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Relevant bibliographies by topics / Quaternionic calculus

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Academic literature on the topic 'Quaternionic calculus'

Author: Grafiati

Published: 29 July 2024

Last updated: 30 July 2024

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Journal articles on the topic "Quaternionic calculus"

1

GHILONI, RICCARDO, VALTER MORETTI, and ALESSANDRO PEROTTI. "CONTINUOUS SLICE FUNCTIONAL CALCULUS IN QUATERNIONIC HILBERT SPACES." Reviews in Mathematical Physics 25, no. 04 (May 2013): 1350006. http://dx.doi.org/10.1142/s0129055x13500062.

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The aim of this work is to define a continuous functional calculus in quaternionic Hilbert spaces, starting from basic issues regarding the notion of spherical spectrum of a normal operator. As properties of the spherical spectrum suggest, the class of continuous functions to consider in this setting is the one of slice quaternionic functions. Slice functions generalize the concept of slice regular function, which comprises power series with quaternionic coefficients on one side and that can be seen as an effective generalization to quaternions of holomorphic functions of one complex variable. The notion of slice function allows to introduce suitable classes of real, complex and quaternionic C*-algebras and to define, on each of these C*-algebras, a functional calculus for quaternionic normal operators. In particular, we establish several versions of the spectral map theorem. Some of the results are proved also for unbounded operators. However, the mentioned continuous functional calculi are defined only for bounded normal operators. Some comments on the physical significance of our work are included.

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Ghamari, Elham, and Dan Kučerovský. "Quaternions and Functional Calculus." Symmetry 11, no. 8 (July 26, 2019): 953. http://dx.doi.org/10.3390/sym11080953.

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In this paper, we develop the notion of generalized characters and a corresponding Gelfand theory for quaternionic C * -algebras. These are C*-algebras whose structure permits an action of the quaternions. Applications are made to functional calculus, and we develop an S-functional calculus related to what we term structural regular functions.

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Bolokhov, Pavel A. "Quaternionic wave function." International Journal of Modern Physics A 34, no. 02 (January 20, 2019): 1950001. http://dx.doi.org/10.1142/s0217751x19500015.

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We argue that quaternions form a natural language for the description of quantum-mechanical wave functions with spin. We use the quaternionic spinor formalism which is in one-to-one correspondence with the usual spinor language. No unphysical degrees of freedom are admitted, in contrast to the majority of literature on quaternions. In this paper, we first build a Dirac Lagrangian in the quaternionic form, derive the Dirac equation and take the nonrelativistic limit to find the Schrödinger’s equation. We show that the quaternionic formalism is a natural choice to start with, while in the transition to the noninteracting nonrelativistic limit, the quaternionic description effectively reduces to the regular complex wave function language. We provide an easy-to-use grammar for switching between the ordinary spinor language and the description in terms of quaternions. As an illustration of the broader range of the formalism, we also derive the Maxwell’s equation from the quaternionic Lagrangian of Quantum Electrodynamics. In order to derive the equations of motion, we develop the variational calculus appropriate for this formalism.

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Alpay, Daniel, Fabrizio Colombo, Jonathan Gantner, and David P. Kimsey. "Functions of the infinitesimal generator of a strongly continuous quaternionic group." Analysis and Applications 15, no. 02 (January 25, 2017): 279–311. http://dx.doi.org/10.1142/s021953051650007x.

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The quaternionic analogue of the Riesz–Dunford functional calculus and the theory of semigroups and groups of linear quaternionic operators have recently been introduced and studied. In this paper, we suppose that [Formula: see text] is the quaternionic infinitesimal generator of a strongly continuous group of operators [Formula: see text] and we show how we can define bounded operators [Formula: see text], where [Formula: see text] belongs to a class of functions that is larger than the one to which the quaternionic functional calculus applies, using the quaternionic Laplace–Stieltjes transform. This class includes functions that are slice regular on the [Formula: see text]-spectrum of [Formula: see text] but not necessarily at infinity. Moreover, we establish the relation between [Formula: see text] and the quaternionic functional calculus and we study the problem of finding the inverse of [Formula: see text].

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Ghiloni, Riccardo, Valter Moretti, and Alessandro Perotti. "Spectral representations of normal operators in quaternionic Hilbert spaces via intertwining quaternionic PVMs." Reviews in Mathematical Physics 29, no. 10 (November 2017): 1750034. http://dx.doi.org/10.1142/s0129055x17500349.

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The possibility of formulating quantum mechanics over quaternionic Hilbert spaces can be traced back to von Neumann’s foundational works in the thirties. The absence of a suitable quaternionic version of spectrum prevented the full development of the theory. The first rigorous quaternionic formulation has started only in 2007 with the definition of the spherical spectrum of a quaternionic operator based on a quadratic version of resolvent operator. The relevance of this notion is proved by the existence of a quaternionic continuous functional calculus and a theory of quaternionic semigroups relying upon it. A problem of the quaternionic formulation is the description of composite quantum systems in the absence of a natural tensor product due to non-commutativity of quaternions. A promising tool towards a solution is a quaternionic projection-valued measure (PVM), making possible a tensor product of quaternionic operators with physical relevance. A notion with this property, called intertwining quaternionic PVM, is presented here. This foundational paper aims to investigate the interplay of this new mathematical object and the spherical spectral features of quaternionic generally unbounded normal operators. We discover, in particular, the existence of other spectral notions equivalent to the spherical ones, but based on a standard non-quadratic notion of resolvent operator.

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Schwartz, Charles. "Calculus with a quaternionic variable." Journal of Mathematical Physics 50, no. 1 (January 2009): 013523. http://dx.doi.org/10.1063/1.3058642.

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7

Condurache, Daniel, Mihail Cojocari, and Ionuţ Popa. "Hypercomplex Quaternions and Higher-Order Analysis of Spatial Kinematic Chains." BULETINUL INSTITUTULUI POLITEHNIC DIN IAȘI. Secția Matematica. Mecanică Teoretică. Fizică 69, no. 1-4 (December 1, 2023): 21–34. http://dx.doi.org/10.2478/bipmf-2023-0002.

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Abstract This paper introduces a novel computational method for analyzing the higher-order acceleration field of spatial kinematics chains. The method is based on vector and quaternionic calculus, as well as dual and multidual algebra. A closed-form coordinate-free solution generated by the morphism between the Lie group of rigid body displacements and the unit multidual quaternions is presented. Presented solution is used for higher-order kinematics investigation of lower-pair serial chains. Additionally, a general method for studying the vector field of arbitrary higher-order accelerations is discribed. The method utilizes the “automatic differentiation” feature of multidual and hyper-multidual functions to obtain the higher-order derivative of a rigid body pose without need in further differentiation of the body pose regarding time. Also is proved that all information regarding the properties of the distribution of higher-order accelerations is contained in the specified unit hyper-multidual quaternion.

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Alpay, Daniel, Fabrizio Colombo, and Irene Sabadini. "Perturbation of the generator of a quaternionic evolution operator." Analysis and Applications 13, no. 04 (April 28, 2015): 347–70. http://dx.doi.org/10.1142/s0219530514500249.

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The theory of slice hyperholomorphic functions, introduced in recent years, has important applications in operator theory. The quaternionic version of this function theory and its Cauchy formula yield to a definition of the quaternionic version of the Riesz–Dunford functional calculus which is based on the notion of S-spectrum. This quaternionic functional calculus allows to define the quaternionic evolution operator which appears in the quaternionic version of quantum mechanics proposed by J. von Neumann and later developed by S. L. Adler. Generation results such as the Hille–Phillips–Yosida theorem have been recently proved. In this paper, we study the perturbation of the generator. The motivation of this study is that, as it happens in the classical case of closed complex linear operators, to verify the generation conditions of the Hille–Phillips–Yosida theorem, in the concrete cases, is often difficult. Thus in this paper we study the generation problem from the perturbation point of view. Precisely, given a quaternionic closed operator T that generates the evolution operator [Formula: see text] we study under which condition a closed operator P is such that T + P generates the evolution operator [Formula: see text]. This paper is addressed to people working in different research areas such as hypercomplex analysis and operator theory.

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Ramesh, G., and P. Santhosh Kumar. "Borel functional calculus for quaternionic normal operators." Journal of Mathematical Physics 58, no. 5 (May 2017): 053501. http://dx.doi.org/10.1063/1.4982047.

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Vasilescu, Florian-Horia. "Spectrum and analytic functional calculus in real and quaternionic frameworks: An overview." AIMS Mathematics 9, no. 1 (2023): 2326–44. http://dx.doi.org/10.3934/math.2024115.

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<abstract><p>An approach to the elementary spectral theory for quaternionic linear operators was presented by the author in a recent paper, quoted and discussed in the Introduction, where, unlike in works by other authors, the construction of the analytic functional calculus used a Riesz-Dunford-Gelfand type kernel, and the spectra were defined in the complex plane. In fact, the present author regards the quaternionic linear operators as a special class of real linear operators, a point of view leading to a simpler and a more natural approach to them. The author's main results in this framework are summarized in the following, and other pertinent comments and remarks are also included in this text. In addition, a quaternionic joint spectrum for pairs of operators is discussed, and an analytic functional calculus which uses a Martinelli type kernel in two variables is recalled.</p></abstract>

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Dissertations / Theses on the topic "Quaternionic calculus"

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Laisant, Charles-Ange. "Applications mécaniques du calcul des quaternions suivi de Sur un nouveau mode de transformation des courbes et des surfaces /." Paris : Bibliothèque universitaire Pierre et Marie Curie (BUPMC), 2009. http://jubil.upmc.fr/sdx/pl/toc.xsp?id=TH_000294_001&fmt=upmc&idtoc=TH_000294_001-pleadetoc&base=fa.

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Masson, Simon. "Algorithmique des courbes destinées au contexte de la cryptographie bilinéaire et post-quantique." Electronic Thesis or Diss., Université de Lorraine, 2020. http://www.theses.fr/2020LORR0151.

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Cette thèse étudie l'algorithmie de plusieurs applications cryptographiques liées aux courbes elliptiques et aux isogénies de courbes elliptiques. D'une part, nous étudions le compromis entre efficacité et sécurité concernant les courbes à couplages pour un niveau de sécurité de "128" bits. La menace des récentes avancées sur le calcul de logarithme discret dans certains corps finis nous oriente vers l'étude de nouvelles courbes à couplage. Nous effectuons une comparaison de l'efficacité de ces nouvelles courbes avec celles utilisées actuellement en estimant le temps de calcul pratique. D'autre part, nous présentons la cryptographie à base d'isogénies de courbes supersingulières, considérées actuellement comme résistantes aux ordinateurs quantiques. Nous portons une attention particulière à la sécurité de ces protocoles en apportant une implémentation des calculs d'idéaux connectants entre ordres maximaux d'algèbres de quaternions. Enfin, nous présentons deux constructions de fonctions à délai vérifiables, basées sur des calculs de couplages et d'évaluations d'isogénies de grand degré friable. Ces dernières ne sont pas considérées comme résistantes aux ordinateurs quantiques, mais apportent plusieurs nouveautés par rapport aux constructions actuelles. Nous analysons leur sécurité et effectuons une comparaison entre toutes ces fonctions à un niveau de sécurité de "128" bits
This thesis studies the algorithmic of several cryptographic applications related to elliptic curves and isogenies of elliptic curves. On the one hand, we study the tradeoff between efficiency and security in pairing-based cryptography at the "128"-bit security level. The threat of the recent improvements on the discrete logarithm computation over specific finite fields lead us to study new pairing-friendly curves. We give a comparison of efficiency between our new curves and the state-of-the-art curves by estimating the measurement in practice. On the other and, we present isogeny-based cryptography, considered to be post-quantum resistant. We look at a concrete implementation of cryptanalysis based on connecting ideals between maximal orders of quaternion algebras. Finally, we present two constructions of verifiable delay functions based on computations of pairings and isogenies of large smooth degree. These functions are not considered to be post-quantum resistant, but bring several new properties compared to the current constructions. We analyse their security and give a comparison of all the known functions at the "128"-bit security level

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Books on the topic "Quaternionic calculus"

1

Morais, João Pedro, Svetlin Georgiev, and Wolfgang Sprößig. Real Quaternionic Calculus Handbook. Basel: Springer Basel, 2014. http://dx.doi.org/10.1007/978-3-0348-0622-0.

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2

Wolfgang, Sprössig, ed. Quaternionic and Clifford calculus for physicists and engineers. Chichester: Wiley, 1997.

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3

Brillouin, Léon. Les tenseurs en mécanique et en élasticité. Sceaux: Gabay, 1987.

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author, Tkachev Vladimir 1963, and Vlăduț, S. G. (Serge G.), 1954- author, eds. Nonlinear elliptic equations and nonassociative algebras. Providence, Rhode Island: American Mathematical Society, 2014.

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

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Sprößig, Wolfgang, Svetlin Georgiev, and João Pedro Morais. Real Quaternionic Calculus Handbook. Birkhauser Verlag, 2014.

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Sprößig, Wolfgang, Svetlin Georgiev, and João Pedro Morais. Real Quaternionic Calculus Handbook. Birkhäuser, 2014.

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8

Hou, Qing Jun. Real Quaternionic Calculus Handbook. Scitus Academics LLC, 2017.

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9

Gürlebeck, Klaus, and Wolfgang Sprössig. Quaternionic and Clifford Calculus for Physicists and Engineers. Wiley & Sons, Incorporated, John, 2000.

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10

Gantner, Jonathan. Operator Theory on One-Sided Quaternion Linear Spaces: Intrinsic $S$-Functional Calculus and Spectral Operators. American Mathematical Society, 2021.

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Book chapters on the topic "Quaternionic calculus"

1

Colombo, Fabrizio, Irene Sabadini, and Daniele C. Struppa. "Quaternionic Functional Calculus." In Noncommutative Functional Calculus, 113–200. Basel: Springer Basel, 2011. http://dx.doi.org/10.1007/978-3-0348-0110-2_4.

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Morais, João Pedro, Svetlin Georgiev, and Wolfgang Sprößig. "An Introduction to Quaternions." In Real Quaternionic Calculus Handbook, 1–34. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0622-0_1.

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Morais, João Pedro, Svetlin Georgiev, and Wolfgang Sprößig. "Monomials, Polynomials and Binomials." In Real Quaternionic Calculus Handbook, 149–64. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0622-0_10.

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Morais, João Pedro, Svetlin Georgiev, and Wolfgang Sprößig. "Solutions." In Real Quaternionic Calculus Handbook, 165–209. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0622-0_11.

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Morais, João Pedro, Svetlin Georgiev, and Wolfgang Sprößig. "Quaternions and Spatial Rotation." In Real Quaternionic Calculus Handbook, 35–51. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0622-0_2.

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Morais, João Pedro, Svetlin Georgiev, and Wolfgang Sprößig. "Quaternion Sequences." In Real Quaternionic Calculus Handbook, 53–67. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0622-0_3.

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Morais, João Pedro, Svetlin Georgiev, and Wolfgang Sprößig. "Quaternion Series and Infinite Products." In Real Quaternionic Calculus Handbook, 69–85. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0622-0_4.

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Morais, João Pedro, Svetlin Georgiev, and Wolfgang Sprößig. "Exponents and Logarithms." In Real Quaternionic Calculus Handbook, 87–105. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0622-0_5.

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Morais, João Pedro, Svetlin Georgiev, and Wolfgang Sprößig. "Trigonometric Functions." In Real Quaternionic Calculus Handbook, 107–16. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0622-0_6.

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Morais, João Pedro, Svetlin Georgiev, and Wolfgang Sprößig. "Hyperbolic Functions." In Real Quaternionic Calculus Handbook, 117–24. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0622-0_7.

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Conference papers on the topic "Quaternionic calculus"

1

Xia, Yili, Cyrus Jahanchahi, Dongpo Xu, and Danilo P. Mandic. "The HC calculus, quaternion derivatives and caylay-hamilton form of quaternion adaptive filters and learning systems." In 2014 International Joint Conference on Neural Networks (IJCNN). IEEE, 2014. http://dx.doi.org/10.1109/ijcnn.2014.6889498.

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Jahanchahi, Cyrus, Clive Cheong Took, and Danilo P. Mandic. "On HR calculus, quaternion valued stochastic gradient, and adaptive three dimensional wind forecasting." In 2010 International Joint Conference on Neural Networks (IJCNN). IEEE, 2010. http://dx.doi.org/10.1109/ijcnn.2010.5596629.

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Takahashi, Kazuhiko, Eri Tano, and Masafumi Hashimoto. "Remarks on Quaternion Multi–Layer Neural Network Based on the Generalised HR Calculus." In 2021 Australian & New Zealand Control Conference (ANZCC). IEEE, 2021. http://dx.doi.org/10.1109/anzcc53563.2021.9628250.

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Xu, Wencui, and Dongpo Xu. "The Modified HR Calculus to Reproducing Kernel Hilbert Space and the Quaternion Kernel Least Mean Square Algorithm." In 2015 Eighth International Conference on Internet Computing for Science and Engineering (ICICSE). IEEE, 2015. http://dx.doi.org/10.1109/icicse.2015.19.

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Takahashi, Kazuhiko, Eri Tano, and Masafumi Hashimoto. "Remarks on Feedforward–Feedback Controller Using a Trained Quaternion Neural Network Based on Generalised ℍℝ Calculus and Its Application to Controlling a Robot Manipulator." In 2021 International Conference on Advanced Mechatronic Systems (ICAMechS). IEEE, 2021. http://dx.doi.org/10.1109/icamechs54019.2021.9661487.

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