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Relevant bibliographies by topics / Derivatives of quaternions

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Academic literature on the topic 'Derivatives of quaternions'

Author: Grafiati

Published: 29 July 2024

Last updated: 30 July 2024

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Journal articles on the topic "Derivatives of quaternions"

1

Kim, Ji-Eun. "Approximation of Directional Step Derivative of Complex-Valued Functions Using a Generalized Quaternion System." Axioms 10, no. 3 (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 th

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2

Montgomery-Smith, Stephen, and Cecil Shy. "Using Lie Derivatives with Dual Quaternions for Parallel Robots." Machines 11, no. 12 (2023): 1056. http://dx.doi.org/10.3390/machines11121056.

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We introduce the notion of the Lie derivative in the context of dual quaternions that represent rigid motions and twists. First we define the wrench in terms of dual quaternions. Then we show how the Lie derivative helps understand how actuators affect an end effector in parallel robots, and make it explicit in the two cases case of Stewart Platforms, and cable-driven parallel robots. We also show how to use Lie derivatives with the Newton-Raphson Method to solve the forward kinematic problem for over constrained parallel actuators. Finally, we derive the equations of motion of the end effecto

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3

Saima, Siddiqui, Bingzhao Li, and Samad Muhammad Adnan. "New Sampling Expansion Related to Derivatives in Quaternion Fourier Transform Domain." Mathematics 10, no. 8 (2022): 1217. http://dx.doi.org/10.3390/math10081217.

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The theory of quaternions has gained a firm ground in recent times and is being widely explored, with the field of signal and image processing being no exception. However, many important aspects of quaternionic signals are yet to be explored, particularly the formulation of Generalized Sampling Expansions (GSE). In the present article, our aim is to formulate the GSE in the realm of a one-dimensional quaternion Fourier transform. We have designed quaternion Fourier filters to reconstruct the signal, using the signal and its derivative. Since derivatives contain information about the edges and

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4

Kim, Ji Eun. "Calculation of Two Types of Quaternion Step Derivatives of Elementary Functions." Mathematics 9, no. 6 (2021): 668. http://dx.doi.org/10.3390/math9060668.

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We aim to get the step derivative of a complex function, as it derives the step derivative in the imaginary direction of a real function. Given that the step derivative of a complex function cannot be derived using i, which is used to derive the step derivative of a real function, we intend to derive the complex function using the base direction of the quaternion. Because many analytical studies on quaternions have been conducted, various examples can be presented using the expression of the elementary function of a quaternion. In a previous study, the base direction of the quaternion was rega

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5

Gogberashvili, Merab. "(2 + 1)-Maxwell Equations in Split Quaternions." Physics 4, no. 1 (2022): 329–63. http://dx.doi.org/10.3390/physics4010023.

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The properties of spinors and vectors in (2 + 2) space of split quaternions are studied. Quaternionic representation of rotations naturally separates two SO(2,1) subgroups of the full group of symmetry of the norms of split quaternions, SO(2,2). One of them represents symmetries of three-dimensional Minkowski space-time. Then, the second SO(2,1) subgroup, generated by the additional time-like coordinate from the basis of split quaternions, can be viewed as the internal symmetry of the model. It is shown that the analyticity condition, applying to the invariant construction of split quaternions

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6

LEO, STEFANO DE. "A ONE-COMPONENT DIRAC EQUATION." International Journal of Modern Physics A 11, no. 21 (1996): 3973–85. http://dx.doi.org/10.1142/s0217751x96001863.

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We develop a relativistic free wave equation on the complexified quaternions, linear in the derivatives. Even if the wave functions are only one-component, we show that four independent solutions, corresponding to those of the Dirac equation, exist. A partial set of translations between complex and complexified quaternionic quantum mechanics may be defined.

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7

ROTELLI, P. "THE DIRAC EQUATION ON THE QUATERNION FIELD." Modern Physics Letters A 04, no. 10 (1989): 933–40. http://dx.doi.org/10.1142/s0217732389001106.

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We develop a relativistic free wave equation on the quaternions, linear in the derivatives. Even if the wave function is only two-component, we show that there exists four complex-independent solutions corresponding to those of the Dirac Equation.

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8

Ghanbarpourasl, Habib. "Attitude reconstruction from strap-down rate gyros using power series." Journal of Navigation 74, no. 4 (2021): 763–81. http://dx.doi.org/10.1017/s0373463321000023.

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AbstractThis paper introduces a power series based method for attitude reconstruction from triad orthogonal strap-down gyros. The method is implemented and validated using quaternions and direction cosine matrix in single and double precision implementation forms. It is supposed that data from gyros are sampled with high frequency and a fitted polynomial is used for an analytical description of the angular velocity vector. The method is compared with the well-known Taylor series approach, and the stability of the coefficients’ norm in higher-order terms for both methods is analysed. It is show

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9

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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10

Weng, Zi-Hua. "Forces in the complex octonion curved space." International Journal of Geometric Methods in Modern Physics 13, no. 06 (2016): 1650076. http://dx.doi.org/10.1142/s0219887816500766.

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The paper aims to extend major equations in the electromagnetic and gravitational theories from the flat space into the complex octonion curved space. Maxwell applied simultaneously the quaternion analysis and vector terminology to describe the electromagnetic theory. It inspires subsequent scholars to study the electromagnetic and gravitational theories with the complex quaternions/octonions. Furthermore Einstein was the first to depict the gravitational theory by means of tensor analysis and curved four-space–time. Nowadays some scholars investigate the electromagnetic and gravitational prop

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