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Relevant bibliographies by topics / Hypercomplex numbers

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Academic literature on the topic 'Hypercomplex numbers'

Author: Grafiati

Published: 29 July 2024

Last updated: 25 July 2025

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Journal articles on the topic "Hypercomplex numbers"

1

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

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Votiakova, Lesia, and Lіudmila Nakonechna. "The Normed Algebra of Binary Numbers." Mathematical and computer modelling. Series: Physical and mathematical sciences 26 (December 26, 2024): 5–19. https://doi.org/10.32626/2308-5878.2024-26.5-19.

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The richness of the theory of functions of a complex variable, the effectiveness of its methods have always served as a stimulus and a source of ideas when constructing a theory of the function of a hypercomplex variable. It should be noted that hypercomplex number systems are an extension of the field of complex numbers. Modern hypercomplex studies can be divided into algebraic and analytical; the latter are often called hypercomplex analysis in the broad sense. Hypercomplex systems are an effective tool in mathematical modeling that allows representing complex multidimensional data and opera

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

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Until now, it was believed that, unlike real and complex numbers, the construction of a commutative algebra of quaternions or octonions with division over the field of real numbers is impossible in principle. No one questioned the existing theoretical assertion that quaternions, octonions, and other hypercomplex numbers cannot have the commutativity property. This article demonstrates the following for the first time: (1) the possibility of constructing a normed commutative algebra of quaternions and octonions with division over the field of real numbers; (2) the possibility of constructing a

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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 e

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Chaitin-Chatelin, F., and T. Meškauskas. "Computation with hypercomplex numbers." Nonlinear Analysis: Theory, Methods & Applications 47, no. 5 (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, s

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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 (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 h

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Da¸sdemir, A. "On Hadamard Product of Hypercomplex Numbers." BULLETIN OF THE KARAGANDA UNIVERSITY-MATHEMATICS 104, no. 4 (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, and Ilwoo Cho. "Operators induced by certain hypercomplex systems." Opuscula Mathematica 43, no. 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-theoreti

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10

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

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Dissertations / Theses on the topic "Hypercomplex numbers"

1

Bushman, Nathan. "Hypercomplex Numbers and Early Vector Systems: A History." The Ohio State University, 2020. http://rave.ohiolink.edu/etdc/view?acc_num=osu1585666516546138.

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Хіцко, Яна Володимирівна. "Математичне моделювання задач криптографії та обробки сигналів з використанням неканонічних гіперкомплексних числових систем". Thesis, НТУУ "КПІ", 2016. https://ela.kpi.ua/handle/123456789/15092.

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Дисертація присвячена математичному моделюванню задач криптографії та обробки сигналів з використанням неканонічних гіперкомплексних числових систем, застосування яких зменшує кількість обчислень при функціонуванні таких моделей та дозволяє оптимізувати їх за окремими характеристиками. Результати моделювання задачі розділення секрету показали, що застосування неканонічних гіперкомплексних числових систем, починаючи з вимірності 4, зменшує кількість потрібних обчислень у порівнянні із застосуванням канонічних гіперкомплексних числових систем. Розроблено методи побудови структур неканонічних г

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Books on the topic "Hypercomplex numbers"

1

Kantor, I. L., and A. S. Solodovnikov. Hypercomplex Numbers. Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3650-4.

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M, Berezanskiĭ I͡U. Harmonic analysis in hypercomplex systems. Kluwer Academic, 1998.

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Kantor, I. L. Hypercomplex numbers: An elementary introduction to algebras. Springer-Verlag, 1989.

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4

Francesco, Catoni, ed. The mathematics of Minkowski space-time: With an introduction to commutative hypercomplex numbers. Birkhäuser Verlag, 2008.

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5

Hypercomplex Numbers. 1989.

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Hypercomplex Numbers. Springer Verlag, 1989.

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7

Chen, Steven. Fractals and hypercomplex numbers. 1997.

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8

Kalyuzhnyi, A. A., and Yu M. Berezansky. Harmonic Analysis in Hypercomplex Systems. Springer, 2014.

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9

Berezansky, Yu M., and A. A. Kalyuzhnyi. Harmonic Analysis in Hypercomplex Systems. Springer, 2013.

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10

Berezansky, Yu M., and A. A. Kalyuzhnyi. Harmonic Analysis in Hypercomplex Systems. Yu M Berezansky, 2010.

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Book chapters on the topic "Hypercomplex numbers"

1

Kantor, I. L., and A. S. Solodovnikov. "Hypercomplex Numbers." In Hypercomplex Numbers. Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3650-4_5.

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Stillwell, John. "Hypercomplex Numbers." In Undergraduate Texts in Mathematics. Springer New York, 2002. http://dx.doi.org/10.1007/978-1-4684-9281-1_20.

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Stillwell, John. "Hypercomplex Numbers." In Undergraduate Texts in Mathematics. Springer New York, 2010. http://dx.doi.org/10.1007/978-1-4419-6053-5_20.

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4

Kantor, I. L., and A. S. Solodovnikov. "Complex Numbers." In Hypercomplex Numbers. Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3650-4_1.

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Kantor, I. L., and A. S. Solodovnikov. "Subspaces." In Hypercomplex Numbers. Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3650-4_10.

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Kantor, I. L., and A. S. Solodovnikov. "Lemma on Homogeneous Systems of Equations." In Hypercomplex Numbers. Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3650-4_11.

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Kantor, I. L., and A. S. Solodovnikov. "Scalar Products." In Hypercomplex Numbers. Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3650-4_12.

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Kantor, I. L., and A. S. Solodovnikov. "Orthonormal Basis. Orthogonal Transformation." In Hypercomplex Numbers. Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3650-4_13.

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9

Kantor, I. L., and A. S. Solodovnikov. "Isomorphic Algebras." In Hypercomplex Numbers. Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3650-4_14.

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Kantor, I. L., and A. S. Solodovnikov. "Subalgebras." In Hypercomplex Numbers. Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3650-4_15.

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Conference papers on the topic "Hypercomplex numbers"

1

Watanabe, Ricardo Augusto, Estevao Esmi Laureano, and Cibele Cristina Trinca Watanabe. "Fuzzy Octonion Numbers and Fuzzy Hypercomplex Numbers." In 2019 IEEE International Conference on Fuzzy Systems (FUZZ-IEEE). IEEE, 2019. http://dx.doi.org/10.1109/fuzz-ieee.2019.8858970.

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Gao, Yun'e, and Xuegang Yu. "Two Kinds of Hypercomplex Numbers." In 2010 International Conference on Computing, Control and Industrial Engineering. IEEE, 2010. http://dx.doi.org/10.1109/ccie.2010.222.

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Schulz, Dominik, Jochen Seitz, and Joao Paulo C. Lustosa da Costa. "Widely linear SIMO filtering for hypercomplex numbers." In 2011 IEEE Information Theory Workshop (ITW). IEEE, 2011. http://dx.doi.org/10.1109/itw.2011.6089486.

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4

Shalygin, K. A. "Using the hypercomplex numbers for instantaneous reactive power compensation." In 2012 IEEE 11th International Conference on Actual Problems of Electronics Instrument Engineering (APEIE). IEEE, 2012. http://dx.doi.org/10.1109/apeie.2012.6629096.

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Senna, Fernando Ribeiro de, and Marcos Eduardo Valle. "Tessarine and Quaternion-Valued Deep Neural Networks for Image Classification." In Encontro Nacional de Inteligência Artificial e Computacional. Sociedade Brasileira de Computação - SBC, 2021. http://dx.doi.org/10.5753/eniac.2021.18266.

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Many image processing and analysis tasks are performed with deep neural networks. Although the vast majority of advances have been made with real numbers, recent works have shown that complex and hypercomplex-valued networks may achieve better results. In this paper, we address quaternion-valued and introduce tessarine-valued deep neural networks, including tessarine-valued 2D convolutions. We also address initialization schemes and hypercomplex batch normalization. Finally, a tessarine-valued ResNet model with hypercomplex batch normalization outperformed the corresponding real and quaternion

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Kunisch, Jurgen. "On the use of hypercomplex numbers for antenna and propagation problems." In 2012 6th European Conference on Antennas and Propagation (EuCAP). IEEE, 2012. http://dx.doi.org/10.1109/eucap.2012.6206734.

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Monteiro, Claudio A., and Fernando M. De Paula Neto. "Diabetes Prediction Using Quantum Neurons with Preprocessing Based on Hypercomplex Numbers." In 2021 IEEE Symposium Series on Computational Intelligence (SSCI). IEEE, 2021. http://dx.doi.org/10.1109/ssci50451.2021.9660028.

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Volianskyi, Roman, Vitaliy Kuznetsov, Valeriy Kuznetsov, Oleksandr Ostapchuk, Viktor Artemchuk, and Nina Volianska. "Modeling of Dynamical Objects with Hypercomplex Numbers for Railway Non Traction Consumers with Renewable Energy Sources." In 2021 International Conference on Electrical, Communication, and Computer Engineering (ICECCE). IEEE, 2021. http://dx.doi.org/10.1109/icecce52056.2021.9514151.

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Nigmatullina, Vera T., Ivan I. Popov, Vadim A. Kozlov, and Anatolii N. Leukhin. "Photon echo as a method of optical processor construction for realization of computing operations above hypercomplex numbers." In SPIE Proceedings, edited by Vitaly V. Samartsev. SPIE, 2008. http://dx.doi.org/10.1117/12.801687.

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Zhang, Shuai, Lina Yao, Lucas Vinh Tran, Aston Zhang, and Yi Tay. "Quaternion Collaborative Filtering for Recommendation." In Twenty-Eighth International Joint Conference on Artificial Intelligence {IJCAI-19}. International Joint Conferences on Artificial Intelligence Organization, 2019. http://dx.doi.org/10.24963/ijcai.2019/599.

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This paper proposes Quaternion Collaborative Filtering (QCF), a novel representation learning method for recommendation. Our proposed QCF relies on and exploits computation with Quaternion algebra, benefiting from the expressiveness and rich representation learning capability of Hamilton products. Quaternion representations, based on hypercomplex numbers, enable rich inter-latent dependencies between imaginary components. This encourages intricate relations to be captured when learning user-item interactions, serving as a strong inductive bias as compared with the real-space inner product. All

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