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Relevant bibliographies by topics / Quaternion calculus

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Academic literature on the topic 'Quaternion calculus'

Author: Grafiati

Published: 29 July 2024

Last updated: 30 July 2024

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Journal articles on the topic "Quaternion calculus"

1

Xu, Dongpo, Cyrus Jahanchahi, Clive C. Took, and Danilo P. Mandic. "Enabling quaternion derivatives: the generalized HR calculus." Royal Society Open Science 2, no. 8 (August 2015): 150255. http://dx.doi.org/10.1098/rsos.150255.

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Quaternion derivatives exist only for a very restricted class of analytic (regular) functions; however, in many applications, functions of interest are real-valued and hence not analytic, a typical case being the standard real mean square error objective function. The recent HR calculus is a step forward and provides a way to calculate derivatives and gradients of both analytic and non-analytic functions of quaternion variables; however, the HR calculus can become cumbersome in complex optimization problems due to the lack of rigorous product and chain rules, a consequence of the non-commutativity of quaternion algebra. To address this issue, we introduce the generalized HR (GHR) derivatives which employ quaternion rotations in a general orthogonal system and provide the left- and right-hand versions of the quaternion derivative of general functions. The GHR calculus also solves the long-standing problems of product and chain rules, mean-value theorem and Taylor's theorem in the quaternion field. At the core of the proposed GHR calculus is quaternion rotation, which makes it possible to extend the principle to other functional calculi in non-commutative settings. Examples in statistical learning theory and adaptive signal processing support the analysis.

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Xu, Dongpo, Hua Gao, and Danilo P. Mandic. "A new proof of the generalized Hamiltonian–Real calculus." Royal Society Open Science 3, no. 9 (September 2016): 160211. http://dx.doi.org/10.1098/rsos.160211.

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The recently introduced generalized Hamiltonian–Real (GHR) calculus comprises, for the first time, the product and chain rules that makes it a powerful tool for quaternion-based optimization and adaptive signal processing. In this paper, we introduce novel dual relationships between the GHR calculus and multivariate real calculus, in order to provide a new, simpler proof of the GHR derivative rules. This further reinforces the theoretical foundation of the GHR calculus and provides a convenient methodology for generic extensions of real- and complex-valued learning algorithms to the quaternion domain.

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Takahashi, Kazuhiko, Eri Tano, and Masafumi Hashimoto. "Feedforward–Feedback Controller Based on a Trained Quaternion Neural Network Using a Generalised HR Calculus with Application to Trajectory Control of a Three-Link Robot Manipulator." Machines 10, no. 5 (May 2, 2022): 333. http://dx.doi.org/10.3390/machines10050333.

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This study derives a learning algorithm for a quaternion neural network using the steepest descent method extended to quaternion numbers. This applies the generalised Hamiltonian–Real calculus to obtain derivatives of a real–valued cost function concerning quaternion variables and designs a feedback–feedforward controller as a control system application using such a network. The quaternion neural network is trained in real-time by introducing a feedback error learning framework to the controller. Thus, the quaternion neural network-based controller functions as an adaptive-type controller. The designed controller is applied to the control problem of a three-link robot manipulator, with the control task of making the robot manipulator’s end effector follow a desired trajectory in the Cartesian space. Computational experiments are conducted to investigate the learning capability and the characteristics of the quaternion neural network used in the controller. The experimental results confirm the feasibility of using the derived learning algorithm based on the generalised Hamiltonian–Real calculus to train the quaternion neural network and the availability of such a network for a control systems application.

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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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Chen, Shenglong, Hong-Li Li, Leimin Wang, Cheng Hu, Haijun Jiang, and Zhiming Li. "Finite-time adaptive synchronization of fractional-order delayed quaternion-valued fuzzy neural networks." Nonlinear Analysis: Modelling and Control 28 (June 19, 2023): 1–20. http://dx.doi.org/10.15388/namc.2023.28.32505.

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Based on direct quaternion method, this paper explores the finite-time adaptive synchronization (FAS) of fractional-order delayed quaternion-valued fuzzy neural networks (FODQVFNNs). Firstly, a useful fractional differential inequality is created, which offers an effective way to investigate FAS. Then two novel quaternion-valued adaptive control strategies are designed. By means of our newly proposed inequality, the basic knowledge about fractional calculus, reduction to absurdity as well as several inequality techniques of quaternion and fuzzy logic, several sufficient FAS criteria are derived for FODQVFNNs. Moreover, the settling time of FAS is estimated, which is in connection with the order and initial values of considered systems as well as the controller parameters. Ultimately, the validity of obtained FAS criteria is corroborated by numerical simulations.

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Li, Yongkun, and Xiaofang Meng. "Existence and Global Exponential Stability of Pseudo Almost Periodic Solutions for Neutral Type Quaternion-Valued Neural Networks with Delays in the Leakage Term on Time Scales." Complexity 2017 (2017): 1–15. http://dx.doi.org/10.1155/2017/9878369.

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We propose a class of neutral type quaternion-valued neural networks with delays in the leakage term on time scales that can unify the discrete-time and the continuous-time neural networks. In order to avoid the difficulty brought by the noncommutativity of quaternion multiplication, we first decompose the quaternion-valued system into four real-valued systems. Then, by applying the exponential dichotomic theory of linear dynamic equations on time scales, Banach’s fixed point theorem, the theory of calculus on time scales, and inequality techniques, we obtain some sufficient conditions on the existence and global exponential stability of pseudo almost periodic solutions for this class of neural networks. Our results are completely new even for both the case of the neural networks governed by differential equations and the case of the neural networks governed by difference equations and show that, under a simple condition, the continuous-time quaternion-valued network and its corresponding discrete-time quaternion-valued network have the same dynamical behavior for the pseudo almost periodicity. Finally, a numerical example is given to illustrate the feasibility of our results.

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Xu, Dongpo, Lina Zhang, and Huisheng Zhang. "LEARNING ALGORITHMS IN QUATERNION NEURAL NETWORKS USING GHR CALCULUS." Neural Network World 27, no. 3 (2017): 271–82. http://dx.doi.org/10.14311/nnw.2017.27.014.

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Pletinckx, Daniel. "Quaternion calculus as a basic tool in computer graphics." Visual Computer 5, no. 1-2 (January 1989): 2–13. http://dx.doi.org/10.1007/bf01901476.

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Danielewski, Marek, and Lucjan Sapa. "Foundations of the Quaternion Quantum Mechanics." Entropy 22, no. 12 (December 17, 2020): 1424. http://dx.doi.org/10.3390/e22121424.

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We show that quaternion quantum mechanics has well-founded mathematical roots and can be derived from the model of the elastic continuum by French mathematician Augustin Cauchy, i.e., it can be regarded as representing the physical reality of elastic continuum. Starting from the Cauchy theory (classical balance equations for isotropic Cauchy-elastic material) and using the Hamilton quaternion algebra, we present a rigorous derivation of the quaternion form of the non- and relativistic wave equations. The family of the wave equations and the Poisson equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic continuum. This is the most general kind of quantum mechanics possessing the same kind of calculus of assertions as conventional quantum mechanics. The problem of the Schrödinger equation, where imaginary ‘i’ should emerge, is solved. This interpretation is a serious attempt to describe the ontology of quantum mechanics, and demonstrates that, besides Bohmian mechanics, the complete ontological interpretations of quantum theory exists. The model can be generalized and falsified. To ensure this theory to be true, we specified problems, allowing exposing its falsity.

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Shen, Shiping, Bing Li, and Yongkun Li. "Anti-Periodic Dynamics of Quaternion-Valued Fuzzy Cellular Neural Networks with Time-Varying Delays on Time Scales." Discrete Dynamics in Nature and Society 2018 (June 28, 2018): 1–14. http://dx.doi.org/10.1155/2018/5290786.

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A class of quaternion-valued fuzzy cellular neural networks with time-varying delays on time scales is proposed. Based on inequality analysis techniques on time scales, a fixed point theorem and the theory of calculus on time scales, the existence, and global exponential stability of anti-periodic solutions for this class of neural networks are established. The obtained results are completely new and supplement to the known results. Finally, a numerical example is given to illustrate the feasibility of our results.

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Dissertations / Theses on the topic "Quaternion calculus"

1

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 "Quaternion 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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5

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

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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9

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

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10

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

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Book chapters on the topic "Quaternion calculus"

1

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 Matrices." In Real Quaternionic Calculus Handbook, 133–47. Basel: Springer Basel, 2013. http://dx.doi.org/10.1007/978-3-0348-0622-0_9.

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Hitzer, Eckhard. "Geometric calculus." In Quaternion and Clifford Fourier Transforms, 105–40. Boca Raton: Chapman and Hall/CRC, 2021. http://dx.doi.org/10.1201/9781003184478-3.

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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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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. "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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Kamberov, George, Peter Norman, Franz Pedit, and Ulrich Pinkall. "Chapter 1. Quaternionic Calculus and Immersions." In Quaternions, Spinors, and Surfaces, 3–54. Providence, Rhode Island: American Mathematical Society, 2002. http://dx.doi.org/10.1090/conm/299/01.

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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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Conference papers on the topic "Quaternion 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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