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Journal articles on the topic 'Quaternions algebra'

Author: Grafiati

Published: 7 June 2025

Last updated: 16 July 2025

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1

ATA, Erhan, and Ümit Ziya SAVCI. "Generalized Quaternions and Matrix Algebra." Afyon Kocatepe University Journal of Sciences and Engineering 23, no. 3 (2023): 638–47. http://dx.doi.org/10.35414/akufemubid.1182145.

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In this paper, we established the connection between generalized quaternion algebra and real (complex) matrix algebras by using Hamilton operators. We obtained real and complex matrices corresponding to the real and complex basis of the generalized quaternions. Also, we investigated the basis features of real and complex matrices. We get Pauli matrices corresponding to generalized quaternions. Then, we have shown that the algebra produced by these matrices is isomorphic to the Clifford algebra Cl(E_αβ^3) produced by generalized space E_αβ^3. Finally, we studied the relations among the symplectic matrices group corresponding to generalized unit quaternions, generalized unitary matrices group, and generalized orthogonal matrices group.

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2

ABRATE, MARCO. "QUADRATIC FORMULAS FOR GENERALIZED QUATERNIONS." Journal of Algebra and Its Applications 08, no. 03 (2009): 289–306. http://dx.doi.org/10.1142/s0219498809003308.

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In this paper we derive explicit formulas for computing the roots of a quadratic polynomial with coefficients in a generalized quaternion algebra over any field 𝔽 with characteristic not 2. We also give some example of applications for the derived formulas, solving equations in the algebra of Hamilton's quaternions ℍ, in the ring M2(ℝ) of 2 × 2 square matrices over ℝ and in quaternion algebras over finite fields.

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3

Jiang, Tongsong, Dong Zhang, Zhenwei Guo, Gang Wang, and V. I. Vasil’ev. "Algebraic Techniques for Canonical Forms and Applications in Split Quaternionic Mechanics." Journal of Mathematics 2023 (November 7, 2023): 1–13. http://dx.doi.org/10.1155/2023/4599585.

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The algebra of split quaternions is a recently increasing topic in the study of theory and numerical computation in split quaternionic mechanics. This paper, by means of a real representation of a split quaternion matrix, studies the problem of canonical forms of a split quaternion matrix and derives algebraic techniques for finding the canonical forms of a split quaternion matrix. This paper also gives two applications for the right eigenvalue and diagonalization in split quaternionic mechanics.

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4

Colombaro, Ivano. "Exterior-algebraic formulation of quaternions with applications." Journal of Physics: Conference Series 3027, no. 1 (2025): 012020. https://doi.org/10.1088/1742-6596/3027/1/012020.

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Abstract The purpose of this paper is to describe the formulation of quaternion algebra by means of exterior algebra and calculus, in a three dimensional time-like spacetime. A formal structure is provided, corroborating the equivalence with existing concepts and formulas known in literature. A first application is thus presented by depicting the description of rotations expressed with exterior-algebraic quaternionic notation. Secondly, a formal equivalence between exterior-algebraic quaternions and the classical theory of electromagnetism is recovered, too.

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5

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 (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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6

Dai, Songsong. "Quaternionic Fuzzy Sets." Axioms 12, no. 5 (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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7

Cao, Wensheng. "Quadratic Equation in Split Quaternions." Axioms 11, no. 5 (2022): 188. http://dx.doi.org/10.3390/axioms11050188.

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Split quaternions are noncommutative and contain nontrivial zero divisors. Generally speaking, it is difficult to solve equations in such an algebra. In this paper, by using the roots of any split quaternions and two real nonlinear systems, we derive explicit formulas for computing the roots of x2+bx+c=0 in split quaternion algebra.

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8

Eri̇şi̇r, Tülay, Gökhan Mumcu, Sezai̇ Kiziltuğ, and Funda Akar. "A New Construction of Rectifying Direction Curves for Quaternionic Space Q." WSEAS TRANSACTIONS ON MATHEMATICS 24 (March 14, 2025): 114–25. https://doi.org/10.37394/23206.2025.24.13.

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Our article focuses on the study of quaternions topic introduced by Hamilton. Quaternions are a generalization of complex numbers and have multiple applications in mathematical physics. Another application of quaternions is robotics because what generalizes the imaginary axis is the family i, j, k modeling Euler angles and rotations in space. The first part of the article we recall the different definitions of how the algebra of quaternions is well constructed. The main results are given in the third part and concern: spatial quaternionics rectifying-direction (sqRD) curves and and spatial quaternionic rectifying-donor (sqRDnr) curves. We study a new tip of unit speed associated curves in E 3 , which is also used in robotic systems and kinematics, like a spatial quaternionic rectifying-direction curve and spatial quaternionic rectifying-donor curve. Then, we achieve qualification for the curves. Moreover, we present applications of spatial quaternionic rectifying-direction to some specific curves like helix, slant helix, Salkowski and anti-Salkowski curves or rectifying curves. In addition, we establish different theorems which generalize the results obtained on the quaternionic curves in Q. Then, we give some examples are finally discussed. Consequently, Our paper is centered around theoretical analysis in geometry rather than experimental investigations.

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9

Eri̇şi̇r, Tülay, and Mehmet Ali̇ Güngör. "On Fibonacci spinors." International Journal of Geometric Methods in Modern Physics 17, no. 04 (2020): 2050065. http://dx.doi.org/10.1142/s0219887820500656.

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Spinors are used in physics quite extensively. Basically, the forms of use include Dirac four-spinors, Pauli three-spinors and quaternions. Quaternions in mathematics are essentially equivalent to Pauli spin matrices which can be generated by regarding a quaternion matrix as compound. The goal of this study is also the spinor structure lying in the basis of the quaternion algebra. In this paper, first, we have introduced spinors mathematically. Then, we have defined Fibonacci spinors using the Fibonacci quaternions. Later, we have established the structure of algebra for these spinors. Finally, we have proved some important formulas such as Binet and Cassini formulas which are given for some series of numbers in mathematics for Fibonacci spinors.

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10

Pogorui, Anatoliy, and Tamila Kolomiiets. "Some algebraic properties of complex Segre quaternoins." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 33 (December 27, 2019): 158–59. http://dx.doi.org/10.37069/1683-4720-2019-33-13.

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This paper deals with the basic properties the algebra of Segre quaternions over the field of complex numbers. We study idempotents, ideals, matrix representation and the Peirce decomposition of this algebra. We also investigate the structure of zeros of a polynomial in Segre complex quaternions by reducing it to the system of four polynomial equations in the complex field. In addition, Cauchy-Riemann type conditions are obtained for the differentiability of a function on the complex Segre quaternionic algebra.

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11

Ge, Q. J. "On the Matrix Realization of the Theory of Biquaternions." Journal of Mechanical Design 120, no. 3 (1998): 404–7. http://dx.doi.org/10.1115/1.2829166.

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This paper describes a matrix algebra realization of Clifford’s theory of biquaternions. By examining 4 × 4 skew-symmetric matrices, the paper shows the connection between infinitesimal screws in elliptic three-space and vector quaternions. By studying the matrix exponential of the skew-symmetric matrices, the paper also shows how finite screws in elliptic three-space lead to matrix realization of quaternions. Finally, it is shown that line transformations in elliptic three-space lead to double quaternions and that a dual quaternion is a limiting case of a double quaternion.

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12

Ibrayev, А. Т. "METHOD FOR CONSTRUCTING THE COMMUTATIVE ALGEBRA OF QUATERNION AND OCTONION." PHYSICO-MATHEMATICAL SERIES 6, no. 334 (2020): 5–12. http://dx.doi.org/10.32014/2020.2518-1726.91.

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In this paper, we solve the problem of constructing a commutative algebra of quaternions and octonions. A proof of the theorem is given that the commutativity of quaternions can be ensured by specifying a set of sign coefficients of the directions of reference of the angles between the radius vectors in the coordinate planes of the vector part of the coordinate system of the quaternion space. The method proposed in the development of quaternions possessing the commutative properties of multiplication is used further to construct a commutative octonion algebra. The results obtained on improving the algebra of quaternions and octonions can be used in the development of new hypercomplex numbers with division over the field of real numbers, and can also find application for solving a number of scientific and technical problems in the areas of field theory, physical electronics, robotics, and digital processing of multidimensional signals.

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13

Chaker, Rachid, and Abdelkarim Boua. "New Results on Generalized Quaternion Algebra Involving Generalized Pell-pell Lucas Quaternions." WSEAS TRANSACTIONS ON MATHEMATICS 23 (July 19, 2024): 480–87. http://dx.doi.org/10.37394/23206.2024.23.50.

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his work presents a new sequence, generalized Pell-Pell-Lucas quaternions, we prove that the set of these elements forms an order of generalized quaternions with 3-parameters kλ1,λ2,λ3 as defined by ring theory. In addition, some properties of these elements are presented. The properties in this article refer to kλ1,λ2,λ3 algebras and sometimes to the 2-parameter algebra H(α, β).

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14

Etzel, K. R., and J. M. McCarthy. "Interpolation of Spatial Displacements Using the Clifford Algebra of E4." Journal of Mechanical Design 121, no. 1 (1999): 39–44. http://dx.doi.org/10.1115/1.2829427.

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In this paper we show that the Clifford Algebra of four dimensional Euclidean space yields a set of hypercomplex numbers called “double quaternions.” Interpolation formulas developed to generate Bezier-style quaternion curves are shown to be applicable to double quaternions by simply interpolating the components separately. The resulting double quaternion curves are independent of the coordinate frame in which the key frames are specified. Double quaternions represent rotations in E4 which we use to approximate spatial displacements. The result is a spatial motion interpolation methodology that is coordinate frame invariant to a desired degree of accuracy within a bounded region of three dimensional space. Examples demonstrate the application of this theory to computing distances between spatial displacement, determining the mid-point between two displacements, and generating the spatial motion interpolating a set of key frames.

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15

Puleko, I. V., O. V. Andreev, O. F. Dubina, V. O. Chumakevych, and A. S. Palamarchuk. "MODEL OF MOTION OF UNMANNED AERIAL VEHICLES BASED ON DUAL QUATERNION ALGEBRA." Проблеми створення, випробування, застосування та експлуатації складних інформаційних систем, no. 23 (December 28, 2022): 52–61. http://dx.doi.org/10.46972/2076-1546.2022.23.04.

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The widespread use of unmanned aerial vehicles during warfare has intensified the problem of their management, especially when they are used in large groups. One of the main tasks is to ensure coordinated movement of the group's aircraft in space. Optimizing the movement of each device of the group in three-dimensional space is expedient to use mathematical models. The movement of any unmanned aerial vehicle can be presented as a combination of translational and rotational movements, and its speed as a combination of translational and rotational velocities. Previously, these movements were modeled separately using a system of differential equations or quaternions. In this article, a mathematical model of rotational and translational movements of an aircraft based on the algebra of dual quaternions is developed. Dual quaternions consisting of eight scalars are a compact representation of rigid transformations in space. Therefore, their properties determine the advantage in the course of motion simulation, as they reduce the amount of calculations. Thus, with the help of one dual quaternion, it is possible to provide both translational and rotational motions at once, and the operation of non-commutative multiplication of dual quaternions is used to simulate the movement. The model assumes that the real part of the dual quaternion determines the orientation of the UAV in space, and the dual part determines its position in three-dimensional space. In order to connect aircraft coordinate systems with the model, expressions for the transition from aircraft orientation angles (roll, yaw, and pitch) to dual quaternion parameters and vice versa are obtained. The functionality of the proposed model was confirmed using the developed software for modeling the coordinated movement of aircraft. The software is adapted for graphical display of a large number of aircraft in web browsers with WebGl support. Keywords: motion modeling; rotational and translational movement; unmanned aerial vehicles; quaternions; dual quaternions; algebra of quaternions.

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16

Kosal, Hidayet Huda, and Murat Tosun. "Some Equivalence Relations and Results over the Commutative Quaternions and Their Matrices." Analele Universitatii "Ovidius" Constanta - Seria Matematica 25, no. 3 (2017): 125–42. http://dx.doi.org/10.1515/auom-2017-0040.

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Abstract In this paper, we give some equivalence relations and results over the commutative quaternions and their matrices. In this sense, consimilarity, semisimilarity, and consemisimilarity over the commutative quaternion algebra and commutative quaternion matrix algebra are established. Equalities of these equivalence relations are explicitly determined. Also Syvester-s-Conjugate commutative quaternion matrix equations are studied by means of real representation of the commutative quaternion matrices and consimilarity of the two commutative quaternion matrices.

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17

Kösal, Hidayet Hüda, Emre Kişi, Mahmut Akyiğit, and Beyza Çelik. "Elliptic Quaternion Matrices: A MATLAB Toolbox and Applications for Image Processing." Axioms 13, no. 11 (2024): 771. http://dx.doi.org/10.3390/axioms13110771.

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In this study, we developed a MATLAB 2024a toolbox that performs advanced algebraic calculations in the algebra of elliptic numbers and elliptic quaternions. Additionally, we introduce color image processing methods, such as principal component analysis, image compression, image restoration, and watermarking, based on singular-value decomposition theory for elliptic quaternion matrices; we added these to the newly developed toolbox. The experimental results demonstrate that elliptic quaternionic methods yield better image analysis and processing performance compared to other hypercomplex number-based methods.

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18

Scheglov, Georgiy. "On the geometric interpretation of quaternions by cones." Geometry & Graphics 10, no. 3 (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 (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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20

Li, Guozhi, Fuhai Zhang, Yili Fu, and Shuguo Wang. "Kinematic calibration of serial robot using dual quaternions." Industrial Robot: the international journal of robotics research and application 46, no. 2 (2019): 247–58. http://dx.doi.org/10.1108/ir-10-2018-0221.

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Purpose The purpose of this paper is to propose an error model for serial robot kinematic calibration based on dual quaternions. Design/methodology/approach The dual quaternions are the combination of dual-number theory and quaternion algebra, which means that they can represent spatial transformation. The dual quaternions can represent the screw displacement in a compact and efficient way, so that they are used for the kinematic analysis of serial robot. The error model proposed in this paper is derived from the forward kinematic equations via using dual quaternion algebra. The full pose measurements are considered to apply the error model to the serial robot by using Leica Geosystems Absolute Tracker (AT960) and tracker machine control (T-MAC) probe. Findings Two kinematic-parameter identification algorithms are derived from the proposed error model based on dual quaternions, and they can be used for serial robot calibration. The error model uses Denavit–Hartenberg (DH) notation in the kinematic analysis, so that it gives the intuitive geometrical meaning of the kinematic parameters. The absolute tracker system can measure the position and orientation of the end-effector (EE) simultaneously via using T-MAC. Originality/value The error model formulated by dual quaternion algebra contains all the basic geometrical parameters of serial robot during the kinematic calibration process. The vector of dual quaternion error can be used as an indicator to represent the trend of error change of robot’s EE between the nominal value and the actual value. The accuracy of the EE is improved after nearly 20 measurements in the experiment conduct on robot SDA5F. The simulation and experiment verify the effectiveness of the error model and the calibration algorithms.

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21

Faldiyan, Muhammad, Ema Carnia, and Asep Kuswandi Supriatna. "Existence of Split Property in Quaternion Algebra Over Composite of Quadratic Fields." CAUCHY: Jurnal Matematika Murni dan Aplikasi 8, no. 2 (2023): 142–56. http://dx.doi.org/10.18860/ca.v8i2.22881.

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Quaternions are extensions of complex numbers that are four-dimensional objects. Quaternion consists of one real number and three complex numbers, commonly denoted by the standard vectors and . Quaternion algebra over the field is an algebra in which the multiplication between standard vectors is non-commutative and the multiplication of standard vector with itself is a member of the field. The field considered in this study is the quadratic field and its extensions are biquadratic and composite. There have been many studies done to show the existence of split properties in quaternion algebras over quadratic fields. The purpose of this research is to prove a theorem about the existence of split properties on three field structures, namely quaternion algebras over quadratic fields, biquadratic fields, and composite of quadratic fields. We propose two theorems about biquadratic fields and composite of quadratic fields refer to theorems about the properties of the split on quadratic fields. The result of this research is a theorem proof of three theorems with different field structures that shows the different conditions of the three field structures. The conclusion is that the split property on quaternion algebras over fields exists if certain conditions can be met.

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22

Danielewski, Marek, and Lucjan Sapa. "Quaternions and Cauchy Classical Theory of Elasticity." Advances in Manufacturing Science and Technology 44, no. 2 (2020): 67–70. http://dx.doi.org/10.2478/amst-2019-0014.

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AbstractDeveloped by French mathematician Augustin-Louis Cauchy, the classical theory of elasticity is the starting point to show the value and the physical reality of quaternions. The classical balance equations for the isotropic, elastic crystal, demonstrate the usefulness of quaternions. The family of wave equations and the diffusion equation are a straightforward consequence of the quaternion representation of the Cauchy model of the elastic solid. Using the quaternion algebra, we present the derivation of the quaternion form of the multiple wave equations. The fundamental consequences of all derived equations and relations for physics, chemistry, and future prospects are presented.

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Rawat, Seema. "CPT Invariance of Quaternion Dirac equation." International Journal for Research in Applied Science and Engineering Technology 10, no. 1 (2022): 961–66. http://dx.doi.org/10.22214/ijraset.2022.39956.

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Abstract: In this paper the invariance of Quaternion Dirac equation under Lorentz Transformation, Charge conjugation, Parity transformation and Time reversal operation has been discussed successfully. The invariance under the combined operation of Charge conjugation, Parity and Time reversal (CPT) has also been discussed and expression for C, P, T and combined CPT operators have been obtained in terms of quaternions. Invariance condition for electric and magnetic field has also been obtained. It has been concluded that the Quaternion Dirac equation dominates over ordinary Dirac equation because of the advantage of algebra of quaternions.

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de Farias, Cristiana Miranda. "A Novel Dual Quaternion Based Cost Effcient Recursive Newton-Euler Inverse Dynamics Algorithm." International Journal of Robotic Computing 1, no. 2 (2019): 144–68. http://dx.doi.org/10.35708/rc1868-126255.

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In this paper, the well known recursive Newton-Euler inverse dynamics algorithm for serial manipulators is reformulated into the context of the algebra of Dual Quaternions. Here we structure the forward kinematic description with screws and line displacements rather than the well established Denavit-Hartemberg parameters, thus accounting better efficiency, compactness and simpler dynamical models. We also present here the closed solution for the dqRNEA, and to do so we formalize some of the algebra for dual quaternion-vectors and dual quaternion-matrices. With a closed formulation of the dqRNEA we also create a dual quaternion based formulation for the computed torque control, a feedback linearization method for controlling a serial manipulator's torques in the joint space. Finally, a cost analysis of the main Dual Quaternions operations and of the Newton-Euler inverse dynamics algorithm as a whole is made and compared with other results in the literature.

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Kansu, Mustafa Emre. "Quaternionic representation of electromagnetism for material media." International Journal of Geometric Methods in Modern Physics 16, no. 07 (2019): 1950105. http://dx.doi.org/10.1142/s0219887819501056.

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In this study, the electromagnetic fields are developed in the presence of both the electric and magnetic induction fields by quaternion algebra. In this sense, the polarization and magnetization effects, which are valid in the material media, gain much importance. Quaternions are one of the most convenient tools for representing electromagnetism with regard to having non-commutative but associative algebraic division ring. By defining the quaternion induction field, the quaternion source term has been obtained in basic and elegant notation for the first time. In addition, one type of Poynting theorem, named as the Minkowski form, has been presented including the permittivity and permeability constants by quaternions.

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Tan, Elif, Diana Savin, and Semih Yılmaz. "A New Class of Leonardo Hybrid Numbers and Some Remarks on Leonardo Quaternions over Finite Fields." Mathematics 11, no. 22 (2023): 4701. http://dx.doi.org/10.3390/math11224701.

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In this paper, we present a new class of Leonardo hybrid numbers that incorporate quantum integers into their components. This advancement presents a broader generalization of the q-Leonardo hybrid numbers. We explore some fundamental properties associated with these numbers. Moreover, we study special Leonardo quaternions over finite fields. In particular, we determine the Leonardo quaternions that are zero divisors or invertible elements in the quaternion algebra over the finite field Zp for special values of prime integer p.

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Lai, Wei Kit, Kiat Tat Qua, and Denis Chee Keong Wong. "On the sum of an idempotent and a tripotent in a quaternion algebra over the ring of integers modulo p." ITM Web of Conferences 67 (2024): 01022. http://dx.doi.org/10.1051/itmconf/20246701022.

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Let H be denoted as quaternions. Quaternions form an algebra over a ring R, as an extension of complex numbers into a four dimensional space, where H = {a0 + a1i + a2j + a3k | a0, a1, a2, a3 ∈ R}. A quaternion algebra, particularly defined over fields of characteristic 0, finds numerous applications in physics. In this article, we explore some properties of the sum of an idempotent and a tripotent in the finite ring H/Zp, adapting the definition of SIT rings that was introduced by Ying et al in 2016. We provide some conditions for H/Zp to be SIT rings and we give some examples of weakly tripotent rings (Breaz and Cimpean, 2018) in H/Zp.

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Chen, Yu. "Quaternion group algebra and representations of the quaternions." Communications in Algebra 47, no. 2 (2019): 749–60. http://dx.doi.org/10.1080/00927872.2018.1498859.

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Chudá, Hana. "Universal approach to derivation of quaternion rotation formulas." MATEC Web of Conferences 292 (2019): 01060. http://dx.doi.org/10.1051/matecconf/201929201060.

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This paper introduces and defines the quaternion with a brief insight into its properties and algebra. The main part of this paper is devoted to the derivation of basic equations of the vector rotation around each rotational x, y, z axis. Then, the equations of generalized quaternion rotation and express the general rotation operator is derived. Finally the utilization of equations is demonstrated on a simple example. For purposes of simplicity the quaternions theory is demonstrated around the z-axis by γ angle. For the purpose of this paper, the fact that the subspace of vector quaternions may be regarded as being equivalent to the ordinary is used.

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TAŞ, Ferhat. "Trajectory curves and surfaces: A new perspective via projective geometric algebra." Communications Faculty Of Science University of Ankara Series A1Mathematics and Statistics 73, no. 1 (2023): 64–75. http://dx.doi.org/10.31801/cfsuasmas.1170867.

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The aim of this work is to define quaternion curves and surfaces and their conjugates via operators in Euclidean projective geometric algebra (EPGA). In this space, quaternions were obtained by the geometric product of vector fields. New vector fields, which we call trajectory curves and surfaces, were obtained by using this new quaternion operator. Moreover, dual quaternion curves are determined by a similar method and then their generated motion is studied. Illustrative examples are given.

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Ghiloni, Riccardo, and Alessandro Perotti. "Lagrange polynomials over Clifford numbers." Journal of Algebra and Its Applications 14, no. 05 (2015): 1550069. http://dx.doi.org/10.1142/s0219498815500693.

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We construct Lagrange interpolating polynomials for a set of points and values belonging to the algebra of real quaternions ℍ ≃ ℝ0,2, or to the real Clifford algebra ℝ0,3. In the quaternionic case, the approach by means of Lagrange polynomials is new, and gives a complete solution of the interpolation problem. In the case of ℝ0,3, such a problem is dealt with here for the first time. Elements of the recent theory of slice regular functions are used. Leaving apart the classical cases ℝ0,0 ≃ ℝ, ℝ0,1 ≃ ℂ and the trivial case ℝ1,0 ≃ ℝ⊕ℝ, the interpolation problem on Clifford algebras ℝp,q with (p,q) ≠ (0,2), (0,3) seems to have some intrinsic difficulties.

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32

Grigoryan, Artyom M., and Alexis A. Gomez. "Commutative Quaternion Algebra with Quaternion Fourier Transform-Based Alpha-Rooting Color Image Enhancement." Computers 14, no. 2 (2025): 37. https://doi.org/10.3390/computers14020037.

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In this paper, we describe the associative and commutative algebra or the (2,2)-model of quaternions with application in color image enhancement. The method of alpha-rooting, which is based on the 2D quaternion discrete Fourier transform (QDFT) is considered. In the (2,2)-model, the aperiodic convolution of quaternion signals can be calculated by the product of their QDFTs. The concept of linear convolution is simple, that is, it is unique, and the reduction of this operation to the multiplication in the frequency domain makes this model very attractive for processing color images. Note that in the traditional quaternion algebra, which is not commutative, the convolution can be chosen in many different ways, and the number of possible QDFTs is infinite. And most importantly, the main property of the traditional Fourier transform that states that the aperiodic convolution is the product of the transform in the frequency domain is not valid. We describe the main property of the (2,2)-model of quaternions, the quaternion exponential functions and convolution. Three methods of alpha-rooting based on the 2D QDFT are presented, and illustrative examples on color image enhancement are given. The image enhancement measures to estimate the quality of the color images are described. Examples of the alpha-rooting enhancement on different color images are given and analyzed with the known histogram equalization and Retinex algorithms. Our experimental results show that the alpha-rooting method in the quaternion space is one of the most effective methods of color image enhancement. Quaternions allow all colors in each pixel to be processed as a whole, rather than individually as is done in traditional processing methods.

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33

Ibrayev, Alpamys T. "Method for Constructing a Commutative Algebra of Hypercomplex Numbers." Symmetry 15, no. 9 (2023): 1652. http://dx.doi.org/10.3390/sym15091652.

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Until now, it was believed that, unlike real and complex numbers, the construction of a commutative algebra of quaternions or octonions with division over the field of real numbers is impossible in principle. No one questioned the existing theoretical assertion that quaternions, octonions, and other hypercomplex numbers cannot have the commutativity property. This article demonstrates the following for the first time: (1) the possibility of constructing a normed commutative algebra of quaternions and octonions with division over the field of real numbers; (2) the possibility of constructing a normed commutative algebra of six-dimensional and ten-dimensional hypercomplex numbers with division over the field of real numbers; (3) a method for constructing a normed commutative algebra of N-dimensional hypercomplex numbers with division over the field of real numbers for even values of N; and (4) the possibility of constructing a normed commutative algebra of other N-dimensional hypercomplex numbers with division over the field of real numbers. The article also shows that when using specific forms of representation of unit vectors, the product of vectors has the property of commutativity. Normed commutative algebras of N-dimensional hypercomplex numbers can be widely used to solve many topical scientific problems in the field of theoretical physics for modeling force fields with various types of symmetry, in cryptography for developing a number of new cryptographic programs using hypercomplex number algebras with different values of dimension, and in many other areas of fundamental and applied sciences.

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34

Berry, Thomas, and Matt Visser. "Relativistic Combination of Non-Collinear 3-Velocities Using Quaternions." Universe 6, no. 12 (2020): 237. http://dx.doi.org/10.3390/universe6120237.

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Quaternions have an (over a century-old) extensive and quite complicated interaction with special relativity. Since quaternions are intrinsically 4-dimensional, and do such a good job of handling 3-dimensional rotations, the hope has always been that the use of quaternions would simplify some of the algebra of the Lorentz transformations. Herein we report a new and relatively nice result for the relativistic combination of non-collinear 3-velocities. We work with the relativistic half-velocities w defined by v=2w1+w2, so that w=v1+1−v2=v2+O(v3), and promote them to quaternions using w=wn^, where n^ is a unit quaternion. We shall first show that the composition of relativistic half-velocities is given by w1⊕2≡w1⊕w2≡(1−w1w2)−1(w1+w2), and then show that this is also equivalent to w1⊕2=(w1+w2)(1−w2w1)−1. Here as usual we adopt units where the speed of light is set to unity. Note that all of the complicated angular dependence for relativistic combination of non-collinear 3-velocities is now encoded in the quaternion multiplication of w1 with w2. This result can furthermore be extended to obtain novel elegant and compact formulae for both the associated Wigner angle Ω and the direction of the combined velocities: eΩ=eΩΩ^=(1−w1w2)−1(1−w2w1), and w^1⊕2=eΩ/2w1+w2|w1+w2|. Finally, we use this formalism to investigate the conditions under which the relativistic composition of 3-velocities is associative. Thus, we would argue, many key results that are ultimately due to the non-commutativity of non-collinear boosts can be easily rephrased in terms of the non-commutative algebra of quaternions.

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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 (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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36

MCKAGUE, MATTHEW. "SUPER-QUANTUM NONLOCAL CORRELATIONS IN QUATERNIONIC QUANTUM THEORY." International Journal of Quantum Information 09, no. 06 (2011): 1355–62. http://dx.doi.org/10.1142/s0219749911008052.

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We consider the nonlocal properties of naive quaternionic quantum theory, in which the complex numbers are replaced by the quaternions as the underlying algebra. Specifically, we show that it is possible to construct a nonlocal box. This allows one to rule out quaternionic quantum theory using assumptions about communication complexity or information causality while also providing a model for a nonlocal box using familiar structures.

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37

Sariyildiz, Emre, Eray Cakiray, and Hakan Temeltas. "A Comparative Study of Three Inverse Kinematic Methods of Serial Industrial Robot Manipulators in the Screw Theory Framework." International Journal of Advanced Robotic Systems 8, no. 5 (2011): 64. http://dx.doi.org/10.5772/45696.

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In this paper, we compare three inverse kinematic formulation methods for the serial industrial robot manipulators. All formulation methods are based on screw theory. Screw theory is an effective way to establish a global description of rigid body and avoids singularities due to the use of the local coordinates. In these three formulation methods, the first one is based on quaternion algebra, the second one is based on dual-quaternions, and the last one that is called exponential mapping method is based on matrix algebra. Compared with the matrix algebra, quaternion algebra based solutions are more computationally efficient and they need less storage area. The method which is based on dual-quaternion gives the most compact and computationally efficient solution. Paden-Kahan sub-problems are used to derive inverse kinematic solutions. 6-DOF industrial robot manipulator's forward and inverse kinematic equations are derived using these formulation methods. Simulation and experimental results are given.

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38

Jia, Zong-Ru, and Qing-Wen Wang. "The General Solution to a System of Tensor Equations over the Split Quaternion Algebra with Applications." Mathematics 13, no. 4 (2025): 644. https://doi.org/10.3390/math13040644.

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This paper presents a systematic investigation into the solvability and the general solution of a tensor equation system within the split quaternion algebra framework. As an extension of classical quaternions with distinctive pseudo-Euclidean properties, split quaternions offer unique advantages in multidimensional signal processing applications. We establish rigorous necessary and sufficient conditions for the existence of solutions to the proposed tensor equation system, accompanied by explicit formulations for general solutions when solvability criteria are satisfied. The theoretical framework is further strengthened by the development of computational algorithms and numerical validations through concrete examples. Notably, we demonstrate the practical implementation of our theoretical findings through encryption/decryption algorithms for color video data.

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39

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 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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40

Bekar, Murat, and Yusuf Yayli. "Semi-Euclidean quasi-elliptic planar motion." International Journal of Geometric Methods in Modern Physics 13, no. 07 (2016): 1650089. http://dx.doi.org/10.1142/s0219887816500894.

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The aim of this paper is to study the algebra of split semi-quaternions with their basic properties. Also, the results of the Euclidean planar motion given by Blaschke and Grünwald is generalized to semi-Euclidean planar motion by using the algebra of split semi-quaternions.

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41

KRIEG, ALOYS. "THETA SERIES OVER THE HURWITZ QUATERNIONS." International Journal of Number Theory 06, no. 01 (2010): 25–36. http://dx.doi.org/10.1142/s1793042110002788.

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There are six theta constants over the Hurwitz quaternions on the quaternion half-space of degree 2. The paper describes the behavior of these theta constants under the transpose mapping, which can be derived from the Fourier expansions. The results are applied to the theta series of the first and second kind.

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42

Andreea-Elena, Țugui (Baias). "Some Properties of Quaternion Algebra over the Sets of Real and Complex Numbers." International Journal of Computational and Applied Mathematics & Computer Science 3 (December 18, 2023): 122–32. http://dx.doi.org/10.37394/232028.2023.3.14.

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In this article, we analyzed complex quaternions and the matrix representations associated with 2x2 complex quaternions. We provided detailed insights into the fundamental properties of quaternions, highlighting essential features of the corresponding matrix representations. Additionally, we examined real quaternions, emphasizing their specific characteristics, and we explored aspects related to matrix representations of quaternions with real coefficients of size 4x4. The central aim of this article is to conduct a detailed comparison between the two types of matrix representations, both in real and complex contexts. In light of the obtained results, the article seeks to make significant contributions to the understanding and application of quaternions in various mathematical domains.

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43

Kalauni, Pushpa, and J. C. A. Barata. "Role of division algebra in seven-dimensional gauge theory." Modern Physics Letters A 30, no. 10 (2015): 1550047. http://dx.doi.org/10.1142/s0217732315500479.

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The algebra of octonions 𝕆 forms the largest normed division algebra over the real numbers ℝ, complex numbers ℂ and quaternions ℍ. The usual three-dimensional vector product is given by quaternions, while octonions produce seven-dimensional vector product. Thus, octonionic algebra is closely related to the seven-dimensional algebra, therefore one can extend generalization of rotations in three dimensions to seven dimensions using octonions. An explicit algebraic description of octonions has been given to describe rotational transformation in seven-dimensional space. We have also constructed a gauge theory based on non-associative algebra to discuss Yang–Mills theory and field equation in seven-dimensional space.

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44

Jakovčević Stor, Nevena, and Ivan Slapničar. "Inverses and Determinants of Arrowhead and Diagonal-Plus-Rank-One Matrices over Associative Algebras." Axioms 13, no. 6 (2024): 409. http://dx.doi.org/10.3390/axioms13060409.

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This article considers arrowhead and diagonal-plus-rank-one matrices in Fn×n where F∈{R,C,H} and where H is a noncommutative algebra of quaternions. We provide unified formulas for fast determinants and inverses for considered matrices. The formulas are unified in the sense that the same formula holds in both commutative and noncommutative associative fields or algebras, with noncommutative examples being matrices of quaternions and block matrices. Each formula requires O(n) arithmetic operations, as does multiplication of such matrices with a vector. The formulas are efficiently implemented using the polymorphism or multiple-dispatch feature of the Julia programming language.

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Alpay, Daniel, Fabrizio Colombo, David P. Kimsey, and Irene Sabadini. "Wiener Algebra for the Quaternions." Mediterranean Journal of Mathematics 13, no. 5 (2015): 2463–82. http://dx.doi.org/10.1007/s00009-015-0634-z.

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46

Duplij, Steven. "Hyperpolyadic Structures." Mathematics 12, no. 15 (2024): 2378. http://dx.doi.org/10.3390/math12152378.

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We introduce a new class of division algebras, the hyperpolyadic algebras, which correspond to the binary division algebras R, C, H, O without considering new elements. First, we use the matrix polyadization procedure proposed earlier which increases the dimension of the algebra. The algebras obtained in this way obey binary addition and a nonderived n-ary multiplication and their subalgebras are division n-ary algebras. For each invertible element, we define a new norm which is polyadically multiplicative, and the corresponding map is a n-ary homomorphism. We define a polyadic analog of the Cayley–Dickson construction which corresponds to the consequent embedding of monomial matrices from the polyadization procedure. We then obtain another series of n-ary algebras corresponding to the binary division algebras which have a higher dimension, which is proportional to the intermediate arities, and which are not isomorphic to those obtained by the previous constructions. Second, a new polyadic product of vectors in any vector space is defined, which is consistent with the polyadization procedure using vectorization. Endowed with this introduced product, the vector space becomes a polyadic algebra which is a division algebra under some invertibility conditions, and its structure constants are computed. Third, we propose a new iterative process (we call it the “imaginary tower”), which leads to nonunital nonderived ternary division algebras of half the dimension, which we call “half-quaternions” and “half-octonions”. The latter are not the subalgebras of the binary division algebras, but subsets only, since they have different arity. Nevertheless, they are actually ternary division algebras, because they allow division, and their nonzero elements are invertible. From the multiplicativity of the introduced “half-quaternion” norm, we obtain the ternary analog of the sum of two squares identity. We show that the ternary division algebra of imaginary “half-octonions” is unitless and totally associative.

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47

Martinez-Terán, Gerardo, Oswaldo Ureña-Ponce, Gerardo Soria-García, Susana Ortega-Cisneros, and Eduardo Bayro-Corrochano. "Fast Study Quadric Interpolation in the Conformal Geometric Algebra Framework." Electronics 11, no. 10 (2022): 1527. http://dx.doi.org/10.3390/electronics11101527.

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Interpolating trajectories of points and geometric entities is an important problem for kinematics. To describe these trajectories, several algorithms have been proposed using matrices, quaternions, dual-quaternions, and the Study quadric; the last one allows the embedding of motors as 8D vectors into projective space P7, where the interpolation of rotations and translations becomes a linear problem. Furthermore, conformal geometric algebra (CGA) is an effective and intuitive framework for representing and manipulating geometric entities in Euclidean spaces, and it allows the use of quaternions and dual-quaternions formulated as Motors. In this paper, a new methodology for accelerating the Study quadric Interpolation based on Conformal Geometric Algebra is presented. This methodology uses General Purpose Graphics Processing Units (GPUs) and it is applied for medical robotics, but it can also be extended to other areas such as aeronautics, robotics, and graphics processing.

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Chaker, Rachid, and Abdelkarim Boua. "Some results on generalized quaternions algebra with generalized Fibonacci quaternions." Annals of the Alexandru Ioan Cuza University - Mathematics 69, no. 2 (2023): 233–46. http://dx.doi.org/10.47743/anstim.2023.00015.

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Chen, Yan, Qing-Wen Wang, and Lv-Ming Xie. "Dual Quaternion Matrix Equation AXB = C with Applications." Symmetry 16, no. 3 (2024): 287. http://dx.doi.org/10.3390/sym16030287.

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Dual quaternions have wide applications in automatic differentiation, computer graphics, mechanics, and others. Due to its application in control theory, matrix equation AXB=C has been extensively studied. However, there is currently limited information on matrix equation AXB=C regarding the dual quaternion algebra. In this paper, we provide the necessary and sufficient conditions for the solvability of dual quaternion matrix equation AXB=C, and present the expression for the general solution when it is solvable. As an application, we derive the ϕ-Hermitian solutions for dual quaternion matrix equation AXAϕ=C, where the ϕ-Hermitian extends the concepts of Hermiticity and η-Hermiticity. Lastly, we present a numerical example to verify the main research results of this paper.

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Li, Guozhi, Shuizhong Zou, Shuxue Din, and Bin Qi. "Modified Hand–Eye Calibration Using Dual Quaternions." Applied Sciences 12, no. 23 (2022): 12480. http://dx.doi.org/10.3390/app122312480.

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This paper presents a modified model for hand–eye calibration based on dual quaternion algebra. By using dual quaternions to represent the rotations and translations of a rigid body simultaneously in the task space, the formulation is elegant for the analysis of the hand–eye equation. The hand–eye transformation derived in this study is represented in a compact manner, which uses a combination of the dual part and the real part of the dual quaternion. Although the hand–eye equation can be solved by using six elements of a dual quaternion without using its scalar parts, the scaler numbers in both the real and dual parts of a dual quaternion contain part of the pose information. The originality is based on the derivation of the construct of the identification algorithm of external parameters of the camera by using all eight elements of a dual quaternion. Then, the data transformation between the cameras of the dual-arm hand–eye robot system is presented. The corresponding results demonstrate that the proposed hand–eye calibration algorithm can process measurement data with noise and can also improve the identification accuracy to verify its efficiency.

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