33 results on '"Complex number"'
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2. ON THE (p, q) --NARAYANA n --DIMENSIONAL RECURRENCES.
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KULOĞLU, BAHAR and ÖZKAN, ENGİN
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COMPLEX numbers - Abstract
In this study, a different perspective was brought to Narayana sequences and one-, two-, three- and n --dimensional recurrence relations of these sequences were created. Then, some identities ranging from one to n --dimensions of these recurrences were created. [ABSTRACT FROM AUTHOR]
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- 2023
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3. FIXED POINTS IN BICOMPLEX VALUED S-METRIC SPACES WITH APPLICATIONS.
- Author
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Siva, G.
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LINEAR equations , *CONTRACTIONS (Topology) , *EXISTENCE theorems , *LINEAR systems , *COMPLEX numbers - Abstract
This article introduces the idea of bicomplex valued S-metric space and deduces some of its features. Additionally, for bicomplex valued S-metric spaces, some fixed point results of contraction maps are shown to meet various categories of rational inequalities. Moreover, these results generalize certain significant, well-known results. An example is provided to highlight our major result. Furthermore, a theorem guaranteeing the existence of the one and only solution to the linear system of equations was developed using our main result. [ABSTRACT FROM AUTHOR]
- Published
- 2023
4. Erros em esquemas de demonstração com números complexos.
- Author
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Damas Beites, Patrícia, Branco, Maria Luísa, and Costa, Cecília
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COMPLEX numbers ,SECONDARY education ,EDUCATION students ,PARALLELOGRAMS ,CONSTRUCTION materials ,READING comprehension - Abstract
Copyright of Educacao e Pesquisa is the property of Faculdade de Educacao da Universidade de Sao Paulo and its content may not be copied or emailed to multiple sites or posted to a listserv without the copyright holder's express written permission. However, users may print, download, or email articles for individual use. This abstract may be abridged. No warranty is given about the accuracy of the copy. Users should refer to the original published version of the material for the full abstract. (Copyright applies to all Abstracts.)
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- 2022
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5. Complex Numbers and Rhythmic Changes.
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Geethamma, V. G., Gopinath, Deepa P., and Daniel, Jacob K.
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SINE waves ,HARMONIC motion ,COMPLEX numbers ,TRIGONOMETRIC functions ,TRIGONOMETRY ,NUMBER concept ,DYNAMIC mechanical analysis - Abstract
The concept of complex numbers (CNs) is used in many disciplines. In many cases, students find it difficult to understand the logic behind CNs. Rotations, vibrations, and oscillations result in sine or cosine waves. Mathematical representation of rotation/vibration/oscillation is done in two ways—trigonometry and complex numbers. But the algebraic calculation is easier if CNs are used instead of trigonometric functions. The use of CNs as an effective representation of sinusoidal variations is discussed in this article. [ABSTRACT FROM AUTHOR]
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- 2021
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6. Low-Complexity High-Precision Method and Architecture for Computing the Logarithm of Complex Numbers.
- Author
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Chen, Hui, Yu, Zongguang, Zhang, Yonggang, Lu, Zhonghai, Fu, Yuxiang, and Li, Li
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COMPLEX numbers , *LOGARITHMS , *SYNTHETIC aperture radar , *SIMULATION software - Abstract
This paper proposes a low-complexity method and architecture to compute the logarithm of complex numbers based on coordinate rotation digital computer (CORDIC). Our method takes advantage of the vector mode of circular CORDIC and hyperbolic CORDIC, which only needs shift-add operations in its hardware implementation. Our architecture has lower design complexity and higher performance compared with conventional architectures. Through software simulation, we show that this method can achieve high precision for logarithm computation, reaching the relative error of 10−7. Finally, we design and implement an example circuit under TSMC 28nm CMOS technology. According to the synthesis report, our architecture has smaller area, lower power consumption, higher precision and wider operation range compared with the alternative architectures. [ABSTRACT FROM AUTHOR]
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- 2021
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7. Generalization of Dempster–Shafer theory: A complex mass function.
- Author
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Xiao, Fuyuan
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DEMPSTER-Shafer theory ,REAL numbers ,COMPLEX numbers ,GENERALIZATION ,ALGORITHMS - Abstract
Dempster–Shafer evidence theory has been widely used in various fields of applications, because of the flexibility and effectiveness in modeling uncertainties without prior information. However, the existing evidence theory is insufficient to consider the situations where it has no capability to express the fluctuations of data at a given phase of time during their execution, and the uncertainty and imprecision which are inevitably involved in the data occur concurrently with changes to the phase or periodicity of the data. In this paper, therefore, a generalized Dempster–Shafer evidence theory is proposed. To be specific, a mass function in the generalized Dempster–Shafer evidence theory is modeled by a complex number, called as a complex basic belief assignment, which has more powerful ability to express uncertain information. Based on that, a generalized Dempster's combination rule is exploited. In contrast to the classical Dempster's combination rule, the condition in terms of the conflict coefficient between the evidences is released in the generalized Dempster's combination rule. Hence, it is more general and applicable than the classical Dempster's combination rule. When the complex mass function is degenerated from complex numbers to real numbers, the generalized Dempster's combination rule degenerates to the classical evidence theory under the condition that the conflict coefficient between the evidences is less than 1. In a word, this generalized Dempster–Shafer evidence theory provides a promising way to model and handle more uncertain information. Thanks to this advantage, an algorithm for decision-making is devised based on the generalized Dempster–Shafer evidence theory. Finally, an application in a medical diagnosis illustrates the efficiency and practicability of the proposed algorithm. [ABSTRACT FROM AUTHOR]
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- 2020
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8. Implementation and Performance Evaluation of the Frequency-Domain-Based Bit Flipping Controller for Stabilizing the Single-Bit High-Order Interpolative Sigma Delta Modulators.
- Author
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Zhai, Huishan and Ling, Bingo Wing-Kuen
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ELECTRONIC modulators ,LOGIC circuits ,FREQUENCY discriminators ,INTEGER programming ,COMPLEX numbers - Abstract
This paper is an extension of the existing works on the frequency-domain-based bit flipping control strategy for stabilizing the single-bit high-order interpolative sigma delta modulator. In particular, this paper proposes the implementation and performs the performance evaluation of the control strategy. For the implementation, a frequency detector is used to detect the resonance frequencies of the input sequence of the sigma delta modulator. Then, a neural-network-based controller is used for finding the solution of the integer programming problem. Finally, the buffers and the combinational logic gates as well as an inverter are used for implementing the proposed control strategy. For the performance evaluation, the stability region in terms of the input dynamical range is evaluated. It is found that the control strategy can significantly increase the input dynamical range from 0.24 to 0.58. Besides, the control strategy can be applied to a wider class of the input signals compared to the clipping method. [ABSTRACT FROM AUTHOR]
- Published
- 2020
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9. Hypercomplex Widely Linear Estimation Through the Lens of Underpinning Geometry.
- Author
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Nitta, Tohru, Kobayashi, Masaki, and Mandic, Danilo P.
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QUATERNIONS , *COMPLEX numbers , *GEOMETRY , *COMPLEX variables , *COMPUTATIONAL complexity , *DEGREES of freedom , *MATHEMATICAL complexes , *DIVISION algebras - Abstract
We provide a rigorous account of the equivalence between the complex-valued widely linear estimation method and the quaternion involution widely linear estimation method with their vector-valued real linear estimation counterparts. This is achieved by an account of degrees of freedom and by providing matrix mappings between a complex variable and an isomorphic bivariate real vector, and a quaternion variable versus a quadri-variate real vector. Furthermore, we show that the parameters in the complex-valued linear estimation method, the complex-valued widely linear estimation method, the quaternion linear estimation method, the quaternion semi-widely linear estimation method, and the quaternion involution widely linear estimation method include distinct geometric structures imposed on complex numbers and quaternions, respectively, whereas the real-valued linear estimation methods do not exhibit any structure. This key difference explains, both in theoretical and practical terms, the advantage of estimation in division algebras (complex, quaternion) over their multivariate real vector counterparts. In addition, we discuss the computational complexities of the estimators of the hypercomplex widely linear estimation methods. [ABSTRACT FROM AUTHOR]
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- 2019
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10. Square-root-extended complex Kalman filter for estimation of symmetrical components in power system.
- Author
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Cui, Bowen
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KALMAN filtering ,ELECTRIC power ,ELECTRIC potential ,COMPLEX numbers ,ACCURACY - Abstract
The paper presents a square-root-extended complex Kalman filter (SRECKF) by decomposing covariance matrix with its square-root forms to improve stability of the filter for estimating complex number. αβ transformation is used to map three-phase instantaneous voltages in the abc phases into instantaneous voltages on the αβ axes, and a non-linear state equation and observation equation of the three-phase voltages are built by introducing a complex vector and defining state variables. Positive symmetrical component, negative symmetrical components, and frequency of the three-phase voltages are estimated using traditional extended complex Kalman filter (ECKF), the estimation results show that the method proposed here are superior to traditional extended complex Kalman filter on estimation accuracy and convergence rate. [ABSTRACT FROM AUTHOR]
- Published
- 2019
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11. Dialectical Multivalued Logic and Probabilistic Theory.
- Author
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Doménech, José Luis Usó, Nescolarde-Selva, Josué Antonio, and Segura-Abad, Lorena
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MANY-valued logic , *QUANTUM mechanics , *TRUTH functions (Mathematical logic) , *COMPLEX numbers , *PROBABILITY theory - Abstract
There are two probabilistic algebras: one for classical probability and the other for quantum mechanics. Naturally, it is the relation to the object that decides, as in the case of logic, which algebra is to be used. From a paraconsistent multivalued logic therefore, one can derive a probability theory, adding the correspondence between truth value and fortuity. [ABSTRACT FROM AUTHOR]
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- 2017
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12. Mathematical programming formulations for the alternating current optimal power flow problem
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Daniel Bienstock, Mauro Escobar, Claudio Gentile, Leo Liberti, Industrial Engineering and Operations Research Department (IEOR Dept), Columbia University [New York], Laboratoire d'informatique de l'École polytechnique [Palaiseau] (LIX), Centre National de la Recherche Scientifique (CNRS)-École polytechnique (X), Istituto di Analisi dei Sistemi ed Informatica 'Antonio Ruberti' [Roma] (IASI), Consiglio Nazionale delle Ricerche (CNR), and Centre National de la Recherche Scientifique (CNRS)
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Mathematical optimization ,Linear programming ,Computer science ,020209 energy ,0211 other engineering and technologies ,General Decision Sciences ,Smart grid ,02 engineering and technology ,Management Science and Operations Research ,Theoretical Computer Science ,Management Information Systems ,law.invention ,law ,FOS: Mathematics ,0202 electrical engineering, electronic engineering, information engineering ,complex numbers ,ACOPF ,Mathematics - Optimization and Control ,021103 operations research ,[INFO.INFO-RO]Computer Science [cs]/Operations Research [cs.RO] ,Electrical grid ,Power (physics) ,Nonlinear system ,Computational Theory and Mathematics ,Flow (mathematics) ,Optimization and Control (math.OC) ,90C90, 90C26 ,Minification ,Alternating current ,Complex number - Abstract
International audience; Power flow refers to the injection of power on the lines of an electrical grid, so that all the injections at the nodes form a consistent flow within the network. Optimality, in this setting, is usually intended as the minimization of the cost of generating power. Current can either be direct or alternating: while the former yields approximate linear programming formulations, the latter yields formulations of a much more interesting sort: namely, nonconvex nonlinear programs in complex numbers. In this technical survey, we derive formulation variants and relaxations of the alternating current optimal power flow problem.
- Published
- 2020
13. Multiplication-Related Classes of Complex Numbers
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Rafał Ziobro
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multiplication ,Applied Mathematics ,68v20 ,40-04 ,Computational Mathematics ,QA1-939 ,Order (group theory) ,complex numbers ,Multiplication ,order ,Arithmetic ,Complex number ,Mathematics - Abstract
Summary The use of registrations is useful in shortening Mizar proofs [1], [2], both in terms of formalization time and article space. The proposed system of classes for complex numbers aims to facilitate proofs involving basic arithmetical operations and order checking. It seems likely that the use of self-explanatory adjectives could also improve legibility of these proofs, which would be an important achievement [3]. Additionally, some potentially useful definitions, following those defined for real numbers, are introduced.
- Published
- 2020
14. Operações com números complexos: análise de erros cometidos por acadêmicos de Engenharia
- Author
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Isolda Gianni de Lima, Thaísa Jacintho Müller, and Cassiano Scott Puhl
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complex numbers ,learning gaps ,engineering students ,error analysis ,Computer science ,Learning object ,números complexos ,Errores ,Error analysis ,Mathematics education ,Operation ,lcsh:LC8-6691 ,Operaciones aritméticas ,lcsh:Special aspects of education ,lcsh:Mathematics ,_Otros temas de matemáticas superiores ,Subtraction ,Comprensión ,General Medicine ,Division (mathematics) ,lacunas de aprendizagem ,lcsh:QA1-939 ,Comprehension ,análise de erros ,Números complejos ,Multiplication ,Complex number ,acadêmicos de engenharia - Abstract
This paper describes a research that took place in a University in Rio Grande do Sul, with the aim at verifying levels of comprehension on Complex Numbers in Engineering scholars. The choice of that specific content is justified since complex numbers are the knowledge foundation for the analysis of alternating current circuits, practiced in Engineering courses. In a specific way, this investigation sought to identify and analyze errors committed by the scholars in basic mathematical operations, as addition, subtraction, multiplication and division with complex numbers. The research had a qualitative nature, being characterized as a multiple and holistic case study, whose instrument was a mixed questionnaire and whose analysis method used was Error Analysis. It is concluded that only a few scholars understand basic concepts and are able to operate with complex numbers. That was verified mainly and almost exclusively when considering addition and subtraction operations though. Even academics who had studied complex numbers in previous courses found it difficult to solve multiplication and division operations, and most have not solved these operations. Therefore, seeking to assist engineering professors and academics, a learning object is being created with the purpose of filling these learning gaps. That material aims to provide prior knowledge for the analysis of alternating current circuits and, thus, to help qualifying the processes of teaching and learning in Engineering., Este artigo apresenta um trabalho de pesquisa que se propôs a verificar o nível de compreensão sobre números complexos em acadêmicos de Engenharia de uma Instituição de Ensino Superior do Rio Grande do Sul. Justifica-se a escolha do conteúdo matemático uma vez que números complexos configuram um conhecimento base para a análise de circuitos elétricos em corrente alternada, praticada em cursos de Engenharia. Em específico, esta investigação buscou identificar e analisar os erros cometidos pelos acadêmicos em operações de matemática básica, como adição, subtração, multiplicação e divisão com números complexos. De cunho qualitativo, a pesquisa caracteriza-se como um estudo de caso múltiplo e holístico, cujos dados, produzidos por um questionário misto, foram analisados a partir de proposições da Análise de Erros. Conclui-se que são poucos os acadêmicos que compreendem conceitos e operações básicas com números complexos, tendo sido constatado certo domínio apenas nas operações de adição e subtração. Mesmo havendo, entre os acadêmicos, participantes que estudaram números complexos no Ensino Superior, a maioria deles apresentou dificuldades ou não resolveu as operações de multiplicação e divisão. Diante disso, em busca de auxiliar acadêmicos de Engenharia e seus professores, está sendo criado um objeto de aprendizagem como recurso de apoio para recuperar as lacunas de conhecimento em números complexos. Esse material tem enfoque na análise de circuitos elétricos em corrente alternada, colaborando para qualificar os processos de ensino e de aprendizagem em Engenharia.
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- 2020
15. Asset Pricing Model Based on Fractional Brownian Motion.
- Author
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Yan, Yu and Wang, Yiming
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BROWNIAN motion , *COMPLEX numbers , *REAL numbers , *MERTON Model , *DECISION making - Abstract
This paper introduces one unique price motion process with fractional Brownian motion. We introduce the imaginary number into the agent's subjective probability for the reason of convergence; further, the result similar to Ito Lemma is proved. As an application, this result is applied to Merton's dynamic asset pricing framework. We find that the four order moment of fractional Brownian motion is entered into the agent's decision-making. The decomposition of variance of economic indexes supports the possibility of the complex number in price movement. [ABSTRACT FROM AUTHOR]
- Published
- 2022
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16. A Hermitian Positive Definite neural network for micro-Doppler complex covariance processing
- Author
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Frédéric Barbaresco, Matthieu Cord, Olivier Schwander, Jean-Yves Schneider, Daniel A. Brooks, Thales LAS France, Machine Learning and Information Access (MLIA), LIP6, Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS)-Sorbonne Université (SU)-Centre National de la Recherche Scientifique (CNRS), Thales Air Systems, and Thales Group [France]
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020301 aerospace & aeronautics ,drone classification ,Artificial neural network ,Covariance matrix ,covariance matrices ,020206 networking & telecommunications ,02 engineering and technology ,Positive-definite matrix ,Covariance ,[INFO.INFO-NE]Computer Science [cs]/Neural and Evolutionary Computing [cs.NE] ,neural networks ,Hermitian matrix ,Synthetic data ,[INFO.INFO-AI]Computer Science [cs]/Artificial Intelligence [cs.AI] ,symbols.namesake ,0203 mechanical engineering ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,complex numbers ,[INFO]Computer Science [cs] ,Complex number ,Gaussian process ,Algorithm ,micro- Doppler ,Mathematics - Abstract
International audience; In its raw form, micro-Doppler radar data takes the form of a complex time-series, which can be seen as multiple realizations of a Gaussian process. As such, a complex covariance matrix constitutes a viable and synthetic representation of such data. In this paper, we introduce a neural network on Hermitian Positive Definite (HPD) matrices, that is complex-valued Symmetric Positive Definite (SPD) matrices, or complex covariance matrices. We validate this new architecture on synthetic data, comparing against previous similar methods.
- Published
- 2019
17. Iterants, Majorana Fermions and the Majorana-Dirac Equation.
- Author
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Kauffman, Louis H.
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DIRAC equation , *CLIFFORD algebras , *MATRICES (Mathematics) , *COMPLEX numbers , *EQUATIONS , *MAJORANA fermions , *SCHRODINGER equation - Abstract
This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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18. Feynman Machine: A Geometric Computational Machinery Based on the Path Integration
- Author
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Yoshiteru Ishida
- Subjects
Computer science ,Computation ,Probabilistic logic ,Propagator ,Spin foam ,quantum computing ,Algebra ,symbols.namesake ,Path integral formulation ,symbols ,General Earth and Planetary Sciences ,Feynman diagram ,complex numbers ,Functional integration ,Primality test ,Fractional quantum mechanics ,Complex number ,path integral ,General Environmental Science ,Quantum computer - Abstract
This paper proposes a framework for a new quantum computation based on the Feynman's path integral. The Feynman's path integral has been studied in quantum physics, however, the computational machinery may be used in universal computations. The light path design will be done in a light geometric automaton. As examples, we will present the primality test of an integer (for the deterministic case of light particle) and the stable marriage problem (for the probabilistic case of light wave).
- Published
- 2016
19. Understanding quaternions.
- Author
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Goldman, Ron
- Subjects
QUATERNIONS ,MULTIPLICATION ,VECTOR algebra ,COMPUTER graphics ,CRITICAL thinking ,ROTATIONAL motion ,DIMENSIONS ,COMPLEX numbers - Abstract
Abstract: Quaternion multiplication can be applied to rotate vectors in 3-dimensions. Therefore in Computer Graphics, quaternions are sometimes used in place of matrices to represent rotations in 3-dimensions. Yet while the formal algebra of quaternions is well-known in the Graphics community, the derivations of the formulas for this algebra and the geometric principles underlying this algebra are not well understood. The goals of this paper are: [i.] To provide a fresh, geometric interpretation of quaternions, appropriate for contemporary Computer Graphics; [ii.] To derive the formula for quaternion multiplication from first principles; [iii.] To present better ways to visualize quaternions, and the effect of quaternion multiplication on points and vectors in 3-dimensions based on insights from the algebra and geometry of multiplication in the complex plane; [iv.] To develop simple, intuitive proofs of the sandwiching formulas for rotation and reflection; [v.] To show how to apply sandwiching to compute perspective projections. In Part I of this paper, we investigate the algebra of quaternion multiplication and focus in particular on topics i and ii. In Part II we apply our insights from Part I to analyze the geometry of quaternion multiplication with special emphasis on topics iii, iv and v. [ABSTRACT FROM AUTHOR]
- Published
- 2011
- Full Text
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20. Identifiability and numerical algebraic geometry
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Jonathan D. Hauenstein, Daniel J. Bates, and Nicolette Meshkat
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Polynomial ,Complex Numbers ,Physiology ,Vector Spaces ,Algebraic geometry ,Pharmacokinetic Analysis ,Polynomials ,Compartment Models ,01 natural sciences ,Number Theory ,Medicine and Health Sciences ,Mathematics ,Numerical Analysis ,0303 health sciences ,Multidisciplinary ,Body Fluids ,Blood ,Ordinary differential equation ,Physical Sciences ,Medicine ,Anatomy ,Algorithms ,Research Article ,Computer and Information Sciences ,Science ,Models, Biological ,03 medical and health sciences ,Humans ,Applied mathematics ,Computer Simulation ,Differential algebra ,0101 mathematics ,Finite set ,030304 developmental biology ,Pharmacology ,010102 general mathematics ,Biology and Life Sciences ,Numerical Analysis, Computer-Assisted ,Computing Methods ,Interpolation ,Algebra ,Pharmacologic Analysis ,Linear Algebra ,Identifiability ,Algebraic Geometry ,Complex number ,Vector space - Abstract
A common problem when analyzing models, such as mathematical modeling of a biological process, is to determine if the unknown parameters of the model can be determined from given input-output data. Identifiable models are models such that the unknown parameters can be determined to have a finite number of values given input-output data. The total number of such values over the complex numbers is called the identifiability degree of the model. Unidentifiable models are models such that the unknown parameters can have an infinite number of values given input-output data. For unidentifiable models, a set of identifiable functions of the parameters are sought so that the model can be reparametrized in terms of these functions yielding an identifiable model. In this work, we use numerical algebraic geometry to determine if a model given by polynomial or rational ordinary differential equations is identifiable or unidentifiable. For identifiable models, we present a novel approach to compute the identifiability degree. For unidentifiable models, we present a novel numerical differential algebra technique aimed at computing a set of algebraically independent identifiable functions. Several examples are used to demonstrate the new techniques.
- Published
- 2019
21. Formulation of Strain Fatigue Criterion Based on Complex Numbers.
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Łagoda, Tadeusz, Głowacka, Karolina, Kurek, Marta, Skibicki, Dariusz, Maletta, Carmine, and Marsavina, Liviu
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COMPLEX numbers , *SHEAR strain , *PHYSICAL constants , *TORSION , *NUMBER systems - Abstract
In the case of many low-cycle multiaxial fatigue criteria, we encounter a mathematical problem of adding vectors of normal and shear strains. Typically, the problem of defining an equivalent strain is solved by weighting factors. Unfortunately, this ignores the fact that these vectors represent other physical quantities: the normal strain is a longitudinal strain, and the shear strain is a rotation angle. Therefore, the goal of the present work was to propose a method of combining different types of strains by adopting a system of complex numbers. The normal strain was defined as the real part and the shear strain was defined as the imaginary part. Using this approach, simple load states, such as pure bending and pure torsion, have been transformed into an expression for equivalent strain identical to the previously proposed criteria defined by Macha. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
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22. A Rabbit Hole between Topology and Geometry
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David G. Glynn
- Subjects
Article Subject ,Geometry ,Pappus ,Topology ,Noncommutative geometry ,planar graphs ,Planar graph ,symbols.namesake ,bundle theorem ,Bundle ,symbols ,complex numbers ,Pappus theorem ,Invariant (mathematics) ,Quaternion ,Commutative property ,Complex number ,Mathematics - Abstract
Copyright © 2013 David G. Glynn. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited., Topology and geometry should be very closely related mathematical subjects dealing with space. However, they deal with different aspects, the first with properties preserved under deformations, and the second with more linear or rigid aspects, properties invariant under translations, rotations, or projections. The present paper shows a way to go between them in an unexpected way that uses graphs on orientable surfaces, which already have widespread applications. In this way infinitely many geometrical properties are found, starting with the most basic such as the bundle and Pappus theorems. An interesting philosophical consequence is that the most general geometry over noncommutative skewfields such as Hamilton's quaternions corresponds to planar graphs, while graphs on surfaces of higher genus are related to geometry over commutative fields such as the real or complex numbers.
- Published
- 2013
23. On an assumption of geometric foundation of numbers
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Giuseppina Anatriello, Francesco Saverio Tortoriello, Giovanni Vincenzi, Anatriello, Giuseppina, Tortoriello, Francesco Saverio, and Vincenzi, Giovanni
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Field (mathematics) ,geometric constructions ,Euclidean geometries ,01 natural sciences ,Education ,Mathematics (miscellaneous) ,Euclidean geometry ,Calculus ,complex numbers ,0101 mathematics ,Axiom ,Pythagorean theorem ,Mathematics ,Structure (mathematical logic) ,Applied Mathematics ,010102 general mathematics ,05 social sciences ,050301 education ,Euclidean geometries, geometric constructions, plane geometry, complex numbers, Pythagorean theorem ,Algebra ,Geometric group theory ,Line (geometry) ,plane geometry ,0503 education ,Complex number - Abstract
n line with the latest positions of Gottlob Frege, this article puts forward the hypothesis that the cognitive bases of mathematics are geometric in nature. Starting from the geometry axioms of the Elements of Euclid, we introduce a geometric theory of proportions along the lines of the one introduced by Grassmann in Ausdehnungslehre in 1844. Assuming as axioms, the cognitive contents of the theorems of Pappus and Desargues, through their configurations, in an Euclidean plane a natural field structure can be identified that reveals the purely geometric nature of complex numbers. Reasoning based on figures is becoming a growing interdisciplinary field in logic, philosophy and cognitive sciences, and is also of considerable interest in the field of education, moreover, recently, it has been emphasized that the mutual assistance that geometry and complex numbers give is poorly pointed out in teaching and that a unitary vision of geometrical aspects and calculation can be clarifying..
- Published
- 2016
24. A complete classification of quintic space curves with rational rotation-minimizing frames
- Author
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Takis Sakkalis and Rida T. Farouki
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Discrete mathematics ,Surface (mathematics) ,Pure mathematics ,Algebra and Number Theory ,Rotation-minimizing frames ,Hopf map ,Tangent ,Orthonormal frame ,Complex numbers ,Characterization (mathematics) ,Polynomial identities ,Quintic function ,Computational Mathematics ,Pythagorean-hodograph curves ,Quaternions ,Algebraic number ,Hopf fibration ,Complex number ,Mathematics - Abstract
An adapted orthonormal frame (f"1,f"2,f"3) on a space curve r(t), where f"1=r^'/|r^'| is the curve tangent, is rotation-minimizing if its angular velocity satisfies @w@?f"1=0, i.e., the normal-plane vectors f"2,f"3 exhibit no instantaneous rotation about f"1. The simplest space curves with rational rotation-minimizing frames (RRMF curves) form a subset of the quintic spatial Pythagorean-hodograph (PH) curves, identified by certain non-linear constraints on the curve coefficients. Such curves are useful in motion planning, swept surface constructions, computer animation, robotics, and related fields. The condition that identifies the RRMF quintics as a subset of the spatial PH quintics requires a rational expression in four quadratic polynomials u(t),v(t),p(t),q(t) and their derivatives to be reducible to an analogous expression in just two polynomials a(t),b(t). This condition has been analyzed, thus far, in the case where a(t),b(t) are also quadratic, the corresponding solutions being called Class I RRMF quintics. The present study extends these prior results to provide a complete categorization of all possible PH quintic solutions to the RRMF condition. A family of Class II RRMF quintics is thereby newly identified, that correspond to the case where a(t),b(t) are linear. Modulo scaling/rotation transformations, Class II curves have five degrees of freedom, as with the Class I curves. Although Class II curves have rational RMFs that are only of degree 6-as compared to degree 8 for Class I curves-their algebraic characterization is more involved than for the latter. Computed examples are used to illustrate the construction and properties of this new class of RRMF quintics. A novel approach for generating RRMF quintics, based on the sum-of-four-squares decomposition of positive real polynomials, is also introduced and briefly discussed.
- Published
- 2012
- Full Text
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25. A Quantum-Based Similarity Method in Virtual Screening
- Author
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Naomie Salim, Ali Ahmed, Mubarak Himmat, Faisal Saeed, and Mohammed Mumtaz Al-Dabbagh
- Subjects
Models, Molecular ,Computer science ,quantum-based similarity ,Drug Evaluation, Preclinical ,Pharmaceutical Science ,Similarity measure ,Quantum spacetime ,Bioinformatics ,Article ,Analytical Chemistry ,lcsh:QD241-441 ,User-Computer Interface ,symbols.namesake ,lcsh:Organic chemistry ,Similarity (network science) ,Drug Discovery ,similarity searching approach ,complex numbers ,Physical and Theoretical Chemistry ,Representation (mathematics) ,Quantum ,Virtual screening ,Molecular Structure ,business.industry ,ligand-based ,Organic Chemistry ,quantum mechanics ,Hilbert space ,Pattern recognition ,virtual screening ,Chemistry (miscellaneous) ,symbols ,Quantum Theory ,Molecular Medicine ,Artificial intelligence ,business ,Complex number - Abstract
One of the most widely-used techniques for ligand-based virtual screening is similarity searching. This study adopted the concepts of quantum mechanics to present as state-of-the-art similarity method of molecules inspired from quantum theory. The representation of molecular compounds in mathematical quantum space plays a vital role in the development of quantum-based similarity approach. One of the key concepts of quantum theory is the use of complex numbers. Hence, this study proposed three various techniques to embed and to re-represent the molecular compounds to correspond with complex numbers format. The quantum-based similarity method that developed in this study depending on complex pure Hilbert space of molecules called Standard Quantum-Based (SQB). The recall of retrieved active molecules were at top 1% and top 5%, and significant test is used to evaluate our proposed methods. The MDL drug data report (MDDR), maximum unbiased validation (MUV) and Directory of Useful Decoys (DUD) data sets were used for experiments and were represented by 2D fingerprints. Simulated virtual screening experiment show that the effectiveness of SQB method was significantly increased due to the role of representational power of molecular compounds in complex numbers forms compared to Tanimoto benchmark similarity measure.
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- 2015
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26. On the concept image of complex numbers
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Edvard Nordlander and Maria Cortas Nordlander
- Subjects
Property (philosophy) ,Applied Mathematics ,media_common.quotation_subject ,Lärande ,Education ,Variety (cybernetics) ,Test (assessment) ,complex numbers ,concept image ,conceptions ,misconceptions ,Identification (information) ,Mathematics (miscellaneous) ,Engineering education ,Concept learning ,Perception ,ComputingMilieux_COMPUTERSANDEDUCATION ,Mathematics education ,Learning ,Complex number ,Mathematics ,media_common - Abstract
A study of how Swedish students understand the concept of complex numbers was performed. A questionnaire was issued reflecting the student view of own perception. Obtained answers show a variety of concept images describing how students adopt the concept of complex numbers. These concept images are classified into four categories in order to clarify the learning situation. Furthermore, this study also revealed a variety of misconceptions regarding this concept, and most of the misconceptions were also possible to refer to the classification system. In addition, results from an identification test show that students have difficulties discerning the basic property of complex numbers, i.e. that any number is a complex number.
- Published
- 2011
27. Erlangen Program at Large-1: Geometry of Invariants
- Author
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Vladimir V. Kisil
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Mathematics - Differential Geometry ,Split-complex number ,Pure mathematics ,semisimple groups ,Geometry ,parabolic ,split-complex numbers ,Representation theory ,analytic function theory ,Clifford algebras ,hyperbolic ,FOS: Mathematics ,complex numbers ,Algebraic number ,Complex Variables (math.CV) ,Representation Theory (math.RT) ,Mathematical Physics ,Mathematics ,double numbers ,Mathematics - Complex Variables ,Group (mathematics) ,30G35, 22E46, 30F45, 32F45 ,lcsh:Mathematics ,Clifford algebra ,Erlangen program ,lcsh:QA1-939 ,Algebra ,Differential Geometry (math.DG) ,Möbius transformations ,dual numbers ,Geometry and Topology ,elliptic ,Complex number ,Mathematics - Representation Theory ,Analysis ,Analytic function - Abstract
This paper presents geometrical foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of analytic function theories based on the representation theory of SL(2,R) group. We describe here geometries of corresponding domains. The principal role is played by Clifford algebras of matching types. In this paper we also generalise the Fillmore-Springer-Cnops construction which describes cycles as points in the extended space. This allows to consider many algebraic and geometric invariants of cycles within the Erlangen program approach. For an easy-reading introduction see arXiv:math/0607387. An outline of the whole approach is given in arXiv:1006.2115., Comment: AMS-LaTeX, 47 p, 80 PS graphics in 19 figures; v2: minor corrections v3: a substantial revision; v4 & v5: small improvements; v6: revised sections on lengths, infinitesimal cycles, parabolic Cayley transform; v7, v8 & v9: numerous minor improvements and updates; v10: the final version published in SIGMA; v11: the reference to Schwerdtfeger's book is added
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- 2010
28. Categorical properties of the complex numbers
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Jamie Vicary
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Discrete mathematics ,Pure mathematics ,Complex conjugate ,General Computer Science ,Categorical quantum mechanics ,Scalar (mathematics) ,Existence theorem ,Symmetric monoidal category ,Theoretical Computer Science ,category theory ,Quantum theory (mathematics) ,Quantum theory ,Mathematics::Category Theory ,complex numbers ,Category theory ,Complex number ,Categorical variable ,Computer Science(all) ,Mathematics - Abstract
Given the success of categorical approaches to quantum theory, it is interesting to consider why the complex numbers are special from a categorical perspective. We describe natural categorical conditions under which the scalars of a monoidal †-category gain many of the features of the complex numbers. Central to our approach are †-limits, certain types of limits which are compatible with the †-functor; we explore their properties and prove an existence theorem for them. Our main theorem is that in a nontrivial monoidal †-category with finite †-limits and simple tensor unit, and in which the self-adjoint scalars satisfy a completeness condition, the scalars are valued in the complex numbers, and scalar involution is exactly complex conjugation.
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- 2010
29. Information lossless full-rate full-diversity trace-orthogonal space-time codes
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Sergio Barbarossa and Antonio Fasano
- Subjects
Block code ,Discrete mathematics ,Coding gains ,Complex numbers ,Constellation (CO) ,Fountain code ,Full Rate ,Linear code ,Algorithm ,Complex number ,Coding gain ,Square (algebra) ,Online codes ,Mathematics - Abstract
Trace-orthogonality is an important property of linear space-time codes that has emerged relatively recently. In this work we propose a quite general procedure for the generation of such codes for any number of transmit antennas and uses of the channel. Simulations suggest that the use of rectangular encoding matrices, in place of square ones, could be beneficial to the coding gain. Moreover we are able to guarantee full-diversity of the proposed codes when information symbols are carved from the quadratic field Q(j). Since the components of any complex number can be approximated with arbitrary precision by rational numbers, our scheme could be used to guarantee full-diversity for any conceivable complex constellation.
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- 2006
30. Linear Reducts of the Complex Field
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James Loveys
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linearity ,Pure mathematics ,Complex field ,Logic ,Linearity ,Combinatorics ,03C45 ,finite covers ,strongly minimal sets ,complex numbers ,reducts ,03C65 ,Complex number ,Mathematics - Abstract
A reduct of a first-order structure is another structure on the same set with perhaps fewer definable predicates. We consider reducts of the complex field which are proper (not essentially the whole field) but nontrivial in a sense to be made precise in the paper. Our main result lists seven kinds of reducts. The list is complete in the sense that every reduct is a finite cover of one of these. We also investigate when two items on our list can be the same, in a couple of natural senses.
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- 2004
31. Pythagorean-hodograph curves in Euclidean spaces of dimension greater than 3
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Takis Sakkalis and Rida T. Farouki
- Subjects
Polynomial ,Octonions ,Applied Mathematics ,Mathematical analysis ,Dimension (graph theory) ,Hopf map ,Parameterization of n-tuples ,Characterization (mathematics) ,Complex numbers ,Combinatorics ,Computational Mathematics ,Number theory ,Euclidean geometry ,Pythagorean-hodograph curves ,Hopf fibration ,Quaternions ,Arc length ,Complex number ,Mathematics - Abstract
A polynomial Pythagorean-hodograph (PH) curve r(t)=(x"1(t),...,x"n(t)) in R^n is characterized by the property that its derivative components satisfy the Pythagorean condition x"1^'^2(t)+...+x"n^'^2(t)=@s^2(t) for some polynomial @s(t), ensuring that the arc length s(t)=@!@s(t)dt is simply a polynomial in the curve parameter t. PH curves have thus far been extensively studied in R^2 and R^3, by means of the complex-number and the quaternion or Hopf map representations, and the basic theory and algorithms for their practical construction and analysis are currently well-developed. However, the case of PH curves in R^n for n>3 remains largely unexplored, due to difficulties with the characterization of Pythagorean (n+1)-tuples when n>3. Invoking recent results from number theory, we characterize the structure of PH curves in dimensions n=5 and n=9, and investigate some of their properties.
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32. Distribution of points on the circle
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Øystein J. Rødseth
- Subjects
Combinatorics ,Unit circle ,Algebra and Number Theory ,Distribution (number theory) ,Interval (graph theory) ,Complex numbers ,Distribution ,Arcs ,Complex number ,Real number ,Connection (mathematics) ,Mathematics - Abstract
In connection with the proof of his celebrated “2.4-Theorem”, Freiman proved that if α 1 , … , α N are real numbers such that each interval [ u , u + 1 / 2 ) contains at most n of the α j mod 1, then | ∑ j = 1 N exp ( 2 π i α j ) | ⩽ 2 n − N . Freiman's result was extended by Moran and Pollington, and recently by Lev. This paper contains further extensions.
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33. The Role of Complex Numbers in Interdisciplinary Integration in Mathematics Teaching
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Katerina Anevska, Risto Malcheski, and Valentina Gogovska
- Subjects
Elementary mathematics ,Computer science ,Interpretation (philosophy) ,Mathematics education ,ComputingMilieux_COMPUTERSANDEDUCATION ,Interdiscilinary integration ,Euclid's plane geometry ,General Materials Science ,complex numbers ,Complex number - Abstract
Complex numbers are an obligatory content of mathematics education. However, almost without exception, in all degrees of education their application is restricted to geometrical interpretation of the complex number and solving algebraic equations. It is well known that opportunities for mathematics teaching integration, offered by specific content topics, are not fully utilized, especially in higher degrees of education. In this paper, we will present program content that can significantly improve integration of teaching, and improve the training of future mathematics teachers. Needles to say, we can achieve the last with the introduction of a new teaching course Geometry of complex numbers , as well as with the study of the mentioned content within the existing courses in elementary mathematics found at the majority of faculties that prepare mathematics teachers.
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