2,920 results on '"Complex number"'
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2. On the Vector Representation of Characteristic Functions
- Author
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Wolf-Dieter Richter
- Subjects
complex number ,imaginary number ,vector representation ,vector exponential function ,characteristic function ,Fourier transformation ,Statistics ,HA1-4737 - Abstract
Based upon the vector representation of complex numbers and the vector exponential function, we introduce the vector representation of characteristic functions and consider some of its elementary properties such as its polar representation and a vector power expansion.
- Published
- 2023
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3. On the Vector Representation of Characteristic Functions.
- Author
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Richter, Wolf-Dieter
- Subjects
BIVECTORS ,COMPLEX numbers ,EXPONENTIAL functions ,VECTOR valued functions ,FOURIER transforms ,CHARACTERISTIC functions - Abstract
Based upon the vector representation of complex numbers and the vector exponential function, we introduce the vector representation of characteristic functions and consider some of its elementary properties such as its polar representation and a vector power expansion. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
4. ON THE (p, q) --NARAYANA n --DIMENSIONAL RECURRENCES.
- Author
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KULOĞLU, BAHAR and ÖZKAN, ENGİN
- Subjects
- *
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]
- Published
- 2023
- Full Text
- View/download PDF
5. FIXED POINTS IN BICOMPLEX VALUED S-METRIC SPACES WITH APPLICATIONS.
- Author
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Siva, G.
- Subjects
- *
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
6. Conceptual Understanding of Complex Analysis Number using Flipped Learning
- Author
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Fariz Setyawan and Siti Nur Rohmah
- Subjects
flipped learning ,handout ,complex number ,Mathematics ,QA1-939 - Abstract
Flipped Learning is one of the alternatives of teaching and learning approach in mathematics classroom. The objective of this study is exploring students’ conceptual understanding about complex number using flipped learning with handout. The subject of the study are the students in 5th semester students of mathematics education department in 2019/2020. The study used qualitative approach to describe the implementation of flipped learning. There are 31,6% of 19 respondents give score very satisfied. This result then observed by using the test with all the students understand with the definition of complex numbers. Besides they can adapt their learning activity using flipped learning with complex analysis handout. As legibility aspect of the handout, there are 52,6% of the respondents gives score satisfied and 26,3% of the respondents are very satisfied. The score indicates that the flipped learning with handout helps students to understand about the complex number concepts.
- Published
- 2021
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7. Essential requirement of complex number for oscillatory phenomenon in intracellular trafficking process
- Author
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Yoshinori Marunaka, M.D., Ph.D. and Katsumi Yagi, Ph.D.
- Subjects
Mathematical analysis ,Complex number ,Oscillation ,Intracellular trafficking ,Biotechnology ,TP248.13-248.65 - Abstract
Intracellular protein trafficking processes consisting of three intracellular states are described by three differential equations. To solve the equations, a quadratic equation is required, and its roots are generally real or complex. The purpose of the present study is to clarify the meanings of roots of real and complex numbers. To clarify the point, we define that: 1) ‘kI’ is the insertion rate from an insertion state trafficking to the plasma membrane state; 2) ‘kE’, the endocytotic rate from the plasma membrane state trafficking to a recycling state; 3) ‘kR’, the recycling rate from the recycling state trafficking to the insertion state. Amounts of proteins in three states are expressed as αelt+βemt+γ with α,β,γ = constant and l and m are roots of a quadratic equation, r2+kI+kE+kRr+kIkE+kIkR+kEkR=0. When l and m are real kI2+kE2+kR2>2kIkE+kEkR+kRkI, amounts of proteins in three states shows no oscillatory change but a monotonic change after a transient increase (or decrease); when l and m are complex kI2+kE2+kR2
- Published
- 2021
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8. Erros em esquemas de demonstração com números complexos.
- Author
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Damas Beites, Patrícia, Branco, Maria Luísa, and Costa, Cecília
- Subjects
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.)
- Published
- 2022
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9. Complex-valued statistical learning for inspecting youth labour force participation in Serbia
- Author
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Tutmez Bulent and Terek Edit
- Subjects
labour force ,complex number ,youth employment ,measurement uncertainty ,Business ,HF5001-6182 - Abstract
Youth employment in labour force has a critical importance in socio-economic planning. It is expected that the use of dynamic and able-bodied work force can increase the quality of the industrial products and it is also necessary to strengthen the economy. More importantly, the contribution to labour force has a crucial importance for public welfare. At this stage, since the role of young women in the total labour force is not considered sufficiently, this study concentrates on the determination of the relative effect of women labour force. For this purpose, first the youth employment is considered as a complex quantity; along with the real component (young men), the contribution of young women is treated as the imaginary component. By using the data derived from 20 different domains (sectors), the problem is stated as a complex value problem and a measurement uncertainty analysis is utilized. The measurement uncertainty of the complex quantity (employment) is expressed by a region in the complex plane. Finally, a confidence ellipse at 95% confidence level is produced. The phase diagrams produced by statistical learning have provided some abnormalities and also potentials.
- Published
- 2020
10. Complex Numbers and Rhythmic Changes.
- Author
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Geethamma, V. G., Gopinath, Deepa P., and Daniel, Jacob K.
- Subjects
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]
- Published
- 2021
- Full Text
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11. De-Moivre and Euler Formulae for Dual-Complex Numbers
- Author
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Mehmet Ali Güngör and Ömer Tetik
- Subjects
complex number ,dual numbers ,Mathematics ,QA1-939 - Abstract
In this study, we generalize the well-known formulae of De-Moivre and Euler of complex numbers to dual-complex numbers. Furthermore, we investigate the roots and powers of a dual-complex number by using these formulae. Consequently, we give some examples to illustrate the main results in this paper.
- Published
- 2019
- Full Text
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12. Several formulas for Bernoulli numbers and polynomials
- Author
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Claudio Pita-Ruiz, Bijan Kumar Patel, and Takao Komatsu
- Subjects
Algebra and Number Theory ,Computer Networks and Communications ,Applied Mathematics ,020206 networking & telecommunications ,Stirling numbers of the second kind ,0102 computer and information sciences ,02 engineering and technology ,01 natural sciences ,Microbiology ,Bernoulli polynomials ,Combinatorics ,symbols.namesake ,010201 computation theory & mathematics ,0202 electrical engineering, electronic engineering, information engineering ,symbols ,Discrete Mathematics and Combinatorics ,Bernoulli number ,Complex number ,Mathematics - Abstract
A generalized Stirling numbers of the second kind \begin{document}$ S_{a,b}\left(p,k\right) $\end{document} , involved in the expansion \begin{document}$ \left(an+b\right)^{p} = \sum_{k = 0}^{p}k!S_{a,b}\left(p,k\right) \binom{n}{k} $\end{document} , where \begin{document}$ a \neq 0, b $\end{document} are complex numbers, have studied in [ 16 ]. In this paper, we show that Bernoulli polynomials \begin{document}$ B_{p}(x) $\end{document} can be written in terms of the numbers \begin{document}$ S_{1,x}\left(p,k\right) $\end{document} , and then use the known results for \begin{document}$ S_{1,x}\left(p,k\right) $\end{document} to obtain several new explicit formulas, recurrences and generalized recurrences for Bernoulli numbers and polynomials.
- Published
- 2023
13. 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
- Subjects
- *
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]
- Published
- 2021
- Full Text
- View/download PDF
14. Asset Pricing Model Based on Fractional Brownian Motion
- Author
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Yu Yan and Yiming Wang
- Subjects
Ito Lemma ,fractional Brownian motion ,asset price ,complex number ,high order moments ,Thermodynamics ,QC310.15-319 ,Mathematics ,QA1-939 ,Analysis ,QA299.6-433 - 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.
- Published
- 2022
- Full Text
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15. Generalization of Dempster–Shafer theory: A complex mass function.
- Author
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Xiao, Fuyuan
- Subjects
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]
- Published
- 2020
- Full Text
- View/download PDF
16. Quasi-Quanta Language Package
- Author
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Emmerson, Parker
- Subjects
additive ,procedure ,homomorphism ,complex number ,domain ,Transform ,Numeric Energy ,group functor ,sharp-logics ,Quasi-Quanta ,Infinity meaning ,charge distribution ,orientability ,transcendental numbers ,logic vector ,Entanglement ,Energy of Number ,quantum field ,gauge ,Vector-Wave ,coordinates ,boundaries ,Language ,Energy Numbers ,manifold ,algebraic law ,element ,coboundary ,field ,multiplicative ,curvature ,range ,metric tensor ,real-valued function ,Quasi-Quanta Extended Operational-Integrable Function ,iteratives ,Fractal ,energy vector ,smooth ,imaginary gauge artefact ,differential ,topological counting ,Morphism ,Geometry ,projection ,hodge dual ,pattern ,connectedness ,embedding ,FOS: Mathematics ,intersection ,algorithm ,Pre-numeric Quasi-Quanta ,algebras ,Cross-fractal ,quantum gravity ,quasi-quanta logic ,cohomology ,Integral Field ,Mathematics ,omega sub lambda, the highest energy level - Abstract
I investigate combinations of quasi-quanta expressions and how they yield alternatesolutions for expressions inMorphic Topology of Numeric Energy: A Fractal Morphism of Topological Counting Shows Real Differentiation of Numeric Energy. For Praising Jehovah, I do publish these mathematical gesturing forms from the infinity meaning of His word. Thanks mom! This quasi-quanta language package outlines methods for combining by topo- logical functor entanglement, symbolic, numeric-energy components. Methods, guidelines and algebraic rules for combining the quasi-quanta into the energy number equivalencies are also notated herein. The Quasi-Quanta Language Package is intended to show the symbolic pat- terns for configuring the quasi quanta symbology into the numeric energy ex- pressions. This should put to rest any doubt that Energy Numbers are indeed a real, logically configured phenomenon a priori to real or complex numbers, but optionally mappable to the real or complex plane. Pre-numeric energy symbol configurations offer a broad language of pat- tern detection and logical symbol operation delineated with particular solving methods herein. This hopefully provides a new way to looking at the branches of mathematics and their inter-operable analog functions. So, inevitably, we decompose the current perspective on numbers and prove a novel method for ordering and combining symbolic orientations.
- Published
- 2023
- Full Text
- View/download PDF
17. 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
- Subjects
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
- Full Text
- View/download PDF
18. Iterants, Majorana Fermions and the Majorana-Dirac Equation
- Author
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Louis H. Kauffman
- Subjects
discrete ,complex number ,iterant ,nilpotent ,Clifford algebra ,spacetime algebra ,Mathematics ,QA1-939 - 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.
- Published
- 2021
- Full Text
- View/download PDF
19. Discrete Fourier transformation processor based on complex radix (−1 + j) number system
- Author
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Anidaphi Shadap and Prabir Saha
- Subjects
Complex binary number system (CNBS) ,Conversion algorithms ,Complex number ,Discrete Fourier transformation (DFT) ,Radix ,Integer ,Engineering (General). Civil engineering (General) ,TA1-2040 - Abstract
Complex radix (−1 + j) allows the arithmetic operations of complex numbers to be done without treating the divide and conquer rules, which offers the significant speed improvement of complex numbers computation circuitry. Design and hardware implementation of complex radix (−1 + j) converter has been introduced in this paper. Extensive simulation results have been incorporated and an application of this converter towards the implementation of discrete Fourier transformation (DFT) processor has been presented. The functionality of the DFT processor have been verified in Xilinx ISE design suite version 14.7 and performance parameters like propagation delay and dynamic switching power consumption have been calculated by Virtuoso platform in Cadence. The proposed DFT processor has been implemented through conversion, multiplication and addition. The performance parameter matrix in terms of delay and power consumption offered a significant improvement over other traditional implementation of DFT processor.
- Published
- 2017
- Full Text
- View/download PDF
20. Square-root-extended complex Kalman filter for estimation of symmetrical components in power system
- Author
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Bowen Cui
- Subjects
covariance matrices ,Kalman filters ,power system state estimation ,matrix decomposition ,nonlinear equations ,vectors ,square-root-extended complex Kalman filter ,complex number ,observation equation ,three-phase voltages ,complex vector ,positive symmetrical component ,negative symmetrical components ,traditional extended complex Kalman filter ,state variables ,three-phase instantaneous voltages ,covariance matrix decomposition ,filter stability ,αβ transformation ,abc phases ,αβ axes ,nonlinear state equation ,ECKF ,convergence rate ,Engineering (General). Civil engineering (General) ,TA1-2040 - 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.
- Published
- 2019
- Full Text
- View/download PDF
21. Formulation of Strain Fatigue Criterion Based on Complex Numbers
- Author
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Tadeusz Łagoda, Karolina Głowacka, Marta Kurek, and Dariusz Skibicki
- Subjects
normal strain ,shear strain ,fatigue criteria ,critical plane ,complex number ,Technology ,Electrical engineering. Electronics. Nuclear engineering ,TK1-9971 ,Engineering (General). Civil engineering (General) ,TA1-2040 ,Microscopy ,QH201-278.5 ,Descriptive and experimental mechanics ,QC120-168.85 - 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.
- Published
- 2021
- Full Text
- View/download PDF
22. Hypercomplex Widely Linear Estimation Through the Lens of Underpinning Geometry.
- Author
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Nitta, Tohru, Kobayashi, Masaki, and Mandic, Danilo P.
- Subjects
- *
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]
- Published
- 2019
- Full Text
- View/download PDF
23. Square-root-extended complex Kalman filter for estimation of symmetrical components in power system.
- Author
-
Cui, Bowen
- Subjects
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
- Full Text
- View/download PDF
24. On the Fundamental Theorem of Algebra and Its Equivalence to the Frobenius Theorem on Division Algebras.
- Author
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Jabbarov, I. Sh. and Hasanova, G. K.
- Subjects
- *
FUNDAMENTAL theorem of algebra , *MATHEMATICAL equivalence , *DIVISION algebras - Abstract
In this article we give a new proof of the Fundamental Theorem of Algebra. Our proof is algebraic. We simplify the known proof of the Fundamental Theorem considering special case of polynomials of odd degree with real coeffcients. This case allows us to apply the method of mathematical induction to get the proof in general case without using infiniteness of the field. [ABSTRACT FROM AUTHOR]
- Published
- 2019
25. A Note on Horadam Hybrinomials
- Author
-
Can Kızılateş
- Subjects
Combinatorics ,Matematik ,Dual number ,Horadam number,Horadam polynomial,Hybrid number ,General Medicine ,Complex number ,algebra_number_theory ,Mathematics - Abstract
This paper ensures an extensive survey of the generalization of the various hybrid numbers and hybrid polynomials especially as part of its enhancing importance in the disciplines of mathematics and physics. In this paper, by using the Horadam polynomials, we define the Horadam hybrid polynomials called Horadam hybrinomials. We obtain some special cases and algebraic properties of the Horadam hybrinomials such as recurrence relation, generating function, exponential generating function, Binet formula, summation formulas, Catalan's identity, Cassini's identity and d'Ocagne's identity, respectively. Moreover, we give some applications related to the Horadam hybrinomials in matrices.
- Published
- 2022
26. Color Image Recovery Using Low-Rank Quaternion Matrix Completion Algorithm
- Author
-
Jifei Miao and Kit Ian Kou
- Subjects
Rank (linear algebra) ,Color image ,Image and Video Processing (eess.IV) ,MathematicsofComputing_NUMERICALANALYSIS ,ComputingMethodologies_IMAGEPROCESSINGANDCOMPUTERVISION ,Matrix norm ,Numerical Analysis (math.NA) ,Electrical Engineering and Systems Science - Image and Video Processing ,Computer Graphics and Computer-Aided Design ,Matrix (mathematics) ,Tensor (intrinsic definition) ,FOS: Mathematics ,FOS: Electrical engineering, electronic engineering, information engineering ,Mathematics - Numerical Analysis ,Representation (mathematics) ,Quaternion ,Complex number ,Algorithm ,Software ,ComputingMethodologies_COMPUTERGRAPHICS ,Mathematics - Abstract
As a new color image representation tool, quaternion has achieved excellent results in color image processing problems. In this paper, we propose a novel low-rank quaternion matrix completion algorithm to recover missing data of a color image. Motivated by two kinds of low-rank approximation approaches (low-rank decomposition and nuclear norm minimization) in traditional matrix-based methods, we combine the two approaches in our quaternion matrix-based model. Furthermore, the nuclear norm of the quaternion matrix is replaced by the sum of the Frobenius norm of its two low-rank factor quaternion matrices. Based on the relationship between the quaternion matrix and its equivalent complex matrix, the problem eventually is converted from the quaternion number domain to the complex number domain. An alternating minimization method is applied to solve the model. Simulation results on color image recovery show the superior performance and efficiency of the proposed algorithm over some tensor-based and quaternion-based ones.
- Published
- 2022
27. Definite integral of the logarithm hyperbolic secant function in terms of the Hurwitz zeta function
- Author
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Robert Reynolds and Allan Stauffer
- Subjects
Pure mathematics ,entries of gradshteyn and ryzhik ,Logarithm ,General Mathematics ,lcsh:Mathematics ,hyperbolic integrals ,Definite integrals ,Function (mathematics) ,Rational function ,hurwitz zeta function ,lcsh:QA1-939 ,bierens de haan ,Hurwitz zeta function ,Complex number ,Mathematics - Abstract
We evaluate definite integrals of the form given by $\int_{0}^{\infty}R(a, x)\log (\cos (\alpha) \text{sech}(x)+1)dx$. The function $R(a, x)$ is a rational function with general complex number parameters. Definite integrals of this form yield closed forms for famous integrals in the books of Bierens de Haan [4] and Gradshteyn and Ryzhik [5].
- Published
- 2021
28. Derivation of some integrals in Gradshteyn and Ryzhik
- Author
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Robert Reynolds and Allan Stauffer
- Subjects
Physics ,Combinatorics ,entries of gradshteyn and ryzhik ,General Mathematics ,lcsh:Mathematics ,hyperbolic integrals ,Definite integrals ,lerch function ,hypergeometric function ,Function (mathematics) ,Hypergeometric function ,lcsh:QA1-939 ,Complex number - Abstract
In this work we present derivations of the formula listed in entry 4.113 in the sixth edition of Gradshteyn and Rhyzik's table of integrals. We evaluate two definite integrals of the form $ \begin{equation*} \int_{0}^{\infty}\frac{e^{-iay}(-iy+\log(z))^k+e^{iay}(iy+\log(z))^k}{\cosh(by)}dy \end{equation*} $ and $ \begin{equation*} \int_{0}^{\infty}\frac{e^{iay}(iy+\log(z))^k-e^{-iay}(-iy+\log(z))^k}{\sinh(b y)}dy \end{equation*} $ in terms of the Lerch function where $ k $, $ a $, $ z $ and $ b $ are arbitrary complex numbers. The entries in the table(s) are obtained as special cases in the paper below.
- Published
- 2021
29. Centrosymmetric universal realizability
- Author
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Yankis R. Linares, Ana I. Julio, and Ricardo Soto
- Subjects
Combinatorics ,Matrix (mathematics) ,Algebra and Number Theory ,Realizability ,Spectrum (functional analysis) ,Canonical form ,Nonnegative matrix ,Centrosymmetric matrix ,Lambda ,Complex number ,Mathematics - Abstract
A list $\Lambda =\{\lambda_{1},\ldots,\lambda_{n}\}$ of complex numbers is said to be realizable, if it is the spectrum of an entrywise nonnegative matrix $A$. In this case, $A$ is said to be a realizing matrix. $\Lambda$ is said to be universally realizable, if it is realizable for each possible Jordan canonical form (JCF) allowed by $\Lambda$. The problem of the universal realizability of spectra is called the universal realizability problem (URP). Here, we study the centrosymmetric URP, that is, the problem of finding a nonnegative centrosymmetric matrix for each JCF allowed by a given list $\Lambda $. In particular, sufficient conditions for the centrosymmetric URP to have a solution are generated.
- Published
- 2021
30. An exceptional G(2) extension of the Standard Model from the correspondence with Cayley–Dickson algebras automorphism groups
- Author
-
Nicolò Masi
- Subjects
High Energy Physics - Theory ,Particle physics ,Pure mathematics ,Science ,FOS: Physical sciences ,Article ,Theoretical particle physics ,High Energy Physics - Phenomenology (hep-ph) ,Gauge group ,Algebraic number ,Physics ,Multidisciplinary ,Group (mathematics) ,Computer Science::Information Retrieval ,Sedenion ,Automorphism ,High Energy Physics - Phenomenology ,Standard Model (mathematical formulation) ,High Energy Physics - Theory (hep-th) ,Homogeneous space ,Medicine ,Phenomenology ,Complex number - Abstract
In this article I propose a new criterion to extend the Standard Model of particle physics from a straightforward algebraic conjecture: the symmetries of physical microscopic forces originate from the automorphism groups of main Cayley–Dickson algebras, from complex numbers to octonions and sedenions. This correspondence leads to a natural enlargement of the Standard Model color sector, from a SU(3) gauge group to an exceptional Higgs-broken G(2) group, following the octonionic automorphism relation guideline. In this picture, an additional ensemble of massive G(2)-gluons emerges, which is separated from the particle dynamics of the Standard Model.
- Published
- 2021
31. Three-point iterative algorithm in the absence of the derivative for solving nonlinear equations and their basins of attraction
- Author
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Mohamed S. M. Bahgat
- Subjects
Discrete mathematics ,Iterative method ,Derivative ,Type (model theory) ,Nonlinear equations ,Free derivative ,Nonlinear system ,Alpha (programming language) ,Order of convergence ,QA1-939 ,Point (geometry) ,Basin of attraction ,Fractal ,Complex number ,Mathematics - Abstract
In this paper, we suggested and analyzed a new higher-order iterative algorithm for solving nonlinear equation $$g(x)=0$$ g ( x ) = 0 , $$g:{\mathbb {R}}\longrightarrow {\mathbb {R}}$$ g : R ⟶ R , which is free from derivative by using the approximate version of the first derivative, and we studied the basins of attraction for the proposed iterative algorithm to find complex roots of complex functions $$g:{\mathbb {C}}\longrightarrow {\mathbb {C}}$$ g : C ⟶ C . To show the effectiveness of the proposed algorithm for the real and the complex domains, the numerical results for the considered examples are given and graphically clarified. The basins of attraction of the existing methods and our algorithm are offered and compared to clarify their performance. The proposed algorithm satisfied the condition such that $$|x_{m}-\alpha | | x m - α | < 1.0 × 10 - 15 , as well as the maximum number of iterations is less than or equal to 3, so the proposed algorithm can be applied to efficiently solve numerous type non-linear equations.
- Published
- 2021
32. On the configuration of the singular fibers of jet schemes of rational double points
- Author
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Yoshimune Koreeda
- Subjects
Surface (mathematics) ,Pure mathematics ,Algebra and Number Theory ,Jet (mathematics) ,Fiber (mathematics) ,Singular point of a curve ,Mathematics - Algebraic Geometry ,Singularity ,Integer ,FOS: Mathematics ,Graph (abstract data type) ,Algebraic Geometry (math.AG) ,14J17 ,Complex number ,Mathematics - Abstract
To each variety $X$ and a nonnegative integer $m$, there is a space $X_m$ over $X$, called the jet scheme of $X$ of order $m$, parametrizing $m$-th jets on $X$. Its fiber over a singular point of $X$ is called a singular fiber. For a surface with a rational double point, Mourtada gave a one-to-one correspondence between the irreducible components of the singular fiber of $X_m$ and the exceptional curves of the minimal resolution of $X$ for $m \gg 0$. In this paper, for a surface $X$ over complex number with a singularity of $A_n$ or $D_4$-type, we study the intersections of irreducible components of the singular fiber and construct a graph using this information. The vertices of the graph correspond to irreducible components of the singular fiber and two vertices are connected when the intersection of the corresponding components is maximal for the inclusion relation. In the case of $A_n$ or $D_4$-type singularity, we show that this graph is isomorphic to the resolution graph for $m \gg 0$., 18 pages
- Published
- 2021
33. A Sparse Algorithm for Computing the DFT Using Its Real Eigenvectors
- Author
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Linda S. DeBrunner, Victor DeBrunner, and Rajesh Thomas
- Subjects
T57-57.97 ,Applied mathematics. Quantitative methods ,Computer science ,Computation ,Fast Fourier transform ,discrete fourier transform ,Dot product ,Discrete Fourier transform ,DFT eigenvectors ,harmonic analysis ,Multiplication ,Complex number ,Algorithm ,Eigenvalues and eigenvectors ,Integer (computer science) - Abstract
Direct computation of the discrete Fourier transform (DFT) and its FFT computational algorithms requires multiplication (and addition) of complex numbers. Complex number multiplication requires four real-valued multiplications and two real-valued additions, or three real-valued multiplications and five real-valued additions, as well as the requisite added memory for temporary storage. In this paper, we present a method for computing a DFT via a natively real-valued algorithm that is computationally equivalent to a N=2k-length DFT (where k is a positive integer), and is substantially more efficient for any other length, N. Our method uses the eigenstructure of the DFT, and the fact that sparse, real-valued, eigenvectors can be found and used to advantage. Computation using our method uses only vector dot products and vector-scalar products.
- Published
- 2021
34. MV-algebras and Partially Cyclically Ordered Groups
- Author
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Gérard Leloup, Laboratoire Manceau de Mathématiques (LMM), and Le Mans Université (UM)
- Subjects
Pure mathematics ,cyclically ordered abelian groups ,[MATH.MATH-AC]Mathematics [math]/Commutative Algebra [math.AC] ,Existential quantification ,partially cyclically ordered abelian groups ,Cyclic group ,Characterization (mathematics) ,Commutative Algebra (math.AC) ,01 natural sciences ,010104 statistics & probability ,Simple (abstract algebra) ,FOS: Mathematics ,0101 mathematics ,Algebra over a field ,Mathematics ,MV-algebras ,Algebra and Number Theory ,[MATH.MATH-RA]Mathematics [math]/Rings and Algebras [math.RA] ,010102 general mathematics ,Cyclic order ,Mathematics - Logic ,Mathematics - Rings and Algebras ,Mathematics - Commutative Algebra ,pseudofinite ,MV-chains ,[MATH.MATH-LO]Mathematics [math]/Logic [math.LO] ,Unimodular matrix ,Computational Theory and Mathematics ,Rings and Algebras (math.RA) ,Geometry and Topology ,Logic (math.LO) ,Complex number - Abstract
We prove that there exists a functorial correspondence between MV-algebras and partially cyclically ordered groups which are wound round of lattice-ordered groups. It follows that some results about cyclically ordered groups can be stated in terms of MV-algebras. For example, the study of groups together with a cyclic order allows to get a first-order characterization of groups of unimodular complex numbers and of finite cyclic groups. We deduce a characterization of pseudofinite MV-chains and of pseudo-simple MV-chains (i.e. which share the same first-order properties as some simple ones). We can generalize these results to some non-lineraly ordered MV-algebras, for example hyper-archimedean MV-algebras.
- Published
- 2021
35. Two Challenging Concepts in Mathematics Education: Subject-Specific Thoughts on the Complex Unit and Angles
- Author
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Philipp Bitzenbauer and Joaquin M. Veith
- Subjects
symbols.namesake ,Point (typography) ,Subject specific ,De Moivre's formula ,Mathematics education ,symbols ,Mathematical content ,Notation ,Complex number ,Education ,Focus (linguistics) ,Unit (housing) - Abstract
In this paper, we focus on two particularly problematic concepts in teaching mathematics: the complex unit i and angles. These concepts are naturally linked via De Moivre’s theorem but are independently misused in numerous contexts. We present definitions, notations, and ways of speaking about these terms from mathematics education that are not valid from a subject-specific point of view. We justify how these incorrectly used definitions, notations, and ways of speaking convey a false picture of the mathematical content and present options for introducing these concepts in classroom practice in a correct way.
- Published
- 2021
36. 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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Huishan Zhai and Bingo Wing-Kuen Ling
- Subjects
high-order interpolative sigma delta modulator ,bit flipping control ,quantization ,fractal ,chaos ,complex number ,Technology ,Engineering (General). Civil engineering (General) ,TA1-2040 ,Biology (General) ,QH301-705.5 ,Physics ,QC1-999 ,Chemistry ,QD1-999 - 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.
- Published
- 2020
- Full Text
- View/download PDF
37. Generators and Relations for Un(Z[1/2,i])
- Author
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Peter Selinger and Xiaoning Bian
- Subjects
FOS: Computer and information sciences ,Physics ,Discrete mathematics ,Quantum Physics ,Computer Science - Logic in Computer Science ,Ring (mathematics) ,Group (mathematics) ,Computer Science - Emerging Technologies ,FOS: Physical sciences ,Unitary matrix ,Subring ,Logic in Computer Science (cs.LO) ,Quantum circuit ,Matrix (mathematics) ,Emerging Technologies (cs.ET) ,Computer Science::Emerging Technologies ,Quantum Physics (quant-ph) ,Complex number ,Quantum computer - Abstract
Consider the universal gate set for quantum computing consisting of the gates X, CX, CCX, omega^dagger H, and S. All of these gates have matrix entries in the ring Z[1/2,i], the smallest subring of the complex numbers containing 1/2 and i. Amy, Glaudell, and Ross proved the converse, i.e., any unitary matrix with entries in Z[1/2,i] can be realized by a quantum circuit over the above gate set using at most one ancilla. In this paper, we give a finite presentation by generators and relations of U_n(Z[1/2,i]), the group of unitary nxn-matrices with entries in Z[1/2,i]., Comment: In Proceedings QPL 2021, arXiv:2109.04886
- Published
- 2021
38. Tensor Network Rewriting Strategies for Satisfiability and Counting
- Author
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Niel de Beaudrap, Aleks Kissinger, and Konstantinos Meichanetzidis
- Subjects
FOS: Computer and information sciences ,Discrete mathematics ,Polynomial (hyperelastic model) ,Quantum Physics ,Diagram ,FOS: Physical sciences ,Computational Complexity (cs.CC) ,Computer Science::Computational Complexity ,Semiring ,Computer Science - Computational Complexity ,Range (mathematics) ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,ComputingMethodologies_SYMBOLICANDALGEBRAICMANIPULATION ,Tensor ,Rewriting ,Quantum Physics (quant-ph) ,Time complexity ,Complex number ,Mathematics - Abstract
We provide a graphical treatment of SAT and #SAT on equal footing. Instances of #SAT can be represented as tensor networks in a standard way. These tensor networks are interpreted by diagrams of the ZH-calculus: a system to reason about tensors over C in terms of diagrams built from simple generators, in which computation may be carried out by transformations of diagrams alone. In general, nodes of ZH diagrams take parameters over C which determine the tensor coefficients; for the standard representation of #SAT instances, the coefficients take the value 0 or 1. Then, by choosing the coefficients of a diagram to range over B, we represent the corresponding instance of SAT. Thus, by interpreting a diagram either over the boolean semiring or the complex numbers, we instantiate either the decision or counting version of the problem. We find that for classes known to be in P, such as 2SAT and #XORSAT, the existence of appropriate rewrite rules allows for efficient simplification of the diagram, producing the solution in polynomial time. In contrast, for classes known to be NP-complete, such as 3SAT, or #P-complete, such as #2SAT, the corresponding rewrite rules introduce hyperedges to the diagrams, in numbers which are not easily bounded above by a polynomial. This diagrammatic approach unifies the diagnosis of the complexity of CSPs and #CSPs and shows promise in aiding tensor network contraction-based algorithms., Comment: In Proceedings QPL 2020, arXiv:2109.01534
- Published
- 2021
39. A Note on Some Integrals by Malmsten and Bierens de Haan
- Author
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Allan Stauffer and Robert Reynolds
- Subjects
Statistics and Probability ,Economics and Econometrics ,Correctness ,Current (mathematics) ,Calculus ,Order (group theory) ,Function (mathematics) ,Statistics, Probability and Uncertainty ,Integral transform ,Complex number ,Connection (mathematics) ,Mathematics ,Vector potential - Abstract
Carl Johan Malmsten (1846) and David Beirens de Haan (1847) published work containing some interesting integrals. While no formal derivations of the integrals in his book Nouvelles Tables d'Integrales Defines are available in current literature deriving and evaluating such formulae are useful in all aspects of science and engineering whenever such formulae are used. Formulae in the book of Bierens de Haan are used in connection with certain potential problems where there is the need to determine the vector potential of two parallel, infinitely long, tubular rectangular conductors carrying cur-rents in opposite directions. In this current work we supply formal derivations for some of these integrals along with deriving some special cases as new integrals in order to expand upon the book of Bierens de haan to aid in potential research where these formulae are applicable. Updating book of integrals is always a useful exercise as it keeps the volume accurate and more useful for potential readers and researchers. Formal derivations are also useful as they help in verifying the correctness of integrals in such volumes. The definite integral we derived in this work is given by (1) in terms of the Lerch function, where the parameters a; k; m; and p are general complex numbers subject to their restrictions. This formal derivation is then used to derive the correct version of a definite integral transform along with new formulae. Some of the results in this work are new.
- Published
- 2021
40. Circuit Theory Based on New Concepts and Its Application to Quantum Theory
- Author
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Hirofumi Sanada, Nobuo Nagai, and Takashi Yahagi
- Subjects
Physics ,Amplitude ,Quantum mechanics ,Phase (waves) ,Refraction (sound) ,Resonance ,Reflection coefficient ,Complex number ,Electromagnetic radiation ,Network analysis - Published
- 2021
41. Аналог теореми Меньшова – Трохимчука для моногенних функцій у тривимірній комутативній алгебрі
- Author
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S. A. Plaksa and M. V. Tkachuk
- Subjects
Work (thermodynamics) ,Pure mathematics ,Mathematics::Complex Variables ,Gâteaux derivative ,Field (mathematics) ,Function (mathematics) ,Commutative algebra ,Complex number ,Mathematics - Abstract
UDC 517.54 The aim of this work is to weaken the conditions of monogeneity for functions that take values in a given three-dimensional commutative algebra over the field of complex numbers. The monogeneity of the function is understood as a combination of its continuity and the existence of the Gateaux derivative.
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- 2021
42. Stability Theory for Nullity and Deficiency of Linear Relations
- Author
-
Silas Luliro Kito and Gerald Wanjala
- Subjects
Pure mathematics ,Article Subject ,All inside ,Applied Mathematics ,MathematicsofComputing_GENERAL ,Banach space ,Perturbation (astronomy) ,Stability (probability) ,TheoryofComputation_MATHEMATICALLOGICANDFORMALLANGUAGES ,Stability theory ,QA1-939 ,Constant (mathematics) ,GeneralLiterature_REFERENCE(e.g.,dictionaries,encyclopedias,glossaries) ,Complex number ,Mathematics ,Analysis - Abstract
Let A and B be two closed linear relations acting between two Banach spaces X and Y , and let λ be a complex number. We study the stability of the nullity and deficiency of A when it is perturbed by λ B . In particular, we show the existence of a constant ρ > 0 for which both the nullity and deficiency of A remain stable under perturbation by λ B for all λ inside the disk λ < ρ .
- Published
- 2021
43. On involutiveness of linear combinations of a quadratic matrix and an arbitrary matrix
- Author
-
Nurgül Kalaycı and Murat Sarduvan
- Subjects
Statistics and Probability ,Matematik ,Algebra and Number Theory ,Block matrix ,Combinatorics ,Matrix (mathematics) ,Quadratic equation ,Quadratic matrix,Partitioned matrix,Linear combination,Diagonalization,Direct sum of matrices ,Geometry and Topology ,Linear combination ,Complex number ,Mathematics ,Analysis - Abstract
We characterize the involutiveness of the linear combinations of the form $a{\bf{A}} + b{\bf{B}}$ when $a,b$ are nonzero complex numbers, ${\bf{A}}$ is a quadratic $n \times n$ nonzero matrix and ${\bf{B}}$ is an arbitrary $n \times n$ nonzero matrix, under certain properties imposed on $\bf{A}$ and $\bf{B}$. Moreover, we give some examples illustrating our main results.
- Published
- 2021
44. Note on a Stieltjes Transform in terms of the Lerch Function
- Author
-
Allan Stauffer and Robert C. Reynolds
- Subjects
Statistics and Probability ,Numerical Analysis ,Algebra and Number Theory ,Logarithm ,Applied Mathematics ,Function (mathematics) ,Theoretical Computer Science ,Combinatorics ,Special functions ,Geometry and Topology ,Connection (algebraic framework) ,Complex number ,Stieltjes transform ,Mathematics - Abstract
In this work the authors derive the Stieltjes transform of the logarithmic function in terms of the Lerch function. This transform is used to derive closed form solutions involving fundamental constants and special functions. Specifically we derive the definite integral given by\[\int_{0}^{\infty} \frac{(1-b x)^m \log ^k(c (1-b x))+(b x+1)^m \log ^k(c (b x+1))}{a+x^2}dx\]where $a,b,c,m$ and $k$ are general complex numbers subject to the restrictions given in connection with the formulas.
- Published
- 2021
45. The Mellin Transform of Logarithmic and Rational Quotient Function in terms of the Lerch Function
- Author
-
Robert Reynolds and Allan Stauffer
- Subjects
Pure mathematics ,Mellin transform ,Current (mathematics) ,Article Subject ,Logarithm ,General Mathematics ,010102 general mathematics ,Definite integrals ,02 engineering and technology ,Function (mathematics) ,Integral transform ,01 natural sciences ,QA1-939 ,0202 electrical engineering, electronic engineering, information engineering ,020201 artificial intelligence & image processing ,0101 mathematics ,Complex number ,Mathematics ,Quotient - Abstract
Upon reading the famous book on integral transforms volume II by Erdeyli et al., we encounter a formula which we use to derive a Mellin transform given by ∫ 0 ∞ x m − 1 log k a x / β 2 + x 2 γ + x d x , where the parameters a , k , β , and γ are general complex numbers. This Mellin transform will be derived in terms of the Lerch function and is not listed in current literature to the best of our knowledge. We will use this transform to create a table of definite integrals which can be used to extend similar tables in current books featuring such formulae.
- Published
- 2021
46. Low-Complexity High-Precision Method and Architecture for Computing the Logarithm of Complex Numbers
- Author
-
Zongguang Yu, Yonggang Zhang, Zhonghai Lu, Li Li, Yuxiang Fu, and Chen Hui
- Subjects
Logarithm ,business.industry ,Computer science ,Computation ,Computational science ,Range (mathematics) ,Software ,CMOS ,Hardware and Architecture ,Approximation error ,Hardware_ARITHMETICANDLOGICSTRUCTURES ,Electrical and Electronic Engineering ,CORDIC ,business ,Complex number - 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.
- Published
- 2021
47. Complex Numbers and Rhythmic Changes
- Author
-
Jacob K. Daniel, V. G. Geethamma, and Deepa P. Gopinath
- Subjects
Quantitative Biology::Neurons and Cognition ,Oscillation ,Computer science ,Physics::Medical Physics ,Mathematical analysis ,Quantitative Biology::Genomics ,Education ,Vibration ,Mathematics::Metric Geometry ,Trigonometric functions ,Sine ,Physics::Chemical Physics ,Algebraic number ,Representation (mathematics) ,Complex number ,Rotation (mathematics) - 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.
- Published
- 2021
48. Hypercomplex Models of Multichannel Images
- Author
-
V. G. Labunets
- Subjects
IMAGE PROCESSING ,Hypercomplex number ,Pixel ,Basis (linear algebra) ,HYPERCOMPLEX ALGEBRAS ,Computer science ,Image processing ,Invariant pattern recognition ,Mathematics (miscellaneous) ,Computer Science::Computer Vision and Pattern Recognition ,Algebra over a field ,Algorithm ,Commutative property ,Complex number ,MULTICHANNEL IMAGES - Abstract
We present a new theoretical approach to the processing of multidimensional and multicomponent images based on the theory of commutative hypercomplex algebras, which generalize the algebra of complex numbers. The main goal of the paper is to show that commutative hypercomplex numbers can be used in multichannel image processing in a natural and effective manner. We suppose that animal brains operate with hypercomplex numbers when processing multichannel retinal images. In our approach, each multichannel pixel is regarded as a $$K$$ -dimensional ( $$K$$ D) hypercomplex number rather than a $$K$$ D vector, where $$K$$ is the number of different optical channels. This creates an effective mathematical basis for various function–number transformations of multichannel images and invariant pattern recognition.
- Published
- 2021
49. A New Construction of Holditch Theorem for Homothetic Motions in <math xmlns='http://www.w3.org/1998/Math/MathML' id='M1'> <msub> <mrow> <mi>C</mi> </mrow> <mrow> <mtext>p</mtext> </mrow> </msub> </math>
- Author
-
Tülay Erişir
- Subjects
Article Subject ,Physics ,QC1-999 ,Applied Mathematics ,010102 general mathematics ,Mathematical analysis ,Geometric representation ,General Physics and Astronomy ,Motion (geometry) ,Kinematics ,Computer Science::Computational Geometry ,01 natural sciences ,Homothetic transformation ,Planar ,0103 physical sciences ,010307 mathematical physics ,0101 mathematics ,Complex plane ,Complex number ,Mathematics - Abstract
In this study, the planar kinematics has been studied in a generalized complex plane which is a geometric representation of the generalized complex number system. Firstly, the planar kinematic formulas with one parameter for homothetic motions in the generalized complex plane have been mentioned briefly. Then, the Steiner area formula given areas of the trajectories drawn by the points taken in a generalized complex plane have been obtained during the one-parameter planar homothetic motion. Finally, the Holditch theorem, which gives the relationship between these areas of trajectories, has been expressed for homothetic motions in a generalized complex plane. So, this theorem obtained in this study is the most general form of all Holditch theorems obtained so far.
- Published
- 2021
50. Uniqueness of difference polynomials
- Author
-
Xiaomei Zhang and Xiang Chen
- Subjects
Polynomial (hyperelastic model) ,Physics ,General Mathematics ,010102 general mathematics ,Zero (complex analysis) ,Order (ring theory) ,uniqueness ,01 natural sciences ,010101 applied mathematics ,Combinatorics ,borel exceptional values ,Difference polynomials ,difference polynomial ,QA1-939 ,Uniqueness ,0101 mathematics ,Complex number ,Mathematics ,Meromorphic function - Abstract
Let $ f(z) $ be a transcendental meromorphic function of finite order and $ c\in\Bbb{C} $ be a nonzero constant. For any $ n\in\Bbb{N}^{+} $, suppose that $ P(z, f) $ is a difference polynomial in $ f(z) $ such as $ P(z, f) = a_{n}f(z+nc)+a_{n-1}f(z+(n-1)c)+\cdots+a_{1}f(z+c)+a_{0}f(z) $, where $ a_{k} (k = 0, 1, 2, \cdots, n) $ are not all zero complex numbers. In this paper, the authors investigate the uniqueness problems of $ P(z, f) $.
- Published
- 2021
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