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Статті в журналах з теми "Hypercomplex number"

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Автор: Grafiati

Опубліковано: 30 травня 2022

Оновлено: 31 травня 2022

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1

Labunets, V. G., E. V. Kokh, and E. Ostheimer. "ALGEBRAIC MODELS AND METHODS OF COMPUTER IMAGE PROCESSING. PART 1. MULTIPLET MODELS OF MULTICHANNEL IMAGES." Computer Optics 42, no. 1 (March 30, 2018): 84–95. http://dx.doi.org/10.18287/2412-6179-2018-42-1-84-95.

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We present a new theoretical framework for multichannel image processing using commutative hypercomplex algebras. Hypercomplex algebras generalize the algebras of complex numbers. The main goal of the work is to show that hypercomplex algebras can be used to solve problems of multichannel (color, multicolor, and hyperspectral) image processing in a natural and effective manner. In this work, we suppose that the animal brain operates with hypercomplex numbers when processing multichannel retinal images. In our approach, each multichannel pixel is considered not as an K–D vector, but as an K–D hypercomplex number, where K is the number of different optical channels. The aim of this part is to present algebraic models of subjective perceptual color, multicolor and multichannel spaces.

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2

Pulver, Sandra. "Quaternions: The hypercomplex number system." Mathematical Gazette 92, no. 525 (November 2008): 431–36. http://dx.doi.org/10.1017/s0025557200183639.

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Are there solutions of the equation x2 + 1 = 0 ? Carl Fredrich Gauss (1777–1855) conjectured that there was a solution and that it was the square root of - 1 . But since the squares of all real numbers, positive or negative, are positive, Gauss introduced a fanciful idea. His solution to this equation was , which he named i. He integrated i with the real numbers to form a set known as , the complex numbers, where each element in that set was of the form a + bi, where a, . Gauss illustrated this on a graph, the horizontal axis became the real axis and represented the real coefficient, while the vertical axis became the imaginary axis and represented the imaginary coefficient.

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3

SHU, JIAN-JUN, and YAJING LI. "HYPERCOMPLEX CROSS-CORRELATION OF DNA SEQUENCES." Journal of Biological Systems 18, no. 04 (December 2010): 711–25. http://dx.doi.org/10.1142/s0218339010003470.

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A hypercomplex representation of DNA is proposed to facilitate comparing DNA sequences with fuzzy composition. With the hypercomplex number representation, the conventional sequence analysis method, such as, dot matrix analysis, dynamic programming, and cross-correlation method have been extended and improved to align DNA sequences with fuzzy composition. The hypercomplex dot matrix analysis can provide more control over the degree of alignment desired. A new scoring system has been proposed to accommodate the hypercomplex number representation of DNA and integrated with dynamic programming alignment method. By using hypercomplex cross-correlation, the match and mismatch alignment information between two aligned DNA sequences are separately stored in the resultant real part and imaginary parts respectively. The mismatch alignment information is very useful to refine consensus sequence based motif scanning.

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4

Kalinovsky, Ya A., Yu E. Boyarinova, Ya V. Khitsko, and A. S. Sukalo. "Use of Methods for Generating Isomorphic Hypercomplex Number Systems to Increase the Efficiency of Multiplying Hypercomplex Numbers." Èlektronnoe modelirovanie 40, no. 5 (October 25, 2018): 27–40. http://dx.doi.org/10.15407/emodel.40.05.027.

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5

Silva Leite, F. "The geometry of hypercomplex matrices." Linear and Multilinear Algebra 34, no. 2 (March 1993): 123–32. http://dx.doi.org/10.1080/03081089308818216.

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6

Кalinovskiy, Ya А., and Yu E. Boiarinova. "Method for Representing an Exponent in a Fifth-dimensional Hypercomplex Number Systems Using a Hypercomplex Computing Software." Èlektronnoe modelirovanie 43, no. 6 (December 6, 2021): 3–18. http://dx.doi.org/10.15407/emodel.43.06.003.

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Анотація:

The structure of method for constructing a representation of an exponential function in hypercomplex number systems (HNS) by the method of solving an associated system of linear differential equations is considered. Brief information about the hypercomplex computing software (HCS) is given. With the use of HCS, the necessary cumbersome operations on symbolic expressions were performed when constructing the representation of the exponent in the fifthdimensional HNS. Fragments of programs in the environment of HCS and results of symbolic calculations are resulted.

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7

SUNDHEIM, PAUL. "A MULTIPLICATIVE DETERMINANT FOR 2m-DIMENSIONAL MATRICES." Journal of Algebra and Its Applications 13, no. 01 (August 20, 2013): 1350067. http://dx.doi.org/10.1142/s0219498813500679.

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A multiplication for a specific nested collection of multidimensional matrices is defined by association with a system of n = 2m-dimensional hypercomplex numbers. A totally symmetric and multiplicative determinant is then derived from the system which extends the Cayley hyperdeterminant to these higher dimensions. The determinant is related to the zero divisors of the system of hypercomplex numbers. Properties of the determinant are then discussed.

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8

Dotti, Isabel G., and Anna Fino. "Hypercomplex eight-dimensional nilpotent Lie groups." Journal of Pure and Applied Algebra 184, no. 1 (October 2003): 41–57. http://dx.doi.org/10.1016/s0022-4049(03)00104-x.

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9

Hauser, Jochem, and Walter Dröscher. "Gravity Beyond Einstein? Part II: Fundamental Physical Principles, Number Systems, Novel Groups, Dark Energy, and Dark Matter, MOND." Zeitschrift für Naturforschung A 74, no. 5 (May 27, 2019): 387–446. http://dx.doi.org/10.1515/zna-2018-0559.

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AbstractThis article attempts to explain the underlying physics of several recent experiments and astrophysical observations that have been mystifying the physics community for quite some time. So far, none of the advanced theories beyond the standard models of particle physics and cosmology have shown sufficient potential to resolve these mysteries. The reason for this failure may lie in the fact that these theories are based on the concept of extra space dimensions that appears to be in conflict with numerous experiments, in particular with recent Large Hadron Collider data. Therefore, the novel idea of extra number systems is introduced, replacing the idea of extra space dimensions. This approach is complemented by a set of fundamental physical principles that provide the constraints and guidelines for a modified physical formulation in agreement with known experimental reality. However, such a theory requires novel physical concepts in conjunction with novel symmetry groups. These groups give rise to additional types of matter, termed hypercomplex masses (which are responsible for the extreme hypercomplex gravitational fields, see below, and are also denoted as matter flavour), including, for instance, particles of negative mass, identified with dark matter. Furthermore, four-dimensional Minkowski spacetime, assumed to be a quasi de Sitter space $dS^{1,3}$dual spacetime, $DdS^{1,3}$, with imaginary time coordinate; that is, time is a complex quantity. The three spatial coordinates are shared by the two spacetimes. Dark matter is assumed to reside in $DdS^{1,3}$ and therefore is principally invisible. On the other hand, its gravitational interaction with ordinary matter (m ≥ 0) in spacetime $dS^{1,3}$ is directly perceptible. The novel group structure predicts the existence of a fourth particle family of negative masses; that is, besides the dark matter particle χ of mass $m_{\chi}\approx-80.77$ GeV/c2, there is the dark neutrino νχ of mass $m_{\nu_{\chi}}\approx-3.23$ eV/c2. Moreover, the hypercomplex group structure of gravity ($SU(2)\times SU(2)$) postulates three gravitational bosons for cosmological fields [resulting from Einstein’s theory of general relativity (GR)], the graviton $\nu_{G_{N}}$ with spin 2, the novel gravitophoton $\nu_{gp}$ with spin 1 (existence of weak gravitomagnetic fields of GR), and the quintessence particle νq with spin 0, which, when present, mediates an interaction between ordinary matter (m ≥ 0) and the ubiquitous scalar field of dark energy. In addition, the existence of extreme gravity fields (hypercomplex gravity) is postulated, based on the second group SU(2), and an interaction between electromagnetism and hypercomplex gravity is predicted, mediated by three additional hypercomplex-gravity bosons. Some long-standing problems of cosmology will be addressed; namely, the Big Bang scenario and the origin of dark energy and the nature of dark matter and their relation to the modified Newtonian dynamics hypothesis will be discussed.

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10

KIM, JI EUN, and KWANG HO SHON. "COSET OF A HYPERCOMPLEX NUMBER SYSTEM IN CLIFFORD ANALYSIS." Bulletin of the Korean Mathematical Society 52, no. 5 (September 30, 2015): 1721–28. http://dx.doi.org/10.4134/bkms.2015.52.5.1721.

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11

Wang, Xingyuan, and Tao Jin. "Hyperdimensional generalized M–J sets in hypercomplex number space." Nonlinear Dynamics 73, no. 1-2 (March 5, 2013): 843–52. http://dx.doi.org/10.1007/s11071-013-0836-5.

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12

Arana-Daniel, Nancy. "Complex and Hypercomplex-Valued Support Vector Machines: A Survey." Applied Sciences 9, no. 15 (July 31, 2019): 3090. http://dx.doi.org/10.3390/app9153090.

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In recent years, the field of complex, hypercomplex-valued and geometric Support Vector Machines (SVM) has undergone immense progress due to the compatibility of complex and hypercomplex number representations with analytic signals, as well as the power of description that geometric entities provide to object descriptors. Thus, several interesting applications can be developed using these types of data and algorithms, such as signal processing, pattern recognition, classification of electromagnetic signals, light, sonic/ultrasonic and quantum waves, chaos in the complex domain, phase and phase-sensitive signal processing and nonlinear filtering, frequency, time-frequency and spatiotemporal domain processing, quantum computation, robotics, control, time series prediction, and visual servoing, among others. This paper presents and discusses the importance, recent progress, prospective applications, and future directions of complex, hypercomplex-valued and geometric Support Vector Machines.

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13

BERKOVICH, Y., and A. SHENKMAN. "HYPERNION NUMBERS AND THEIR USE IN THE ANALYSIS OF NETWORKS DRIVEN BY NONSINUSOIDAL SOURCES." Journal of Circuits, Systems and Computers 13, no. 01 (February 2004): 65–76. http://dx.doi.org/10.1142/s0218126604001192.

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A method of using hypercomplex numbers for the analysis of linear electric circuits with nonsinusoidal voltages and currents has been proposed. Similar to the complex number method for circuits with sinusoidal voltages and currents, the proposed method reduces the analysis of nonsinusoidal circuits to the analysis of direct-current circuits. A special system of hypercomplex numbers, called hypernions, has been created in order to obtain a new efficient method for analyzing nonsinusoidal networks. This system is interesting in that it expands the concept of numbers and attaches meaning to equations and transformations involving discontinuous functions and makes it possible to describe various non-Euclidean spaces. It is shown that the proposed method of analysis of linear nonsinusoidal electric networks makes it possible to carry out numerical calculations for complex circuits by using standard software.

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14

Shpakivskyi, Vitalii. "Hypercomplex method for solving linear PDEs." Proceedings of the Institute of Applied Mathematics and Mechanics NAS of Ukraine 32 (December 28, 2018): 160–81. http://dx.doi.org/10.37069/1683-4720-2018-32-16.

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Algebraic-analytic approach to constructing solutions for given partial differential equations were investigated in many papers. In particular, in papers [1-14]. It involves solving two problems. Problem (P 1) is to describe all the sets of vectors \( e_1, e_2, \ldots, e_d \), which satisfy the characteristic equation (or specify the procedure by which they can be found). And the problem (P 2) is to describe all the components of monogenic (i.e., continuous and differentiable in sense Gateaux) functions. In particular, for the equation (4) we must describe the components of the function \( \Phi(\zeta) = \exp \zeta \). Note that in the papers [15, 16] a constructive description of all analytic functions with values is obtained in an arbitrary finite-dimensional commutative associative algebra over the field \(\mathbb{C}\). The Theorem 5.1 of the paper [17] states that it is enough to limit the study of monogenic functions in algebras with the basis of \( \{1, \eta_1, \eta_2, \ldots, \eta_{n-1} \} \), where \( \eta_1, \eta_2, \ldots, \eta_{n-1} \) are nilpotents. In addition, in [18] it is showed that in each algebra with a basis of the form \( \{1, \eta_1, \eta_2, \ldots, \eta_{n-1} \} \) the equation (3) has solutions. That is, the problems (P 1) and (P 2) are completely solved on the classes of commutative associative algebras with the basis \( \{1, \eta_1, \eta_2, \ldots, \eta_{n-1}\} \). It is worth noting that in a finite-dimensional algebra a decomposition of monogenic functions has a finite number of components, and therefore, it generates a finite number of solutions of a given partial differential equations. In this paper, we propose a procedure for constructing an infinite number of families of solutions of given linear differential equations with partial derivatives with constant coefficients. We use monogenic functions that are defined on some sequences of commutative associative algebras over the field of complex numbers. To achieve this goal, we first study the solutions of the so-called characteristic equation on a given sequence of algebras. Further, we investigate monogenic functions on the sequence of algebras and study their relation with solutions of partial deferential equations. The proposed method is used to construct solutions of some equations of mathematical physics. In particular, for the three-dimensional Laplace equation and the wave equation, for the equation of transverse oscillations of the elastic rod and the conjugate equation, a generalized biharmonic equation and the two-dimensional Helmholtz equation. We note that this method yields all analytic solutions of the two-dimensional Laplace equation and the two-dimensional biharmonic equation (Goursat formula).

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15

SHU, J. "Pairwise alignment of the DNA sequence using hypercomplex number representation." Bulletin of Mathematical Biology 66, no. 5 (September 2004): 1423–38. http://dx.doi.org/10.1016/j.bulm.2004.01.005.

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16

Kuba, G. "On the Distribution of Squares of Hypercomplex Integers." Journal of Number Theory 88, no. 2 (June 2001): 313–34. http://dx.doi.org/10.1006/jnth.2000.2644.

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17

KALINOVSKY, Ya A., and Yu E. BOYARINOVA. "The Metod for Research of Isomorphism of Indecomposable Hypercomplex Number Systems." Èlektronnoe modelirovanie 39, no. 3 (June 6, 2017): 61–76. http://dx.doi.org/10.15407/emodel.39.03.061.

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18

CARIOW, Aleksandr. "An unified approach for developing rationalized algorithms for hypercomplex number multiplication." PRZEGL�D ELEKTROTECHNICZNY 1, no. 2 (February 5, 2015): 38–41. http://dx.doi.org/10.15199/48.2015.02.09.

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19

Nag, S., J. A. Hillman, and B. Datta. "Characterisation theorems for compact hypercomplex manifolds." Journal of the Australian Mathematical Society. Series A. Pure Mathematics and Statistics 43, no. 2 (October 1987): 231–45. http://dx.doi.org/10.1017/s1446788700029372.

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AbstractWe have defined and studied some pseudogroups of local diffeomorphisms which generalise the complex analytic pseudogroups. A 4-dimensional (or 8-dimensional) manifold modelled on these ‘Further pseudogroups’ turns out to be a quaternionic (respectively octonionic) manifold.We characterise compact Further manifolds as being products of compact Riemann surfaces with appropriate dimensional spheres. It then transpires that a connected compact quaternionic (H) (respectively O) manifold X, minus a finite number of circles (its ‘real set’), is the orientation double covering of the product Y × P2, (respectively Y×P6), where Y is a connected surface equipped with a canonical conformal structure and Pn is n-dimensonal real projective space.A corollary is that the only simply-connected compact manifolds which can allow H (respectively O) structure are S4 and S2 × S2 (respectively S8 and S2×S6).Previous authors, for example Marchiafava and Salamon, have studied very closely-related classes of manifolds by differential geometric methods. Our techniques in this paper are function theoretic and topological.

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20

Bill, Jeremiah, Lance Champagne, Bruce Cox, and Trevor Bihl. "Meta-Heuristic Optimization Methods for Quaternion-Valued Neural Networks." Mathematics 9, no. 9 (April 23, 2021): 938. http://dx.doi.org/10.3390/math9090938.

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In recent years, real-valued neural networks have demonstrated promising, and often striking, results across a broad range of domains. This has driven a surge of applications utilizing high-dimensional datasets. While many techniques exist to alleviate issues of high-dimensionality, they all induce a cost in terms of network size or computational runtime. This work examines the use of quaternions, a form of hypercomplex numbers, in neural networks. The constructed networks demonstrate the ability of quaternions to encode high-dimensional data in an efficient neural network structure, showing that hypercomplex neural networks reduce the number of total trainable parameters compared to their real-valued equivalents. Finally, this work introduces a novel training algorithm using a meta-heuristic approach that bypasses the need for analytic quaternion loss or activation functions. This algorithm allows for a broader range of activation functions over current quaternion networks and presents a proof-of-concept for future work.

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21

Gu, Yingqiu. "Clifford Algebra and Hypercomplex Number as well as Their Applications in Physics." Journal of Applied Mathematics and Physics 10, no. 04 (2022): 1375–93. http://dx.doi.org/10.4236/jamp.2022.104097.

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22

Pukhtaievych, Roman, and Sergiy Plaksa. "On logarithmic residue of monogenic functions in a three-dimensional commutative algebra with one-dimensional radical." Analele Universitatii "Ovidius" Constanta - Seria Matematica 25, no. 3 (December 20, 2017): 167–82. http://dx.doi.org/10.1515/auom-2017-0043.

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Abstract We consider monogenic functions taking values in a three-dimensional commutative algebra A2 over the field of complex numbers with one- dimensional radical. We calculate the logarithmic residues of monogenic functions acting from a three-dimensional real subspace of A2 into A2. It is shown that the logarithmic residue depends not only on zeros and singular points of a function but also on points at which the function takes values in ideals of A2, and, in general case, is a hypercomplex number.

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23

Hauser, Jochem, and Walter Dröscher. "Gravity beyond Einstein? Part III: numbers and coupling constants, contradictory experiments, hypercomplex gravity like-fields, propellantless space propulsion." Zeitschrift für Naturforschung A 77, no. 1 (November 4, 2021): 13–86. http://dx.doi.org/10.1515/zna-2021-0147.

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Abstract This article, the last in a series of three articles, attempts to unravel the underlying physics of recent experiments regarding the contradictory properties of the neutron lifetime that has been a complete riddle for quite some time. So far, none of the advanced theories beyond the Standard Models (SMs) of particle physics and cosmology have shown sufficient potential to resolve this mystery. We also try to explain the blatant contradiction between the predictions of particle physics and experiments concerning the nature and properties of the (so far undetected) dark matter and dark energy particles. To this end the novel concepts of both negative and hypercomplex matter (giving rise to the concept of matter flavor) are introduced, replacing the field of real numbers by hypercomplex numbers. This extension of the number system in physics leads to both novel internal symmetries requiring new elementary particles – as outlined in Part I and II, and to novel types of matter. Hypercomplex numbers are employed in place of the widely accepted (but never observed) concept of extra space dimensions – and, hence, also to question the corresponding concept of supersymmetry. To corroborate this claim, we report on the latest experimental searches for novel and supersymmetric elementary particles by direct searches at the Large Hadron Collider (LHC) and other colliders as well as numerous other dedicated experiments that all have come up empty handed. The same holds true for the dark matter search at European Council for Nuclear Research (CERN) [CERN Courier Team, “Funky physics at KIT,” in CERN Courier, 2020, p. 11]. In addition, new experiments looking for dark or hidden photons (e.g., FUNK at Karlsruhe Institute of Technology, CAST at CERN, and ALPS at Desy, Hamburg) are discussed that all produced negative results for the existence of the hitherto unseen but nevertheless gravitationally noticeably dark matter. In view of this contradicting outcome, we suggest a four-dimensional Minkowski spacetime, assumed to be a quasi de Sitter space, dS 1,3, complemented by a dual spacetime, denoted by DdS 1,3, in which the dark matter particles that are supposed to be of negative mass reside. This space is endowed with an imaginary time coordinate, −it and an imaginary speed of light, ic. This means that time is considered a complex quantity, but energy m(ic)2 > 0. With this construction visible and dark matter both represent positive energies, and hence gravitation makes no distinction between these two types of matter. As dark matter is supposed to reside in dual space DdS 1,3, it is principally undetectable in our spacetime. That this is evident has been confirmed by numerous astrophysical observations. As the concept of matter flavor may possibly resolve the contradictory experimental results concerning the lifetime of the neutron [J. T. Wilson, “Space based measurement of the neutron lifetime using data from the neutron spectrometer on NASA’s messenger mission,” Phys. Rev. Res., vol. 2, p. 023216, 2020] this fact could be considered as a first experimental hint for the actual existence of hypercomplex matter. In canonical gravity the conversion of electromagnetic into gravity-like fields (as surmised by Faraday and Einstein) should be possible, but not in cosmological gravity (hence these attempts did not succeed), and thus these conversion fields are outside general relativity. In addition, the concept of hypercomplex mass in conjunction with magnetic monopoles emerging from spin ice materials is discussed that may provide the enabling technology for long sought propellantless space propulsion.

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24

Perez-Alvarado, Fernando, M. A. Hussein, and T. Becker. "Active contour and hypercomplex number for detecting Fusarium infestation and damages on barley grains." New Biotechnology 29 (September 2012): S22—S23. http://dx.doi.org/10.1016/j.nbt.2012.08.055.

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25

DEMİR, SÜLEYMAN, MURAT TANIŞLI, and TÜLAY TOLAN. "OCTONIC GRAVITATIONAL FIELD EQUATIONS." International Journal of Modern Physics A 28, no. 21 (August 20, 2013): 1350112. http://dx.doi.org/10.1142/s0217751x13501121.

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Generalized field equations of linear gravity are formulated on the basis of octons. When compared to the other eight-component noncommutative hypercomplex number systems, it is demonstrated that associative octons with scalar, pseudoscalar, pseudovector and vector values present a convenient and capable tool to describe the Maxwell–Proca-like field equations of gravitoelectromagnetism in a compact and simple way. Introducing massive graviton and gravitomagnetic monopole terms, the generalized gravitational wave equation and Klein–Gordon equation for linear gravity are also developed.

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26

Klipkov, S. I. "On a new approach to the construction of hypercomplex number systems of rank two over the field of complex numbers." Ukrainian Mathematical Journal 63, no. 1 (June 2011): 158–68. http://dx.doi.org/10.1007/s11253-011-0494-z.

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27

Et. al., Dr Indrajit Patra ,. "Shifts in the Foundation: The Continual Modification and Generalization of Axioms and the Search for the Mathematical Principles that Underlie our Reality." Turkish Journal of Computer and Mathematics Education (TURCOMAT) 12, no. 2 (April 11, 2021): 1095–106. http://dx.doi.org/10.17762/turcomat.v12i2.1126.

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The study shall seek to explore the deep, underlying correspondence between the mathematical world of pure numbers and our physical reality. The study begins by pointing out that while the familiar, one-dimensional real numbers quantify many aspects of our day-to-day reality, complex numbers provide the mathematical foundations of quantum mechanics and also describe the behavior of more complicated quantum networks and multi-party correlations, and quaternions underlie Einsteinian special theory of relativity, and then poses the question whether the octonions could play a similar role in constructing a grander theory of our universe. The study then points out that by increasing the level of abstraction and generalization of axiomatic assumptions, we could construct a more powerful number system based on octonions, the seditions, or even other hypercomplex numbers so that we may more accurately describe the universe in its totality.

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28

ISOKAWA, TEIJIRO, HARUHIKO NISHIMURA, NAOTAKE KAMIURA, and NOBUYUKI MATSUI. "ASSOCIATIVE MEMORY IN QUATERNIONIC HOPFIELD NEURAL NETWORK." International Journal of Neural Systems 18, no. 02 (April 2008): 135–45. http://dx.doi.org/10.1142/s0129065708001440.

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Associative memory networks based on quaternionic Hopfield neural network are investigated in this paper. These networks are composed of quaternionic neurons, and input, output, threshold, and connection weights are represented in quaternions, which is a class of hypercomplex number systems. The energy function of the network and the Hebbian rule for embedding patterns are introduced. The stable states and their basins are explored for the networks with three neurons and four neurons. It is clarified that there exist at most 16 stable states, called multiplet components, as the degenerated stored patterns, and each of these states has its basin in the quaternionic networks.

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29

Kalinovsky, J. A., Y. E. Boyarinova, and J. V. Khitsko. "Method of Selecting Hypercomplex Number Systems for Modeling Digital Reversing Filters of the 3rd and 4th Orders." Èlektronnoe modelirovanie 41, no. 4 (August 8, 2019): 03–18. http://dx.doi.org/10.15407/emodel.41.04.003.

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30

Ibrayev, А. Т. "METHOD FOR CONSTRUCTING THE COMMUTATIVE ALGEBRA OF QUATERNION AND OCTONION." PHYSICO-MATHEMATICAL SERIES 6, no. 334 (December 15, 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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31

Fernandez-Guasti, Manuel. "Powers of Elliptic Scator Numbers." Axioms 10, no. 4 (October 7, 2021): 250. http://dx.doi.org/10.3390/axioms10040250.

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Elliptic scator algebra is possible in 1+n dimensions, n∈N. It is isomorphic to complex algebra in 1 + 1 dimensions, when the real part and any one hypercomplex component are considered. It is endowed with two representations: an additive one, where the scator components are represented as a sum; and a polar representation, where the scator components are represented as products of exponentials. Within the scator framework, De Moivre’s formula is generalized to 1+n dimensions in the so called Victoria equation. This novel formula is then used to obtain compact expressions for the integer powers of scator elements. A scator in S1+n can be factored into a product of n scators that are geometrically represented as its projections onto n two dimensional planes. A geometric interpretation of scator multiplication in terms of rotations with respect to the scalar axis is expounded. The powers of scators, when the ratio of their director components is a rational number, lie on closed curves. For 1 + 2 dimensional scators, twisted curves in a three dimensional space are obtained. Collecting previous results, it is possible to evaluate the exponential of a scator element in 1 + 2 dimensions.

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32

Antonov, Alexander Alexandrovich. "Hypothesis of the Hidden Multiverse Explains Dark Matter and Dark Energy." Applied Physics Research 9, no. 2 (February 16, 2017): 30. http://dx.doi.org/10.5539/apr.v9n2p30.

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There are currently a large number of Multiverse hypotheses, which are, however, non-verifiable, i.e. they can be neither confirmed nor refuted experimentally even in the distant future. In contrast, the hypothesis of the hidden Multiverse considered in the article is verifiable and therefore has a right to be called a theory. The theory uses the principle of physical reality of imaginary numbers discovered 500 years ago, including complex and hypercomplex numbers, as fundamental and proved by the author theoretically and experimentally. This principle has allowed revealing a number of serious mistakes in the special theory of relativity. An adjusted version of the special theory of relativity has been proposed and the theory of the hidden Multiverse has been developed on its basis. The Multiverse has been referred to as hidden, because parallel universes it contains are mutually invisible. The nature of their invisibility is explained in the article. It is shown that dark matter and dark energy are other universes of the hidden Multiverse apart from ours. Analysis of data from WMAP and Planck spacecrafts has shown that the hidden Multiverse has quaternion structure comprising four pairs of universes and antiverses (i.e., four pairs of matter and antimatter).

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33

Gu, Ying-Qiu. "A Note on the Representation of Clifford Algebras." Journal of Geometry and Symmetry in Physics 62 (2021): 29–52. http://dx.doi.org/10.7546/jgsp-62-2021-29-52.

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In this note we construct explicit complex and real faithful matrix representations of the Clifford algebras $\Cl_{p,q}$. The representation is based on Pauli matrices and has an elegant structure similar to the fractal geometry. In the cases $p+q=4m$, the representation is unique in equivalent sense, and the $1+3$ dimensional space-time corresponds to the simplest and best case. Besides, the relation between the curvilinear coordinate frame and the local orthonormal basis in the curved space-time is discussed in detail, the covariant derivatives of the spinor and tensors are derived, and the connection of the orthogonal basis in tangent space is calculated. These results are helpful for both theoretical analysis and practical calculation. The basis matrices are the faithful representation of Clifford algebras in any $p+q$ dimensional Minkowski space-time or Riemann space, and the Clifford calculus converts the complicated relations in geometry and physics into simple and concise algebraic operations. Clifford numbers over any number field $\mathbb{F}$ expressed by this matrix basis form a well-defined $2^n$ dimensional hypercomplex number system. Therefore, we can expect that Clifford algebras will complete a large synthesis in science.

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34

Quadling, Douglas, I. L. Kantor, A. S. Solodovnikov, and A. Shenitzer. "Hypercomplex Numbers." Mathematical Gazette 74, no. 470 (December 1990): 399. http://dx.doi.org/10.2307/3618163.

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35

Schmitz, Georg J. "Quantitative mereology: An essay to align physics laws with a philosophical concept." Physics Essays 33, no. 4 (December 25, 2020): 479–88. http://dx.doi.org/10.4006/0836-1398-33.4.479.

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Mereology stands for the philosophical concept of parthood and is based on a sound set of fundamental axioms and relations. One of these axioms relates to the “existence of a universe as a thing having part all other things.” The present article formulates this logical expression first as an algebraic inequality and eventually as an algebraic equation reading in words: “The universe equals the sum of all things.” “All things” here are quantified by a “number of things.” Eventually, this algebraic equation is normalized leading to an expression: “The whole equals the sum of all fractions.” This introduces “1” or “100%” as a quantitative—numerical—value describing the “whole.” The resulting “basic equation” can then be subjected to a number of algebraic operations. Especially squaring this equation leads to correlation terms between the things implying that the whole is more than just the sum of its parts. Multiplying the basic equation (or its square) by a scalar allows for the comparison to and aligning with physics equations like the entropy equation, the ideal gas equation, an equation for the Lorentz-factor, conservation laws for mass and energy, the energy-mass equivalence, the Boltzmann statistics, and the energy levels in a Hydrogen atom. It further leads to a “contrast equation,” which may form the basis for the definition of a length and a time scale. Multiplying the basic equation with vectors, pseudovectors, pseudoscalars, and eventually hypercomplex numbers opens up the realm of possibilities to generate many further equations.

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36

Chaitin-Chatelin, F., and T. Meškauskas. "Computation with hypercomplex numbers." Nonlinear Analysis: Theory, Methods & Applications 47, no. 5 (August 2001): 3391–400. http://dx.doi.org/10.1016/s0362-546x(01)00454-0.

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37

Da¸sdemir, A. "On Hadamard Product of Hypercomplex Numbers." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 104, no. 4 (December 30, 2021): 68–73. http://dx.doi.org/10.31489/2021m4/68-73.

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Certain product rules take various forms in the set of hypercomplex numbers. In this paper, we introduce a new multiplication form of the hypercomplex numbers that will be called «the Hadamard product», inspired by the analogous product in the real matrix space, and investigate some algebraic properties of that, including the norm of inequality. In particular, we extend our new definition and its applications to the complex matrix theory.

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38

Kisil, Vladimir V. "Induced Representations and Hypercomplex Numbers." Advances in Applied Clifford Algebras 23, no. 2 (October 12, 2012): 417–40. http://dx.doi.org/10.1007/s00006-012-0373-1.

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39

Catoni, Francesco, Roberto Cannata, Enrico Nichelatti, and Paolo Zampetti. "Commutative hypercomplex numbers and functions of hypercomplex variable: a matrix study." Advances in Applied Clifford Algebras 15, no. 2 (October 2005): 183–212. http://dx.doi.org/10.1007/s00006-005-0011-2.

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40

Malek, Sami, Peter Ouillette, Whitney Wright, Amanda Dressel, Judy Karp, Paula Bockenstedt, Ammar Al-Zoubi, et al. "Comprehensive Ultra High Resolution Analysis of Copy Number and Allele Status Including Novel Microdeletions Spanning NF1 in Acute Myelogenous Leukemia." Blood 112, no. 11 (November 16, 2008): 3107. http://dx.doi.org/10.1182/blood.v112.11.3107.3107.

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Abstract Karyotypic abnormalities are of dominant importance in AML risk prognostication and therapy selection. A comprehensive description of subchromosomal genomic copy number changes and allele status together with gene mutation analysis and identification of chromosomal translocations is needed to fully harvest the prognostic and biological power of genomic changes in AML. We have analyzed DNA from 96 AML-derived pure blast populations (purified using column-based multi parameter negative selection followed by multi-gated FACS sorting) compared with paired buccal DNA using the Affymetrix 6.0 SNP platform. To support data analysis and display, we have developed the software tools PLUT and LOH tool version 2 and have refined dChipSNP. Data were supplemented with blast karyotypes and mutation status of Flt3, NPM1, p53, N-ras and K-ras. Results: AML cases carried between zero and thirty-four subchromosomal losses and gains. Approximately 23% of all cases had ≥3 subchromosomal lesions and approximately 50% of all cases had no such lesions. Of the 22 cases with complex and hypercomplex genomic changes by SNP profiling, ~50% had mutations in p53 exons 5–9 and ~50% were p53 wild-type by sequence analysis. Monoallelic deletion of p53 as part of various deletions at 17p was found in 7 of 96 (7%) cases. Importantly, 7/96=7% additional cases demonstrated UPD at 17p which spanned the p53 locus. Of all 14/96=15% of cases with LOH at 17p (with and without copy loss) 9/14=64% carried p53 mutations and 11/14=79% had complex karyotypes. Of the AML cases with NPM1 mutations (19 of 96 cases or 20%), 75% carried no detectable subchromosomal lesions, while 25% carried between 1 and 4 such lesions. Previously unidentified microdeletions resulting in monoallelic NF1 loss were identified on 17p in a total of 10 of 96 (10%) cases, thus suggesting activation of the Ras pathway independent of Ras mutations (all affected cases had wild-type N-ras and K-ras) in a substantial subset of AML cases. Mutational analysis of all coding exons of NF1 on the retained allele in the affected cases is ongoing. Additional recurrent microdeletions were identified on chromosomes 3p and 3q as well as 12p, the latter encompassing the genes ETV6 and p27, as previously reported. Finally, analysis of the prognostically and therapeutically important deletions 5q and 7q identified large deletions without recurrent microdeletions. Regions of minimal loss on 5q have been delineated.

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41

SUNDHEIM, PAUL. "AN INFINITE SYSTEM OF HYPERCOMPLEX NUMBERS." Journal of Algebra and Its Applications 11, no. 06 (November 14, 2012): 1250117. http://dx.doi.org/10.1142/s0219498812501174.

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A system of hypercomplex numbers is defined for each dimension that is a power of 2. In each of these dimensions, the numbers are associative and commutative but their advantages lie in the ease of multiplication, how closely their properties mirror the properties of the complex numbers in 2 dimensions as well as the location and properties of the zero divisors. In addition, the basis forms a group under multiplication providing a multiplication table that relates all of the roots of –1 and real and non-real roots of 1.

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42

Brewer, Sky. "Projective Cross-ratio on Hypercomplex Numbers." Advances in Applied Clifford Algebras 23, no. 1 (June 6, 2012): 1–14. http://dx.doi.org/10.1007/s00006-012-0335-7.

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43

Cheng, Daizhan, Zhengping Ji, Jun-e. Feng, Shihua Fu, and Jianli Zhao. "Perfect hypercomplex algebras: Semi-tensor product approach." Mathematical Modelling and Control 1, no. 4 (2021): 177–87. http://dx.doi.org/10.3934/mmc.2021017.

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<abstract><p>The set of associative and commutative hypercomplex numbers, called the perfect hypercomplex algebras (PHAs) is investigated. Necessary and sufficient conditions for an algebra to be a PHA via semi-tensor product (STP) of matrices are reviewed. The zero sets are defined for non-invertible hypercomplex numbers in a given PHA, and characteristic functions are proposed for calculating zero sets. Then PHA of various dimensions are considered. First, classification of $ 2 $-dimensional PHAs are investigated. Second, all the $ 3 $-dimensional PHAs are obtained and the corresponding zero sets are calculated. Finally, $ 4 $- and higher dimensional PHAs are also considered.</p></abstract>

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44

Kisil, Vladimir V. "Symmetry, Geometry and Quantization with Hypercomplex Numbers." Geometry, Integrability and Quantization 18 (2017): 11–76. http://dx.doi.org/10.7546/giq-18-2017-11-76.

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45

Hase-Liu, Matthew, and Adam Sheffer. "Sum–product phenomena for planar hypercomplex numbers." European Journal of Combinatorics 89 (October 2020): 103162. http://dx.doi.org/10.1016/j.ejc.2020.103162.

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46

Catoni, Francesco, Roberto Cannata, Vincenzo Catoni, and Paolo Zampetti. "N-dimensional geometries generated by hypercomplex numbers." Advances in Applied Clifford Algebras 15, no. 1 (March 2005): 1–25. http://dx.doi.org/10.1007/s00006-005-0001-4.

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47

Mikhailichenko, G. G., and R. M. Muradov. "Hypercomplex numbers in the theory of physical structures." Russian Mathematics 52, no. 10 (October 2008): 20–24. http://dx.doi.org/10.3103/s1066369x08100034.

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48

Vanwormhoudt, M. C. "Rings of hypercomplex numbers for NT Fourier transforms." Signal Processing 67, no. 2 (June 1998): 189–98. http://dx.doi.org/10.1016/s0165-1684(98)00036-x.

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49

Kolodezhnov, V. N. "Three-component hypercomplex numbers on two-dimensional plane." Automation and Remote Control 71, no. 6 (June 2010): 1275–82. http://dx.doi.org/10.1134/s0005117910060287.

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50

Hamilton, J. J. "Hypercomplex numbers and the description of spin states." Journal of Mathematical Physics 38, no. 10 (October 1997): 4914–28. http://dx.doi.org/10.1063/1.531926.

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