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Axioms, Volume 6, Issue 1 (March 2017)

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Editorial

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Open AccessEditorial Acknowledgement to Reviewers of Axioms in 2016
Axioms 2017, 6(1), 2; doi:10.3390/axioms6010002
Received: 11 January 2017 / Accepted: 11 January 2017 / Published: 11 January 2017
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Research

Jump to: Editorial

Open AccessArticle Norm Retrieval and Phase Retrieval by Projections
Axioms 2017, 6(1), 6; doi:10.3390/axioms6010006
Received: 27 January 2017 / Revised: 1 March 2017 / Accepted: 2 March 2017 / Published: 4 March 2017
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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 W 2 = { 0 } , then W 1 W 2 . Full article
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
Open AccessArticle Cuntz Semigroups of Compact-Type Hopf C*-Algebras
Axioms 2017, 6(1), 1; doi:10.3390/axioms6010001
Received: 29 November 2016 / Revised: 22 December 2016 / Accepted: 26 December 2016 / Published: 4 January 2017
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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. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2016)
Open AccessArticle Discrete Frames on Finite Dimensional Left Quaternion Hilbert Spaces
Axioms 2017, 6(1), 3; doi:10.3390/axioms6010003
Received: 8 December 2016 / Revised: 9 February 2017 / Accepted: 17 February 2017 / Published: 21 February 2017
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Abstract An introductory theory of frames on finite dimensional left quaternion Hilbert spaces is demonstrated along the lines of their complex counterpart. Full article
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
Open AccessArticle Kullback-Leibler Divergence and Mutual Information of Experiments in the Fuzzy Case
Axioms 2017, 6(1), 5; doi:10.3390/axioms6010005
Received: 30 January 2017 / Accepted: 1 March 2017 / Published: 3 March 2017
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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. Full article
Open AccessArticle Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefficients
Axioms 2017, 6(1), 4; doi:10.3390/axioms6010004
Received: 12 January 2017 / Accepted: 20 February 2017 / Published: 22 February 2017
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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
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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. Full article
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
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