Open AccessReview
The SIC Question: History and State of Play
Axioms 2017, 6(3), 21; doi:10.3390/axioms6030021 -
Abstract
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
[...] Read more.
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. Full article
Figures

Figure 1

Open AccessArticle
Quincunx Fundamental Refinable Functions in Arbitrary Dimensions
Axioms 2017, 6(3), 20; doi:10.3390/axioms6030020 -
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
[...] Read more.
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. Full article
Open AccessArticle
Assigning Numerical Scores to Linguistic Expressions
Axioms 2017, 6(3), 19; doi:10.3390/axioms6030019 -
Abstract
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
[...] Read more.
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. Full article
Figures

Figure 1

Open AccessArticle
Fractional Integration and Differentiation of the Generalized Mathieu Series
Axioms 2017, 6(3), 18; doi:10.3390/axioms6030018 -
Abstract
We aim to present some formulas for the Saigo hypergeometric fractional integral and differential operators involving the generalized Mathieu series Sμ(r), which are expressed in terms of the Hadamard product of the generalized Mathieu series Sμ(
[...] Read more.
We aim to present some formulas for the Saigo hypergeometric fractional integral and differential operators involving the generalized Mathieu series Sμ(r), which are expressed in terms of the Hadamard product of the generalized Mathieu series Sμ(r) and the Fox–Wright function pΨq(z). 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. Full article
Open AccessArticle
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 -
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
[...] Read more.
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. Full article
Open AccessArticle
An Evaluation of ARFIMA (Autoregressive Fractional Integral Moving Average) Programs
Axioms 2017, 6(2), 16; doi:10.3390/axioms6020016 -
Abstract
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
[...] Read more.
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. Full article
Figures

Figure 1

Open AccessArticle
Scalable and Fully Distributed Localization in Large-Scale Sensor Networks
Axioms 2017, 6(2), 15; doi:10.3390/axioms6020015 -
Abstract
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
[...] Read more.
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. Full article
Figures

Figure 1

Open AccessArticle
Tsallis Entropy and Generalized Shannon Additivity
Axioms 2017, 6(2), 14; doi:10.3390/axioms6020014 -
Abstract
The Tsallis entropy given for a positive parameter α can be considered as a generalization of the classical Shannon entropy. For the latter, corresponding to α=1, there exist many axiomatic characterizations. One of them based on the well-known Khinchin-Shannon axioms
[...] Read more.
The Tsallis entropy given for a positive parameter α can be considered as a generalization of the classical Shannon entropy. For the latter, corresponding to α=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 α=1,2 discussed separately. Full article
Open AccessArticle
No Uncountable Polish Group Can be a Right-Angled Artin Group
Axioms 2017, 6(2), 13; doi:10.3390/axioms6020013 -
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<ω, then lg(x)lg(xk
[...] Read more.
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<ω, then lg(x)lg(xk); (ii) if lg(y)<k<ω and xk=y, then x=e, then there exists a subgroup G* of G of size b (the bounding number) such that G* is not embeddable in A. In particular, we prove that the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes analogous results for free and free abelian uncountable groups. Full article
Open AccessArticle
Toward Measuring Network Aesthetics Based on Symmetry
Axioms 2017, 6(2), 12; doi:10.3390/axioms6020012 -
Abstract
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
[...] Read more.
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. Full article
Figures

Figure 1

Open AccessArticle
Orientation Asymmetric Surface Model for Membranes: Finsler Geometry Modeling
Axioms 2017, 6(2), 10; doi:10.3390/axioms6020010 -
Abstract
We study triangulated surface models with nontrivial surface metrices for membranes. The surface model is defined by a mapping r from a two-dimensional parameter space M to the three-dimensional Euclidean space R3. The metric variable gab, which is
[...] Read more.
We study triangulated surface models with nontrivial surface metrices for membranes. The surface model is defined by a mapping r from a two-dimensional parameter space M to the three-dimensional Euclidean space R3. The metric variable gab, which is always fixed to the Euclidean metric δab, 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. Full article
Figures

Figure 1

Open AccessArticle
Multivariate Extended Gamma Distribution
Axioms 2017, 6(2), 11; doi:10.3390/axioms6020011 -
Abstract
In this paper, I consider multivariate analogues of the extended gamma density, which will provide multivariate extensions to Tsallis statistics and superstatistics. By making use of the pathway parameter β, multivariate generalized gamma density can be obtained from the model considered here.
[...] Read more.
In this paper, I consider multivariate analogues of the extended gamma density, which will provide multivariate extensions to Tsallis statistics and superstatistics. By making use of the pathway parameter β, multivariate generalized gamma density can be obtained from the model considered here. Some of its special cases and limiting cases are also mentioned. Conditional density, best predictor function, regression theory, etc., connected with this model are also introduced. Full article
Figures

Figure 1

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

Figure 1

Open AccessArticle
Fourier Series for Singular Measures
Axioms 2017, 6(2), 7; doi:10.3390/axioms6020007 -
Abstract
Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure μ on [0,1), every fL2(μ) possesses a Fourier series of the form f(x)=n
[...] Read more.
Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure μ on [0,1), every fL2(μ) possesses a Fourier series of the form f(x)=n=0cne2πinx. We show that the coefficients cn can be computed in terms of the quantities f^(n)=01f(x)e2πinxdμ(x). We also demonstrate a Shannon-type sampling theorem for functions that are in a sense μ-bandlimited. Full article
Open AccessArticle
Norm Retrieval and Phase Retrieval by Projections
Axioms 2017, 6(1), 6; doi:10.3390/axioms6010006 -
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
[...] Read more.
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 W1,W2 and W1W2={0}, then W1W2. Full article
Open AccessArticle
Kullback-Leibler Divergence and Mutual Information of Experiments in the Fuzzy Case
Axioms 2017, 6(1), 5; doi:10.3390/axioms6010005 -
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
[...] Read more.
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. Full article
Open AccessArticle
Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefficients
Axioms 2017, 6(1), 4; doi:10.3390/axioms6010004 -
Abstract
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
[...] Read more.
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. Full article
Figures

Figure 1

Open AccessArticle
Discrete Frames on Finite Dimensional Left Quaternion Hilbert Spaces
Axioms 2017, 6(1), 3; doi:10.3390/axioms6010003 -
Abstract An introductory theory of frames on finite dimensional left quaternion Hilbert spaces is demonstrated along the lines of their complex counterpart. Full article
Open AccessEditorial
Acknowledgement to Reviewers of Axioms in 2016
Axioms 2017, 6(1), 2; doi:10.3390/axioms6010002 -