Quadratic Forms and Space-Time Block Codes from Generalized Quaternion and Biquaternion Algebras

In the context of space-time block codes (STBCs), the theory of generalized quaternion and biquaternion algebras (i.e., tensor products of two quaternion algebras) over arbitrary base fields is presented, as well as quadratic form theoretic criteria to check if such algebras are division algebras. For base fields relevant to STBCs, these criteria are exploited, via Springer's theorem, to construct several explicit infinite families of (bi-)quaternion division algebras. These are used to obtain new 2\x 2 and 4\x 4 STBCs.Comment: 8 pages, final versio

Hadron models and related New Energy issues

The present book covers a wide-range of issues from alternative hadron models to their likely implications in New Energy research, including alternative interpretation of lowenergy reaction (coldfusion) phenomena. The authors explored some new approaches to describe novel phenomena in particle physics. M Pitkanen introduces his nuclear string hypothesis derived from his Topological Geometrodynamics theory, while E. Goldfain discusses a number of nonlinear dynamics methods, including bifurcation, pattern formation (complex GinzburgLandau equation) to describe elementary particle masses. Fu Yuhua discusses a plausible method for prediction of phenomena related to New Energy development. F. Smarandache discusses his unmatter hypothesis, and A. Yefremov et al. discuss Yang-Mills field from Quaternion Space Geometry. Diego Rapoport discusses theoretical link between Torsion fields and Hadronic Mechanic. A.H. Phillips discusses semiconductor nanodevices, while V. and A. Boju discuss Digital Discrete and Combinatorial methods and their likely implications in New Energy research. Pavel Pintr et al. describe planetary orbit distance from modified Schrödinger equation, and M. Pereira discusses his new Hypergeometrical description of Standard Model of elementary particles. The present volume will be suitable for researchers interested in New Energy issues, in particular their link with alternative hadron models and interpretation

Quaternion Algebras

This open access textbook presents a comprehensive treatment of the arithmetic theory of quaternion algebras and orders, a subject with applications in diverse areas of mathematics. Written to be accessible and approachable to the graduate student reader, this text collects and synthesizes results from across the literature. Numerous pathways offer explorations in many different directions, while the unified treatment makes this book an essential reference for students and researchers alike. Divided into five parts, the book begins with a basic introduction to the noncommutative algebra underlying the theory of quaternion algebras over fields, including the relationship to quadratic forms. An in-depth exploration of the arithmetic of quaternion algebras and orders follows. The third part considers analytic aspects, starting with zeta functions and then passing to an idelic approach, offering a pathway from local to global that includes strong approximation. Applications of unit groups of quaternion orders to hyperbolic geometry and low-dimensional topology follow, relating geometric and topological properties to arithmetic invariants. Arithmetic geometry completes the volume, including quaternionic aspects of modular forms, supersingular elliptic curves, and the moduli of QM abelian surfaces. Quaternion Algebras encompasses a vast wealth of knowledge at the intersection of many fields. Graduate students interested in algebra, geometry, and number theory will appreciate the many avenues and connections to be explored. Instructors will find numerous options for constructing introductory and advanced courses, while researchers will value the all-embracing treatment. Readers are assumed to have some familiarity with algebraic number theory and commutative algebra, as well as the fundamentals of linear algebra, topology, and complex analysis. More advanced topics call upon additional background, as noted, though essential concepts and motivation are recapped throughout

Employing data fusion & diversity in the applications of adaptive signal processing

The paradigm of adaptive signal processing is a simple yet powerful method for the class of system identification problems. The classical approaches consider standard one-dimensional signals whereby the model can be formulated by flat-view matrix/vector framework. Nevertheless, the rapidly increasing availability of large-scale multisensor/multinode measurement technology has render no longer sufficient the traditional way of representing the data. To this end, the author, who from this point onward shall be referred to as `we', `us', and `our' to signify the author myself and other supporting contributors i.e. my supervisor, my colleagues and other overseas academics specializing in the specific pieces of research endeavor throughout this thesis, has applied the adaptive filtering framework to problems that employ the techniques of data diversity and fusion which includes quaternions, tensors and graphs. At the first glance, all these structures share one common important feature: invertible isomorphism. In other words, they are algebraically one-to-one related in real vector space. Furthermore, it is our continual course of research that affords a segue of all these three data types. Firstly, we proposed novel quaternion-valued adaptive algorithms named the n-moment widely linear quaternion least mean squares (WL-QLMS) and c-moment WL-LMS. Both are as fast as the recursive-least-squares method but more numerically robust thanks to the lack of matrix inversion. Secondly, the adaptive filtering method is applied to a more complex task: the online tensor dictionary learning named online multilinear dictionary learning (OMDL). The OMDL is partly inspired by the derivation of the c-moment WL-LMS due to its parsimonious formulae. In addition, the sequential higher-order compressed sensing (HO-CS) is also developed to couple with the OMDL to maximally utilize the learned dictionary for the best possible compression. Lastly, we consider graph random processes which actually are multivariate random processes with spatiotemporal (or vertex-time) relationship. Similar to tensor dictionary, one of the main challenges in graph signal processing is sparsity constraint in the graph topology, a challenging issue for online methods. We introduced a novel splitting gradient projection into this adaptive graph filtering to successfully achieve sparse topology. Extensive experiments were conducted to support the analysis of all the algorithms proposed in this thesis, as well as pointing out potentials, limitations and as-yet-unaddressed issues in these research endeavor.Open Acces

Algèbres hypercomplexes pour le Calcul

Dans les domaines mathématique ou applicatif, la multiplication de nombres possède un rôle clef pour le Calcul. En Science et en Ingénierie, la nonlinéarité offre de grands défis de modélisation mais aussi de résolution. Notre approche vise, via la multiplication, l'étude de certains phénomènes non linéaires que l'on retrouve fréquemment dans le domaine de la Science et de l'Industrie. Pour cela, nous étudions dans cette thèse la multiplication de nombres multidimensionnels, associée à des structures algébriques en dimension finie appelées algèbres hypercomplexes. Nous utilisons la multiplication comme lien entre les divisions apparentes des différents domaines théorique et pratique que nous abordons par une approche transdisciplinaire. Nous effectuons une analyse comparative entre les algèbres hypercomplexes et les principaux outils de Calcul, approche qui n’est pas développée dans la littérature existante. Nous présentons une synthèse des applications existantes (par ex. robotique, modélisation 3D, électromagnétisme) et des principaux avantages des algèbres hypercomplexes, pour la Science et l’Ingénierie. A partir des conséquences de l’utilisation des structures alternatives (autres que réelles ou complexes), nous proposons une extension nouvelle de la théorie spectrale présentée sous le nom de couplage spectral. Grâce aux algèbres hypercomplexes et à la théorie du couplage spectral, nous présentons des applications inédites à la mécanique et à la chimie ainsi que des perspectives pour le domaine du calcul quantique. Pour les domaines d’applications présentés, existants ou inédits, nous étudions les aspects de modélisation théorique et aussi d’analyse numérique. Nous montrons que suivant les cas d'étude, les aspects numériques avantageux découlent d'un choix judicieux des modèles et des algèbres hypercomplexes associées. Ces avantages sont principalement dus à la manière de définir la multiplication dans les algèbres concernées. Dans les domaines applicatifs abordés, une grande partie des modèles théoriques et numériques repose actuellement sur l’utilisation des nombres réels ou complexes ainsi que sur l’algèbre linéaire. Nous montrons dans cette thèse que les algèbres hypercomplexes sont complémentaires des outils algébriques actuellement utilisés et possèdent un vaste potentiel théorique et pratique, grandement sous-exploité pour le Calcul