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Journal articles on the topic 'Quaternion calculus'

Author: Grafiati

Published: 29 July 2024

Last updated: 30 July 2024

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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 (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-commutati

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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 (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

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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 (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

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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 (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 i

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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 derive

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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

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7

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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8

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

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9

Danielewski, Marek, and Lucjan Sapa. "Foundations of the Quaternion Quantum Mechanics." Entropy 22, no. 12 (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

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10

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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Arthurs, A. M., and G. R. Walsh. "On Hammersley's minimum problem for a rolling sphere." Mathematical Proceedings of the Cambridge Philosophical Society 99, no. 3 (1986): 529–34. http://dx.doi.org/10.1017/s0305004100064471.

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AbstractThe problem posed by Hammersley (1983) of finding the shortest path along which a sphere can roll from one prescribed state to another is formulated by using quaternion calculus of variations and optimal control theory. This leads to a system of coupled nonlinear differential equations with prescribed end conditions. From the resulting expression for the curvature, it is shown that the differential equation of the required path in intrinsic coordinates is the same as the equation of motion of a simple pendulum, giving a solution in terms of elliptic integrals.

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12

Chelaru, Teodor-Viorel, Valentin Pană, and Costin Ene. "Performance Evaluation for Launcher Testing Vehicle." Aerospace 9, no. 9 (2022): 504. http://dx.doi.org/10.3390/aerospace9090504.

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The paper’s purpose is to present a calculus model for a testing vehicle that can be used to validate guidance, navigation and control systems for reusable launchers in all flight phases. The technical solution is based on a throttleable engine with thrust vectoring control and a reaction control system (RCS) used for roll. For calculus, we will develop a nonlinear model with six degrees of freedom, based on quaternion, extended with nonlinear equations that use pulse modulation in order to control roll. In order to synthesize the controller, we also develop a linear model similar to the launc

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Mengüç, Engin Cemal. "Novel quaternion‐valued least‐mean kurtosis adaptive filtering algorithm based on the GHR calculus." IET Signal Processing 12, no. 4 (2018): 487–95. http://dx.doi.org/10.1049/iet-spr.2017.0340.

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14

Szelmanowski, Andrzej, Andrzej Pazur, Mariusz Żokowski, Paweł Janik, and Wojciech Paterek. "Computer modeling of electrical coil characteristics dedicated for helmet-mounted cueing systems with magnetic method." AUTOBUSY – Technika, Eksploatacja, Systemy Transportowe 20, no. 1-2 (2019): 352–59. http://dx.doi.org/10.24136/atest.2019.065.

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Paper presents the importance of the helmet-mounted cueing systems in the field of flight safety increasing and mission execution effectiveness. There are presented the selected helmet systems used in the foreign and domestic solutions, utilizing the magnetic method. An original method of pilot’s helmet angular position determination utilizing the magnetic field from single flat coil and quaternion calculus is presented. The valuable elements are given mathematical relationships describing the characteristics of the magnetic field generated by selected types of electric coils as well as the re

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CASTELLANI, L., R. CATENACCI, M. DEBERNARDI, and C. PAGANI. "NONCOMMUTATIVE DE RHAM COHOMOLOGY OF FINITE GROUPS." International Journal of Modern Physics A 19, no. 12 (2004): 1961–86. http://dx.doi.org/10.1142/s0217751x04018403.

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We study de Rham cohomology for various differential calculi on finite groups G up to order 8. These include the permutation group S3, the dihedral group D4 and the quaternion group Q. Poincaré duality holds in every case, and under some assumptions (essentially the existence of a top form) we find that it must hold in general. A short review of the bicovariant (noncommutative) differential calculus on finite G is given for selfconsistency. Exterior derivative, exterior product, metric, Hodge dual, connections, torsion, curvature, and biinvariant integration can be defined algebraically. A pro

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16

Ghamari, Elham, and Dan Kučerovský. "Quaternions and Functional Calculus." Symmetry 11, no. 8 (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 (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 transi

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18

GHILONI, RICCARDO, VALTER MORETTI, and ALESSANDRO PEROTTI. "CONTINUOUS SLICE FUNCTIONAL CALCULUS IN QUATERNIONIC HILBERT SPACES." Reviews in Mathematical Physics 25, no. 04 (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.

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19

Popa, Călin-Adrian. "Synchronization of Clifford-valued neural networks with leakage, time-varying, and infinite distributed delays on time scales." AIMS Mathematics 9, no. 7 (2024): 18796–823. http://dx.doi.org/10.3934/math.2024915.

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<abstract><p>Neural networks (NNs) with values in multidimensional domains have lately attracted the attention of researchers. Thus, complex-valued neural networks (CVNNs), quaternion-valued neural networks (QVNNs), and their generalization, Clifford-valued neural networks (ClVNNs) have been proposed in the last few years, and different dynamic properties were studied for them. On the other hand, time scale calculus has been proposed in order to jointly study the properties of continuous time and discrete time systems, or any hybrid combination between the two, and was also success

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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 (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 re

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21

Salnikov, Nikolay, Serhiy Melnychuk, and Vyacheslav Gubarev. "Ellipsoidal estimation of parameters of rotational and translational motion of a non-cooperative space vehicle from visual information." International Scientific Technical Journal "Problems of Control and Informatics" 68, no. 6 (2023): 35–63. http://dx.doi.org/10.34229/1028-0979-2023-6-3.

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 The use of near-Earth space is currently complicated by the presence of space debris objects in Earthʼs orbit, which include spent stages of launch vehicles, inoperative spacecraft, and other large and small objects associated with human activity in space. One of the elements of solving the problem of space debris is the docking and capture of an uncontrolled non-cooperative space object or spacecraft by a so-called on-orbit servicing spacecraft to carry out further actions to repair it, refuel or change its orbit. The situation is complicated by the fact that, under the influence of va

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22

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 (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 transfo

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23

Enslin, J. H. R. "Ortogonale voorstelling van drywing deur middel van kwaternione." Suid-Afrikaanse Tydskrif vir Natuurwetenskap en Tegnologie 9, no. 1 (1990): 11–14. http://dx.doi.org/10.4102/satnt.v9i1.433.

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The increased use of power control equipment resulted in the distortion of the excitation and response functions in power systems from simple sinusoids. These systems resulted in discrepancies in the general definition of power. This distortion has a negative effect on the accurate definition and representation of power in a contaminated power system. The mathematical representation of power with the aid of the theory of quaternions in vector calculus is investigated to obtain a generalized definition of power in all power systems, especially in power systems where the excitation and response

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24

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

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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 (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

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Colombo, Fabrizio, and Jonathan Gantner. "Formulations of the -functional calculus and some consequences." Proceedings of the Royal Society of Edinburgh: Section A Mathematics 146, no. 3 (2016): 509–45. http://dx.doi.org/10.1017/s0308210515000645.

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In this paper we introduce the two possible formulations of the -functional calculus that are based on the Fueter–Sce mapping theorem in integral form and we introduce the pseudo--resolvent equation. In the case of dimension 3 we prove the -resolvent equation and we study the analogue of the Riesz projectors associated with this calculus. The case of dimension 3 is also useful to study the quaternionic version of the -functional calculus.

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27

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 t

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

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Colombo, Fabrizio, and Irene Sabadini. "On the formulations of the quaternionic functional calculus." Journal of Geometry and Physics 60, no. 10 (2010): 1490–508. http://dx.doi.org/10.1016/j.geomphys.2010.05.014.

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Colombo, Fabrizio, and Irene Sabadini. "On Some Properties of the Quaternionic Functional Calculus." Journal of Geometric Analysis 19, no. 3 (2009): 601–27. http://dx.doi.org/10.1007/s12220-009-9075-x.

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31

Gover, A. Rod, and Jan Slovák. "Invariant local twistor calculus for quaternionic structures and related geometries." Journal of Geometry and Physics 32, no. 1 (1999): 14–56. http://dx.doi.org/10.1016/s0393-0440(99)00018-2.

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Gantner, Jonathan. "Operator Theory on One-Sided Quaternionic Linear Spaces: Intrinsic S-Functional Calculus and Spectral Operators." Memoirs of the American Mathematical Society 267, no. 1297 (2020): 0. http://dx.doi.org/10.1090/memo/1297.

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Hertrich-Jeromin, Udo. "Supplement on curved flats in the space of point pairs and isothermic surfaces: A quaternionic calculus." Documenta Mathematica 2 (1997): 335–50. http://dx.doi.org/10.4171/dm/33.

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Gürlebeck, Klaus, Dmitrii Legatiuk, and Kemmar Webber. "Operator Calculus Approach to Comparison of Elasticity Models for Modelling of Masonry Structures." Mathematics 10, no. 10 (2022): 1670. http://dx.doi.org/10.3390/math10101670.

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The solution of any engineering problem starts with a modelling process aimed at formulating a mathematical model, which must describe the problem under consideration with sufficient precision. Because of heterogeneity of modern engineering applications, mathematical modelling scatters nowadays from incredibly precise micro- and even nano-modelling of materials to macro-modelling, which is more appropriate for practical engineering computations. In the field of masonry structures, a macro-model of the material can be constructed based on various elasticity theories, such as classical elasticit

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Kraußhar, Rolf Sören. "Applications of the Quaternionic Calculus to the Convective Stationary MHD Equations in $${\mathbb{R}^3}$$ R 3." Advances in Applied Clifford Algebras 24, no. 4 (2014): 1047–58. http://dx.doi.org/10.1007/s00006-014-0481-1.

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Richards, Joan L. "Generations of Reason: A Family's Search for Meaning in Post-Newtonian England." Perspectives on Science and Christian Faith 75, no. 1 (2023): 63–65. http://dx.doi.org/10.56315/pscf3-23richards.

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GENERATIONS OF REASON: A Family's Search for Meaning in Post-Newtonian England by Joan L. Richards. New Haven, CT: Yale University Press, 2021. 456 pages, with 21 b/w illustrations, 1,218 endnotes, and a 35-page index. Hardcover; $45.00. ISBN: 9780300255492. *The title gives no clue who this book is about. Nor does the publisher's description on its website, the abbreviated blurb inside the book jacket, the four endorsements posted on the jacket's back ("beautifully written," "epic masterpiece," "magnificent study," "compelling and wide-ranging"), or even the chapter titles. The reader first l

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Alpay, Daniel, Fabrizio Colombo, Tao Qian, and Irene Sabadini. "The H∞ functional calculus based on the S-spectrum for quaternionic operators and for n-tuples of noncommuting operators." Journal of Functional Analysis 271, no. 6 (2016): 1544–84. http://dx.doi.org/10.1016/j.jfa.2016.06.009.

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Crilly, Tony. "Real quaternionic calculus handbook by João Pedro Morais, Svetlin Georgiev, Wolfgang Sprößig, p. 216, £54.99, ISBN 978-3-0348-0621-3, Birkhäuser (2014)." Mathematical Gazette 105, no. 563 (2021): 370–71. http://dx.doi.org/10.1017/mag.2021.90.

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Gudrun Kalmbach HE. "Interactions, space presentations, blocks and cross products." GSC Advanced Research and Reviews 6, no. 2 (2021): 061–73. http://dx.doi.org/10.30574/gscarr.2021.6.2.0012.

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Physics counts four basic forces, the electromagnetic EMI, weak WI, strong SI interactions and gravity GR. The first three are provided with a unified theory which partly needs revision and has the symmetry U(1)xSU(2)xSU(3). In this article their space presentations are described in order to inlcude a theory for gravity which cannot be added directly to the standrd model. There are many instances of gravitational actions which are different from the other three interactions. Gravity uses geometrical models beside spactime, often projective, including stereographic and spiralic orthogonal subsp

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40

Khedhiri, Hedi, and Taher Mkademi. "Foundational aspects of a new matrix holomorphic structure." Arab Journal of Mathematical Sciences, March 21, 2024. http://dx.doi.org/10.1108/ajms-08-2023-0002.

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PurposeIn this paper we talk about complex matrix quaternions (biquaternions) and we deal with some abstract methods in mathematical complex matrix analysis.Design/methodology/approachWe introduce and investigate the complex space HC consisting of all 2 × 2 complex matrices of the form ξ=z1+iw1z2+iw2−z‾2−iw‾2z‾1+iw‾1, (z1,w1,z2,w2)∈C4.FindingsWe develop on HC a new matrix holomorphic structure for which we provide the fundamental operational calculus properties.Originality/valueWe give sufficient and necessary conditions in terms of Cauchy–Riemann type quaternionic differential equations for h

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"Collection of Three Monographs Pertaining to Quaternionic Analysis." Journal of Electrical Electronics Engineering 3, no. 3 (2024): 01–42. http://dx.doi.org/10.33140/jeee.03.03.001.

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This particular submission contains (inter alia) copies of three (3) monographs, whose purpose is to further elaborate upon various topics having been enunciated in the author’s previous set of submissions, namely – (a) “A Supplementary Discourse on the Classification and Calculus of Quaternion Hypercomplex Functions - PARTS 1/10 to 10/10.” (b) “Supplementary Notes pertaining to a Specific Quaternion Analogue of the Cauchy-Goursat Theorem.” Which have been published under the ‘VIXRA’ Mathematics subheading: - ‘Functions and Analysis’. II. Copy of Author’s Original Monograph No.1. (1) Title of

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Li, Qing, and Zhida Ren. "A new fractional-order augmented quaternion-valued approach for degradation prognostics of bearings using generalized Hamilton-real calculus." IEEE Transactions on Instrumentation and Measurement, 2022, 1. http://dx.doi.org/10.1109/tim.2022.3218546.

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Popa, Călin-Adrian. "Stability and synchronization of octonion-valued neural networks with leakage and mixed delays on time scales." Computational and Applied Mathematics 43, no. 5 (2024). http://dx.doi.org/10.1007/s40314-024-02820-5.

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AbstractThere has been a great deal of interest in the last few years for neural networks (NNs) with values in multidimensional domains. The most popular models are complex-valued neural networks (CVNNs), followed by quaternion-valued neural networks (QVNNs), and, more recently, by Clifford-valued neural networks (ClVNNs). However, also very recently, a different type of NNs were put forward, namely octonion-valued neural networks (OVNNs). OVNNs are defined on the 8D octonion algebra, and they are not a special type of ClVNNs, because Clifford numbers are associative, whereas octonions are not

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Colombaro, Ivano. "Time-like definition of quaternions in exterior algebra." Ricerche di Matematica, August 22, 2023. http://dx.doi.org/10.1007/s11587-023-00810-z.

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AbstractA formal description of quaternions by means of exterior calculus is presented. Considering a three-dimensional space-time characterized by three time-like coordinates, we have been able to consistently recover a suitable formulation of quaternions by means of the properties arising from exterior algebra and calculus. As an application, it is also illustrated how rotations may be written in terms of quaternions, in accordance with definition provided in exterior algebra.

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Montgomery-Smith, Stephen. "Functional Calculus for Dual Quaternions." Advances in Applied Clifford Algebras 33, no. 3 (2023). http://dx.doi.org/10.1007/s00006-023-01282-y.

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"Real quaternionic calculus handbook." Choice Reviews Online 52, no. 05 (2014): 52–2594. http://dx.doi.org/10.5860/choice.185180.

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Ceyhan, Hazal, Zehra Özdemir, and Ismail Gök. "Multiplicative generalized tube surfaces with multiplicative quaternions algebra." Mathematical Methods in the Applied Sciences, April 3, 2024. http://dx.doi.org/10.1002/mma.10065.

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Along with other types of calculus, multiplicative calculus brings an entirely new perspective. Geometry now has a new field as a result of this new understanding. In this study, multiplicative differential geometry was used to explore peculiar surfaces. Multiplicative quaternions are also used to depict surfaces. Additionally, multiplicative differential geometry was used to generate the accretive surface subject, which is a developing subject. The derived surfaces' perspective silhouette curve equation is provided. The Bishop multiplicative frame was also established and applied when express

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Cerejeiras, P., U. Kähler, and R. S. Kraußhar. "Variational Principles in Quaternionic Analysis with Applications to the Stationary MHD Equations." Complex Analysis and Operator Theory 18, no. 3 (2024). http://dx.doi.org/10.1007/s11785-023-01455-4.

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AbstractIn this paper we aim to combine tools from variational calculus with modern techniques from quaternionic analysis that involve Dirac type operators and related hypercomplex integral operators. The aim is to develop new methods for showing geometry independent explicit global existence and uniqueness criteria as well as new computational methods with special focus to the stationary incompressible viscous magnetohydrodynamic equations. We first show how to specifically apply variational calculus in the quaternionic setting. To this end we explain how the mountain pass theorem can be succ

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Vasilescu, Florian-Horia. "Quaternionic Regularity via Analytic Functional Calculus." Integral Equations and Operator Theory 92, no. 2 (2020). http://dx.doi.org/10.1007/s00020-020-2574-7.

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Colombo, Fabrizio, Stefano Pinton, and Peter Schlosser. "The $$H^\infty $$-Functional Calculi for the Quaternionic Fine Structures of Dirac Type." Milan Journal of Mathematics, March 6, 2024. http://dx.doi.org/10.1007/s00032-024-00392-x.

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AbstractIn recent works, various integral representations have been proposed for specific sets of functions. These representations are derived from the Fueter–Sce extension theorem, considering all possible factorizations of the Laplace operator in relation to both the Cauchy–Fueter operator (often referred to as the Dirac operator) and its conjugate. The collection of these function spaces, along with their corresponding functional calculi, are called the quaternionic fine structures within the context of the S-spectrum. In this paper, we utilize these integral representations of functions to

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