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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

An introduction to the edition of two Lemaître’s original manuscripts

Maere C. de, Lambert D.

Demonstratio Mathematica

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2017

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Vol. 50, nr 1

2--27

EN

The aim of this paper is to explain the contributions of G. Lemaître to Spinor Theory. At the end of the paper, we edited also, for the first time two short manuscripts: Spineurs et Quanta and Les spineurs et la physique quantique, written by Lemaître in December 1955 and in January 1956. This edition is a way of honouring Professor Michael Heller because he was the first, with Professor Odon Godart, who discovered, classified and published unedited manuscripts of G. Lemaître.

4

Studies on complex and hypercomplex multidimensional analytic signals

Snopek K. M.

Prace Naukowe Politechniki Warszawskiej. Elektronika

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2013

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z. 190

1--169

EN

Within the last twenty years, the theory of complex and hypercomplex multidimensional signals has been still developing and finding new interesting applications in various fields This monograph, devoted to analytic complex and hypercomplex signals, presents mutual relations between these two approaches in signal- and frequency domains. Complex and hypercomplex signals are n-dimerisional (n-D) generalizations of corresponding complex and hypercomplex numbers belonging to Cayley-Dickson and Clifford algebras. The brief look into the properties of the Cayley-Dickson algebras of quaternions, octonions and sedenions and also into the Clifford algebras of biquaternions and bioctonions is presented. The tables of multiplication of imaginary units in all considered algebras have been constructed. The main part of the book concerns complex and hypercomplex multidimensional signals with spectra defined using various Fourier transformations (FTs). The n-D Clifford FT is recalled and the new n-D Cayley-Dickson FT is introduced. For n = 2, both are equivalent and known as the Quaternion FT (QFT). For n = 3, the new Cayley-Dickson FT, named the Octonion FT (OFT), is introduced The relation between the 3-D complex FT and the OFT is derived basing on multiplication rules in the algebra of octonions. The n-D complex and hypercomplex analytic signals are defined in the signal domain using the n-D Convolution with the corresponding n-D delta distribution. Such a idea was proposed by Hahn in 1992 for multidimensional complex signals with single-orthant spectra. Here, it is applied for quaternion and octonion analytic signals. Their spectra are defined in analogy to the Hahn's approach, as the inverse QFT (resp. inverse OFT) of a single-quadrant (resp. single-octant) spectrum. The complete set of definitions of 2-D and 3-D analytic signals with spectra in all orthants is included. It has been noticed that there are some conjugate pairs between them and basing on this fact, the lower rank signals have been defined. Basing on definitions of complex, quaternion and octonion analytic signals, it is possible to define the local amplitudes and local phase functions of a signal. As the dimension of a signal is higher, the number of polar components increases. It has been shown that in 2-D, a signal is completely defined by two amplitudes and two phase functions in the complex case and in the quaternion case, by a single amplitude and three phases represented by Euler angles. In case of separable signals, the number of components is reduced. In 3-D, the complex analytic signal is represented by four amplitudes and four phase functions, while the octonion analytic signal has a single amplitude and seven phase functions. In this work, the hypothesis concerning the mutual relations between the complex and octonion polar representation is discussed.

5

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.

6

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.

7

Transformacja Darboux-Baecklunda i algebry Clifforda

Cieśliński J.

Postępy Fizyki

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2000

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T. 51, z. dodatk

89-90