Axioms

15 pages, 400 KiB

Open AccessArticle

Norm Retrieval and Phase Retrieval by Projections

by Peter G. Casazza, Dorsa Ghoreishi, Shani Jose and Janet C. Tremain

Axioms 2017, 6(1), 6; https://doi.org/10.3390/axioms6010006 - 4 Mar 2017

Cited by 15 | Viewed by 5458

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.

W_{1}, W_{2}

and

W_{1} \cap W_{2} = {0}

, then

W_{1} ⊥ W_{2}

. Full article

(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)

13 pages, 294 KiB

Open AccessArticle

Kullback-Leibler Divergence and Mutual Information of Experiments in the Fuzzy Case

by Dagmar Markechová

Axioms 2017, 6(1), 5; https://doi.org/10.3390/axioms6010005 - 3 Mar 2017

Cited by 9 | Viewed by 4834

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. Full article

21 pages, 5393 KiB

Open AccessArticle

Sparse Wavelet Representation of Differential Operators with Piecewise Polynomial Coefﬁcients

by Dana Černá and Václav Finĕk

Axioms 2017, 6(1), 4; https://doi.org/10.3390/axioms6010004 - 22 Feb 2017

Cited by 15 | Viewed by 5108

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. Full article

(This article belongs to the Special Issue Wavelet and Frame Constructions, with Applications)

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15 pages, 250 KiB

Open AccessArticle

Discrete Frames on Finite Dimensional Left Quaternion Hilbert Spaces

by M. Khokulan, K. Thirulogasanthar and S. Srisatkunarajah

Axioms 2017, 6(1), 3; https://doi.org/10.3390/axioms6010003 - 21 Feb 2017

Cited by 11 | Viewed by 4342

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)

2 pages, 155 KiB

Open AccessEditorial

Acknowledgement to Reviewers of Axioms in 2016

by Axioms Editorial Office

Axioms 2017, 6(1), 2; https://doi.org/10.3390/axioms6010002 - 11 Jan 2017

Viewed by 3103

Abstract

The editors of Axioms would like to express their sincere gratitude to the following reviewers for assessing manuscripts in 2016.[...] Full article

21 pages, 463 KiB

Open AccessArticle

Cuntz Semigroups of Compact-Type Hopf C*-Algebras

by Dan Kučerovský

Axioms 2017, 6(1), 1; https://doi.org/10.3390/axioms6010001 - 4 Jan 2017

Cited by 1 | Viewed by 3920

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. Full article

(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2016)

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

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