Authors:John Herr, Eric Weber First page: 7 Abstract: Using the Kaczmarz algorithm, we prove that for any singular Borel probability measure μ on [ 0 , 1 ) , every f ∈ L 2 ( μ ) possesses a Fourier series of the form f ( x ) = ∑ n = 0 ∞ c n e 2 π i n x . We show that the coefficients c n can be computed in terms of the quantities f ^ ( n ) = ∫ 0 1 f ( x ) e − 2 π i n x d μ ( x ) . We also demonstrate a Shannon-type sampling theorem for functions that are in a sense μ -bandlimited. PubDate: 2017-03-28 DOI: 10.3390/axioms6020007 Issue No:Vol. 6, No. 2 (2017)

Authors:David Galas, Gregory Dewey, James Kunert-Graf, Nikita Sakhanenko First page: 8 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 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. PubDate: 2017-04-01 DOI: 10.3390/axioms6020008 Issue No:Vol. 6, No. 2 (2017)

Authors:Jianzhong Wang First page: 9 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 ) T a → ( z ) = 1 , an s × s Laurent polynomial matrix A ( z ) (together with its inverse A - 1 ( z ) ) is called a symmetric Laurent polynomial matrix extension of the dual pair ( a → ( z ) , b → ( z ) ) if A ( z ) has similar symmetry, the inverse A - 1 ( Z ) also is a Laurent polynomial matrix, the first column of A ( z ) is a → ( z ) and the first row of A - 1 ( z ) is ( b → ( z ) ) T . In this paper, we introduce the Euclidean symmetric division and the symmetric elementary matrices in the Laurent polynomial ring and reveal their relation. Based on the Euclidean symmetric division algorithm in the Laurent polynomial ring, we develop a novel and effective algorithm for symmetric Laurent polynomial matrix extension. We also apply the algorithm in the construction of multi-band symmetric perfect reconstruction filter banks. PubDate: 2017-04-20 DOI: 10.3390/axioms6020009 Issue No:Vol. 6, No. 2 (2017)

Authors:Evgenii Proutorov, Hiroshi Koibuchi First page: 10 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 R 3 . The metric variable g a b , which is always fixed to the Euclidean metric δ a b , can be extended to a more general non-Euclidean metric on M in the continuous model. The problem we focus on in this paper is whether such an extension is well defined or not in the discrete model. We find that a discrete surface model with a nontrivial metric becomes well defined if it is treated in the context of Finsler geometry (FG) modeling, where triangle edge length in M depends on the direction. It is also shown that the discrete FG model is orientation asymmetric on invertible surfaces in general, and for this reason, the FG model has a potential advantage for describing real physical membranes, which are expected to have some asymmetries for orientation-changing transformations. PubDate: 2017-04-25 DOI: 10.3390/axioms6020010 Issue No:Vol. 6, No. 2 (2017)

Authors:Dhannya Joseph First page: 11 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. 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. PubDate: 2017-04-24 DOI: 10.3390/axioms6020011 Issue No:Vol. 6, No. 2 (2017)

Authors:Zengqiang Chen, Matthias Dehmer, Frank Emmert-Streib, Mowshowitz, Yongtang Shi First page: 12 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 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. PubDate: 2017-05-06 DOI: 10.3390/axioms6020012 Issue No:Vol. 6, No. 2 (2017)

Authors:Gianluca Paolini, Saharon Shelah First page: 13 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 l g ( x ) ≤ l g ( x k ) ; (ii) if l g ( y ) < k < ω and x k = y , then x = e , then there exists a subgroup G * of G of size b (the bounding number) such that G * is not embeddable in A. In particular, we prove that the automorphism group of a countable structure cannot be an uncountable right-angled Artin group. This generalizes analogous results for free and free abelian uncountable groups. PubDate: 2017-05-11 DOI: 10.3390/axioms6020013 Issue No:Vol. 6, No. 2 (2017)

Authors:Sonja Jäckle, Karsten Keller First page: 14 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 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. PubDate: 2017-06-14 DOI: 10.3390/axioms6020014 Issue No:Vol. 6, No. 2 (2017)

Authors:Miao Jin, Su Xia, Hongyi Wu, Xianfeng Gu First page: 15 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 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. PubDate: 2017-06-15 DOI: 10.3390/axioms6020015 Issue No:Vol. 6, No. 2 (2017)

Authors:Dan Kučerovský First page: 1 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. 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. PubDate: 2017-01-04 DOI: 10.3390/axioms6010001 Issue No:Vol. 6, No. 1 (2017)

Authors:M. Khokulan, K. Thirulogasanthar, S. Srisatkunarajah First page: 3 Abstract: An introductory theory of frames on finite dimensional left quaternion Hilbert spaces is demonstrated along the lines of their complex counterpart. PubDate: 2017-02-21 DOI: 10.3390/axioms6010003 Issue No:Vol. 6, No. 1 (2017)

Authors:Dana Černá, Václav Finĕk First page: 4 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 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. PubDate: 2017-02-22 DOI: 10.3390/axioms6010004 Issue No:Vol. 6, No. 1 (2017)

Authors:Dagmar Markechová First page: 5 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 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. PubDate: 2017-03-03 DOI: 10.3390/axioms6010005 Issue No:Vol. 6, No. 1 (2017)

Authors:Peter Casazza, Dorsa Ghoreishi, Shani Jose, Janet Tremain First page: 6 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 into two subsets with spans W 1 , W 2 and W 1 ∩ W 2 = { 0 } , then W 1 ⊥ W 2 . PubDate: 2017-03-04 DOI: 10.3390/axioms6010006 Issue No:Vol. 6, No. 1 (2017)