2075-1680/2/3/311
We study the asymptotics in L2 for complexity penalized least squares regression for the discrete approximation of finite-dimensional signals on continuous domains—e.g., images—by piecewise smooth functions. We introduce a fairly general setting, which comprises most of the presently popular partitions of signal or image domains, like interval, wedgelet or related partitions, as well as Delaunay triangulations. Then, we prove consistency and derive convergence rates. Finally, we illustrate by way of relevant examples that the abstract results are useful for many applications.Axioms2013-07-0923Article10.3390/axioms20303113113442075-16802013-07-09doi: 10.3390/axioms2030311Laurent DemaretFelix FriedrichVolkmar LiebscherGerhard Winkler<![CDATA[Axioms, Vol. 2, Pages 286-310: Some Notes on the Use of the Windowed Fourier Transform for Spectral Analysis of Discretely Sampled Data]]>
http://www.mdpi.com/2075-1680/2/3/286
The properties of the Gabor and Morlet transforms are examined with respect to the Fourier analysis of discretely sampled data. Forward and inverse transform pairs based on a fixed window with uniform sampling of the frequency axis can satisfy numerically the energy and reconstruction theorems; however, transform pairs based on a variable window or nonuniform frequency sampling in general do not. Instead of selecting the shape of the window as some function of the central frequency, we propose constructing a single window with unit energy from an arbitrary set of windows that is applied over the entire frequency axis. By virtue of using a fixed window with uniform frequency sampling, such a transform satisfies the energy and reconstruction theorems. The shape of the window can be tailored to meet the requirements of the investigator in terms of time/frequency resolution. The algorithm extends naturally to the case of nonuniform signal sampling without modification beyond identification of the Nyquist interval.Axioms2013-06-2423Article10.3390/axioms20302862863102075-16802013-06-24doi: 10.3390/axioms2030286Robert Johnson<![CDATA[Axioms, Vol. 2, Pages 271-285: Change Detection Using Wavelets in Solution Monitoring Data for Nuclear Safeguards]]>
http://www.mdpi.com/2075-1680/2/2/271
Wavelet analysis is known to be a good option for change detection in many contexts. Detecting changes in solution volumes that are measured with both additive and relative error is an important aspect of safeguards for facilities that process special nuclear material. This paper qualitatively compares wavelet-based change detection to a lag-one differencing option using realistic simulated solution volume data for which the true change points are known. We then show quantitatively that Haar wavelet-based change detection is effective for finding the approximate location of each change point, and that a simple piecewise linear optimization step is effective to refine the initial wavelet-based change point estimate.Axioms2013-06-1822Article10.3390/axioms20202712712852075-16802013-06-18doi: 10.3390/axioms2020271Claire LongoTom BurrKary Myers<![CDATA[Axioms, Vol. 2, Pages 224-270: Quantitative Hahn-Banach Theorems and Isometric Extensions for Wavelet and Other Banach Spaces]]>
http://www.mdpi.com/2075-1680/2/2/224
We introduce and study Clarkson, Dol’nikov-Pichugov, Jacobi and mutual diameter constants reflecting the geometry of a Banach space and Clarkson, Jacobi and Pichugov classes of Banach spaces and their relations with James, self-Jung, Kottman and Schäffer constants in order to establish quantitative versions of Hahn-Banach separability theorem and to characterise the isometric extendability of Hölder-Lipschitz mappings. Abstract results are further applied to the spaces and pairs from the wide classes IG and IG+ and non-commutative Lp-spaces. The intimate relation between the subspaces and quotients of the IG-spaces on one side and various types of anisotropic Besov, Lizorkin-Triebel and Sobolev spaces of functions on open subsets of an Euclidean space defined in terms of differences, local polynomial approximations, wavelet decompositions and other means (as well as the duals and the lp-sums of all these spaces) on the other side, allows us to present the algorithm of extending the main results of the article to the latter spaces and pairs. Special attention is paid to the matter of sharpness. Our approach is quasi-Euclidean in its nature because it relies on the extrapolation of properties of Hilbert spaces and the study of 1-complemented subspaces of the spaces under consideration.Axioms2013-05-2322Article10.3390/axioms20202242242702075-16802013-05-23doi: 10.3390/axioms2020224Sergey Ajiev<![CDATA[Axioms, Vol. 2, Pages 208-223: Using the Choquet Integral in the Fuzzy Reasoning Method of Fuzzy Rule-Based Classification Systems]]>
http://www.mdpi.com/2075-1680/2/2/208
In this paper we present a new fuzzy reasoning method in which the Choquet integral is used as aggregation function. In this manner, we can take into account the interaction among the rules of the system. For this reason, we consider several fuzzy measures, since it is a key point on the subsequent success of the Choquet integral, and we apply the new method with the same fuzzy measure for all the classes. However, the relationship among the set of rules of each class can be different and therefore the best fuzzy measure can change depending on the class. Consequently, we propose a learning method by means of a genetic algorithm in which the most suitable fuzzy measure for each class is computed. From the obtained results it is shown that our new proposal allows the performance of the classical fuzzy reasoning methods of the winning rule and additive combination to be enhanced whenever the fuzzy measure is appropriate for the tackled problem.Axioms2013-04-2322Article10.3390/axioms20202082082232075-16802013-04-23doi: 10.3390/axioms2020208Edurne BarrenecheaHumberto BustinceJavier FernandezDaniel PaternainJosé Sanz<![CDATA[Axioms, Vol. 2, Pages 182-207: Time Scale Analysis of Interest Rate Spreads and Output Using Wavelets]]>
http://www.mdpi.com/2075-1680/2/2/182
This paper adds to the literature on the information content of different spreads for real activity by explicitly taking into account the time scale relationship between a variety of monetary and financial indicators (real interest rate, term and credit spreads) and output growth. By means of wavelet-based exploratory data analysis we obtain richer results relative to the aggregate analysis by identifying the dominant scales of variation in the data and the scales and location at which structural breaks have occurred. Moreover, using the “double residuals” regression analysis on a scale-by-scale basis, we find that changes in the spread in several markets have different information content for output at different time frames. This is consistent with the idea that allowing for different time scales of variation in the data can provide a fruitful understanding of the complex dynamics of economic relationships between variables with non-stationary or transient components, certainly richer than those obtained using standard time domain methods.Axioms2013-04-2322Article10.3390/axioms20201821822072075-16802013-04-23doi: 10.3390/axioms2020182Marco GallegatiJames RamseyWilli Semmler<![CDATA[Axioms, Vol. 2, Pages 142-181: A Sequential, Implicit, Wavelet-Based Solver for Multi-Scale Time-Dependent Partial Differential Equations]]>
http://www.mdpi.com/2075-1680/2/2/142
This paper describes and tests a wavelet-based implicit numerical method for solving partial differential equations. Intended for problems with localized small-scale interactions, the method exploits the form of the wavelet decomposition to divide the implicit system created by the time-discretization into multiple smaller systems that can be solved sequentially. Included is a test on a basic non-linear problem, with both the results of the test, and the time required to calculate them, compared with control results based on a single system with fine resolution. The method is then tested on a non-trivial problem, its computational time and accuracy checked against control results. In both tests, it was found that the method requires less computational expense than the control. Furthermore, the method showed convergence towards the fine resolution control results.Axioms2013-04-2322Article10.3390/axioms20201421421812075-16802013-04-23doi: 10.3390/axioms2020142Donald McLarenLucy CampbellRémi Vaillancourt<![CDATA[Axioms, Vol. 2, Pages 122-141: Construction of Multiwavelets on an Interval]]>
http://www.mdpi.com/2075-1680/2/2/122
Boundary functions for wavelets on a finite interval are often constructed as linear combinations of boundary-crossing scaling functions. An alternative approach is based on linear algebra techniques for truncating the infinite matrix of the DiscreteWavelet Transform to a finite one. In this article we show how an algorithm of Madych for scalar wavelets can be generalized to multiwavelets, given an extra assumption. We then develop a new algorithm that does not require this additional condition. Finally, we apply results from a previous paper to resolve the non-uniqueness of the algorithm by imposing regularity conditions (including approximation orders) on the boundary functions.Axioms2013-04-1722Article10.3390/axioms20201221221412075-16802013-04-17doi: 10.3390/axioms2020122Ahmet AltürkFritz Keinert<![CDATA[Axioms, Vol. 2, Pages 100-121: Divergence-Free Multiwavelets on the Half Plane]]>
http://www.mdpi.com/2075-1680/2/2/100
We use the biorthogonal multiwavelets related by differentiation constructed in previous work to construct compactly supported biorthogonal multiwavelet bases for the space of vector fields on the upper half plane R2 + such that the reconstruction wavelets are divergence-free and have vanishing normal components on the boundary of R2 +. Such wavelets are suitable to study the Navier–Stokes equations on a half plane when imposing a Navier boundary condition.Axioms2013-04-1122Article10.3390/axioms20201001001212075-16802013-04-11doi: 10.3390/axioms2020100Joseph LakeyPhan Nguyen<![CDATA[Axioms, Vol. 2, Pages 85-99: On the q-Analogues of Srivastava’s Triple Hypergeometric Functions]]>
http://www.mdpi.com/2075-1680/2/2/85
We find Euler integral formulas, summation and reduction formulas for q-analogues of Srivastava’s three triple hypergeometric functions. The proofs use q-analogues of Picard’s integral formula for the first Appell function, a summation formula for the first Appell function based on the Bayley–Daum formula, and a general triple series reduction formula of Karlsson. Many of the formulas are purely formal, since it is difficult to find convergence regions for these functions of several complex variables. We use the Ward q-addition to describe the known convergence regions of q-Appell and q-Lauricella functions.Axioms2013-04-1122Article10.3390/axioms202008585992075-16802013-04-11doi: 10.3390/axioms2020085Thomas Ernst<![CDATA[Axioms, Vol. 2, Pages 67-84: Mollification Based on Wavelets]]>
http://www.mdpi.com/2075-1680/2/2/67
The mollification obtained by truncating the expansion in wavelets is studied, where the wavelets are so chosen that noise is reduced and the Gibbs phenomenon does not occur. The estimations of the error of approximation of the mollification are given for the case when the fractional derivative of a function is calculated. Noting that the estimations are applicable even when the orthogonality of the wavelets is not satisfied, we study mollifications using unorthogonalized wavelets, as well as those using orthogonal wavelets.Axioms2013-03-2522Article10.3390/axioms202006767842075-16802013-03-25doi: 10.3390/axioms2020067Tohru MoritaKen-ichi Sato<![CDATA[Axioms, Vol. 2, Pages 58-66: Golden Ratio and a Ramanujan-Type Integral]]>
http://www.mdpi.com/2075-1680/2/1/58
In this paper, we give a pedagogical introduction to several beautiful formulas discovered by Ramanujan. Using these results, we evaluate a Ramanujan-type integral formula. The result can be expressed in terms of the Golden Ratio.Axioms2013-03-2021Article10.3390/axioms201005858662075-16802013-03-20doi: 10.3390/axioms2010058Hei-Chi Chan<![CDATA[Axioms, Vol. 2, Pages 44-57: Signal Estimation Using Wavelet Analysis of Solution Monitoring Data for Nuclear Safeguards]]>
http://www.mdpi.com/2075-1680/2/1/44
Wavelets are explored as a data smoothing (or de-noising) option for solution monitoring data in nuclear safeguards. In wavelet-smoothed data, the Gibbs phenomenon can obscure important data features that may be of interest. This paper compares wavelet smoothing to piecewise linear smoothing and local kernel smoothing, and illustrates that the Haar wavelet basis is effective for reducing the Gibbs phenomenon.Axioms2013-03-2021Article10.3390/axioms201004444572075-16802013-03-20doi: 10.3390/axioms2010044Tom BurrClaire Longo<![CDATA[Axioms, Vol. 2, Pages 20-43: Some Modular Relations Analogues to the Ramanujan’s Forty Identities with Its Applications to Partitions]]>
http://www.mdpi.com/2075-1680/2/1/20
Recently, the authors have established a large class of modular relations involving the Rogers-Ramanujan type functions J(q) and K(q) of order ten. In this paper, we establish further modular relations connecting these two functions with Rogers-Ramanujan functions, Göllnitz-Gordon functions and cubic functions, which are analogues to the Ramanujan’s forty identities for Rogers-Ramanujan functions. Furthermore, we give partition theoretic interpretations of some of our modular relations.Axioms2013-02-1821Article10.3390/axioms201002020432075-16802013-02-18doi: 10.3390/axioms2010020Chandrashekar AdigaNasser Bulkhali<![CDATA[Axioms, Vol. 2, Pages 10-19: Generalized q-Stirling Numbers and Their Interpolation Functions]]>
http://www.mdpi.com/2075-1680/2/1/10
In this paper, we define the generating functions for the generalized q-Stirling numbers of the second kind. By applying Mellin transform to these functions, we construct interpolation functions of these numbers at negative integers. We also derive some identities and relations related to q-Bernoulli numbers and polynomials and q-Stirling numbers of the second kind.Axioms2013-02-0821Article10.3390/axioms201001010192075-16802013-02-08doi: 10.3390/axioms2010010Hacer OzdenIsmail CangulYilmaz Simsek<![CDATA[Axioms, Vol. 2, Pages 1-9: On the Content Bound for Real Quadratic Field Extensions]]>
http://www.mdpi.com/2075-1680/2/1/1
Let K be a finite extension of ℚ and let S = {ν} denote the collection of K normalized absolute values on K. Let V K + denote the additive group of adeles over K and let c : V K + → ℝ ≥0 denote the content map defined as c( { a v } ) = ∏ v∈s v( a v ) for { a v }∈ V K + . A classical result of J. W. S. Cassels states that there is a constant c &gt; 0 depending only on the field K with the following property: if { a v }∈ V K + with c( { a v } ) &gt; c , then there exists a non-zero element b ∈ K for which v(b)≤v( a v ), ∀v∈ S . Let cK be the greatest lower bound of the set of all c that satisfy this property. In the case that K is a real quadratic extension there is a known upper bound for cK due to S. Lang. The purpose of this paper is to construct a new upper bound for cK in the case that K has class number one. We compare our new bound with Lang’s bound for various real quadratic extensions and find that our new bound is better than Lang’s in many instances.Axioms2012-12-2821Article10.3390/axioms2010001192075-16802012-12-28doi: 10.3390/axioms2010001Robert Underwood<![CDATA[Axioms, Vol. 1, Pages 395-403: Generating Functions for q-Apostol Type Frobenius–Euler Numbers and Polynomials]]>
http://www.mdpi.com/2075-1680/1/3/395
The aim of this paper is to construct generating functions, related to nonnegative real parameters, for q-Eulerian type polynomials and numbers (or q-Apostol type Frobenius–Euler polynomials and numbers). We derive some identities for these polynomials and numbers based on the generating functions and functional equations. We also give multiplication formula for the generalized Apostol type Frobenius–Euler polynomials.Axioms2012-12-0713Article10.3390/axioms10303953954032075-16802012-12-07doi: 10.3390/axioms1030395Yilmaz Simsek<![CDATA[Axioms, Vol. 1, Pages 384-394: On the Equilibria of Generalized Dynamical Systems]]>
http://www.mdpi.com/2075-1680/1/3/384
This research work presents original properties of the equilibrium critical (ideal) points sets for an important class of generalized dynamical systems. The existence and significant results regarding such points are specified. Strong connections with the Vector Optimization by the Efficiency and the Potential Theory together with its applications following Choquet’s boundaries are provided.Axioms2012-12-0613Article10.3390/axioms10303843843942075-16802012-12-06doi: 10.3390/axioms1030384Vasile Postolică<![CDATA[Axioms, Vol. 1, Pages 372-383: The Cranks for 5-Core Partitions]]>
http://www.mdpi.com/2075-1680/1/3/372
It is well known that the number of 5-core partitions of 5kn + 5k − 1 is a multiple of 5k. In [1] a statistic called a crank was developed to sort the 5-core partitions of 5n + 4 and 25n + 24 into 5 and 25 classes of equal size, respectively. In this paper we will develop the cranks that can be used to sort the 5-core partitions of 5kn + 5k − 1 into 5k classes of equal size.Axioms2012-12-0313Article10.3390/axioms10303723723832075-16802012-12-03doi: 10.3390/axioms1030372Louis Kolitsch<![CDATA[Axioms, Vol. 1, Pages 365-371: New Curious Bilateral q-Series Identities]]>
http://www.mdpi.com/2075-1680/1/3/365
By applying a classical method, already employed by Cauchy, to a terminating curious summation by one of the authors, a new curious bilateral q-series identity is derived. We also apply the same method to a quadratic summation by Gessel and Stanton, and to a cubic summation by Gasper, respectively, to derive a bilateral quadratic and a bilateral cubic summation formula.Axioms2012-10-3113Article10.3390/axioms10303653653712075-16802012-10-31doi: 10.3390/axioms1030365Frédéric JouhetMichael J. Schlosser<![CDATA[Axioms, Vol. 1, Pages 324-364: Frobenius–Schur Indicator for Categories with Duality]]>
http://www.mdpi.com/2075-1680/1/3/324
We introduce the Frobenius–Schur indicator for categories with duality to give a category-theoretical understanding of various generalizations of the Frobenius–Schur theorem including that for semisimple quasi-Hopf algebras, weak Hopf C*-algebras and association schemes. Our framework also clarifies a mechanism of how the “twisted” theory arises from the ordinary case. As a demonstration, we establish twisted versions of the Frobenius–Schur theorem for various algebraic objects. We also give several applications to the quantum SL2.Axioms2012-10-2313Article10.3390/axioms10303243243642075-16802012-10-23doi: 10.3390/axioms1030324Kenichi Shimizu<![CDATA[Axioms, Vol. 1, Pages 291-323: The Hecke Bicategory]]>
http://www.mdpi.com/2075-1680/1/3/291
We present an application of the program of groupoidification leading up to a sketch of a categorification of the Hecke algebroid—the category of permutation representations of a finite group. As an immediate consequence, we obtain a categorification of the Hecke algebra. We suggest an explicit connection to new higher isomorphisms arising from incidence geometries, which are solutions of the Zamolodchikov tetrahedron equation. This paper is expository in style and is meant as a companion to Higher Dimensional Algebra VII: Groupoidification and an exploration of structures arising in the work in progress, Higher Dimensional Algebra VIII: The Hecke Bicategory, which introduces the Hecke bicategory in detail.Axioms2012-10-0913Communication10.3390/axioms10302912913232075-16802012-10-09doi: 10.3390/axioms1030291Alexander E. Hoffnung<![CDATA[Axioms, Vol. 1, Pages 259-290: The Sum of a Finite Group of Weights of a Hopf Algebra]]>
http://www.mdpi.com/2075-1680/1/3/259
Motivated by the orthogonality relations for irreducible characters of a finite group, we evaluate the sum of a finite group of linear characters of a Hopf algebra, at all grouplike and skew-primitive elements. We then discuss results for products of skew-primitive elements. Examples include groups, (quantum groups over) Lie algebras, the small quantum groups of Lusztig, and their variations (by Andruskiewitsch and Schneider).Axioms2012-10-0513Article10.3390/axioms10302592592902075-16802012-10-05doi: 10.3390/axioms1030259Apoorva Khare<![CDATA[Axioms, Vol. 1, Pages 238-258: A Class of Extended Fractional Derivative Operators and Associated Generating Relations Involving Hypergeometric Functions]]>
http://www.mdpi.com/2075-1680/1/3/238
Recently, an extended operator of fractional derivative related to a generalized Beta function was used in order to obtain some generating relations involving the extended hypergeometric functions [1]. The main object of this paper is to present a further generalization of the extended fractional derivative operator and apply the generalized extended fractional derivative operator to derive linear and bilinear generating relations for the generalized extended Gauss, Appell and Lauricella hypergeometric functions in one, two and more variables. Some other properties and relationships involving the Mellin transforms and the generalized extended fractional derivative operator are also given.Axioms2012-10-0513Article10.3390/axioms10302382382582075-16802012-10-05doi: 10.3390/axioms1030238H. M. SrivastavaRakesh K. ParmarPurnima Chopra<![CDATA[Axioms, Vol. 1, Pages 226-237: Hopf Algebra Symmetries of an Integrable Hamiltonian for Anyonic Pairing]]>
http://www.mdpi.com/2075-1680/1/2/226
Since the advent of Drinfel’d’s double construction, Hopf algebraic structures have been a centrepiece for many developments in the theory and analysis of integrable quantum systems. An integrable anyonic pairing Hamiltonian will be shown to admit Hopf algebra symmetries for particular values of its coupling parameters. While the integrable structure of the model relates to the well-known six-vertex solution of the Yang–Baxter equation, the Hopf algebra symmetries are not in terms of the quantum algebra Uq(sl(2)). Rather, they are associated with the Drinfel’d doubles of dihedral group algebras D(Dn).Axioms2012-09-2012Article10.3390/axioms10202262262372075-16802012-09-20doi: 10.3390/axioms1020226Jon Links<![CDATA[Axioms, Vol. 1, Pages 201-225: Bundles over Quantum RealWeighted Projective Spaces]]>
http://www.mdpi.com/2075-1680/1/2/201
The algebraic approach to bundles in non-commutative geometry and the definition of quantum real weighted projective spaces are reviewed. Principal U(1)-bundles over quantum real weighted projective spaces are constructed. As the spaces in question fall into two separate classes, the negative or odd class that generalises quantum real projective planes and the positive or even class that generalises the quantum disc, so do the constructed principal bundles. In the negative case the principal bundle is proven to be non-trivial and associated projective modules are described. In the positive case the principal bundles turn out to be trivial, and so all the associated modules are free. It is also shown that the circle (co)actions on the quantum Seifert manifold that define quantum real weighted projective spaces are almost free.Axioms2012-09-1712Article10.3390/axioms10202012012252075-16802012-09-17doi: 10.3390/axioms1020201Tomasz BrzezińskiSimon A. Fairfax<![CDATA[Axioms, Vol. 1, Pages 186-200: From Coalgebra to Bialgebra for the Six-Vertex Model: The Star-Triangle Relation as a Necessary Condition for Commuting Transfer Matrices]]>
http://www.mdpi.com/2075-1680/1/2/186
Using the most elementary methods and considerations, the solution of the star-triangle condition (a2+b2-c2)/2ab = ((a’)^2+(b’)^2-(c’))^2/2a’b’ is shown to be a necessary condition for the extension of the operator coalgebra of the six-vertex model to a bialgebra. A portion of the bialgebra acts as a spectrum-generating algebra for the algebraic Bethe ansatz, with which higher-dimensional representations of the bialgebra can be constructed. The star-triangle relation is proved to be necessary for the commutativity of the transfer matrices T(a, b, c) and T(a’, b’, c’).Axioms2012-08-2712Article10.3390/axioms10201861862002075-16802012-08-27doi: 10.3390/axioms1020186Jeffrey R. Schmidt<![CDATA[Axioms, Vol. 1, Pages 173-185: The Duality between Corings and Ring Extensions]]>
http://www.mdpi.com/2075-1680/1/2/173
We study the duality between corings and ring extensions. We construct a new category with a self-dual functor acting on it, which extends that duality. This construction can be seen as the non-commutative case of another duality extension: the duality between finite dimensional algebras and coalgebra. Both these duality extensions have some similarities with the Pontryagin-van Kampen duality theorem.Axioms2012-08-1012Article10.3390/axioms10201731731852075-16802012-08-10doi: 10.3390/axioms1020173Florin F. NichitaBartosz Zielinski<![CDATA[Axioms, Vol. 1, Pages 155-172: Quasitriangular Structure of Myhill–Nerode Bialgebras]]>
http://www.mdpi.com/2075-1680/1/2/155
In computer science the Myhill–Nerode Theorem states that a set L of words in a finite alphabet is accepted by a finite automaton if and only if the equivalence relation ∼L, defined as x ∼L y if and only if xz ∈ L exactly when yz ∈ L, ∀z, has finite index. The Myhill–Nerode Theorem can be generalized to an algebraic setting giving rise to a collection of bialgebras which we call Myhill–Nerode bialgebras. In this paper we investigate the quasitriangular structure of Myhill–Nerode bialgebras.Axioms2012-07-2412Article10.3390/axioms10201551551722075-16802012-07-24doi: 10.3390/axioms1020155Robert G. Underwood<![CDATA[Axioms, Vol. 1, Pages 149-154: Hasse-Schmidt Derivations and the Hopf Algebra of Non-Commutative Symmetric Functions]]>
http://www.mdpi.com/2075-1680/1/2/149
Let NSymm be the Hopf algebra of non-commutative symmetric functions (in an infinity of indeterminates): . It is shown that an associative algebra A with a Hasse-Schmidt derivation ) on it is exactly the same as an NSymm module algebra. The primitives of NSymm act as ordinary derivations. There are many formulas for the generators in terms of the primitives (and vice-versa). This leads to formulas for the higher derivations in a Hasse-Schmidt derivation in terms of ordinary derivations, such as the known formulas of Heerema and Mirzavaziri (and also formulas for ordinary derivations in terms of the elements of a Hasse-Schmidt derivation). These formulas are over the rationals; no such formulas are possible over the integers. Many more formulas are derivable.Axioms2012-07-1612Communication10.3390/axioms10201491491542075-16802012-07-16doi: 10.3390/axioms1020149Michiel Hazewinkel<![CDATA[Axioms, Vol. 1, Pages 111-148: Valued Graphs and the Representation Theory of Lie Algebras]]>
http://www.mdpi.com/2075-1680/1/2/111
Quivers (directed graphs), species (a generalization of quivers) and their representations play a key role in many areas of mathematics including combinatorics, geometry, and algebra. Their importance is especially apparent in their applications to the representation theory of associative algebras, Lie algebras, and quantum groups. In this paper, we discuss the most important results in the representation theory of species, such as Dlab and Ringel’s extension of Gabriel’s theorem, which classifies all species of finite and tame representation type. We also explain the link between species and K-species (where K is a field). Namely, we show that the category of K -species can be viewed as a subcategory of the category of species. Furthermore, we prove two results about the structure of the tensor ring of a species containing no oriented cycles. Specifically, we prove that two such species have isomorphic tensor rings if and only if they are isomorphic as “crushed” species, and we show that if K is a perfect field, then the tensor algebra of a K -species tensored with the algebraic closure of K is isomorphic to, or Morita equivalent to, the path algebra of a quiver.Axioms2012-07-0412Article10.3390/axioms10201111111482075-16802012-07-04doi: 10.3390/axioms1020111Joel Lemay<![CDATA[Axioms, Vol. 1, Pages 99-110: Fat Triangulations, Curvature and Quasiconformal Mappings]]>
http://www.mdpi.com/2075-1680/1/2/99
We investigate the interplay between the existence of fat triangulations, P L approximations of Lipschitz–Killing curvatures and the existence of quasiconformal mappings. In particular we prove that if there exists a quasiconformal mapping between two P L or smooth n-manifolds, then their Lipschitz–Killing curvatures are bilipschitz equivalent. An extension to the case of almost Riemannian manifolds, of a previous existence result of quasimeromorphic mappings on manifolds due to the first author is also given.Axioms2012-07-0412Article10.3390/axioms1020099991102075-16802012-07-04doi: 10.3390/axioms1020099Emil SaucanMeir Katchalski<![CDATA[Axioms, Vol. 1, Pages 74-98: Gradings, Braidings, Representations, Paraparticles: Some Open Problems]]>
http://www.mdpi.com/2075-1680/1/1/74
A research proposal on the algebraic structure, the representations and the possible applications of paraparticle algebras is structured in three modules: The first part stems from an attempt to classify the inequivalent gradings and braided group structures present in the various parastatistical algebraic models. The second part of the proposal aims at refining and utilizing a previously published methodology for the study of the Fock-like representations of the parabosonic algebra, in such a way that it can also be directly applied to the other parastatistics algebras. Finally, in the third part, a couple of Hamiltonians is proposed, suitable for modeling the radiation matter interaction via a parastatistical algebraic model.Axioms2012-06-1511Communication10.3390/axioms101007474982075-16802012-06-15doi: 10.3390/axioms1010074Konstantinos Kanakoglou<![CDATA[Axioms, Vol. 1, Pages 38-73: Foundations of Inference]]>
http://www.mdpi.com/2075-1680/1/1/38
We present a simple and clear foundation for finite inference that unites and significantly extends the approaches of Kolmogorov and Cox. Our approach is based on quantifying lattices of logical statements in a way that satisfies general lattice symmetries. With other applications such as measure theory in mind, our derivations assume minimal symmetries, relying on neither negation nor continuity nor differentiability. Each relevant symmetry corresponds to an axiom of quantification, and these axioms are used to derive a unique set of quantifying rules that form the familiar probability calculus. We also derive a unique quantification of divergence, entropy and information.Axioms2012-06-1511Article10.3390/axioms101003838732075-16802012-06-15doi: 10.3390/axioms1010038Kevin H. KnuthJohn Skilling<![CDATA[Axioms, Vol. 1, Pages 33-37: Introduction to the Yang-Baxter Equation with Open Problems]]>
http://www.mdpi.com/2075-1680/1/1/33
The Yang-Baxter equation first appeared in theoretical physics, in a paper by the Nobel laureate C. N. Yang, and in statistical mechanics, in R. J. Baxter’s work. Later, it turned out that this equation plays a crucial role in: quantum groups, knot theory, braided categories, analysis of integrable systems, quantum mechanics, non-commutative descent theory, quantum computing, non-commutative geometry, etc. Many scientists have found solutions for the Yang-Baxter equation, obtaining qualitative results (using the axioms of various algebraic structures) or quantitative results (usually using computer calculations). However, the full classification of its solutions remains an open problem. In this paper, we present the (set-theoretical) Yang-Baxter equation, we sketch the proof of a new theorem, we state some problems, and discuss about directions for future research.Axioms2012-04-2611Communication10.3390/axioms101003333372075-16802012-04-26doi: 10.3390/axioms1010033Florin Nichita<![CDATA[Axioms, Vol. 1, Pages 21-32: Axiomatic of Fuzzy Complex Numbers]]>
http://www.mdpi.com/2075-1680/1/1/21
Fuzzy numbers are fuzzy subsets of the set of real numbers satisfying some additional conditions. Fuzzy numbers allow us to model very difficult uncertainties in a very easy way. Arithmetic operations on fuzzy numbers have also been developed, and are based mainly on the crucial Extension Principle. When operating with fuzzy numbers, the results of our calculations strongly depend on the shape of the membership functions of these numbers. Logically, less regular membership functions may lead to very complicated calculi. Moreover, fuzzy numbers with a simpler shape of membership functions often have more intuitive and more natural interpretations. But not only must we apply the concept and the use of fuzzy sets, and its particular case of fuzzy number, but also the new and interesting mathematical construct designed by Fuzzy Complex Numbers, which is much more than a correlate of Complex Numbers in Mathematical Analysis. The selected perspective attempts here that of advancing through axiomatic descriptions.Axioms2012-04-2011Article10.3390/axioms101002121322075-16802012-04-20doi: 10.3390/axioms1010021Angel Garrido<![CDATA[Axioms, Vol. 1, Pages 9-20: Discrete Integrals and Axiomatically Defined Functionals]]>
http://www.mdpi.com/2075-1680/1/1/9
Several discrete universal integrals on finite universes are discussed from an axiomatic point of view. We start from the first attempt due to B. Riemann and cover also most recent approaches based on level dependent capacities. Our survey includes, among others, the Choquet and the Sugeno integral and general copula-based integrals.Axioms2012-04-2011Article10.3390/axioms10100099202075-16802012-04-20doi: 10.3390/axioms1010009Erich Peter KlementRadko Mesiar<![CDATA[Axioms, Vol. 1, Pages 4-8: An Itô Formula for an Accretive Operator]]>
http://www.mdpi.com/2075-1680/1/1/4
We give an Itô formula associated to a non-linear semi-group associated to a m-accretive operator.Axioms2012-03-2111Communication10.3390/axioms1010004482075-16802012-03-21doi: 10.3390/axioms1010004Rémi Léandre<![CDATA[Axioms, Vol. 1, Pages 1-3: Another Journal on Mathematical Logic and Mathematical Physics?]]>
http://www.mdpi.com/2075-1680/1/1/1
It is my great pleasure to welcome you to Axioms: Mathematical Logic and Mathematical Physics, a new open access journal, which is dedicated to the foundations (structure and axiomatic basis, in particular) of mathematical and physical theories, not only on crisp or strictly classical sense, but also on fuzzy and generalized sense. This includes the more innovative current scientific trends, devoted to discover and solving new, defying problems. Our new journal does not try to be the same as those journals already dedicated to this field. Below we highlight what makes Axioms: Mathematical Logic and Mathematical Physics different. [...]Axioms2011-09-0111Editorial10.3390/axioms1010001132075-16802011-09-01doi: 10.3390/axioms1010001Angel Garrido