Relevant bibliographies by topics / Quaternion analysis

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  2. Dissertations / Theses
  3. Books
  4. Book chapters
  5. Conference papers

Academic literature on the topic 'Quaternion analysis'

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Published: 29 July 2024
Last updated: 30 July 2024

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Journal articles on the topic "Quaternion analysis"

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Dai, Songsong. "Quaternionic Fuzzy Sets." Axioms 12, no. 5 (May 18, 2023): 490. http://dx.doi.org/10.3390/axioms12050490.

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A novel concept of quaternionic fuzzy sets (QFSs) is presented in this paper. QFSs are a generalization of traditional fuzzy sets and complex fuzzy sets based on quaternions. The novelty of QFSs is that the range of the membership function is the set of quaternions with modulus less than or equal to one, of which the real and quaternionic imaginary parts can be used for four different features. A discussion is made on the intuitive interpretation of quaternion-valued membership grades and the possible applications of QFSs. Several operations, including quaternionic fuzzy complement, union, intersection, and aggregation of QFSs, are presented. Quaternionic fuzzy relations and their composition are also investigated. QFS is designed to maintain the advantages of traditional FS and CFS, while benefiting from the properties of quaternions. Cuts of QFSs and rotational invariance of quaternionic fuzzy operations demonstrate the particularity of quaternion-valued grades of membership.
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Miškinis, P. "ON THE POSSIBLE EXISTENCE OF NEW FERMIONIC DEGREES OF FREEDOM IN D = 6." Mathematical Modelling and Analysis 8, no. 2 (June 30, 2003): 155–64. http://dx.doi.org/10.3846/13926292.2003.9637220.

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The bispinors formed by quaternion in D = 6 dimensional space‐time are proposed to be treated as new fermionic fields. The gauge nonabelian field is formulated by the quaternions. A new kind of physical object, an extended relativistic quaternionic membrane in D = 6, is discussed.
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Kim, Ji Eun, and Kwang Ho Shon. "The Regularity of Functions on Dual Split Quaternions in Clifford Analysis." Abstract and Applied Analysis 2014 (2014): 1–8. http://dx.doi.org/10.1155/2014/369430.

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This paper shows some properties of dual split quaternion numbers and expressions of power series in dual split quaternions and provides differential operators in dual split quaternions and a dual split regular function onΩ⊂ℂ2×ℂ2that has a dual split Cauchy-Riemann system in dual split quaternions.
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Chelnokov, Y. N. "Quaternion methods and models of regular celestial mechanics and astrodynamics." Applied Mathematics and Mechanics 43, no. 1 (January 2022): 21–80. http://dx.doi.org/10.1007/s10483-021-2797-9.

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AbstractThis paper is a review, which focuses on our work, while including an analysis of many works of other researchers in the field of quaternionic regularization. The regular quaternion models of celestial mechanics and astrodynamics in the Kustaanheimo-Stiefel (KS) variables and Euler (Rodrigues-Hamilton) parameters are analyzed. These models are derived by the quaternion methods of mechanics and are based on the differential equations of the perturbed spatial two-body problem and the perturbed spatial central motion of a point particle. This paper also covers some applications of these models. Stiefel and Scheifele are known to have doubted that quaternions and quaternion matrices can be used efficiently to regularize the equations of celestial mechanics. However, the author of this paper and other researchers refuted this point of view and showed that the quaternion approach actually leads to efficient solutions for regularizing the equations of celestial mechanics and astrodynamics.This paper presents convenient geometric and kinematic interpretations of the KS transformation and the KS bilinear relation proposed by the present author. More general (compared with the KS equations) quaternion regular equations of the perturbed spatial two-body problem in the KS variables are presented. These equations are derived with the assumption that the KS bilinear relation was not satisfied. The main stages of the quaternion theory of regularizing the vector differential equation of the perturbed central motion of a point particle are presented, together with regular equations in the KS variables and Euler parameters, derived by the aforementioned theory. We also present the derivation of regular quaternion equations of the perturbed spatial two-body problem in the Levi-Civita variables and the Euler parameters, developed by the ideal rectangular Hansen coordinates and the orientation quaternion of the ideal coordinate frame.This paper also gives new results using quaternionic methods in the perturbed spatial restricted three-body problem.
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Gadetska, Svitlana, Valeriy Dubnitskiy, Yuri Kushneruk, and Alexander Khodyrev. "Calculation of the results of basic arithmetic operations with quaternions defined in the interval form." Advanced Information Systems 7, no. 4 (December 3, 2023): 21–32. http://dx.doi.org/10.20998/2522-9052.2023.4.03.

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The goal of the work. Creation of an EXCEL-oriented calculator for determining the results of basic arithmetic operations with quaternions, which are given by hyperbolic interval numbers. The subject of research is the set of quaternions and arithmetic operations defined on this set. Research method: Algorithmic and numerical analysis of procedures for performing basic arithmetic operations with quaternions given by hyperbolic interval numbers. The obtained results. The definition of a quaternion, the coefficients of which are interval numbers, is given. It was obtained the ratios for determining in the interval form the following characteristics: quaternion norm, quaternion modulus, quaternion vector part modulus, quaternion argument, quaternion sign was obtained. It was obtained the ratios for determining in the interval form the following basic arithmetic operations: sum of quaternions, difference of quaternions, multiplication of a constant value by a quaternion, scalar product of quaternions, product of vector parts of quaternions, product of quaternions, vector product of quaternions, left and right division of quaternions. The ratio for calculating the determinants of the second order, the elements of which are defined in the interval form, is obtained. The ratio for the approximate calculation of determinants of arbitrary order, the elements of which are defined in the interval form, is obtained. It is shown that the operations of multiplication (division) of interval numbers and raising them to an integer power are expedient to perform when they have a hyperbolic form. It is advisable to perform the addition (subtraction) operation with interval numbers given in the classic form or in the CENTER-RADIUS system. It is better to use the last form in the case of determining the coefficients of the quaternions based on the results of technological calculations. Screenshots of the formulas for determining the vector product of quaternions are given, provided that the coefficients at their orths are represented by interval numbers.
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Kim, Ji-Eun. "Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System." Axioms 10, no. 3 (August 30, 2021): 206. http://dx.doi.org/10.3390/axioms10030206.

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The step derivative of a complex function can be defined with various methods. The step direction defines a basis that is distinct from that of a complex number; the derivative can then be treated by using Taylor series expansion in this direction. In this study, we define step derivatives based on complex numbers and quaternions that are orthogonal to the complex basis while simultaneously being distinct from it. Considering previous studies, the step derivative defined using quaternions was insufficient for applying the properties of quaternions by setting a quaternion basis distinct from the complex basis or setting the step direction to which only a part of the quaternion basis was applied. Therefore, in this study, we examine the definition of quaternions and define the step derivative in the direction of a generalized quaternion basis including a complex basis. We find that the step derivative based on the definition of a quaternion has a relative error in some domains; however, it can be used as a substitute derivative in specific domains.
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Scheglov, Georgiy. "On the geometric interpretation of quaternions by cones." Geometry & Graphics 10, no. 3 (January 9, 2023): 23–34. http://dx.doi.org/10.12737/2308-4898-2023-10-3-23-34.

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The geometric interpretation of quaternions is considered. The visualization complexity of quaternions is due to the fact that these objects have four independent parameters. A literature analysis shows that the problem of geometric interpretation of quaternions has not been completely solved to date. The first section provides general provisions on quaternions and the necessary notations. The second section describes the classical geometric interpretation of quaternions by arcs on a unit sphere. The third section describes a new geometric interpretation and its application to the problem of a vector finite rotation. The geometric interpretation of the quaternion as the surface of a right circular cone is presented. This representation allow demonstrating it as a holistic object in which the scalar and vector parts are interconnected, taking into account their modules and signs. For the considered normalized quaternion, it is easy to understanding an important entity, the quaternion versor: in general, it is a cone, which in the limiting case of a pure scalar quaternion transform into a sphere, and in the limiting case of a pure vector quaternion transform into an ordinary vector. This distinctive feature of the proposed geometric interpretation makes it possible, even when projected onto a plane, to clearly distinguish visualization of the quaternions with a nonzero scalar part from pure vector quaternions, which is difficult to do in the other known interpretations. The representation of quaternions by cones clearly demonstrates the need for a double quaternion product, when the vector is rotated around an arbitrary axis. Images of quaternions as cones, spheres and vectors can be useful in the study of quaternion algebra, which is currently finding increasing use in engineering.
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Scheglov, Georgiy. "On the geometric interpretation of quaternions by cones." Geometry & Graphics 10, no. 3 (September 26, 2022): 23–34. http://dx.doi.org/10.12737/2308-4898-2022-10-3-23-34.

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The geometric interpretation of quaternions is considered. The visualization complexity of quaternions is due to the fact that these objects have four independent parameters. A literature analysis shows that the problem of geometric interpretation of quaternions has not been completely solved to date. The first section provides general provisions on quaternions and the necessary notations. The second section describes the classical geometric interpretation of quaternions by arcs on a unit sphere. The third section describes a new geometric interpretation and its application to the problem of a vector finite rotation. The geometric interpretation of the quaternion as the surface of a right circular cone is presented. This representation allow demonstrating it as a holistic object in which the scalar and vector parts are interconnected, taking into account their modules and signs. For the considered normalized quaternion, it is easy to understanding an important entity, the quaternion versor: in general, it is a cone, which in the limiting case of a pure scalar quaternion transform into a sphere, and in the limiting case of a pure vector quaternion transform into an ordinary vector. This distinctive feature of the proposed geometric interpretation makes it possible, even when projected onto a plane, to clearly distinguish visualization of the quaternions with a nonzero scalar part from pure vector quaternions, which is difficult to do in the other known interpretations. The representation of quaternions by cones clearly demonstrates the need for a double quaternion product, when the vector is rotated around an arbitrary axis. Images of quaternions as cones, spheres and vectors can be useful in the study of quaternion algebra, which is currently finding increasing use in engineering.
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Dargys, Adolfas, and Artūras Acus. "Exponential and logarithm of multivector in low-dimensional (n = p + q < 3) Clifford algebras." Nonlinear Analysis: Modelling and Control 27, no. 6 (October 19, 2022): 1129–49. http://dx.doi.org/10.15388/namc.2022.27.29528.

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The aim of the paper is to give a uniform picture of complex, hyperbolic, and quaternion algebras from a perspective of the applied Clifford geometric algebra. Closed form expressions for a multivector exponential and logarithm are presented in real geometric algebras Clp;q when n = p + q = 1 (complex and hyperbolic numbers) and n = 2 (Hamilton, split, and conectorine quaternions). Starting from Cl0;1 and Cl1;0 algebras wherein square of a basis vector is either –1 or +1, we have generalized exponential and logarithm formulas to 2D quaternionic algebras Cl0;2, Cl1;1, and Cl2;0. The sectors in the multivector coefficient space, where 2D logarithm exists are found. They are related with a square root of the multivector.
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Kou, Kit Ian, Jian-Yu Ou, and Joao Morais. "On Uncertainty Principle for Quaternionic Linear Canonical Transform." Abstract and Applied Analysis 2013 (2013): 1–14. http://dx.doi.org/10.1155/2013/725952.

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We generalize the linear canonical transform (LCT) to quaternion-valued signals, known as the quaternionic linear canonical transform (QLCT). Using the properties of the LCT we establish an uncertainty principle for the QLCT. This uncertainty principle prescribes a lower bound on the product of the effective widths of quaternion-valued signals in the spatial and frequency domains. It is shown that only a 2D Gaussian signal minimizes the uncertainty.
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Dissertations / Theses on the topic "Quaternion analysis"

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Ha, Vu Thi Ngoc. "Helmholtz operator in quaternionic analysis." [S.l. : s.n.], 2005. http://www.diss.fu-berlin.de/2005/61/index.html.

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Aparicio, Conrado. "Implementation of a quaternion-based Kalman filter for human body motion tracking using MARG sensors." Thesis, Monterey, Calif. : Springfield, Va. : Naval Postgraduate School ; Available from National Technical Information Service, 2004. http://library.nps.navy.mil/uhtbin/hyperion/04Sep%5FAparicio.pdf.

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Martori, Amanda Lynn. "A Wearable Motion Analysis System to Evaluate Gait Deviations." Scholar Commons, 2013. http://scholarcommons.usf.edu/etd/4724.

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A Wearable Motion Analysis System (WMAS) was developed to evaluate gait, particularly parameters that are indicative of mild traumatic brain injury. The WMAS consisted on six Opal IMUs attached on the sternum, waist, left and right thigh and left and right shank. Algorithms were developed to calculate the knee flexion angle, stride length and cadence parameters during slow, normal and fast gait speeds. The WMAS was validated for repeatability using a robotic arm and accuracy using the Vicon motion capture system, the gold standard for gait analysis. The WMAS calculated the gait parameters to within a clinically acceptable range and is a powerful tool for gait analysis and potential concussion diagnosis outside of a laboratory setting.
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Le, Thu Hoai. "Hyperholomorphic structures and corresponding explicit orthogonal function systems in 3D and 4D." Doctoral thesis, Technische Universitaet Bergakademie Freiberg Universitaetsbibliothek "Georgius Agricola", 2014. http://nbn-resolving.de/urn:nbn:de:bsz:105-qucosa-150508.

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Die Reichhaltigkeit und breite Anwendbarkeit der Theorie der holomorphen Funktionen in der komplexen Ebene ist stark motivierend eine ähnliche Theorie für höhere Dimensionen zu entwickeln. Viele Forscher waren und sind in diese Aufgaben involviert, insbesondere in der Entwicklung der Quaternionenanalysis. In den letzten Jahren wurde die Quaternionenanalysis bereits erfolgreich auf eine Vielzahl von Problemen der mathematischen Physik angewandt. Das Ziel der Dissertation besteht darin, holomorphe Strukturen in höheren Dimensionen zu studieren. Zunächst wird ein neues Holomorphiekonzept vorgelegt, was auf der Theorie rechtsinvertierbarer Operatoren basiert und nicht auf Verallgemeinerungen des Cauchy-Riemann-Systems wie üblich. Dieser Begriff umfasst die meisten der gut bekannten holomorphen Strukturen in höheren Dimensionen. Unter anderem sind die üblichen Modelle für reelle und komplexe quaternionenwertige Funktionen sowie Clifford-algebra-wertige Funktionen enthalten. Außerdem werden holomorphe Funktionen mittels einer geeignete Formel vom Taylor-Typ durch spezielle Funktionen lokal approximiert. Um globale Approximationen für holomorphe Funktionen zu erhalten, werden im zweiten Teil der Arbeit verschiedene Systeme holomorpher Basisfunktionen in drei und vier Dimensionen mittels geeigneter Fourier-Entwicklungen explizit konstruiert. Das Konzept der Holomorphie ist verbunden mit der Lösung verallgemeinerter Cauchy-Riemann Systeme, deren Funktionswerte reellen Quaternionen bzw. reduzierte Quaternionen sind. In expliziter Form werden orthogonale holomorphe Funktionensysteme konstruiert, die Lösungen des Riesz-Systems bzw. des Moisil-Teodorescu Systems über zylindrischen Gebieten im R3, sowie Lösungen des Riesz-Systems in Kugeln des R4 sind. Um konkrete Anwendungen auf Randwertprobleme realisieren zu können wird eine orthogonale Zerlegung eines Rechts-Quasi-Hilbert-Moduls komplex-quaternionischer Funktionen unter gegebenen Bedingungen studiert. Die Ergebnisse werden auf die Behandlung von Maxwell-Gleichungen mit zeitvariabler elektrischer Dielektrizitätskonstante und magnetischer Permeabilität angewandt
The richness and widely applicability of the theory of holomorphic functions in complex analysis requires to perform a similar theory in higher dimensions. It has been developed by many researchers so far, especially in quaternionic analysis. Over the last years, it has been successfully applied to a vast array of problems in mathematical physics. The aim of this thesis is to study the structure of holomorphy in higher dimensions. First, a new concept of holomorphy is introduced based on the theory of right invertible operators, and not by means of an analogue of the Cauchy-Riemann operator as usual. This notion covers most of the well-known holomorphic structures in higher dimensions including real, complex, quaternionic, Clifford analysis, among others. In addition, from our operators a local approximation of a holomorphic function is attained by the Taylor type formula. In order to obtain the global approximation for holomorphic functions, the second part of the thesis deals with the construction of different systems of basis holomorphic functions in three and four dimensions by means of Fourier analysis. The concept of holomorphy is related to the null-solutions of generalized Cauchy-Riemann systems, which take either values in the reduced quaternions or real quaternions. We obtain several explicit orthogonal holomorphic function systems: solutions to the Riesz and Moisil-Teodorescu systems over cylindrical domains in R3, and solutions to the Riesz system over spherical domains in R4. Having in mind concrete applications to boundary value problems, we investigate an orthogonal decomposition of complex-quaternionic functions over a right quasi-Hilbert module under given conditions. It is then applied to the treatment of Maxwell’s equations with electric permittivity and magnetic permeability depending on the time variable
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Morchid, Mohamed. "Représentations robustes de documents bruités dans des espaces homogènes." Thesis, Avignon, 2014. http://www.theses.fr/2014AVIG0202/document.

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En recherche d’information, les documents sont le plus souvent considérés comme des "sacs-de-mots". Ce modèle ne tient pas compte de la structure temporelle du document et est sensible aux bruits qui peuvent altérer la forme lexicale. Ces bruits peuvent être produits par différentes sources : forme peu contrôlée des messages des sites de micro-blogging, messages vocaux dont la transcription automatique contient des erreurs, variabilités lexicales et grammaticales dans les forums du Web. . . Le travail présenté dans cette thèse s’intéresse au problème de la représentation de documents issus de sources bruitées.La thèse comporte trois parties dans lesquelles différentes représentations des contenus sont proposées. La première partie compare une représentation classique utilisant la fréquence des mots à une représentation de haut-niveau s’appuyant sur un espace de thèmes. Cette abstraction du contenu permet de limiter l’altération de la forme de surface du document bruité en le représentant par un ensemble de caractéristiques de haut-niveau. Nos expériences confirment que cette projection dans un espace de thèmes permet d’améliorer les résultats obtenus sur diverses tâches de recherche d’information en comparaison d’une représentation plus classique utilisant la fréquence des mots.Le problème majeur d’une telle représentation est qu’elle est fondée sur un espace de thèmes dont les paramètres sont choisis empiriquement.La deuxième partie décrit une nouvelle représentation s’appuyant sur des espaces multiples et permettant de résoudre trois problèmes majeurs : la proximité des sujets traités dans le document, le choix difficile des paramètres du modèle de thèmes ainsi que la robustesse de la représentation. Partant de l’idée qu’une seule représentation des contenus ne peut pas capturer l’ensemble des informations utiles, nous proposons d’augmenter le nombre de vues sur un même document. Cette multiplication des vues permet de générer des observations "artificielles" qui contiennent des fragments de l’information utile. Une première expérience a validé cette approche multi-vues de la représentation de textes bruités. Elle a cependant l’inconvénient d’être très volumineuse,redondante, et de contenir une variabilité additionnelle liée à la diversité des vues. Dans un deuxième temps, nous proposons une méthode s’appuyant sur l’analyse factorielle pour fusionner les vues multiples et obtenir une nouvelle représentation robuste,de dimension réduite, ne contenant que la partie "utile" du document tout en réduisant les variabilités "parasites". Lors d’une tâche de catégorisation de conversations,ce processus de compression a confirmé qu’il permettait d’augmenter la robustesse de la représentation du document bruité.Cependant, lors de l’élaboration des espaces de thèmes, le document reste considéré comme un "sac-de-mots" alors que plusieurs études montrent que la position d’un terme au sein du document est importante. Une représentation tenant compte de cette structure temporelle du document est proposée dans la troisième partie. Cette représentation s’appuie sur les nombres hyper-complexes de dimension appelés quaternions. Nos expériences menées sur une tâche de catégorisation ont montré l’efficacité de cette méthode comparativement aux représentations classiques en "sacs-de-mots"
In the Information Retrieval field, documents are usually considered as a "bagof-words". This model does not take into account the temporal structure of thedocument and is sensitive to noises which can alter its lexical form. These noisescan be produced by different sources : uncontrolled form of documents in microbloggingplatforms, automatic transcription of speech documents which are errorprone,lexical and grammatical variabilities in Web forums. . . The work presented inthis thesis addresses issues related to document representations from noisy sources.The thesis consists of three parts in which different representations of content areavailable. The first one compares a classical representation based on a term-frequencyrepresentation to a higher level representation based on a topic space. The abstractionof the document content allows us to limit the alteration of the noisy document byrepresenting its content with a set of high-level features. Our experiments confirm thatmapping a noisy document into a topic space allows us to improve the results obtainedduring different information retrieval tasks compared to a classical approach based onterm frequency. The major problem with such a high-level representation is that it isbased on a space theme whose parameters are chosen empirically.The second part presents a novel representation based on multiple topic spaces thatallow us to solve three main problems : the closeness of the subjects discussed in thedocument, the tricky choice of the "right" values of the topic space parameters and therobustness of the topic-based representation. Based on the idea that a single representationof the contents cannot capture all the relevant information, we propose to increasethe number of views on a single document. This multiplication of views generates "artificial"observations that contain fragments of useful information. The first experimentvalidated the multi-view approach to represent noisy texts. However, it has the disadvantageof being very large and redundant and of containing additional variability associatedwith the diversity of views. In the second step, we propose a method based onfactor analysis to compact the different views and to obtain a new robust representationof low dimension which contains only the informative part of the document whilethe noisy variabilities are compensated. During a dialogue classification task, the compressionprocess confirmed that this compact representation allows us to improve therobustness of noisy document representation.Nonetheless, during the learning process of topic spaces, the document is consideredas a "bag-of-words" while many studies have showed that the word position in a7document is useful. A representation which takes into account the temporal structureof the document based on hyper-complex numbers is proposed in the third part. Thisrepresentation is based on the hyper-complex numbers of dimension four named quaternions.Our experiments on a classification task have showed the effectiveness of theproposed approach compared to a conventional "bag-of-words" representation
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Whelchel, Pamela Jean. "Analysis on a hyperplane of the quaternions." CSUSB ScholarWorks, 1995. https://scholarworks.lib.csusb.edu/etd-project/1221.

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Lesesvre, Didier. "Arithmetic Statistics for Quaternion Algebras." Thesis, Sorbonne Paris Cité, 2018. http://www.theses.fr/2018USPCD040.

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Les formes automorphes sont des objets centraux en théorie des nombres. En dépit de leur omniprésence, elles demeurent mystérieuses et leur comportement est loin d'être entièrement compris. Considérer ces formes automorphes au sein de familles a un effet régularisant, et ouvre la voie aux résultats en moyenne : voilà l'esprit des statistiques arithmétiques. La famille de toutes les représentations automorphes d'un groupe réductif donné, appelée famille universelle du groupe, est particulièrement importante. Dans le cas des formes intérieures de GL(2), autrement dit les groupes d'unités d'algèbres de quaternions, la formule des traces de Selberg est une puissante méthode d'approche. Il existe une notion de taille sur les formes automorphes, le conducteur analytique, permettant de tronquer la famille universelle en un ensemble fini pour lequel ces problèmes de statistiques arithmétiques ont un sens.Une loi de comptage pour la famille universelle tronquée est établie, avec un terme d'erreur gagnant par une puissance dans le cas totalement défini, et une constante à forte teneur géométrique. Cette loi de Weyl est généralisée en un résultat d'équirépartition par rapport à une mesure explicite, et mène à vérifier les conjectures de Sato-Tate dans ce cadre. Des statistiques sur les petits zéros des fonctions L associées sont établies, menant à dévoiler partiellement le type de symétrie des algèbres de quaternions.Plusieurs indices sont mentionnés laissant à croire que d'autres groupes sont abordables par les mêmes méthodes, et les lois de comptage conjecturales pour certains groupes unitaires et symplectiques de petits rangs sont énoncées
Automorphic forms are central objects in modern number theory. Despite their ubiquity, they remain mysterious and their behavior is far from understood. Embedding them in wider families has a smoothing effect, allowing results on average: these are the aims of arithmetic statistics. The whole family of automorphic representations of a given reductive group, referred to as its universal family, is of fundamental importance. In the case of inner forms of GL(2), that is to say groups of units of quaternion algebras, the Selberg trace formula is a powerful method to handle it. There is a way to define a suitable notion of size, the analytic conductor, allowing to truncate the universal family to a finite one amenable to arithmetic statistics methods. A counting law for the truncated universal family is established, with a power savings error term in the totally definite case and a geometrically meaningful constant. This Weyl's law is generalized to an equidistribution result with respect to an explicit measure, and leads to answer the Sato-Tate conjectures in this case. Statistics on low-lying zeros are provided, leading to uncover part of the type of symmetry of quaternion algebras.Strong evidence is provided that further ground groups should be amenable as well to the same methods and conjectural counting laws are given in the case of symplectic and unitary groups of low ranks
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Flamant, Julien. "Une approche générique pour l'analyse et le filtrage des signaux bivariés." Thesis, Ecole centrale de Lille, 2018. http://www.theses.fr/2018ECLI0008/document.

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Les signaux bivariés apparaissent dans de nombreuses applications (optique, sismologie, océanographie, EEG, etc.) dès lors que l'analyse jointe de deux signaux réels est nécessaire. Les signaux bivariés simples ont une interprétation naturelle sous la forme d'une ellipse dont les propriétés (taille, forme, orientation) peuvent évoluer dans le temps. Cette propriété géométrique correspondant à la notion de polarisation en physique est fondamentale pour la compréhension et l'analyse des signaux bivariés. Les approches existantes n'apportent cependant pas de description directe des signaux bivariés ou des opérations de filtrage en termes de polarisation. Cette thèse répond à cette limitation par l'introduction d'une nouvelle approche générique pour l'analyse et le filtrage des signaux bivariés. Celle-ci repose sur deux ingrédients essentiels : (i) le plongement naturel des signaux bivariés -- vus comme signaux à valeurs complexes -- dans le corps des quaternions H et (ii) la définition d'une transformée de Fourier quaternionique associée pour une représentation spectrale interprétable de ces signaux. L'approche proposée permet de définir les outils de traitement de signal usuels tels que la notion de densité spectrale, de filtrage linéaire ou encore de spectrogramme ayant une interprétation directe en termes d'attributs de polarisation. Nous montrons la validité de l'approche grâce à des garanties mathématiques et une implémentation numériquement efficace des outils proposés. Diverses expériences numériques illustrent l'approche. En particulier, nous démontrons son potentiel pour la caractérisation de la polarisation des ondes gravitationnelles
Bivariate signals appear in a broad range of applications (optics, seismology, oceanography, EEG, etc.) where the joint analysis of two real-valued signals is required. Simple bivariate signals take the form of an ellipse, whose properties (size, shape, orientation) may evolve with time. This geometric feature of bivariate signals has a natural physical interpretation called polarization. This notion is fundamental to the analysis and understanding of bivariate signals. However, existing approaches do not provide straightforward descriptions of bivariate signals or filtering operations in terms of polarization or ellipse properties. To this purpose, this thesis introduces a new and generic approach for the analysis and filtering of bivariate signals. It essentially relies on two key ingredients: (i) the natural embedding of bivariate signals -- viewed as complex-valued signals -- into the set of quaternions H and (ii) the definition of a dedicated quaternion Fourier transform to enable a meaningful spectral representation of bivariate signals. The proposed approach features the definition of standard signal processing quantities such as spectral densities, linear time-invariant filters or spectrograms that are directly interpretable in terms of polarization attributes. More importantly, the framework does not sacrifice any mathematical guarantee and the newly introduced tools admit computationally fast implementations. Numerical experiments support throughout our theoretical developments. We also demonstrate the potential of the approach for the nonparametric characterization of the polarization of gravitational waves
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9

Gezgin, Erkin Alizade Rasim. "Biokinematic analysis of human arm." [s.l.]: [s.n.], 2006. http://library.iyte.edu.tr/tezler/master/makinamuh/T000547.pdf.

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Thesis (Master)--İzmir Institute of Technology, İzmir, 2006.
Keywords: Biokinematic analysis, quaternionlar, structural groups, mechanism theory,robot manipulators. Includes bibliographical references (leaves 78-85).
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Kumar, Indra E. "Graphical Applications of Complex and Quaternionic Fractional Linear Transformations." Scholarship @ Claremont, 2016. http://scholarship.claremont.edu/scripps_theses/888.

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The geometric properties of the complex numbers and the quaternions, particularly their behavior under fractional linear transformations, make them very useful for modeling certain types of geometric objects. In this thesis, we will connect the characteristics of fractional linear transformations of both the complex numbers and quaternions to the problem of developing and modifying discrete surfaces for problems in computer graphics and engineering.
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Books on the topic "Quaternion analysis"

1

Hitzer, Eckhard. Quaternion and Clifford Fourier Transforms and Wavelets. Basel: Springer Basel, 2013.

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Conway, John Horton. On quaternions and octonions: Their geometry, arithmetic, and symmetry. Natick, Mass: AK Peters, 2003.

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Kravchenko, Vladislav V. Applied quaternionic analysis. Lemgo, Germany: Heldermann, 2003.

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1957-, Kamberov George, ed. Quaternions, spinors, and surfaces. Providence, R.I: American Mathematical Society, 2002.

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Gürlebeck, Klaus, and Wolfgang Sprößig. Quaternionic Analysis and Elliptic Boundary Value Problems. Basel: Birkhäuser Basel, 1990. http://dx.doi.org/10.1007/978-3-0348-7295-9.

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Gürlebeck, Klaus. Quaternionic analysis and elliptic boundary value problems. Berlin: Akademie-Verlag, 1989.

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Wolfgang, Sprössig, Gürlebeck Klaus, and Volkswagenstiftung, eds. Proceedings of the symposium: Analytical and numerical methods in quaternionic and Clifford analysis : Seiffen 1996. [S.l: s.n., 1996.

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Freguglia, Paolo. Dalle equipollenze ai sistemi lineari: Il contributo italiano al calcolo geometrico. Urbino: QuattroVenti, 1992.

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Quaternionic Analysis: Functions of one quaternionic variable. Independent, 2023.

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Hitzer, Eckhard. Quaternion and Clifford Fourier Transforms. Taylor & Francis Group, 2021.

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Book chapters on the topic "Quaternion analysis"

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Traversoni, Leonardo. "Image Analysis Using Quaternion Wavelets." In Geometric Algebra with Applications in Science and Engineering, 326–45. Boston, MA: Birkhäuser Boston, 2001. http://dx.doi.org/10.1007/978-1-4612-0159-5_16.

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Zhu, Guangyu, Shuicheng Yan, Tony X. Han, and Changsheng Xu. "Generative Group Activity Analysis with Quaternion Descriptor." In Lecture Notes in Computer Science, 1–11. Berlin, Heidelberg: Springer Berlin Heidelberg, 2011. http://dx.doi.org/10.1007/978-3-642-17829-0_1.

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Kajiwara, Joji, Xiao Dong Li, and Kwang Ho Shon. "Regeneration in Complex, Quaternion and Clifford Analysis." In Finite or Infinite Dimensional Complex Analysis and Applications, 287–98. Boston, MA: Springer US, 2004. http://dx.doi.org/10.1007/978-1-4613-0221-6_19.

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Piórek, Michał. "Analysis of Chaos for Quaternion Time Series." In Studies in Systems, Decision and Control, 73–88. Cham: Springer International Publishing, 2018. http://dx.doi.org/10.1007/978-3-319-94887-4_7.

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Sheng, Gong. "Harmonic Analysis in Classical Domains on Quaternion Field." In Harmonic Analysis on Classical Groups, 246–58. Berlin, Heidelberg: Springer Berlin Heidelberg, 1991. http://dx.doi.org/10.1007/978-3-642-58189-2_14.

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Gürlebeck, Klaus. "On Weighted Spaces of Monogenic Quaternion-valued Functions." In Clifford Analysis and Its Applications, 81–89. Dordrecht: Springer Netherlands, 2001. http://dx.doi.org/10.1007/978-94-010-0862-4_8.

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Georgiev, S., and J. Morais. "Bochner’s Theorems in the Framework of Quaternion Analysis." In Quaternion and Clifford Fourier Transforms and Wavelets, 85–104. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0603-9_5.

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Rajchakit, Grienggrai, Praveen Agarwal, and Sriraman Ramalingam. "Stability of Discrete-Time Stochastic Quaternion-Valued Neural Networks." In Stability Analysis of Neural Networks, 311–39. Singapore: Springer Singapore, 2021. http://dx.doi.org/10.1007/978-981-16-6534-9_10.

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Guerra, Reynolds León, Edel B. García Reyes, Annette M. González Quevedo, and Heydi Méndez Vázquez. "FqSD: Full-Quaternion Saliency Detection in Images." In Progress in Pattern Recognition, Image Analysis, Computer Vision, and Applications, 462–69. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-13469-3_54.

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Chen, Meng, Chenxia Wang, Xiao Meng, and Zhifang Wang. "Quaternion Principal Component Analysis for Multi-modal Fusion." In Advances in Intelligent Systems and Computing, 11–19. Cham: Springer International Publishing, 2015. http://dx.doi.org/10.1007/978-3-319-23207-2_2.

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Conference papers on the topic "Quaternion analysis"

1

Lianggui, Feng, Theodore E. Simos, George Psihoyios, and Ch Tsitouras. "Quaternionic Numerical Ranges of Normal Quaternion Matrices." In NUMERICAL ANALYSIS AND APPLIED MATHEMATICS: International Conference on Numerical Analysis and Applied Mathematics 2009: Volume 1 and Volume 2. AIP, 2009. http://dx.doi.org/10.1063/1.3241220.

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Bahri, Mawardi, and Ryuichi Ashino. "Relationship between quaternion linear canonical and quaternion fourier transforms." In 2014 International Conference on Wavelet Analysis and Pattern Recognition (ICWAPR). IEEE, 2014. http://dx.doi.org/10.1109/icwapr.2014.6961301.

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Ge, Qiaode Jeffrey, Zihan Yu, Mona Arbab, and Mark Langer. "On the Computation of the Average of Spatial Displacements." In ASME 2022 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2022. http://dx.doi.org/10.1115/detc2022-90156.

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Abstract Many applications in biomechanics and medical imaging call for the analysis of the kinematic errors in a group of patients statistically using the average displacement and the standard deviations from the average. This paper studies the problem of computing the average displacement from a set of given spatial displacements using three types of parametric representations: Euler angles and translation vectors, unit quaternions and translation vectors, and dual quaternions. It has been shown that the use of Euclidean norm in the space of unit quaternions reduces the problem to that of computing the average for each quaternion component separately and independently. While the resulting algorithm is simple, the change of the sign of a unit quaternion could lead to an incorrect result. A novel kinematic measure based on dual quaternions is introduced to capture the separation between two spatial displacement. This kinematic measure is then used to formulate a constrained least squares minimization problem. It has been shown that the problem decomposes into that of finding the optimal translation vector and the optimal unit quaternion. The former is simply the centroid of the set of given translation vectors and the latter can be obtained as the eigenvector corresponding to the least eigenvalue of a 4 × 4 positive definite symmetric matrix. It is found that the weight factor used in combining rotations and translations in the formulation does not play a role in the final outcome. Examples are provided to show the comparisons of these methods.
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"Clifford and Quaternion Analysis." In Proceedings of the 6th International ISAAC Congress. WORLD SCIENTIFIC, 2009. http://dx.doi.org/10.1142/9789812837332_others03.

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"Clifford and quaternion analysis." In Proceedings of the 7th International ISAAC Congress. WORLD SCIENTIFIC, 2010. http://dx.doi.org/10.1142/9789814313179_others04.

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Bahri, Mawardi, Ryuichi Ashino, and Remi Vaillancourt. "Two-dimensional quaternion Fourier transform of type II and quaternion wavelet transform." In 2012 International Conference on Wavelet Analysis and Pattern Recognition (ICWAPR). IEEE, 2012. http://dx.doi.org/10.1109/icwapr.2012.6294808.

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Traversoni, Leonardo, Yi Xu, Theodore E. Simos, George Psihoyios, and Ch Tsitouras. "Velocity and Object Detection Using Quaternion Wavelets." In Numerical Analysis and Applied Mathematics. AIP, 2007. http://dx.doi.org/10.1063/1.2790268.

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Wang, Shi-Yuan, and Cui-Ming Zou. "Quaternion Elastic Net for Signal Recovery." In 2023 International Conference on Wavelet Analysis and Pattern Recognition (ICWAPR). IEEE, 2023. http://dx.doi.org/10.1109/icwapr58546.2023.10336986.

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Kong, Xianwen, Jingjun Yu, and Duanling Li. "Reconfiguration Analysis of a 2-DOF 3-4R Parallel Manipulator With Planar Base and Platform." In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-46219.

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This paper deals with a 2-DOF 3-4R parallel manipulator (PM) with planar base and platform — a novel PM with multiple operation mode (or disassembly-free reconfigurable PM) with minimum number of actuated joints. At first, a set of constraint equations of the 3-4R PM is derived with the orientation of the moving platform represented using a Euler parameter quaternion (also Euler-Rodrigues quaternion) and then solved using the algebraic geometry method. It is found that this 3-4R PM has six 2-DOF operation modes, including the two expected spherical translation mode and sphere-on-sphere rolling mode when the PM was synthesized. The motion characteristics of the moving platform are obtained using the kinematic interpretation of Euler parameter quaternions with certain number of constant zero components, which was presented in a recent paper by the first author of this paper, instead of the eigenspace based approach in the literature. The transition configurations, which are constraint singular configurations, among different operation modes are also presented. This work provides a solid foundation to the development and control of the 2-DOF 3-4R parallel manipulator (PM) with both 2-DOF spherical translation mode and 2-DOF sphere-on-sphere rolling mode.
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Ahmed, A. El-Sayed, Aydah Ahmadi, Theodore E. Simos, George Psihoyios, Ch Tsitouras, and Zacharias Anastassi. "On Weighted Bloch Spaces of Quaternion-Valued Functions." In NUMERICAL ANALYSIS AND APPLIED MATHEMATICS ICNAAM 2011: International Conference on Numerical Analysis and Applied Mathematics. AIP, 2011. http://dx.doi.org/10.1063/1.3636720.

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