Relevant bibliographies by topics / Hypercomplex number

Academic literature on the topic 'Hypercomplex number'

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Published: 30 May 2022
Last updated: 31 May 2022

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Journal articles on the topic "Hypercomplex number"

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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 (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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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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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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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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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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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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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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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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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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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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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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Dissertations / Theses on the topic "Hypercomplex number"

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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 гіперкомплексних числових систем, що задовольняють критеріям побудови цифрового фільтра. Побудовано цифровий фільтр з коефіцієнтами у неканонічних гіперкомплексних числових системах та проведена його оптимізація за параметричною чутливістю.
The thesis is devoted to mathematical modeling of cryptography and signal problems using non-canonical hypercomplex numerical systems, which reduces the calculations amount during these models functioning and allows their optimization by individual characteristics. The modelling results of secret sharing scheme have shown that the use of non-canonical hypercomplex numerical systems starting from dimension 4 reduces the computation amount required in comparison with the use of canonical hypercomplex numerical systems. The methods for synthesis the noncanonical hypercomplex numerical system structures that satisfy the criteria for building a digital filter are developed. The digital filter is developed with the coefficients in noncanonical hypercomplex numerical systems and optimized by the parametric sensitivity.
Диссертация посвящена математическому моделированию задач криптографии и обработки сигналов с использованием неканонических гиперкомплексных числовых систем (ГЧС). Разработаны методы и способы представления и обработки данных в неканонических ГЧС, применение которых упрощает вид математических моделей, уменьшает количество вычислений при их функционировании и позволяет производить их оптимизацию по отдельным признакам. Анализ результатов работ последнего десятилетия по применению гиперкомплексных числовых систем в решении задач криптографии и обработки сигналов показал следущее: 1) применение канонических ГЧС к задаче разделения секрета повышает криптографическую стойкость, но вместе с тем увеличивает количество операций, требуемых для реализации такой задачи. Применение неканонических ГЧС дает возможность минимизировать количество вычислений за счет меньшей размерности системы; 2) синтез цифрового фильтра с использованием канонических ГЧС дает результаты по оптимизации его параметрической чувствительности, но поскольку выбор таких систем ограничен, неканонические ГЧС дают большие возможности по оптимизации чувствительности. В работе совершенствуются методы построения структур ГЧС заданной размерности, в том числе получения множества структур неканонических ГЧС, заданных в общем виде и неканонических гиперкомплексных числовых систем, изоморфных диагональной системе. Эти методы учитывают заданные ограничения представления данных в неканонических ГЧС для моделирования практических задач. Предлагается метод построения некоторых классов изоморфизма для неканонических ГЧС размерности 2. Изоморфные системы используются для минимизации вычислений при таком представления данных. В работе совершенствуются методы определения единичного элемента, нормы, сопряжения и делителей нуля для неканонических гиперкомплексных числовых систем; методы выполнения операций в таких системах. Впервые предлагается метод вычисления вычетов в неканонических ГЧС, который применяется в моделировании задачи разделения секрета и учитывает структурные особенности неканонических гиперкомплексных числовых систем. Предлагается модификация модулярной схемы разделения секрета, которая отличается от существующей представлением информации остатками в неканонических ГЧС по совокупности неканонических гиперкомплексных модулей. Реализована компьютерная модель задачи разделения секрета для неканонических ГЧС третьей и четвертой размерности в системе символьных вычислений MAPLE. Приведены результаты работы такой модели и сравнительные характеристики количества операций в части преобразования данных, непосредственно разделения секрета и восстановления данных. Анализ полученных результатов показал, что в целом, применение неканонических ГЧС к данной модели позволяет использовать меньшую размерность в зависимости от выбора констант при структурных единицах в таблице умножения системы, для обеспечения такой же криптостойкости, как и с использованием канонических ГЧС. Использование неканонической ГЧС размерности 3 для обеспечения такой же криптостойкости, как и при использовании канонической ГЧС размерности 4, не дает нужного эффекта для уменьшения количества вычислений, так как среднее количество операций увеличивается на 92%. Но уже при использовании неканонической ГЧС размерности 4 с 9-ю составными ячейками в таблице умножения с целыми коэффициентами из диапазона {-4,4}, для обеспечения такой же криптостойкости, как и при использовании канонической ГЧС размерности 6, количество требуемых вычислений уменьшается в среднем на 44%. Для успешного восстановления секрета, необходимо использовать числовые системы без делителей нуля и обладающих свойством мультипликативности нормы. В диссертационной работе впервые предлагается метод синтеза неканонических ГЧС, которые могут быть использованы при построении цифрового фильтра. Создана математическая модель рекурсивного цифрового фильтра с гиперкомплексными коэффициентами в полученных неканонических ГЧС третьей размерности. Впервые предлагается метод оптимизации суммарной параметрической чувствительности фильтра, построенного с использованием неканонических ГЧС который позволяет существенно уменьшить параметрическую чувствительность эквивалентного фильтра с вещественными коэффициентами (до ~50%) и существующих фильтров с гиперкомплексными коэффициентами (до ~40%). В работе описано расширение аналитически-программного инструментария в системе символьных вычислений MAPLE, который реализует предложенные модели и методы с учетом структурных особенностей неканонических ГЧС, а именно: определение основных свойств и выполнение операций над неканоническими гиперкомплексными числами; выполнение модулярных операций над неканоническими гиперкомплексными числами; построение структур неканонических ГЧС согласно заданным критериям, в том числе, критерию построения цифрового фильтра; реализация модели задачи разделения секрета в неканонических ГЧС и метода оптимизации параметрической чувствительности цифрового фильтра. Листинги кода приведены в приложениях.
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Books on the topic "Hypercomplex number"

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Kantor, I. L., and A. S. Solodovnikov. Hypercomplex Numbers. New York, NY: 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. Dordrecht: Kluwer Academic, 1998.

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

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Hypercomplex Numbers. Springer Verlag, 1989.

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Chen, Steven. Fractals and hypercomplex numbers. 1997.

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

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

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Nwokolo, Emmanuel, and I. L. Kantor. Hypercomplex Numbers: An Elementary Introduction to Algebras. Springer, 2011.

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Berezansky, Y. M., and A. A. Kalyuzhnyi. Harmonic Analysis in Hypercomplex Systems (Mathematics and Its Applications). Springer, 1998.

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

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Book chapters on the topic "Hypercomplex number"

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Schlote, Karl-Heinz. "Hermann Günther Grassmann and the Theory of Hypercomplex Number Systems." In Boston Studies in the Philosophy of Science, 165–73. Dordrecht: Springer Netherlands, 1996. http://dx.doi.org/10.1007/978-94-015-8753-2_14.

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Kalinovsky, Yakiv, Yuliya Boyarinova, Iana Khitsko, and Liubov Oleshchenko. "Digital Filters Optimization Modelling with Non-canonical Hypercomplex Number Systems." In Advances in Computer Science for Engineering and Education II, 448–58. Cham: Springer International Publishing, 2019. http://dx.doi.org/10.1007/978-3-030-16621-2_42.

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Kantor, I. L., and A. S. Solodovnikov. "Hypercomplex Numbers." In Hypercomplex Numbers, 35–39. New York, NY: 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, 382–403. New York, NY: 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, 415–37. New York, NY: Springer New York, 2010. http://dx.doi.org/10.1007/978-1-4419-6053-5_20.

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Kantor, I. L., and A. S. Solodovnikov. "Alternate Arithmetics on the Numbers a + bi." In Hypercomplex Numbers, 9–13. New York, NY: Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3650-4_2.

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Kantor, I. L., and A. S. Solodovnikov. "Conclusion." In Hypercomplex Numbers, 159–66. New York, NY: Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3650-4_21.

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Kantor, I. L., and A. S. Solodovnikov. "The Doubling Procedure. Cayley Numbers." In Hypercomplex Numbers, 41–51. New York, NY: Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3650-4_6.

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Kantor, I. L., and A. S. Solodovnikov. "Algebras." In Hypercomplex Numbers, 53–63. New York, NY: Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3650-4_7.

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Kantor, I. L., and A. S. Solodovnikov. "The N-Dimensional Vector Space An." In Hypercomplex Numbers, 69–74. New York, NY: Springer New York, 1989. http://dx.doi.org/10.1007/978-1-4612-3650-4_8.

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Conference papers on the topic "Hypercomplex number"

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Dimiev, Stancho, Peter Stoev, and Vladimir Todorov. "Cyclic hypercomplex number systems." In APPLICATIONS OF MATHEMATICS IN ENGINEERING AND ECONOMICS (AMEE '12): Proceedings of the 38th International Conference Applications of Mathematics in Engineering and Economics. AIP, 2012. http://dx.doi.org/10.1063/1.4766806.

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Ma, Wenting, Wei Wu, Zhaohua Xiong, and Xin Xu. "A hypercomplex number-based approach to PolSAR image matching." In the 3rd International Conference. New York, New York, USA: ACM Press, 2017. http://dx.doi.org/10.1145/3162957.3162959.

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Lee, Chung-Ching. "Applications of the 4D Geometric Algebra to Dimensional Mobility Criteria of Delassus-Parallelogram and Bennett Paradoxical Linkages." In ASME 2015 International Design Engineering Technical Conferences and Computers and Information in Engineering Conference. American Society of Mechanical Engineers, 2015. http://dx.doi.org/10.1115/detc2015-46667.

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Geometric algebra is also termed Clifford-Grassmann algebra or hypercomplex number. It allows studying space geometric problems in an easy and compact way. Transforming three-dimensional (3D) Euclidean geometric entities to actual elements of four-dimensional (4D) geometric algebra (abbreviated to g4) through a methodical approach of geometric algebra, one can describe motion displacements as even elements of g4. This article relies on the combined rotation and translation in g4 to establish the dimensional constraints of two non-exceptional overconstrained paradoxical linkages. Firstly, fundamentals of geometric algebra are recalled. Then, the single finite rotation and the composition of two successive finite rotations are introduced. After that, a general rigid-body motion in g4 is revealed for a possible application in exploring paradoxical chains using the geometric algebra. Finally, the metric or dimensional mobility criteria of Delassus-parallelogram four-screw and Bennett four-revolute paradoxical linkages are algebraically verified.
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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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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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6

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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Abstract:
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-valued networks on the CIFAR dataset.
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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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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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9

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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