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1

Structures of spinors fiber bundles with special relativity of Dirac operator using the Clifford algebra

Hassan Yousif Atyeib Ibrahim

Demonstratio Mathematica

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2021

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Vol. 54, nr 1

410--424

EN

The purpose of this article is to demonstrate how to use the mathematics of spinor bundles and their category. We have used the methods of principle fiber bundles obey thorough solid harmonic treatment of pseudo-Riemannian manifolds and spinor structures with Clifford algebras, which couple with Dirac operator to study important applications in cohomology theory.

2

Implementation and evaluation of medical imaging techniques based on conformal geometric algebra

Franchini Silvia, Gentile Antonio, Vassallo Giorgio, Vitabile Salvatore

International Journal of Applied Mathematics and Computer Science

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2020

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Vol. 30, no. 3

415--433

EN

Medical imaging tasks, such as segmentation, 3D modeling, and registration of medical images, involve complex geometric problems, usually solved by standard linear algebra and matrix calculations. In the last few decades, conformal geometric algebra (CGA) has emerged as a new approach to geometric computing that offers a simple and efficient representation of geometric objects and transformations. However, the practical use of CGA-based methods for big data image processing in medical imaging requires fast and efficient implementations of CGA operations to meet both real-time processing constraints and accuracy requirements. The purpose of this study is to present a novel implementation of CGA-based medical imaging techniques that makes them effective and practically usable. The paper exploits a new simplified formulation of CGA operators that allows significantly reduced execution times while maintaining the needed result precision. We have exploited this novel CGA formulation to re-design a suite of medical imaging automatic methods, including image segmentation, 3D reconstruction and registration. Experimental tests show that the re-formulated CGA-based methods lead to both higher precision results and reduced computation times, which makes them suitable for big data image processing applications. The segmentation algorithm provides the Dice index, sensitivity and specificity values of 98.14%, 98.05% and 97.73%, respectively, while the order of magnitude of the errors measured for the registration methods is 10-5.

3

Projective coordinates and compactification in elliptic, parabolic and hyperbolic 2-D geometry

Biswas D.

Journal of Applied Analysis

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2012

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Vol. 18, nr 1

145-158

EN

The result that the upper half plane is not preserved in the hyperbolic case has implications in physics, geometry and analysis. We discuss in details the introduction of projective coordinates for the EPH cases. We also introduce an appropriate compactification for all the three EPH cases, which results in a sphere in the elliptic case, a cylinder in the parabolic case and a crosscap in the hyperbolic case.

4

Operators in Banach modules and the Weyl functional calculus for generators of n-parameter C0-groups

Malejki M.

Opuscula Mathematica

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2001

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Vol. 21

65-92

EN

In the paper, I am going to work with unbounded operators in special Banach modules over real Clifford algebras. I propose the definition of n-spectrum for these operators. The mentioned definition is similar to one considered by A. Mclntosh and A. Pryde for bounded operators. The main aim of this work is to describe the support of the Weyl functional calculus T(A) for several operators (formulas for T(A) is according to Anderson) using the Clifford algebras methods. A = (A1,... , Am) is a tuple of operators in a Banach space. The operators Aj are not necessarily bounded but it will be assumed that the tuple A is a generator of the m-parameter C0-group, which satisfies a polynomial growth condition. The final result is the formula suppT(A) = σ(n)(A) ∩ Rm, where σ(n)(A) is the n-spectrum for an operator associated to the tuple A and n ≤ m is an integer.

PL

W tej pracy zajmuję się badaniem nieograniczonych operatorów działających w pewnych modułach Banacha nad algebrami Clifforda. Proponuję tu definicję pewnego rodzaju widma dla takich operatorów, zwanego n-widmem. Definicja ta jest wzorowana na pracy A. Mclntosha i A. Pryde'a, którzy postawili ją dla operatorów ograniczonych. Głównym celem tej pracy jest jednak opisanie nośnika dla rachunku funkcyjnego Weyla T(A) (dla abstrakcyjnych operatorów - wprowadzonego przez W. O. Andersona) dla układów operatorów. Udaje się to wykonać dzięki wykorzystaniu analizy cliffordowskiej. Układ A = (A1,... , Am) jest generatorem m-parametrowej silnie ciągłej grupy operatorów w przestrzeni Banacha. Będę zakładać, że wzrost tej grupy jest ograniczony wielomianowo. Głównym rezultatem tej pracy jest wykazanie równości suppT(A) = σ(n)(A) ∩Rm, gdzie σ(n)(A) jest wspomnianym n-widmem dla pewnego operatora związanego z układem A, natomiast n ≤ m jest pewną liczbą naturalną.

5

Combining Clifford algebraic computing and term-rewriting for geometric theorem proving

Fevre S., Wang D.

Fundamenta Informaticae

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1999

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Vol. 39, Nr 1,2

85-104

EN

A general approach we have proposed for automatically proving geometric theorems requires both Clifford algebraic reduction and term-rewriting. This paper shows how efficient techniques and software tools developed in the areas of algebraic computation and term-rewriting can be combined for our purpose of theorem proving in geometry. Some investigations and experiments for concrete cases have been carried out by combining routines implemented in Maple V and Objective Caml. The experiments together with several examples illustrate the suitability and performance of our approach.