No Uncountable Polish Group Can be a Right-Angled Artin Group*Axioms* **2017**, *6*(2), 13; doi:10.3390/axioms6020013 - 11 May 2017**Abstract **

We prove that if *G* is a Polish group and *A* a group admitting a system of generators whose associated length function satisfies: (i) if $0<k<\omega $ , then $lg\left(x\right)\le lg({x}^{k}$

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We prove that if *G* is a Polish group and *A* a group admitting a system of generators whose associated length function satisfies: (i) if $0<k<\omega $ , then $lg\left(x\right)\le lg\left({x}^{k}\right)$ ; (ii) if $lg\left(y\right)<k<\omega $ and ${x}^{k}=y$ , then $x=e$ , then there exists a subgroup ${G}^{*}$ of *G* of size $\mathfrak{b}$ (the bounding number) such that ${G}^{*}$ is not embeddable in *A*. In particular, we prove that the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes analogous results for free and free abelian uncountable groups.
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Toward Measuring Network Aesthetics Based on Symmetry*Axioms* **2017**, *6*(2), 12; doi:10.3390/axioms6020012 - 6 May 2017**Abstract **

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In this exploratory paper, we discuss quantitative graph-theoretical measures of network aesthetics. Related work in this area has typically focused on geometrical features (e.g., line crossings or edge bendiness) of drawings or visual representations of graphs which purportedly affect an observer’s perception. Here

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In this exploratory paper, we discuss quantitative graph-theoretical measures of network aesthetics. Related work in this area has typically focused on geometrical features (e.g., line crossings or edge bendiness) of drawings or visual representations of graphs which purportedly affect an observer’s perception. Here we take a very different approach, abandoning reliance on geometrical properties, and apply information-theoretic measures to abstract graphs and networks directly (rather than to their visual representaions) as a means of capturing classical appreciation of structural symmetry. Examples are used solely to motivate the approach to measurement, and to elucidate our symmetry-based mathematical theory of network aesthetics.
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Orientation Asymmetric Surface Model for Membranes: Finsler Geometry Modeling*Axioms* **2017**, *6*(2), 10; doi:10.3390/axioms6020010 - 25 April 2017**Abstract **

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We study triangulated surface models with nontrivial surface metrices for membranes. The surface model is defined by a mapping $\mathbf{r}$ from a two-dimensional parameter space *M* to the three-dimensional Euclidean space ${\mathbf{R}}^{3}$ . The metric variable ${g}_{ab}$ , which is

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We study triangulated surface models with nontrivial surface metrices for membranes. The surface model is defined by a mapping $\mathbf{r}$ from a two-dimensional parameter space *M* to the three-dimensional Euclidean space ${\mathbf{R}}^{3}$ . The metric variable ${g}_{ab}$ , which is always fixed to the Euclidean metric ${\delta}_{ab}$ , can be extended to a more general non-Euclidean metric on *M* in the continuous model. The problem we focus on in this paper is whether such an extension is well defined or not in the discrete model. We find that a discrete surface model with a nontrivial metric becomes well defined if it is treated in the context of Finsler geometry (FG) modeling, where triangle edge length in *M* depends on the direction. It is also shown that the discrete FG model is orientation asymmetric on invertible surfaces in general, and for this reason, the FG model has a potential advantage for describing real physical membranes, which are expected to have some asymmetries for orientation-changing transformations.
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Multivariate Extended Gamma Distribution*Axioms* **2017**, *6*(2), 11; doi:10.3390/axioms6020011 - 24 April 2017**Abstract **

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In this paper, I consider multivariate analogues of the extended gamma density, which will provide multivariate extensions to Tsallis statistics and superstatistics. By making use of the pathway parameter $\beta $ , multivariate generalized gamma density can be obtained from the model considered here.

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In this paper, I consider multivariate analogues of the extended gamma density, which will provide multivariate extensions to Tsallis statistics and superstatistics. By making use of the pathway parameter $\beta $ , multivariate generalized gamma density can be obtained from the model considered here. Some of its special cases and limiting cases are also mentioned. Conditional density, best predictor function, regression theory, etc., connected with this model are also introduced.
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Euclidean Algorithm for Extension of Symmetric Laurent Polynomial Matrix and Its Application in Construction of Multiband Symmetric Perfect Reconstruction Filter Bank*Axioms* **2017**, *6*(2), 9; doi:10.3390/axioms6020009 - 20 April 2017**Abstract **

For a given pair of *s*-dimensional real Laurent polynomials $(\overrightarrow{a}\left(z\right),\overrightarrow{b}\left(z\right))$ , which has a certain type of symmetry and satisfies the dual condition ${\overrightarrow{b}\left(z\right)}^{T}\stackrel{}{a}$

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For a given pair of *s*-dimensional real Laurent polynomials $(\overrightarrow{a}\left(z\right),\overrightarrow{b}\left(z\right))$ , which has a certain type of symmetry and satisfies the dual condition ${\overrightarrow{b}\left(z\right)}^{T}\overrightarrow{a}\left(z\right)=1$ , an $s\times s$ Laurent polynomial matrix $A\left(z\right)$ (together with its inverse ${A}^{-1}\left(z\right)$ ) is called a symmetric Laurent polynomial matrix extension of the dual pair $(\overrightarrow{a}\left(z\right),\overrightarrow{b}\left(z\right))$ if $A\left(z\right)$ has similar symmetry, the inverse ${A}^{-1}\left(Z\right)$ also is a Laurent polynomial matrix, the first column of $A\left(z\right)$ is $\overrightarrow{a}\left(z\right)$ and the first row of ${A}^{-1}\left(z\right)$ is ${\left(\overrightarrow{b}\left(z\right)\right)}^{T}$ . In this paper, we introduce the Euclidean symmetric division and the symmetric elementary matrices in the Laurent polynomial ring and reveal their relation. Based on the Euclidean symmetric division algorithm in the Laurent polynomial ring, we develop a novel and effective algorithm for symmetric Laurent polynomial matrix extension. We also apply the algorithm in the construction of multi-band symmetric perfect reconstruction filter banks.
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Expansion of the Kullback-Leibler Divergence, and a New Class of Information Metrics*Axioms* **2017**, *6*(2), 8; doi:10.3390/axioms6020008 - 1 April 2017**Abstract **

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Inferring and comparing complex, multivariable probability density functions is fundamental to problems in several fields, including probabilistic learning, network theory, and data analysis. Classification and prediction are the two faces of this class of problem. This study takes an approach that simplifies many

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Inferring and comparing complex, multivariable probability density functions is fundamental to problems in several fields, including probabilistic learning, network theory, and data analysis. Classification and prediction are the two faces of this class of problem. This study takes an approach that simplifies many aspects of these problems by presenting a structured, series expansion of the Kullback-Leibler divergence—a function central to information theory—and devise a distance metric based on this divergence. Using the Möbius inversion duality between multivariable entropies and multivariable interaction information, we express the divergence as an additive series in the number of interacting variables, which provides a restricted and simplified set of distributions to use as approximation and with which to model data. Truncations of this series yield approximations based on the number of interacting variables. The first few terms of the expansion-truncation are illustrated and shown to lead naturally to familiar approximations, including the well-known Kirkwood superposition approximation. Truncation can also induce a simple relation between the multi-information and the interaction information. A measure of distance between distributions, based on Kullback-Leibler divergence, is then described and shown to be a true metric if properly restricted. The expansion is shown to generate a hierarchy of metrics and connects this work to information geometry formalisms. An example of the application of these metrics to a graph comparison problem is given that shows that the formalism can be applied to a wide range of network problems and provides a general approach for systematic approximations in numbers of interactions or connections, as well as a related quantitative metric.
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Fourier Series for Singular Measures*Axioms* **2017**, *6*(2), 7; doi:10.3390/axioms6020007 - 28 March 2017**Abstract **

Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure $\mu $ on $[0,1)$ , every $f\in {L}^{2}\left(\mu \right)$ possesses a Fourier series of the form $f\left(x\right)={\sum}_{n}^{}$

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Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure $\mu $ on $[0,1)$ , every $f\in {L}^{2}\left(\mu \right)$ possesses a Fourier series of the form $f\left(x\right)={\sum}_{n=0}^{\infty}{c}_{n}{e}^{2\pi inx}$ . We show that the coefficients ${c}_{n}$ can be computed in terms of the quantities $\widehat{f}\left(n\right)={\int}_{0}^{1}f\left(x\right){e}^{-2\pi inx}d\mu \left(x\right)$ . We also demonstrate a Shannon-type sampling theorem for functions that are in a sense $\mu $ -bandlimited.
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Norm Retrieval and Phase Retrieval by Projections*Axioms* **2017**, *6*(1), 6; doi:10.3390/axioms6010006 - 4 March 2017**Abstract **

We make a detailed study of norm retrieval. We give several classification theorems for norm retrieval and give a large number of examples to go with the theory. One consequence is a new result about Parseval frames: If a Parseval frame is divided

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We make a detailed study of norm retrieval. We give several classification theorems for norm retrieval and give a large number of examples to go with the theory. One consequence is a new result about Parseval frames: If a Parseval frame is divided into two subsets with spans ${W}_{1},{W}_{2}$ and ${W}_{1}\cap {W}_{2}=\left\{0\right\}$ , then ${W}_{1}\perp {W}_{2}$ .
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Kullback-Leibler Divergence and Mutual Information of Experiments in the Fuzzy Case*Axioms* **2017**, *6*(1), 5; doi:10.3390/axioms6010005 - 3 March 2017**Abstract **

The main aim of this contribution is to define the notions of Kullback-Leibler divergence and conditional mutual information in fuzzy probability spaces and to derive the basic properties of the suggested measures. In particular, chain rules for mutual information of fuzzy partitions and

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The main aim of this contribution is to define the notions of Kullback-Leibler divergence and conditional mutual information in fuzzy probability spaces and to derive the basic properties of the suggested measures. In particular, chain rules for mutual information of fuzzy partitions and for Kullback-Leibler divergence with respect to fuzzy P-measures are established. In addition, a convexity of Kullback-Leibler divergence and mutual information with respect to fuzzy P-measures is studied.
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Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefﬁcients*Axioms* **2017**, *6*(1), 4; doi:10.3390/axioms6010004 - 22 February 2017**Abstract **

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We propose a construction of a Hermite cubic spline-wavelet basis on the interval and hypercube. The basis is adapted to homogeneous Dirichlet boundary conditions. The wavelets are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Therefore, the wavelets

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We propose a construction of a Hermite cubic spline-wavelet basis on the interval and hypercube. The basis is adapted to homogeneous Dirichlet boundary conditions. The wavelets are orthogonal to piecewise polynomials of degree at most seven on a uniform grid. Therefore, the wavelets have eight vanishing moments, and the matrices arising from discretization of differential equations with coefﬁcients that are piecewise polynomials of degree at most four on uniform grids are sparse. Numerical examples demonstrate the efﬁciency of an adaptive wavelet method with the constructed wavelet basis for solving the one-dimensional elliptic equation and the two-dimensional Black–Scholes equation with a quadratic volatility.
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Discrete Frames on Finite Dimensional Left Quaternion Hilbert Spaces*Axioms* **2017**, *6*(1), 3; doi:10.3390/axioms6010003 - 21 February 2017**Abstract **
An introductory theory of frames on finite dimensional left quaternion Hilbert spaces is demonstrated along the lines of their complex counterpart.
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Acknowledgement to Reviewers of *Axioms* in 2016*Axioms* **2017**, *6*(1), 2; doi:10.3390/axioms6010002 - 11 January 2017**Abstract **
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Cuntz Semigroups of Compact-Type Hopf C*-Algebras*Axioms* **2017**, *6*(1), 1; doi:10.3390/axioms6010001 - 4 January 2017**Abstract **

The classical Cuntz semigroup has an important role in the study of C*-algebras, being one of the main invariants used to classify recalcitrant C*-algebras up to isomorphism. We consider C*-algebras that have Hopf algebra structure, and find additional structure in their Cuntz semigroups.

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The classical Cuntz semigroup has an important role in the study of C*-algebras, being one of the main invariants used to classify recalcitrant C*-algebras up to isomorphism. We consider C*-algebras that have Hopf algebra structure, and find additional structure in their Cuntz semigroups. We show that in many cases, isomorphisms of Cuntz semigroups that respect this additional structure can be lifted to Hopf algebra (bi)isomorphisms, up to a possible flip of the co-product. This shows that the Cuntz semigroup provides an interesting invariant of C*-algebraic quantum groups.
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Operational Solution of Non-Integer Ordinary and Evolution-Type Partial Differential Equations*Axioms* **2016**, *5*(4), 29; doi:10.3390/axioms5040029 - 13 December 2016**Abstract **

A method for the solution of linear differential equations (DE) of non-integer order and of partial differential equations (PDE) by means of inverse differential operators is proposed. The solutions of non-integer order ordinary differential equations are obtained with recourse to the integral transforms

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A method for the solution of linear differential equations (DE) of non-integer order and of partial differential equations (PDE) by means of inverse differential operators is proposed. The solutions of non-integer order ordinary differential equations are obtained with recourse to the integral transforms and the exponent operators. The generalized forms of Laguerre and Hermite orthogonal polynomials as members of more general Appèl polynomial family are used to find the solutions. Operational definitions of these polynomials are used in the context of the operational approach. Special functions are employed to write solutions of DE in convolution form. Some linear partial differential equations (PDE) are also explored by the operational method. The Schrödinger and the Black–Scholes-like evolution equations and solved with the help of the operational technique. Examples of the solution of DE of non-integer order and of PDE are considered with various initial functions, such as polynomial, exponential, and their combinations.
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Operational Approach and Solutions of Hyperbolic Heat Conduction Equations*Axioms* **2016**, *5*(4), 28; doi:10.3390/axioms5040028 - 12 December 2016**Abstract **

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We studied physical problems related to heat transport and the corresponding differential equations, which describe a wider range of physical processes. The operational method was employed to construct particular solutions for them. Inverse differential operators and operational exponent as well as operational definitions

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We studied physical problems related to heat transport and the corresponding differential equations, which describe a wider range of physical processes. The operational method was employed to construct particular solutions for them. Inverse differential operators and operational exponent as well as operational definitions and operational rules for generalized orthogonal polynomials were used together with integral transforms and special functions. Examples of an electric charge in a constant electric field passing under a potential barrier and of heat diffusion were compared and explored in two dimensions. Non-Fourier heat propagation models were studied and compared with each other and with Fourier heat transfer. Exact analytical solutions for the hyperbolic heat equation and for its extensions were explored. The exact analytical solution for the Guyer-Krumhansl type heat equation was derived. Using the latter, the heat surge propagation and relaxation was studied for the Guyer-Krumhansl heat transport model, for the Cattaneo and for the Fourier models. The comparison between them was drawn. Space-time propagation of a power–exponential function and of a periodic signal, obeying the Fourier law, the hyperbolic heat equation and its extended Guyer-Krumhansl form were studied by the operational technique. The role of various terms in the equations was explored and their influence on the solutions demonstrated. The accordance of the solutions with maximum principle is discussed. The application of our theoretical study for heat propagation in thin films is considered. The examples of the relaxation of the initial laser flash, the wide heat spot, and the harmonic function are considered and solved analytically.
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Discrete Geometry—From Theory to Applications: A Case Study*Axioms* **2016**, *5*(4), 27; doi:10.3390/axioms5040027 - 9 December 2016**Abstract **
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Forman-Ricci Flow for Change Detection in Large Dynamic Data Sets*Axioms* **2016**, *5*(4), 26; doi:10.3390/axioms5040026 - 10 November 2016**Abstract **

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We present a viable geometric solution for the detection of dynamic effects in complex networks. Building on Forman’s discretization of the classical notion of Ricci curvature, we introduce a novel geometric method to characterize different types of real-world networks with an emphasis on

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We present a viable geometric solution for the detection of dynamic effects in complex networks. Building on Forman’s discretization of the classical notion of Ricci curvature, we introduce a novel geometric method to characterize different types of real-world networks with an emphasis on peer-to-peer networks. We study the classical Ricci-flow in a network-theoretic setting and introduce an analytic tool for characterizing dynamic effects. The formalism suggests a computational method for change detection and the identification of fast evolving network regions and yields insights into topological properties and the structure of the underlying data.
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Quantum Quasigroups and the Quantum Yang–Baxter Equation*Axioms* **2016**, *5*(4), 25; doi:10.3390/axioms5040025 - 9 November 2016**Abstract **

Quantum quasigroups are algebraic structures providing a general self-dual framework for the nonassociative extension of Hopf algebra techniques. They also have one-sided analogues, which are not self-dual. The paper presents a survey of recent work on these structures, showing how they furnish various

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Quantum quasigroups are algebraic structures providing a general self-dual framework for the nonassociative extension of Hopf algebra techniques. They also have one-sided analogues, which are not self-dual. The paper presents a survey of recent work on these structures, showing how they furnish various solutions to the quantum Yang–Baxter equation.
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On the *q*-Laplace Transform and Related Special Functions*Axioms* **2016**, *5*(3), 24; doi:10.3390/axioms5030024 - 6 September 2016**Abstract **

Motivated by statistical mechanics contexts, we study the properties of the *q*-Laplace transform, which is an extension of the well-known Laplace transform. In many circumstances, the kernel function to evaluate certain integral forms has been studied. In this article, we establish relationships

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Motivated by statistical mechanics contexts, we study the properties of the *q*-Laplace transform, which is an extension of the well-known Laplace transform. In many circumstances, the kernel function to evaluate certain integral forms has been studied. In this article, we establish relationships between *q*-exponential and other well-known functional forms, such as Mittag–Leffler functions, hypergeometric and *H*-function, by means of the kernel function of the integral. Traditionally, we have been applying the Laplace transform method to solve differential equations and boundary value problems. Here, we propose an alternative, the *q*-Laplace transform method, to solve differential equations, such as as the fractional space-time diffusion equation, the generalized kinetic equation and the time fractional heat equation.
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The Universe in Leśniewski’s Mereology: Some Comments on Sobociński’s Reflections*Axioms* **2016**, *5*(3), 23; doi:10.3390/axioms5030023 - 6 September 2016**Abstract **

Stanisław Leśniewski’s mereology was originally conceived as a theory of foundations of mathematics and it is also for this reason that it has philosophical connotations. The ‘philosophical significance’ of mereology was upheld by Bolesław Sobociński who expressed the view in his correspondence with

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Stanisław Leśniewski’s mereology was originally conceived as a theory of foundations of mathematics and it is also for this reason that it has philosophical connotations. The ‘philosophical significance’ of mereology was upheld by Bolesław Sobociński who expressed the view in his correspondence with J.M. Bocheński. As he wrote to Bocheński in 1948: “[...] it is interesting that, being such a simple deductive theory, mereology may prove a number of very general theses reminiscent of metaphysical ontology”. The theses which Sobociński had in mind were related to the mereological notion of “the Universe”. Sobociński listed them in the letter adding his philosophical commentary but he did not give proofs for them and did not specify precisely the theory lying behind them. This is what we want to supply in the first part of our paper. We indicate some connections between the notion of the universe and other specific mereological notions. Motivated by Sobociński’s informal suggestions showing his preference for mereology over the axiomatic set theory in application to philosophy we propose to consider Sobociński’s formalism in a new frame which is the ZFM theory—an extension of Zermelo-Fraenkel set theory by mereological axioms, developed by A. Pietruszczak. In this systematic part we investigate reasons of ’philosophical hopes’ mentioned by Sobociński, pinned on the mereological concept of “the Universe”.
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