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Axioms, Volume 6, Issue 1 (March 2017) – 6 articles

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Article
Norm Retrieval and Phase Retrieval by Projections
Axioms 2017, 6(1), 6; https://doi.org/10.3390/axioms6010006 - 04 Mar 2017
Cited by 4 | Viewed by 2492
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 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)
Article
Kullback-Leibler Divergence and Mutual Information of Experiments in the Fuzzy Case
Axioms 2017, 6(1), 5; https://doi.org/10.3390/axioms6010005 - 03 Mar 2017
Cited by 7 | Viewed by 2318
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
Article
Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefficients
Axioms 2017, 6(1), 4; https://doi.org/10.3390/axioms6010004 - 22 Feb 2017
Cited by 15 | Viewed by 2408
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
(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)
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Article
Discrete Frames on Finite Dimensional Left Quaternion Hilbert Spaces
Axioms 2017, 6(1), 3; https://doi.org/10.3390/axioms6010003 - 21 Feb 2017
Cited by 2 | Viewed by 1617
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)
Editorial
Acknowledgement to Reviewers of Axioms in 2016
Axioms 2017, 6(1), 2; https://doi.org/10.3390/axioms6010002 - 11 Jan 2017
Viewed by 1451
Abstract
The editors of Axioms would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2016.[...] Full article
Article
Cuntz Semigroups of Compact-Type Hopf C*-Algebras
Axioms 2017, 6(1), 1; https://doi.org/10.3390/axioms6010001 - 04 Jan 2017
Cited by 2 | Viewed by 1625
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. [...] Read more.
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)
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