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Axioms, Volume 6, Issue 2 (June 2017)

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Research

Open AccessArticle Fourier Series for Singular Measures
Axioms 2017, 6(2), 7; doi:10.3390/axioms6020007
Received: 21 February 2017 / Revised: 23 March 2017 / Accepted: 24 March 2017 / Published: 28 March 2017
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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
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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. Full article
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
Open AccessArticle Expansion of the Kullback-Leibler Divergence, and a New Class of Information Metrics
Axioms 2017, 6(2), 8; doi:10.3390/axioms6020008
Received: 31 January 2017 / Revised: 25 March 2017 / Accepted: 27 March 2017 / Published: 1 April 2017
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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
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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. Full article
(This article belongs to the Special Issue Entropy and Information Theory)
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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
Received: 9 March 2017 / Revised: 14 April 2017 / Accepted: 17 April 2017 / Published: 20 April 2017
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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
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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. Full article
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
Open AccessArticle Orientation Asymmetric Surface Model for Membranes: Finsler Geometry Modeling
Axioms 2017, 6(2), 10; doi:10.3390/axioms6020010
Received: 21 March 2017 / Revised: 14 April 2017 / Accepted: 19 April 2017 / Published: 25 April 2017
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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
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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. Full article
(This article belongs to the Special Issue Discrete Geometry and its Applications)
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Open AccessArticle Multivariate Extended Gamma Distribution
Axioms 2017, 6(2), 11; doi:10.3390/axioms6020011
Received: 7 June 2016 / Revised: 7 April 2017 / Accepted: 13 April 2017 / Published: 24 April 2017
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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.
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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 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
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Open AccessArticle Toward Measuring Network Aesthetics Based on Symmetry
Axioms 2017, 6(2), 12; doi:10.3390/axioms6020012
Received: 20 March 2017 / Revised: 3 May 2017 / Accepted: 3 May 2017 / Published: 6 May 2017
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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
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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. Full article
(This article belongs to the Special Issue Entropy and Information Theory)
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Open AccessArticle No Uncountable Polish Group Can be a Right-Angled Artin Group
Axioms 2017, 6(2), 13; doi:10.3390/axioms6020013
Received: 28 March 2017 / Revised: 20 April 2017 / Accepted: 4 May 2017 / Published: 11 May 2017
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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
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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 < 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. Full article
(This article belongs to the collection Topological Groups)
Open AccessArticle Tsallis Entropy and Generalized Shannon Additivity
Axioms 2017, 6(2), 14; doi:10.3390/axioms6020014
Received: 19 May 2017 / Revised: 8 June 2017 / Accepted: 10 June 2017 / Published: 14 June 2017
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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
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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 Scalable and Fully Distributed Localization in Large-Scale Sensor Networks
Axioms 2017, 6(2), 15; doi:10.3390/axioms6020015
Received: 30 March 2017 / Revised: 23 May 2017 / Accepted: 8 June 2017 / Published: 15 June 2017
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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
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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. Full article
(This article belongs to the Special Issue Discrete Geometry and its Applications)
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Open AccessArticle An Evaluation of ARFIMA (Autoregressive Fractional Integral Moving Average) Programs
Axioms 2017, 6(2), 16; doi:10.3390/axioms6020016
Received: 13 March 2017 / Revised: 3 May 2017 / Accepted: 14 June 2017 / Published: 17 June 2017
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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
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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. Full article
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