The SIC Question: History and State of Play*Axioms* **2017**, *6*(3), 21; doi:10.3390/axioms6030021 - 18 July 2017**Abstract **

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Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through

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Recent years have seen significant advances in the study of symmetric informationally complete (SIC) quantum measurements, also known as maximal sets of complex equiangular lines. Previously, the published record contained solutions up to dimension 67, and was with high confidence complete up through dimension 50. Computer calculations have now furnished solutions in all dimensions up to 151, and in several cases beyond that, as large as dimension 844. These new solutions exhibit an additional type of symmetry beyond the basic definition of a SIC, and so verify a conjecture of Zauner in many new cases. The solutions in dimensions 68 through 121 were obtained by Andrew Scott, and his catalogue of distinct solutions is, with high confidence, complete up to dimension 90. Additional results in dimensions 122 through 151 were calculated by the authors using Scott’s code. We recap the history of the problem, outline how the numerical searches were done, and pose some conjectures on how the search technique could be improved. In order to facilitate communication across disciplinary boundaries, we also present a comprehensive bibliography of SIC research.
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Quincunx Fundamental Refinable Functions in Arbitrary Dimensions*Axioms* **2017**, *6*(3), 20; doi:10.3390/axioms6030020 - 6 July 2017**Abstract **

In this paper, we generalize the family of Deslauriers–Dubuc’s interpolatory masks from dimension one to arbitrary dimensions with respect to the quincunx dilation matrices, thereby providing a family of quincunx fundamental refinable functions in arbitrary dimensions. We show that a family of unique

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In this paper, we generalize the family of Deslauriers–Dubuc’s interpolatory masks from dimension one to arbitrary dimensions with respect to the quincunx dilation matrices, thereby providing a family of quincunx fundamental refinable functions in arbitrary dimensions. We show that a family of unique quincunx interpolatory masks exists and such a family of masks is of real value and has the full-axis symmetry property. In dimension $d=2$ , we give the explicit form of such unique quincunx interpolatory masks, which implies the nonnegativity property of such a family of masks.
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Assigning Numerical Scores to Linguistic Expressions*Axioms* **2017**, *6*(3), 19; doi:10.3390/axioms6030019 - 6 July 2017**Abstract **

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In this paper, we study different methods of scoring linguistic expressions defined on a finite set, in the search for a linear order that ranks all those possible expressions. Among them, particular attention is paid to the canonical extension, and its representability through

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In this paper, we study different methods of scoring linguistic expressions defined on a finite set, in the search for a linear order that ranks all those possible expressions. Among them, particular attention is paid to the canonical extension, and its representability through distances in a graph plus some suitable penalization of imprecision. The relationship between this setting and the classical problems of numerical representability of orderings, as well as extension of orderings from a set to a superset is also explored. Finally, aggregation procedures of qualitative rankings and scorings are also analyzed.
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Fractional Integration and Differentiation of the Generalized Mathieu Series*Axioms* **2017**, *6*(3), 18; doi:10.3390/axioms6030018 - 27 June 2017**Abstract **

We aim to present some formulas for the Saigo hypergeometric fractional integral and differential operators involving the generalized Mathieu series ${S}_{\mu}\left(r\right)$ , which are expressed in terms of the Hadamard product of the generalized Mathieu series ${S}_{\mu}($

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We aim to present some formulas for the Saigo hypergeometric fractional integral and differential operators involving the generalized Mathieu series ${S}_{\mu}\left(r\right)$ , which are expressed in terms of the Hadamard product of the generalized Mathieu series ${S}_{\mu}\left(r\right)$ and the Fox–Wright function ${}_{p}{\mathsf{\Psi}}_{q}\left(z\right)$ . Corresponding assertions for the classical Riemann–Liouville and Erdélyi–Kober fractional integral and differential operators are deduced. Further, it is emphasized that the results presented here, which are for a seemingly complicated series, can reveal their involved properties via the series of the two known functions.
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An Independent Set of Axioms of MV-Algebras and Solutions of the Set-Theoretical Yang–Baxter Equation*Axioms* **2017**, *6*(3), 17; doi:10.3390/axioms6030017 - 22 June 2017**Abstract **

The aim of this paper is to give a new equivalent set of axioms for *MV-*algebras, and to show that the axioms are independent. In addition to this, we handle Yang–Baxter equation problem. In conclusion, we construct a new set-theoretical solution for

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The aim of this paper is to give a new equivalent set of axioms for *MV-*algebras, and to show that the axioms are independent. In addition to this, we handle Yang–Baxter equation problem. In conclusion, we construct a new set-theoretical solution for the Yang–Baxter equation by using *MV-*algebras.
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An Evaluation of ARFIMA (Autoregressive Fractional Integral Moving Average) Programs*Axioms* **2017**, *6*(2), 16; doi:10.3390/axioms6020016 - 17 June 2017**Abstract **

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Strong coupling between values at different times that exhibit properties of long range dependence, non-stationary, spiky signals cannot be processed by the conventional time series analysis. The autoregressive fractional integral moving average (ARFIMA) model, a fractional order signal processing technique, is the generalization

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Strong coupling between values at different times that exhibit properties of long range dependence, non-stationary, spiky signals cannot be processed by the conventional time series analysis. The autoregressive fractional integral moving average (ARFIMA) model, a fractional order signal processing technique, is the generalization of the conventional integer order models—autoregressive integral moving average (ARIMA) and autoregressive moving average (ARMA) model. Therefore, it has much wider applications since it could capture both short-range dependence and long range dependence. For now, several software programs have been developed to deal with ARFIMA processes. However, it is unfortunate to see that using different numerical tools for time series analysis usually gives quite different and sometimes radically different results. Users are often puzzled about which tool is suitable for a specific application. We performed a comprehensive survey and evaluation of available ARFIMA tools in the literature in the hope of benefiting researchers with different academic backgrounds. In this paper, four aspects of ARFIMA programs concerning simulation, fractional order difference filter, estimation and forecast are compared and evaluated, respectively, in various software platforms. Our informative comments can serve as useful selection guidelines.
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Scalable and Fully Distributed Localization in Large-Scale Sensor Networks*Axioms* **2017**, *6*(2), 15; doi:10.3390/axioms6020015 - 15 June 2017**Abstract **

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This work proposes a novel connectivity-based localization algorithm, well suitable for large-scale sensor networks with complex shapes and a non-uniform nodal distribution. In contrast to current state-of-the-art connectivity-based localization methods, the proposed algorithm is highly scalable with linear computation and communication costs with

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This work proposes a novel connectivity-based localization algorithm, well suitable for large-scale sensor networks with complex shapes and a non-uniform nodal distribution. In contrast to current state-of-the-art connectivity-based localization methods, the proposed algorithm is highly scalable with linear computation and communication costs with respect to the size of the network; and fully distributed where each node only needs the information of its neighbors without cumbersome partitioning and merging process. The algorithm is theoretically guaranteed and numerically stable. Moreover, the algorithm can be readily extended to the localization of networks with a one-hop transmission range distance measurement, and the propagation of the measurement error at one sensor node is limited within a small area of the network around the node. Extensive simulations and comparison with other methods under various representative network settings are carried out, showing the superior performance of the proposed algorithm.
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Tsallis Entropy and Generalized Shannon Additivity*Axioms* **2017**, *6*(2), 14; doi:10.3390/axioms6020014 - 14 June 2017**Abstract **

The Tsallis entropy given for a positive parameter $\alpha $ can be considered as a generalization of the classical Shannon entropy. For the latter, corresponding to $\alpha =1$ , there exist many axiomatic characterizations. One of them based on the well-known Khinchin-Shannon axioms

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The Tsallis entropy given for a positive parameter $\alpha $ can be considered as a generalization of the classical Shannon entropy. For the latter, corresponding to $\alpha =1$ , there exist many axiomatic characterizations. One of them based on the well-known Khinchin-Shannon axioms has been simplified several times and adapted to Tsallis entropy, where the axiom of (generalized) Shannon additivity is playing a central role. The main aim of this paper is to discuss this axiom in the context of Tsallis entropy. We show that it is sufficient for characterizing Tsallis entropy, with the exceptions of cases $\alpha =1,2$ discussed separately.
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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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