Artigos de revistas: "Quaternions space"

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Miškinis, P. "ON THE POSSIBLE EXISTENCE OF NEW FERMIONIC DEGREES OF FREEDOM IN D = 6". Mathematical Modelling and Analysis 8, n.º 2 (30 de junho de 2003): 155–64. http://dx.doi.org/10.3846/13926292.2003.9637220.

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The bispinors formed by quaternion in D = 6 dimensional space‐time are proposed to be treated as new fermionic fields. The gauge nonabelian field is formulated by the quaternions. A new kind of physical object, an extended relativistic quaternionic membrane in D = 6, is discussed.

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Ge, Q. J. "On the Matrix Realization of the Theory of Biquaternions". Journal of Mechanical Design 120, n.º 3 (1 de setembro de 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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Dzwonkowski, Mariusz, e Roman Rykaczewski. "Quaternion Feistel Cipher with an Infinite Key Space Based on Quaternion Julia Sets". Journal of Telecommunications and Information Technology, n.º 4 (30 de dezembro de 2015): 15–21. http://dx.doi.org/10.26636/jtit.2015.4.979.

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In this paper Quaternion Feistel Cipher (QFC) with an infinite key space based on quaternion Julia sets is proposed. The basic structure of the algorithm is based on the scheme proposed in 2012 by Sastry and Kumar. The proposed algorithm uses special properties of quaternions to perform rotations of data sequences in 3D space for each of the cipher rounds. It also uses Julia sets to form an infinite key space. The plaintext is divided into two square matrices of equal size and written using Lipschitz quaternions. A modular arithmetic was implemented for operations with quaternions. A computer-based analysis has been carried out and obtained results are shown at the end of this paper.

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Puleko, I. V., O. V. Andreev, O. F. Dubina, V. O. Chumakevych e A. S. Palamarchuk. "MODEL OF MOTION OF UNMANNED AERIAL VEHICLES BASED ON DUAL QUATERNION ALGEBRA". Проблеми створення, випробування, застосування та експлуатації складних інформаційних систем, n.º 23 (28 de dezembro de 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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KUMAR, AWNIYA, SUNIL KUMAR SINGH e SHEO KUMAR SINGH. "A Note on Moritoh Transforms". Creative Mathematics and Informatics 33, n.º 2 (14 de maio de 2024): 185–201. http://dx.doi.org/10.37193/cmi.2024.02.05.

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Some fundamental properties of the Moritoh wavelet are discussed in this paper. The Mori- toh transform is approximated for ultra-distributions in generalised Sobolev space. The adjoint formula of the Fourier transform is extended to the Moritoh transform. The convolution for quaternion-valued functions is de- fined for a modified representation of quaternions. Furthermore, the quaternionic Moritoh transform is defined with the help of convolution. The inner product relation and the uncertainty principle are also established for the quaternionic Moritoh wavelet transform.

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Gogberashvili, Merab. "(2 + 1)-Maxwell Equations in Split Quaternions". Physics 4, n.º 1 (17 de março de 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, is equivalent to some system of differential equations for quaternionic spinors and vectors. Assuming that the derivatives by extra time-like coordinate generate triality (supersymmetric) rotations, the analyticity equation is reduced to the exact Dirac–Maxwell system in three-dimensional Minkowski space-time.

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ATASOY, Ali, e Faik BABADA˘G. "A new Approach to Hyper Dual Split Quaternions with Different Polar Representation". General Letters in Mathematics 14, n.º 3 (setembro de 2024): 75–82. http://dx.doi.org/10.31559/glm2024.14.3.4.

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Hamilton first introduced quaternions in 1843 as a way to represent rotations in three dimensional space, and since then, they have become the important tool in many fields. One advantage of quaternions over other methods of representing rotations is their ability to avoid the problem of gimbal lock, which can occur when using Euler angles. Quaternions also have a relatively simple algebraic structure and can be efficiently implemented in computer algorithms. In recent years, quaternions have been used in the development of virtual reality systems and computer games, where they are used to represent orientations of objects in three-dimensional space. They have also been applied in robotics, control theory, and signal processing. Overall, quaternions have become the valuable tools in many areas of mathematics and engineering, and their usage continue to expand. Sangwine and Bihan introduced a quaternion polar representation that draws inspiration from the Cayley-Dickson form. In their formulation, they express quaternions using a complex modulus and argument. The Cayley-Dickson construction is a mathematical procedure that extends the concept of complex numbers to higher dimensions, paving the way for the development of quaternions. On the other hand, the complex argument represents the direction or orientation of the quaternion in a manner analogous to the argument of a complex number. This approach provides a concise and insightful way to represent quaternions, offering a geometric interpretation that aligns with the principles of complex analysis.

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Etzel, K. R., e J. M. McCarthy. "Interpolation of Spatial Displacements Using the Clifford Algebra of E4". Journal of Mechanical Design 121, n.º 1 (1 de março de 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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Weng, Zi-Hua. "Forces in the complex octonion curved space". International Journal of Geometric Methods in Modern Physics 13, n.º 06 (15 de junho de 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 properties making use of the complex quaternion/octonion curved space. From the orthogonality of two complex quaternions, it is possible to define the covariant derivative of the complex quaternion curved space, describing the gravitational properties in the complex quaternion curved space. Further it is possible to define the covariant derivative of the complex octonion curved space by means of the orthogonality of two complex octonions, depicting simultaneously the electromagnetic and gravitational properties in the complex octonion curved space. The result reveals that the connection coefficient and curvature of the complex octonion curved space will exert an influence on the field strength and field source of the electromagnetic and gravitational fields, impacting the linear momentum, angular momentum, torque, energy, and force and so forth.

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Cansu, Gizem, Yusuf Yaylı e İsmail Gök. "A new quaternion valued frame of curves with an application". Filomat 35, n.º 1 (2021): 315–30. http://dx.doi.org/10.2298/fil2101315c.

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The aim of the paper is to obtain a new version of Serret-Frenet formulae for a quaternionic curve in R4 by using the method given by Bharathi and Nagaraj. Then, we define quaternionic helices in H named as quaternionic right and left X-helix with the help of given a unit vector field X. Since the quaternion product is not commutative, the authors ([4], [7]) have used by one-sided multiplication to find a space curve related to a given quaternionic curve in previous studies. Firstly, we obtain new expressions by using the right product and the left product for quaternions. Then, we generalized the construction of Serret-Frenet formulae of quaternionic curves. Finally, as an application, we obtain an example that supports the theory of this paper.

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Weng, Zi-Hua. "Color Confinement and Spatial Dimensions in the Complex-Sedenion Space". Advances in Mathematical Physics 2017 (2017): 1–26. http://dx.doi.org/10.1155/2017/9876464.

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The paper aims to apply the complex-sedenions to explore the wave functions and field equations of non-Abelian gauge fields, considering the spatial dimensions of a unit vector as the color degrees of freedom in the complex-quaternion wave functions, exploring the physical properties of the color confinement essentially. J. C. Maxwell was the first to employ the quaternions to study the electromagnetic fields. His method inspires subsequent scholars to introduce the quaternions, octonions, and sedenions to research the electromagnetic field, gravitational field, and nuclear field. The application of complex-sedenions is capable of depicting not only the field equations of classical mechanics, but also the field equations of quantum mechanics. The latter can be degenerated into the Dirac equation and Yang-Mills equation. In contrast to the complex-number wave function, the complex-quaternion wave function possesses three new degrees of freedom, that is, three color degrees of freedom. One complex-quaternion wave function is equivalent to three complex-number wave functions. It means that the three spatial dimensions of unit vector in the complex-quaternion wave function can be considered as the “three colors”; naturally the color confinement will be effective. In other words, in the complex-quaternion space, the “three colors” are only the spatial dimensions, rather than any property of physical substance.

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ATA, Erhan, e Ümit Ziya SAVCI. "Generalized Quaternions and Matrix Algebra". Afyon Kocatepe University Journal of Sciences and Engineering 23, n.º 3 (22 de junho de 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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Czaplewski, Bartosz, Mariusz Dzwonkowski e Roman Rykaczewski. "Digital Fingerprinting Based on Quaternion Encryption Scheme for Gray-Tone Images". Journal of Telecommunications and Information Technology, n.º 2 (30 de junho de 2014): 3–11. http://dx.doi.org/10.26636/jtit.2014.2.1016.

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In this paper a new idea of digital images fingerprinting is proposed. The method is based on quaternion encryption in the Cipher Block Chaining (CBC) mode. Quaternions are hyper-complex numbers of rank 4 and thus often applied to mechanics in three-dimensional space. The encryption algorithm described in the paper is designed for graytone images but can easily be adopted for color ones. For the encryption purpose, the algorithm uses the rotation of data vectors presented as quaternions in a three-dimensional space around another quaternion (key). On the receiver’s side, a small amount of unnoticeable by human eye errors occurs in the decrypted images. These errors are used as a user’s digital fingerprint for the purpose of traitor tracing in case of copyright violation. A computer-based simulation was performed to scrutinize the potential presented quaternion encryption scheme for the implementation of digital fingerprinting. The obtained results are shown at the end of this paper.

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HANLON, B. E., e G. C. JOSHI. "SPONTANEOUS CP VIOLATION FROM A QUATERNIONIC KALUZA–KLEIN THEORY". International Journal of Modern Physics A 08, n.º 19 (30 de julho de 1993): 3263–83. http://dx.doi.org/10.1142/s0217751x93001314.

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Motivated by the isomorphism between the universal covering group of the six-dimensional Lorentz group and the special linear group over the quaternions, a locally quaternionic covariant theory is postulated to exist in six space–time dimensions. Compactifying onto the space–time M4 ⊗ S2 a complex theory is retrieved on the four-dimensional Minkowski space with the essential quaternionic nature confined to S2. Quaternionic spinors are introduced and a dimensionally reduced theory recovered which exhibits a CP-violating effect via spontaneous symmetry breaking.

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KRIEG, ALOYS. "THETA SERIES OVER THE HURWITZ QUATERNIONS". International Journal of Number Theory 06, n.º 01 (fevereiro de 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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Yuan, Jianping, Xianghao Hou, Chong Sun e Yu Cheng. "Fault-tolerant pose and inertial parameters estimation of an uncooperative spacecraft based on dual vector quaternions". Proceedings of the Institution of Mechanical Engineers, Part G: Journal of Aerospace Engineering 233, n.º 4 (16 de janeiro de 2018): 1250–69. http://dx.doi.org/10.1177/0954410017751766.

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Estimating the parameters of an unknown free-floating tumbling spacecraft is an essential task for the on-orbit servicing missions. This paper proposes a dual vector quaternion based fault-tolerant pose and inertial parameters estimation algorithm of an uncooperative space target using two formation flying small satellites. Firstly, by utilizing the dual vector quaternions to model the kinematics and dynamics of the system, not only the representation of the model is concise and compacted, but also the translational and rotational coupled effects are considered. By using this modeling technique along with the measurements from the on-board vision-based sensors, a dual vector quaternion based extended Kalman filter for each of the two small satellites is designed. Secondly, both of the estimations from each small satellite will be used as inputs of the fault-tolerant algorithm. This algorithm is based on the fault-tolerant federal extended Kalman filter strategy to overcome the estimation errors caused by the faulty measurements, the unknown space environment and the computing errors by setting the appropriate ratios of the two estimations from the first step dual vector quaternions extended Kalman filter. Together with the first and second steps, a novel fault-tolerant dual vector quaternions federal extended Kalman filter using two formation flying small satellites is proposed by this paper to estimate the pose and inertial parameters of a free-floating tumbling space target. By utilizing the estimation algorithm, a good prior knowledge of the unknown space target can be achieved. Finally, the proposed dual vector quaternion federal extended Kalman filter is validated by mathematical simulations to show its robust performances.

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Pu, Yasong, Yaoyao Shi, Xiaojun Lin, Yuan Hu e Zhishan Li. "C2-Continuous Orientation Planning for Robot End-Effector with B-Spline Curve Based on Logarithmic Quaternion". Mathematical Problems in Engineering 2020 (22 de julho de 2020): 1–16. http://dx.doi.org/10.1155/2020/2543824.

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Smooth orientation planning is beneficial for the working performance and service life of industrial robots, keeping robots from violent impacts and shocks caused by discontinuous orientation planning. Nevertheless, the popular used quaternion interpolations can hardly guarantee C2 continuity for multiorientation interpolation. Aiming at the problem, an efficient quaternion interpolation methodology based on logarithmic quaternion was proposed. Quaternions of more than two key orientations were expressed in the exponential forms of quaternion. These four-dimensional quaternions in space S3, when logarithms were taken for them, could be converted to three-dimensional points in space R3 so that B-spline interpolation could be applied freely to interpolate. The core formulas that B-spline interpolated points were mapped to quaternion were founded since B-spline interpolated point vectors were decomposed to the product of unitized forms and exponents were taken for them. The proposed methodology made B-spline curve applicable to quaternion interpolation through dimension reduction and the high-order continuity of the B-spline curve remained when B-spline interpolated points were mapped to quaternions. The function for reversely finding control points of B-spline curve with zero curvature at endpoints was derived, which helped interpolation curve become smoother and sleeker. The validity and rationality of the principle were verified by the study case. For comparison, the study case was also analyzed by the popular quaternion interpolations, Spherical Linear Interpolation (SLERP) and Spherical and Quadrangle (SQUAD). The comparison results demonstrated the proposed methodology had higher smoothness than SLERP and SQUAD and thus would provide better protection for robot end-effector from violent impacts led by unreasonable multiorientation interpolation. It should be noted that the proposed methodology can be extended to multiorientation quaternion interpolation with higher continuity than the second order.

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Ibrayev, А. Т. "METHOD FOR CONSTRUCTING THE COMMUTATIVE ALGEBRA OF QUATERNION AND OCTONION". PHYSICO-MATHEMATICAL SERIES 6, n.º 334 (15 de dezembro de 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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Dargys, Adolfas, e Artūras Acus. "Exponential and logarithm of multivector in low-dimensional (n = p + q < 3) Clifford algebras". Nonlinear Analysis: Modelling and Control 27, n.º 6 (19 de outubro de 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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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, n.º 1 (23 de outubro de 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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Hashim, Hafizzuddin Firdaus Bin, e Takehiko Ogawa. "Estimation of Forearm Motion Based on EMG Using Quaternion Neural Network". Journal of Advanced Computational Intelligence and Intelligent Informatics 26, n.º 3 (20 de maio de 2022): 269–78. http://dx.doi.org/10.20965/jaciii.2022.p0269.

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Quaternions are useful for representing data in three-dimensional space, and the quaternion neural network is effective for learning data in this context. On the other hand, estimating biological motion based on myopotential can be performed directly using electromyogram (EMG) signals as the computer interface. The trajectory of human forearm movement within the three-dimensional space can provide important information. In this study, the relationship between the myopotential of the upper arm muscles and the forearm motion was estimated and investigated using a quaternion neural network.

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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, n.º 2 (1 de dezembro de 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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LUKIERSKI, J., e A. NOWICKI. "QUATERNIONIC SIX-DIMENSIONAL (SUPER)TWISTOR FORMALISM AND COMPOSITE (SUPER)SPACES". Modern Physics Letters A 06, n.º 03 (30 de janeiro de 1991): 189–97. http://dx.doi.org/10.1142/s0217732391000154.

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We extend by real quaternions the D=4 twistor and supertwistor formalism. The notion of quaternionic D=4 composite superspaces is considered. The construction of D=6 real composite space-time variables as well as D=6 real composite superspaces is shown.

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Ali, Sajid, Sara Shakil Qureshi e Syed Ali Hassan. "Quaternion Codes in MIMO System of Dual-Polarized Antennas". Applied Sciences 11, n.º 7 (1 de abril de 2021): 3131. http://dx.doi.org/10.3390/app11073131.

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The use of quaternion orthogonal designs (QODs) to describe point-to-point communication among dual-polarized antennas has the potential to provide higher rate orthogonal and quasi-orthogonal complex designs exploiting polarization diversity among space and time diversities. Furthermore, it is essential to have a space time block code (STBC) which offers a linear and decoupled decoder which quasi-orthogonal designs fail to attain. In this paper, we show how the realm of quaternions unexpectedly offers us a possible solution and codes obtained from quaternion designs mostly achieve both linear and decoupled decoders. This motivated us to perform an indispensable search for QODs such that the code rate is bounded below by 1/2 and does not sharply decrease as the number of transmit antennas increases. It is shown that three famous recursive techniques do not satisfy this criteria and their code rates decrease rather rapidly. Therefore, we propose another method of constructing quaternion designs suitable for any number of transmit antennas and verify that these attain linear and decoupled decoders with the system model based on quaternionic channel. It is shown that such designs outperform others in terms of transmit diversity, code rates and the optimality of the proposed decoder is validated through simulation results.

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LAMOTKIN, ALEXEY, NATALIA MISYURA e EVGENII MITYUSHOV. "THE TRAJECTORIES CONSTRUCTION OF THE UNIVERSAL JOINT MOVEMENT IN THE CONFIGURATION SPACE IN ℝ3". Computational Nanotechnology 10, n.º 1 (30 de março de 2023): 60–66. http://dx.doi.org/10.33693/2313-223x-2023-10-1-60-66.

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In this paper, the study of the movement of the universal joint using the quaternion formalism was carried out, the law of movement of the cross of the universal joint was established with a known law of rotation of the drive shaft. A method of visual interpretation of the law of motion of the crosspiece is proposed, using the mapping of unit quaternions into a three-dimensional ball with radius 2π.

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Valverde, Alfredo, e Panagiotis Tsiotras. "Spacecraft Robot Kinematics Using Dual Quaternions". Robotics 7, n.º 4 (12 de outubro de 2018): 64. http://dx.doi.org/10.3390/robotics7040064.

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In recent years, there has been a growing interest in servicing orbiting satellites. In most cases, in-orbit servicing relies on the use of spacecraft-mounted robotic manipulators to carry out complicated mission objectives. Dual quaternions, a mathematical tool to conveniently represent pose, has recently been adopted within the space industry to tackle complex control problems during the stages of proximity operations and rendezvous, as well as for the dynamic modeling of robotic arms mounted on a spacecraft. The objective of this paper is to bridge the gap in the use of dual quaternions that exists between the fields of spacecraft control and fixed-base robotic manipulation. In particular, we will cast commonly used tools in the field of robotics as dual quaternion expressions, such as the Denavit-Hartenberg parameterization, or the product of exponentials formula. Additionally, we provide, via examples, a study of the kinematics of different serial manipulator configurations, building up to the case of a completely free-floating robotic system. We provide expressions for the dual velocities of the different types of joints that commonly arise in industrial robots, and we end by providing a collection of results that cast convex constraints commonly encountered by space robots during proximity operations in terms of dual quaternions.

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Lai, Wei Kit, Kiat Tat Qua e 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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Yefremov, Alexander P. "Physical theories in hypercomplex geometric description". International Journal of Geometric Methods in Modern Physics 11, n.º 06 (julho de 2014): 1450062. http://dx.doi.org/10.1142/s0219887814500625.

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Compact description is given of algebras of poly-numbers: quaternions, bi-quaternions, double (split-complex) and dual numbers. All units of these (and exceptional) algebras are shown to be represented by direct products of 2D vectors of a local basis defined on a fundamental surface. In this math medium a series of equalities identical or similar to known formulas of physical laws is discovered. In particular, a condition of the algebras' stability with respect to transformations of the 2D-basis turns out equivalent to the spinor (Schrödinger–Pauli and Hamilton–Jacobi) equations of mechanics. It is also demonstrated that isomorphism of SO(3, 1) and SO(3, ℂ) groups leads to formulation of a quaternion relativity theory predicting all effects of special relativity but simplifying solutions of relativistic problems in non-inertial frames. Finely it is shown that the Cauchy–Riemann type equations written for functions of quaternion variable repeat vacuum Maxwell equations of electrodynamics, while a quaternion space with non-metricity comprises main relations of Yang–Mills field theory.

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Stanfield, Kyl, e Ahmad Bani Younes. "Dual-Quaternion Analytic LQR Control Design for Spacecraft Proximity Operations". Sensors 21, n.º 11 (21 de maio de 2021): 3597. http://dx.doi.org/10.3390/s21113597.

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Proximity operations offer aggregate capability for a spacecraft operating in close proximity to another spacecraft, to perform on-orbit satellite servicing, or to a space object to perform debris removal. To utilize a spacecraft performing such advanced maneuvering operations and perceiving of the relative motion of a foreign spacecraft, these trajectories must be modeled accurately based on the coupled translational and rotational dynamics models. This paper presents work towards exploiting the dual-quaternion representations of spacecraft relative dynamics for proximity operations and developing a sub-optimal control law for efficient and robust maneuvers. A linearized model using dual-quaternions for the proximity operation was obtained, and its stability was verified using Monte Carlo simulations for the linear quadratic regulator solution. A sub-optimal control law using generalized higher order feedback gains in dual-quaternion form was developed based on small error approximations for the proximity operation and also verified through Monte Carlo simulations. Necessary information needed to understand the theory behind the use of the dual-quaternion is also overviewed within this paper, including the validity of using the dual-quaternions against their Cartesian or quaternion equivalents.

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GONGOPADHYAY, KRISHNENDU, ABHISHEK MUKHERJEE e SUJIT KUMAR SARDAR. "TEST MAP AND DISCRETENESS IN SL(2, ℍ)". Glasgow Mathematical Journal 61, n.º 03 (7 de agosto de 2018): 523–33. http://dx.doi.org/10.1017/s0017089518000332.

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AbstractLet ℍ be the division ring of real quaternions. Let SL(2, ℍ) be the group of 2 × 2 quaternionic matrices $A={\scriptsize{(\begin{array}{l@{\quad}l} a & b \\ c & d \end{array})}}$ with quaternionic determinant det A = |ad − aca−1b| = 1. This group acts by the orientation-preserving isometries of the five-dimensional real hyperbolic space. We obtain discreteness criteria for Zariski-dense subgroups of SL(2, ℍ).

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Li, Guozhi, Shuizhong Zou, Shuxue Din e Bin Qi. "Modified Hand–Eye Calibration Using Dual Quaternions". Applied Sciences 12, n.º 23 (6 de dezembro de 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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NEMES, F., e B. MIKÓCZI. "PERTURBED KEPLER PROBLEM IN GENERAL RELATIVITY WITH QUATERNIONS". International Journal of Modern Physics D 22, n.º 08 (21 de junho de 2013): 1350040. http://dx.doi.org/10.1142/s0218271813500405.

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The motion of binary star systems is re-examined in the presence of perturbations from the theory of general relativity. To handle the singularity of the Kepler problem, the equation of motion is regularized and linearized with quaternions. In this way first-order perturbation results are derived using the quaternion-based approach.

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Zanetti, Renato. "Rotations, Transformations, Left Quaternions, Right Quaternions?" Journal of the Astronautical Sciences 66, n.º 3 (2 de abril de 2019): 361–81. http://dx.doi.org/10.1007/s40295-018-00151-2.

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Sbitnev, Valeriy. "Relativistic Fermion and Boson Fields: Bose-Einstein Condensate as a Time Crystal". Symmetry 15, n.º 2 (18 de janeiro de 2023): 275. http://dx.doi.org/10.3390/sym15020275.

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In a basis of the space-time coordinate frame four quaternions discovered by Hamilton can be used. For subsequent reproduction of the coordinate frame these four quaternions are expanded to four 4 × 4 matrices with real-valued matrix coefficients −0 and 1. This group set is isomorphic to the SU(2) group. Such a matrix basis introduces extra six degrees of freedom of matter motion in space-time. There are three rotations about three space axes and three boosts along these axes. Next one declares the differential generating operators acting on the energy-momentum density tensor written in the above quaternion basis. The subsequent actions of this operator together with its transposed one on the above tensor lead to the emergence of the gravitomagnetic equations that are like the Maxwell equations. Wave equations extracted from the gravitomagnetic ones describe the propagation of energy density waves and their vortices through space. The Dirac equations and their reduction to two equations with real-valued functions, the quantum Hamilton-Jacobi equations and the continuity equations, are considered. The Klein-Gordon equations arising on the mass shell hints to the alternation of the paired fermion fields and boson ones. As an example, a Feynman diagram of an electron–positron time crystal is illustrated.

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Caliskan, Abdussamet, e Süleyman Şenyurt. "The dual spatial quaternionic expression of ruled surfaces". Thermal Science 23, Suppl. 1 (2019): 403–11. http://dx.doi.org/10.2298/tsci181125053c.

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In this paper, the ruled surface which corresponds to a curve on dual unit sphere is rederived with the help of dual spatial quaternions. We extend the term of dual expression of ruled surface using dual spatial quaternionic method. The correspondences in dual space of closed ruled surfaces are quaternionically expressed. As a consequence, the integral invariants of these surfaces and the relationships between these invariants are shown

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Martinez-Terán, Gerardo, Oswaldo Ureña-Ponce, Gerardo Soria-García, Susana Ortega-Cisneros e Eduardo Bayro-Corrochano. "Fast Study Quadric Interpolation in the Conformal Geometric Algebra Framework". Electronics 11, n.º 10 (11 de maio de 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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Kartiwa, Alit, Asep K. Supriatna, Endang Rusyaman e Jumat Sulaiman. "Review of Quaternion Differential Equations: Historical Development, Applications, and Future Direction". Axioms 12, n.º 5 (16 de maio de 2023): 483. http://dx.doi.org/10.3390/axioms12050483.

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Quaternion is a four-dimensional and an extension of the complex number system. It is often viewed from various fields, such as analysis, algebra, and geometry. Several applications of quaternions are related to an object’s rotation and motion in three-dimensional space in the form of a differential equation. In this paper, we do a systematic literature review on the development of quaternion differential equations. We utilize PRISMA (preferred reporting items for systematic review and meta-analyses) framework in the review process as well as content analysis. The expected result is a state-of-the-art and the gap of concepts or problems that still need to develop or answer. It was concluded that there are still some opportunities to develop a quaternion differential equation using a quaternion function domain.

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Mansouri, Imed, e Mohammed Ouali. "QUATERNION REPRESENTATION OF THE POWER MANIPULABILITY". Transactions of the Canadian Society for Mechanical Engineering 35, n.º 2 (junho de 2011): 309–36. http://dx.doi.org/10.1139/tcsme-2011-0018.

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In this paper, a quaternion formulation of the power manipulability is developed. Quantifying the mechanical transmissibility of a mechanism, this parameter was previously introduced as a new homogeneous performance index of robot manipulators; however, its evaluation requires complex manipulations, particularly those of quadri-vectors. Furthermore, the quadri-vector form fits exactly with the structure of quaternions. Hence, a quaternion based method may be useful for power performance indices evaluation. It will be shown that this method leads to a recursive algorithm that manipulates kinematics’ parameters as matrices defined over the real quaternion space. Therefore, a new mostly simplified procedure is elaborated, which is suitable to robots performance analysis related issues. The resultant algorithm is largely simpler and more efficient to evaluate the power based performance indices.

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Li, Gui San. "Quaternions and Vector Mapping in Three-Dimensional Space". Applied Mechanics and Materials 235 (novembro de 2012): 101–6. http://dx.doi.org/10.4028/www.scientific.net/amm.235.101.

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On the basis of analyzing the relations between vector operation and properties of multiplication operation of quaternions, the geometric representation and mapping of quaternions and vectors defined in three-dimensional space are established. The mathmatic operation of space vector, in addition to any space unit vector, rotating around three coordinate axises is carried based on the utilization of quaternions.The main purpose of this paper is to solve the issues of mathmatic tools introduced in spatial mechanism analysises and integrations by making use of quatnions.

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Purwar, Anurag, e Qiaode Jeffrey Ge. "On the Effect of Dual Weights in Computer Aided Design of Rational Motions". Journal of Mechanical Design 127, n.º 5 (27 de janeiro de 2005): 967–72. http://dx.doi.org/10.1115/1.1906263.

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In recent years, it has become well known that rational Bézier and B-spline curves in the space of dual quaternions correspond to rational Bézier and B-spline motions. However, the influence of weights of these dual quaternion curves on the resulting rational motions has been largely unexplored. In this paper, we present a thorough mathematical exposition on the influence of dual-number weights associated with dual quaternions for rational motion design. By deriving the explicit equations for the point trajectories of the resulting motion, we show that the effect of real weights on the resulting motion is similar to that of a rational Bézier curve and how the change in dual part of a dual-number weight affects the translational component of the motion. We also show that a rational Bézier motion can be reparameterized in a manner similar to a rational Bézier curve. Several examples are presented to illustrate the effects of the weights on rational motions.

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de Oliveira, Andre Schneider, Edson Roberto De Pieri e Ubirajara Franco Moreno. "A new method of applying differential kinematics through dual quaternions". Robotica 35, n.º 4 (24 de novembro de 2015): 907–21. http://dx.doi.org/10.1017/s0263574715000880.

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SUMMARYDifferential kinematics is a traditional approach to linearize the mapping between the workspace and joint space. However, a Jacobian matrix cannot be inverted directly in redundant systems or in configurations where kinematic singularities occur. This work presents a novel approach to the solution of differential kinematics through the use of dual quaternions. The main advantage of this approach is to reduce “drift” error in differential kinematics and to ignore the kinematic singularities. An analytical dual-quaternionic Jacobian is defined, which allows for the application of this approach in any robotic system.

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Çanakcı, Zeynep, O. Oğulcan Tuncer, İsmai̇l Gök e Yusuf Yaylı. "The construction of circular surfaces with quaternions". Asian-European Journal of Mathematics 12, n.º 07 (18 de novembro de 2019): 1950091. http://dx.doi.org/10.1142/s1793557119500918.

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In this paper, we obtain equations of circular surfaces by using unit quaternions and express these surfaces in terms of homothetic motions. Furthermore, we introduce new roller coaster surfaces constructed by the spherical indicatrices of a spatial curve in Euclidean [Formula: see text]-space. Then, we express parametric equations of roller coaster surfaces by means of unit quaternions and orthogonal matrices corresponding to these quaternions. Moreover, we present some illustrated examples.

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Chelnokov, Yu N. "Quaternion Methods and Regular Models of Celestial and Space Flight Mechanic: Using Euler (Rodrigues-Hamilton) Parameters to Describe Orbital (Trajectory) Motion. II: Perturbed Spatial Restricted Three-Body Problem". Известия Российской академии наук. Механика твердого тела, n.º 1 (1 de janeiro de 2023): 3–32. http://dx.doi.org/10.31857/s0572329922600293.

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The article considers the problem of regularizing the features of the classical equations of celestial mechanics and space flight mechanics (astrodynamics), which use variables that characterize the shape and size of the instantaneous orbit (trajectory) of the moving body under study, and Euler angles that describe the orientation of the used rotating (intermediate) coordinate system or the orientation of the instantaneous orbit, or the plane of the orbit of a moving body in an inertial coordinate system. Singularity-type features (division by zero) of these classical equations are generated by Euler angles and complicate the analytical and numerical study of orbital motion problems. These singularities are effectively eliminated by using the four-dimensional Euler (Rodrigues-Hamilton) parameters and Hamiltonian rotation quaternions. In this (second) part of the work, new regular quaternion models of celestial mechanics and astrodynamics are obtained that do not have the above features and are built within the framework of a perturbed spatial limited three-body problem (for example, the Earth, the Moon (or the Sun) and a spacecraft (or an asteroid)): equations of trajectory motion written in non-holonomic or orbital or ideal coordinate systems, for the description of the rotational motion of which the Euler (Rodrigues-Hamilton) parameters and quaternions of Hamilton rotations are used. New regular quaternion equations of the perturbed spatial restricted three-body problem are also obtained, constructed using two-dimensional ideal rectangular Hansen coordinates, Euler parameters and quaternion variables, as well as using complex compositions of Hansen coordinates and Euler parameters (Cayley-Klein parameters). The advantage of the proposed orbital motion equations constructed using the Euler parameters over the equations constructed using the Euler angles is due to the well-known advantages of the quaternion kinematic equations in the Euler parameters included in the proposed equations over the kinematic equations in the Euler angles included in the classical equations.

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Dahm, Rolf. "Spin-Flavour Symmetry and Contractions Towards Classical Space-Time Symmetry". International Journal of Modern Physics A 12, n.º 01 (10 de janeiro de 1997): 147–52. http://dx.doi.org/10.1142/s0217751x97000190.

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A classification scheme of hadrons is proposed on the basis of the division algebra H of quaternions and an appropriate geometry. This scheme suggests strongly to understand flavour symmetry in another manner than from standard symmetry schemes. In our approach, we do not start from "exact" symmetry groups like SU(2) × SU(2) chiral symmetry and impose various symmetry breaking mechanisms which collide with theorems wellknown from quantum field theory. On the contrary, the approximate symmetry properties of the hadron spectrum at low energies, usually classified by "appropriately" broken compact flavour groups, emerge very naturally as a low energy reduction of the noncompact (dynamical) symmetry group Sl(2,H). This quaternionic approach not only avoids most of the wellknown conceptual problems of Chiral Dynamics but it also allows for a general treatment of relativistic flavour symmetries as well as it yields a direct connection towards classical relativistic symmetry.

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RAJA, M., Ugur GUVEN e Kartikay SINGH. "Design and Implementation of a 3-Dimensional Attitudes Estimator Device using Low Cost Accelerometer & Gyroscope with Microcontroller IDE". INCAS BULLETIN 12, n.º 2 (5 de junho de 2020): 151–61. http://dx.doi.org/10.13111/2066-8201.2020.12.2.13.

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Navigation and guidance systems for most automobile as well as aerospace applications require a coupled chip setup known as Inertial Measurement Units (IMU) which, depending on the degree of freedoms, contains a Gyroscope (for maintaining orientation and angular velocity), Accelerometers (to determine acceleration in the respective direction) and a Magnetometer (to determine the respective magnetic fields). In the three-dimensional space, any required rotation analysis is limited to the coordinate systems and all subtended angles in either direction must be defined by a fixed axis to effectively estimate the stability and to define all the attitude estimates needed to compile different rotations and orientations. The Quaternions are mathematical notations used for defining rotations and orientation in three-dimensional space. The simplest terms Quaternions are impossible to visualize in a three-dimensional space; the first three terms will be identical to the coordinate system, but through Quaternions another vector quantity is added into the equations, which may in fact underline how we can account for all rotational quantities. The fundamental analysis of these components different applications for various fields is proposed.

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Cao, Zongsheng, Qianqian Xu, Zhiyong Yang, Xiaochun Cao e Qingming Huang. "Dual Quaternion Knowledge Graph Embeddings". Proceedings of the AAAI Conference on Artificial Intelligence 35, n.º 8 (18 de maio de 2021): 6894–902. http://dx.doi.org/10.1609/aaai.v35i8.16850.

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In this paper, we study the problem of learning representations of entities and relations in the knowledge graph for the link prediction task. Our idea is based on the observation that the vast majority of the related work only models the relation as a single geometric operation such as translation or rotation, which limits the representation power of the underlying models and makes it harder to match the complicated relations existed in real-world datasets. To embrace a richer set of relational information, we propose a new method called dual quaternion knowledge graph embedding (DualE), which introduces dual quaternions into knowledge graph embeddings. Specifically, a dual quaternion behaves like a “complex quaternion” with its real and imaginary part all being quaternary. The core of DualE lies a specific design of dual-quaternion-based multiplication, which universally models relations as the compositions of a series of translation and rotation operations. The major merits of DualE are three-fold:1) it is the first unified framework embracing both rotation based and translation-based models, 2) it expands the embedding space to the dual quaternion space with a more intuitive physical and geometric interpretation, 3) it satisfies the key patterns and the multiple relations pattern of relational representation learning. Experimental results on four real-world datasets demonstrate the effectiveness of our DualE method.

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Xiang, Chao, Cong Fu, Deng Cai e Xiaofei He. "Modeling Noncommutative Composition of Relations for Knowledge Graph Embedding". Electronics 12, n.º 6 (12 de março de 2023): 1348. http://dx.doi.org/10.3390/electronics12061348.

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Knowledge Graph Embedding (KGE) is a powerful way to express Knowledge Graphs (KGs), which can help machines learn patterns hidden in the KGs. Relation patterns are useful hidden patterns, and they usually assist machines to predict unseen facts. Many existing KGE approaches can model some common relation patterns like symmetry/antisymmetry, inversion, and commutative composition patterns. However, most of them are weak in modeling noncommutative composition patterns. It means these approaches can not distinguish a lot of composite relations like “father’s mother” and “mother’s father”. In this work, we propose a new KGE method called QuatRotatScalE (QRSE) to overcome this weakness, since it utilizes rotation and scaling transformations of quaternions to design the relation embedding. Specifically, we embed the relations and entities into a quaternion vector space under the difference norm KGE framework. Since the multiplication of quaternions does not satisfy the commutative law, QRSE can model noncommutative composition patterns naturally. The experimental results on the synthetic dataset also support that QRSE has this ability. In addition, the experimental results on real-world datasets show that QRSE reaches state-of-the-art in link prediction problem.

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BUCHANAN, WALTER, JAGANNATHAN GOMATAM e BONNIE STEVES. "GENERALIZED MANDELBROT SETS FOR MEROMORPHIC COMPLEX AND QUATERNIONIC MAPS". International Journal of Bifurcation and Chaos 12, n.º 08 (agosto de 2002): 1755–77. http://dx.doi.org/10.1142/s0218127402005443.

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The concepts of the Mandelbrot set and the definition of the stability regions of cycles for rational maps require careful investigation. The standard definition of the Mandelbrot set for the map f : z → z2+ c (the set of c values for which the iteration of the critical point at 0 remains bounded) is inappropriate for meromorphic maps such as the inverse square map. The notion of cycle sets, introduced by Brooks and Matelski [1978] for the quadratic map and applied to meromorphic maps by Yin [1994], facilitates a precise definition of the Mandelbrot parameter space for these maps. Close scrutiny of the cycle sets of these maps reveals generic fractal structures, echoing many of the features of the Mandelbrot set. Computer representations confirm these features and allow the dynamical comparison with the Mandelbrot set. In the parameter space, a purely algebraic result locates the stability regions of the cycles as the zeros of characteristic polynomials. These maps are generalized to quaternions. The powerful theoretical support that exists for complex maps is not generally available for quaternions. However, it is possible to construct and analyze cycle sets for a class of quaternionic rational maps (QRM). Three-dimensional sections of the cycle sets of QRM are nontrivial extensions of the cycle sets of complex maps, while sharing many of their features.

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Stoppato, Caterina. "Regular Moebius transformations of the space of quaternions". Annals of Global Analysis and Geometry 39, n.º 4 (27 de novembro de 2010): 387–401. http://dx.doi.org/10.1007/s10455-010-9238-9.

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Kalauni, Pushpa, e J. C. A. Barata. "Role of division algebra in seven-dimensional gauge theory". Modern Physics Letters A 30, n.º 10 (25 de março de 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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