Artykuły w czasopismach: „Hypercomplex numbers”

1

Quadling, Douglas, I. L. Kantor, A. S. Solodovnikov i A. Shenitzer. "Hypercomplex Numbers". Mathematical Gazette 74, nr 470 (grudzień 1990): 399. http://dx.doi.org/10.2307/3618163.

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Ibrayev, Alpamys T. "Method for Constructing a Commutative Algebra of Hypercomplex Numbers". Symmetry 15, nr 9 (26.08.2023): 1652. http://dx.doi.org/10.3390/sym15091652.

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

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Gu, Ying-Qiu. "Clifford Algebras, Hypercomplex Numbers and Nonlinear Equations in Physics". Geometry, Integrability and Quantization 25 (2023): 47–72. http://dx.doi.org/10.7546/giq-25-2023-47-72.

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Hypercomplex number systems are vector algebras with the definition of multiplication and division of vectors, satisfying the associativity and distributive law. In this paper, some new types of hypercomplex numbers and their fundamental properties are introduced, the Clifford algebra formalisms of hydrodynamics and gauge field equations are established, and some novel consistent conditions helpful to understand the properties of solutions to nonlinear physical equations are derived. The coordinate transformation and covariant derivatives of hypercomplex numbers are also discussed. The basis elements of the hypercomplex numbers have group-like properties and satisfy a structure equation $\A^2=n\A$. The hypercomplex number system integrates the advantages of algebra, geometry and analysis, and provides a unified, standard and elegant language and tool for scientific theories and engineering technology, so it is easy to learn and use. The description of mathematical, physical and engineering problems by hypercomplex numbers is of neat formalism, symmetric structure and standard derivation, which is especially suitable for the efficient processing of the higher dimensional complicated systems.

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4

Chaitin-Chatelin, F., i T. Meškauskas. "Computation with hypercomplex numbers". Nonlinear Analysis: Theory, Methods & Applications 47, nr 5 (sierpień 2001): 3391–400. http://dx.doi.org/10.1016/s0362-546x(01)00454-0.

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Gu, Ying-Qiu. "Hypercomplex Numbers and Roots of Algebraic Equation". Journal of Geometry and Symmetry in Physics 64 (2022): 9–22. http://dx.doi.org/10.7546/jgsp-64-2022-9-22.

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By means of hypercomplex numbers, in this paper we discuss algebraic equations and obtain some interesting relations. A structure equation $A^2=nA$ of a group is derived. The matrix representation of a group constitutes the basis elements of a hypercomplex number system. By a canonical real matrix representation of a cyclic group, we define the cyclic number system, which is exactly the solution space of the higher order algebraic equations, and thus can be used to solve the roots of algebraic equations. Hypercomplex numbers are linear algebras with definition of multiplication and division, satisfying the associativity and distributive law, which provide a unified, standard, and elegant language for many complex mathematical and physical objects. So, we have one more proof that the hypercomplex numbers are worthy of application in teaching and scientific research.

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Labunets, V. G., E. V. Kokh i E. Ostheimer. "ALGEBRAIC MODELS AND METHODS OF COMPUTER IMAGE PROCESSING. PART 1. MULTIPLET MODELS OF MULTICHANNEL IMAGES". Computer Optics 42, nr 1 (30.03.2018): 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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7

Da¸sdemir, A. "On Hadamard Product of Hypercomplex Numbers". BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 104, nr 4 (30.12.2021): 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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Alpay, Daniel, i Ilwoo Cho. "Operators induced by certain hypercomplex systems". Opuscula Mathematica 43, nr 3 (2023): 275–333. http://dx.doi.org/10.7494/opmath.2023.43.3.275.

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In this paper, we consider a family $\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}$ of rings of hypercomplex numbers, indexed by the real numbers, which contain both the quaternions and the split-quaternions. We consider natural Hilbert-space representations $\{(\mathbb{C}^{2},\pi_{t})\}_{t\in\mathbb{R}}$ of the hypercomplex system $\{ \mathbb{H}_{t}\}_{t\in\mathbb{R}}$, and study the realizations $\pi_{t}(h)$ of hypercomplex numbers $h \in \mathbb{H}_{t}$, as $(2\times 2)$-matrices acting on $\mathbb{C}^{2}$, for an arbitrarily fixed scale $t\in\mathbb{R}$. Algebraic, operator-theoretic, spectral-analytic, and free-probabilistic properties of them are considered.

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9

Kisil, Vladimir V. "Induced Representations and Hypercomplex Numbers". Advances in Applied Clifford Algebras 23, nr 2 (12.10.2012): 417–40. http://dx.doi.org/10.1007/s00006-012-0373-1.

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Catoni, Francesco, Roberto Cannata, Enrico Nichelatti i Paolo Zampetti. "Commutative hypercomplex numbers and functions of hypercomplex variable: a matrix study". Advances in Applied Clifford Algebras 15, nr 2 (październik 2005): 183–212. http://dx.doi.org/10.1007/s00006-005-0011-2.

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Cheng, Daizhan, Zhengping Ji, Jun-e. Feng, Shihua Fu i Jianli Zhao. "Perfect hypercomplex algebras: Semi-tensor product approach". Mathematical Modelling and Control 1, nr 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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12

SUNDHEIM, PAUL. "A MULTIPLICATIVE DETERMINANT FOR 2m-DIMENSIONAL MATRICES". Journal of Algebra and Its Applications 13, nr 01 (20.08.2013): 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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SUNDHEIM, PAUL. "AN INFINITE SYSTEM OF HYPERCOMPLEX NUMBERS". Journal of Algebra and Its Applications 11, nr 06 (14.11.2012): 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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14

Brewer, Sky. "Projective Cross-ratio on Hypercomplex Numbers". Advances in Applied Clifford Algebras 23, nr 1 (6.06.2012): 1–14. http://dx.doi.org/10.1007/s00006-012-0335-7.

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Pulver, Sandra. "Quaternions: The hypercomplex number system". Mathematical Gazette 92, nr 525 (listopad 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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Kalinovsky, Ya A., Yu E. Boyarinova, Ya V. Khitsko i A. S. Sukalo. "Use of Methods for Generating Isomorphic Hypercomplex Number Systems to Increase the Efficiency of Multiplying Hypercomplex Numbers". Èlektronnoe modelirovanie 40, nr 5 (25.10.2018): 27–40. http://dx.doi.org/10.15407/emodel.40.05.027.

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

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Catoni, Francesco, Roberto Cannata, Vincenzo Catoni i Paolo Zampetti. "N-dimensional geometries generated by hypercomplex numbers". Advances in Applied Clifford Algebras 15, nr 1 (marzec 2005): 1–25. http://dx.doi.org/10.1007/s00006-005-0001-4.

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BERKOVICH, Y., i A. SHENKMAN. "HYPERNION NUMBERS AND THEIR USE IN THE ANALYSIS OF NETWORKS DRIVEN BY NONSINUSOIDAL SOURCES". Journal of Circuits, Systems and Computers 13, nr 01 (luty 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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21

Cação, Isabel, Maria Irene Falcão i Helmuth Malonek. "Hypercomplex Polynomials, Vietoris’ Rational Numbers and a Related Integer Numbers Sequence". Complex Analysis and Operator Theory 11, nr 5 (25.02.2017): 1059–76. http://dx.doi.org/10.1007/s11785-017-0649-5.

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Ibrayev, A., A. Alkhan i A. Toktar. "PROBLEMS OF BUILDING AND APPLICATION OF HYPERCOMPLEX NUMBERS". EurasianUnionScientists 4, nr 2(83) (2.04.2021): 33–37. http://dx.doi.org/10.31618/esu.2413-9335.2021.4.83.1261.

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The article is dedicated to the problems of using multidimensional numbers for mathematical and computer modeling of complex physical processes and the design of knowledge-intensive devices, including digital image processing. The emphasis is on the issues of building the methods for processing three-dimensional signals. It is proposed to use three-dimensional variables presented in the form of hypercomplex numbers to formulate the three-dimensional Fourier transformation forms, which allows to analyze and process three-dimensional signals.

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L. Negulescu, Vlad. "The Vector Hypercomplex Numbers and the Matter Waves". International Journal of Applied Physics 9, nr 2 (30.08.2022): 12–16. http://dx.doi.org/10.14445/23500301/ijap-v9i2p103.

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Mikhailichenko, G. G., i R. M. Muradov. "Hypercomplex numbers in the theory of physical structures". Russian Mathematics 52, nr 10 (październik 2008): 20–24. http://dx.doi.org/10.3103/s1066369x08100034.

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Kolodezhnov, V. N. "Three-component hypercomplex numbers on two-dimensional plane". Automation and Remote Control 71, nr 6 (czerwiec 2010): 1275–82. http://dx.doi.org/10.1134/s0005117910060287.

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26

Hamilton, J. J. "Hypercomplex numbers and the description of spin states". Journal of Mathematical Physics 38, nr 10 (październik 1997): 4914–28. http://dx.doi.org/10.1063/1.531926.

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Vanwormhoudt, M. C. "Rings of hypercomplex numbers for NT Fourier transforms". Signal Processing 67, nr 2 (czerwiec 1998): 189–98. http://dx.doi.org/10.1016/s0165-1684(98)00036-x.

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Votiakova, Lesia, i Viktoriia Bodenchuk. "Matrix Algebra В as Euclidean Space". Mathematical and computer modelling. Series: Physical and mathematical sciences 24 (5.12.2023): 5–13. http://dx.doi.org/10.32626/2308-5878.2023-24.5-13.

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Representation of information by means of hypercomplex numerical systems is used in various problems of science and technology: in classical mechanics, solid body mechanics, electrodynamics, radio electronics, computer animation, and others [1]. Often a hypercomplex system (that is, a system whose elements are considered to be hypercomplex numbers) is understood as any finite-dimensional algebra over a field. An important place among such algebraic structures is occupied by matrix algebras. The impossibility of constructing algebras with division does not at all mean the impossibility of constructing algebras without division, but their properties are close to the first ones (use of defined division). Since each algebra of finite rank can be monomorphically immersed in some complete matrix algebra, this caused, so to speak, an inverse approach to the construction of new algebras. A certain subalgebra stands out from a complete matrix algebra, which is a matrix representation of an algebra of finite rank. It is the implementation of such an approach that makes it possible to endow elements of algebra of finite rank with matrix characteristics, in particular, a canonical representation of algebra elements is constructed through the spectral representation of a matrix, and the algebra itself is endowed with a topological structure through one of the matrix norms. At the same time, an additional condition is often imposed, that it be an algebra over the field of real or complex numbers. The article constructs a real algebra of finite rank, the elements of which are matrices of the second order with the same sum of rows and columns. We endowed it with a norm and a scalar product, demonstrating that it is a Euclidean space. This algebra is a matrix representation of the algebra of hypercomplex numbers, which we called binary in our research [4].

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Yefremov, Alexander P. "Physical theories in hypercomplex geometric description". International Journal of Geometric Methods in Modern Physics 11, nr 06 (lipiec 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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Borio, Daniele. "A Vector Representation of Multicomplex Numbers and Its Application to Radio Frequency Signals". Axioms 13, nr 5 (14.05.2024): 324. http://dx.doi.org/10.3390/axioms13050324.

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Hypercomplex numbers, which are multi-dimensional extensions of complex numbers, have been proven beneficial in the development of advanced signal processing algorithms, including multi-dimensional filter design, linear regression and classification. We focus on multicomplex numbers, sets of hypercomplex numbers with commutative products, and introduce a vector representation allowing one to isolate the hyperbolic real and imaginary parts of a multicomplex number. The orthogonal decomposition of a multicomplex number is also discussed, and its connection with Hadamard matrices is highlighted. Finally, a multicomplex polar representation is provided. These properties are used to extend the standard complex baseband signal representation to the multi-dimensional case. It is shown that a set of 2n Radio Frequency (RF) signals can be represented as the real part of a single multicomplex signal modulated by several frequencies. The signal RFs are related through a Hadamard matrix to the modulating frequencies adopted in the multicomplex baseband representation. Moreover, an orthogonal decomposition is provided for the obtained multicomplex baseband signal as a function of the complex baseband representations of the input RF signals.

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Pavlov, Stanislav, Dmitry Kozlov, Mikhail Bakulin, Aleksandr Zuev, Andrey Latyshev i Alexander Beliaev. "Generalization of Neural Networks on Second-Order Hypercomplex Numbers". Mathematics 11, nr 18 (19.09.2023): 3973. http://dx.doi.org/10.3390/math11183973.

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The vast majority of existing neural networks operate by rules set within the algebra of real numbers. However, as theoretical understanding of the fundamentals of neural networks and their practical applications grow stronger, new problems arise, which require going beyond such algebra. Various tasks come to light when the original data naturally have complex-valued formats. This situation is encouraging researchers to explore whether neural networks based on complex numbers can provide benefits over the ones limited to real numbers. Multiple recent works have been dedicated to developing the architecture and building blocks of complex-valued neural networks. In this paper, we generalize models by considering other types of hypercomplex numbers of the second order: dual and double numbers. We developed basic operators for these algebras, such as convolution, activation functions, and batch normalization, and rebuilt several real-valued networks to use them with these new algebras. We developed a general methodology for dual and double-valued gradient calculations based on Wirtinger derivatives for complex-valued functions. For classical computer vision (CIFAR-10, CIFAR-100, SVHN) and signal processing (G2Net, MusicNet) classification problems, our benchmarks show that the transition to the hypercomplex domain can be helpful in reaching higher values of metrics, compared to the original real-valued models.

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Zaripov, R. G. "Representations of a group of difference information functions in the extended parastatistics of non-extensive systems". Izvestiya vysshikh uchebnykh zavedenii. Fizika, nr 7 (2022): 9–15. http://dx.doi.org/10.17223/00213411/65/7/9.

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Algebraic and matrix representations of the group of information functions of the difference of nonextensive systems for three types of conformally generalized hypercomplex numbers are given. The corresponding geometries with metric functions are global Finsler geometries.

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Kyrov, Vladimir Aleksandrovich. "Hypercomplex numbers in some geometries of two sets. II". Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, nr 5 (2020): 39–54. http://dx.doi.org/10.26907/0021-3446-2020-5-39-54.

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Mikhailichenko, G. G., i V. A. Kyrov. "Hypercomplex numbers in some geometries of two sets. I". Russian Mathematics 61, nr 7 (20.05.2017): 15–24. http://dx.doi.org/10.3103/s1066369x17070039.

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Kyrov, V. A. "Hypercomplex Numbers in Some Geometries of Two Sets. II". Russian Mathematics 64, nr 5 (maj 2020): 31–48. http://dx.doi.org/10.3103/s1066369x20050047.

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Kamal, Aliaa, Moumen El-Melegy, Hassan El-Hawary i Khaled Hussein. "Face Recognition by Principal Component Regression using Hypercomplex Numbers". Assiut University Journal of Multidisciplinary Scientific Research 51, nr 3 (1.09.2022): 268–78. http://dx.doi.org/10.21608/aunj.2022.131391.1006.

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Delsuc, Marl A. "Spectral representation of 2D NMR spectra by hypercomplex numbers". Journal of Magnetic Resonance (1969) 77, nr 1 (marzec 1988): 119–24. http://dx.doi.org/10.1016/0022-2364(88)90036-4.

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Catoni, Francesco, Roberto Cannata, Vincenzo Catoni i Paolo Zampetti. "Two-dimensional hypercomplex numbers and related trigonometries and geometries". Advances in Applied Clifford Algebras 14, nr 1 (marzec 2004): 47–68. http://dx.doi.org/10.1007/s00006-004-0008-2.

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GÜRSES, NURTEN, i GÜLSÜM YELİZ ŞENTÜRK. "MATRIX THEORY OVER DGC NUMBERS". Journal of Science and Arts 23, nr 1 (30.03.2023): 209–28. http://dx.doi.org/10.46939/j.sci.arts-23.1-a17.

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Classical matrix theory for real, complex and hypercomplex numbers is a well-known concept. Is it possible to construct matrix theory over dual-generalized complex (DGC) matrices? The answer to this question is given in this paper. The paper is constructed as follows. Firstly, the fundamental concepts for DGC matrices are introduced and DGC special matrices are defined. Then, theoretical results related to eigenvalues/eigenvectors are obtained and universal similarity factorization equality (USFE) regarding to the dual fundamental matrix are presented. Also, spectral theorems for Hermitian and unitary matrices are introduced. Finally, due to the importance of unitary matrices, a method for finding a DGC unitary matrix is stated and examples for spectral theorem are given.

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GARANT–PELLETIER, V., i D. ROCHON. "ON A GENERALIZED FATOU–JULIA THEOREM IN MULTICOMPLEX SPACES". Fractals 17, nr 03 (wrzesień 2009): 241–55. http://dx.doi.org/10.1142/s0218348x09004326.

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In this article we introduce the hypercomplex 3D fractals generated from Multicomplex Dynamics. We generalize the well known Mandelbrot and filled-in Julia sets for the multicomplex numbers (i.e. bicomplex, tricomplex, etc.). In particular, we give a multicomplex version of the so-called Fatou-Julia theorem. More precisely, we present a complete topological characterization in ℝ2n of the multicomplex filled-in Julia set for a quadratic polynomial in multicomplex numbers of the form w2 + c. We also point out the symmetries between the principal 3D slices of the generalized Mandelbrot set for tricomplex numbers.

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41

Sangwine, S. J. "Fourier transforms of colour images using quaternion or hypercomplex, numbers". Electronics Letters 32, nr 21 (1996): 1979. http://dx.doi.org/10.1049/el:19961331.

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Hauser, Jochem, i 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, nr 1 (4.11.2021): 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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Martinez, C. A. P., André L. M. Martinez, Manoel Ferreira Borges Neto i Emerson Vitor Castelani. "SQUARE OF THE ERROR OCTONIONIC THEOREM AND HYPERCOMPLEX FOURIER SERIES". TEMA (São Carlos) 14, nr 3 (24.11.2013): 483. http://dx.doi.org/10.5540/tema.2013.014.03.0483.

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<div class="page" title="Page 1"><div class="layoutArea"><div class="column"><p><span>The focus of this paper is to address some classical results for a class of hypercomplex numbers. More specifically we present an extension of the Square of the Error Theorem and a Bessel inequality for octonions. </span></p></div></div></div>

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Bill, Jeremiah, Lance Champagne, Bruce Cox i Trevor Bihl. "Meta-Heuristic Optimization Methods for Quaternion-Valued Neural Networks". Mathematics 9, nr 9 (23.04.2021): 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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Yefremov, Alexander P. "Structure of Hypercomplex Units and Exotic Numbers as Sections of Bi-Quaternions". Advanced Science Letters 3, nr 4 (1.12.2010): 537–42. http://dx.doi.org/10.1166/asl.2010.1135.

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Kurdgelaidze, D. F. "Secondary quantization of the Dirac free field in hypercomplex system of numbers". Soviet Physics Journal 32, nr 10 (październik 1989): 840–45. http://dx.doi.org/10.1007/bf00898319.

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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, nr 2 (11.04.2021): 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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Pukhtaievych, Roman, i 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, nr 3 (20.12.2017): 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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Anwar, Aidah Nabilah, Amir Kamal Amir i Nurdin Hinding. "The Construction of Armendariz Ring using Formal Triangle Matrix Ring". Jurnal Matematika, Statistika dan Komputasi 19, nr 2 (5.01.2023): 304–16. http://dx.doi.org/10.20956/j.v19i2.23263.

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Trinion and Quaternion numbers are one of the hypercomplex numbers which is an extensions of the complex number. From Trinion and Quaternion numbers, a bimodule can be formed which is an ordered pair of Trinion and Quaternion. Furthermore, Trinion number, Quaternion number, and their bimodule can be formed into a Formal Triangle Matrix. The Formal Triangle Matrix is better known as the Upper Triangle Matrix. Since Trinion number, Quaternion number and their bimodule are rings, then the Formal Triangle Matrix can be called as the Formal Triangular Matrix Ring. The purpose of this study is to construct the Armendariz Ring using the Formal Triangular Matrix Ring. The obtained results will show that the Formal Triangular Matrix Rings are the -Skew Armendariz Ring and the -Skew -Armendariz Ring, where is a Ring Endomorphism and is -derivation.

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Fernandez-Guasti, Manuel. "Powers of Elliptic Scator Numbers". Axioms 10, nr 4 (7.10.2021): 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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