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Axioms, Volume 3, Issue 4 (December 2014) , Pages 342-379

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Open AccessArticle A Simplified Algorithm for Inverting Higher Order Diffusion Tensors
Axioms 2014, 3(4), 369-379; https://doi.org/10.3390/axioms3040369
Received: 20 April 2014 / Revised: 5 November 2014 / Accepted: 7 November 2014 / Published: 14 November 2014
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Abstract
In Riemannian geometry, a distance function is determined by an inner product on the tangent space. In Riemann–Finsler geometry, this distance function can be determined by a norm. This gives more freedom on the form of the so-called indicatrix or the set of [...] Read more.
In Riemannian geometry, a distance function is determined by an inner product on the tangent space. In Riemann–Finsler geometry, this distance function can be determined by a norm. This gives more freedom on the form of the so-called indicatrix or the set of unit vectors. This has some interesting applications, e.g., in medical image analysis, especially in diffusion weighted imaging (DWI). An important application of DWI is in the inference of the local architecture of the tissue, typically consisting of thin elongated structures, such as axons or muscle fibers, by measuring the constrained diffusion of water within the tissue. From high angular resolution diffusion imaging (HARDI) data, one can estimate the diffusion orientation distribution function (dODF), which indicates the relative diffusivity in all directions and can be represented by a spherical polynomial. We express this dODF as an equivalent spherical monomial (higher order tensor) to directly generalize the (second order) diffusion tensor approach. To enable efficient computation of Riemann–Finslerian quantities on diffusion weighted (DW)-images, such as the metric/norm tensor, we present a simple and efficient algorithm to invert even order spherical monomials, which extends the familiar inversion of diffusion tensors, i.e., symmetric matrices. Full article
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Open AccessCommunication The Yang-Baxter Equation, (Quantum) Computers and Unifying Theories
Axioms 2014, 3(4), 360-368; https://doi.org/10.3390/axioms3040360
Received: 22 September 2014 / Revised: 25 October 2014 / Accepted: 4 November 2014 / Published: 14 November 2014
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Abstract
Quantum mechanics has had an important influence on building computers;nowadays, quantum mechanics principles are used for the processing and transmission ofinformation. The Yang-Baxter equation is related to the universal gates from quantumcomputing and it realizes a unification of certain non-associative structures. Unifyingstructures could [...] Read more.
Quantum mechanics has had an important influence on building computers;nowadays, quantum mechanics principles are used for the processing and transmission ofinformation. The Yang-Baxter equation is related to the universal gates from quantumcomputing and it realizes a unification of certain non-associative structures. Unifyingstructures could be seen as structures which comprise the information contained in other(algebraic) structures. Recently, we gave the axioms of a structure which unifies associativealgebras, Lie algebras and Jordan algebras. Our paper is a review and a continuation of thatapproach. It also contains several geometric considerations. Full article
(This article belongs to the Special Issue Hopf Algebras, Quantum Groups and Yang-Baxter Equations 2014)
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Open AccessArticle Weak n-Ary Relational Products in Allegories
Axioms 2014, 3(4), 342-359; https://doi.org/10.3390/axioms3040342
Received: 16 August 2014 / Revised: 15 October 2014 / Accepted: 17 October 2014 / Published: 30 October 2014
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Abstract
Allegories are enriched categories generalizing a category of sets and binary relations. Accordingly, relational products in an allegory can be viewed as a generalization of Cartesian products. There are several definitions of relational products currently in the literature. Interestingly, definitions for binary products [...] Read more.
Allegories are enriched categories generalizing a category of sets and binary relations. Accordingly, relational products in an allegory can be viewed as a generalization of Cartesian products. There are several definitions of relational products currently in the literature. Interestingly, definitions for binary products do not generalize easily to n-ary ones. In this paper, we provide a new definition of an n-ary relational product, and we examine its properties. Full article
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