120 results on '"Complex number"'
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2. On the Vector Representation of Characteristic Functions
- Author
-
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
- Full Text
- View/download PDF
3. On the Vector Representation of Characteristic Functions.
- Author
-
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
-
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. Robustness of convergence demonstrated byparametric-guiding andcomplex-root-tunneling algorithms for Bratu’s problem
- Author
-
Liu, Zhi, Shih, Tienmo, and Chen, Zhong
- Published
- 2022
- Full Text
- View/download PDF
6. FIXED POINTS IN BICOMPLEX VALUED S-METRIC SPACES WITH APPLICATIONS.
- Author
-
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
7. Conceptual Understanding of Complex Analysis Number using Flipped Learning
- Author
-
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
- Full Text
- View/download PDF
8. 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
- Full Text
- View/download PDF
9. 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
- Full Text
- View/download PDF
10. EXCEL VBA ALKALMAZÁS FEJLESZTÉSE KOMPLEX SZÁM GYÖKEINEK KISZÁMÍTÁSÁRA.
- Author
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Zoltán, Fabulya
- Abstract
Copyright of Current Social & Economic Processes / Jelenkori Társadalmi és Gazdasági Folyamatok is the property of University of Szeged, Faculty of Engineering 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
- Full Text
- View/download PDF
11. Complex-valued statistical learning for inspecting youth labour force participation in Serbia
- Author
-
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
12. Complex Numbers and Rhythmic Changes.
- Author
-
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
- View/download PDF
13. 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
- View/download PDF
14. A Study on Dual-Generalized Complex and Hyperbolic-Generalized Complex Numbers.
- Author
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GURSES, Nurten, SENTURK, Gulsum Yeliz, and YUCE, Salim
- Subjects
- *
ALGEBRAIC numbers , *COMPLEX numbers - Abstract
This work is intended to introduce the theories of dual-generalized complex and hyperbolicgeneralized complex numbers. The algebraic properties of these numbers are taken into consideration. Besides, dual-generalized complex and hyperbolic-generalized complex valued functions are defined and different matrix representations of these numbers are examined. Moreover, a remarkable classification are given for special cases and the set of complexgeneralized complex numbers are mentioned. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
15. 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
16. Asset Pricing Model Based on Fractional Brownian Motion
- Author
-
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
- View/download PDF
17. Generalization of Dempster–Shafer theory: A complex mass function.
- Author
-
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
18. Implementation and Performance Evaluation of the Frequency-Domain-Based Bit Flipping Controller for Stabilizing the Single-Bit High-Order Interpolative Sigma Delta Modulators.
- Author
-
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
19. A Novel Transformation Method for Solving Complex Interval Matrix.
- Author
-
Babakordi, F.
- Subjects
- *
COMPLEX matrices , *COMPLEX numbers , *SUBTRACTION (Mathematics) - Abstract
Since complex interval matrix have many applications in different fields of science, in this paper interval complex matrix system as [W][Z] = [K] in which [W]; [K] are n × n known interval complex matrices and [Z] is n × n unknown interval complex matrix is studied. Using operations on interval complex numbers and matrices and defining a theorem, two auxiliary addition and subtraction complex systems are introduced and proved. Then, using the equality property of two complex numbers, the auxiliary interval complex systems are transformed to real crisp systems. Then the new system is solved and [Z] is achieved. Finally, some numerical examples are given to illustrate the applicability and ability of the proposed approach. [ABSTRACT FROM AUTHOR]
- Published
- 2020
20. One Approach for Solving Trigonometric Equations Using Complex in the Mathematical Education.
- Author
-
Andreev, Ivo, Georgiev, Ivan, and Varbanova, Margarita
- Subjects
- *
COMPLEX numbers , *EQUATIONS , *SCHOOL children , *MATHEMATICS , *MATHEMATICAL complex analysis - Abstract
The goal of this development is introducing a reader the solution of one class comprising trigonometric equations in the Teaching Course of Mathematics by using trigonometric form of the complex numbers. An exemplary approach to solving these equations, suitable for students from 11th to 12th grade, as well as for pupils participating in mathematical camps, olympiads, mathematical competitions, computer mathematics olympiads is considered. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
21. Iterants, Majorana Fermions and the Majorana-Dirac Equation
- Author
-
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
22. Discrete Fourier transformation processor based on complex radix (−1 + j) number system
- Author
-
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
23. Square-root-extended complex Kalman filter for estimation of symmetrical components in power system
- Author
-
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
24. Formulation of Strain Fatigue Criterion Based on Complex Numbers
- Author
-
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
25. Hypercomplex Widely Linear Estimation Through the Lens of Underpinning Geometry.
- Author
-
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
26. Novel elegant fuzzy genetic algorithms in classification problems.
- Author
-
Venkatanareshbabu, K., Nisheel, S., Sakthivel, R., and Muralitharan, K.
- Subjects
- *
FUZZY algorithms , *CLASSIFICATION algorithms , *GENETIC algorithms , *DATA structures , *FUZZY logic , *PARETO principle - Abstract
In this paper, we propose three novel algorithms such as Novel genetic algorithm complex-valued backpropagation neural network (GA-CVBNN), Novel elegant fuzzy genetic algorithm (EFGA) and elegant fuzzy genetic algorithm-based complex-valued backpropagation neural network (EFGA-CVBNN) for classification of accuracy in datasets. In GA-CVBNN, classical Genetic Algorithm has been used for selecting appropriate initial weights for CVBNN. The EFGA is developed to resolve the drawback of classical GA by employing fuzzy logic to control parameters and selective pressure of GA. The EFGA uses a Min-Heap data structure and Pareto principle to improve the classical genetic algorithm. The EFGA-CVBNN resolves the drawbacks of classical CVBNN by employing EFGA at the time of initial weight selection. From the simulation result, the GA-CVBNN performs better than existing CVBNN and it is not efficient. To enhance the performance of GA-CVBNN, we have developed EFGA-CVBNN. Experimental results on various synthetic datasets and benchmark datasets taken from UCI machine learning repository shows that EFGA-CVBNN outperforms PSO-CVBNN in terms of classification accuracy and time. Statistical t test has been used to validate the obtained results. [ABSTRACT FROM AUTHOR]
- Published
- 2019
- Full Text
- View/download PDF
27. 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
28. On the Fundamental Theorem of Algebra and Its Equivalence to the Frobenius Theorem on Division Algebras.
- Author
-
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
29. Why the power of diversity does not always produce better groups and societies.
- Author
-
Takefuji, Yoshiyasu
- Subjects
- *
SWARM intelligence , *PREDICTION theory , *COMPLEX numbers , *REAL numbers , *ARTIFICIAL intelligence , *RANDOM forest algorithms - Abstract
Diversity is supposed to create better groups and societies but sometimes fails. It is explained why the power of diversity may not create better groups in the current diversity prediction theory. Diversity may hurt civic life and introduce distrust. This is because the current diversity prediction theory is based on real numbers that ignore individual abilities. Its diversity prediction theory maximizes performance with infinite population size. Contrary to this, collective intelligence or swarm intelligence is not maximized by infinite population size, but by population size. The extended diversity prediction theory using the complex number allows us to express individual abilities or qualities. The diversity of complex numbers always produces better groups and societies. The wisdom of crowds, collective intelligence, swarm intelligence or nature-inspired intelligence is implemented in the current machine learning or artificial intelligence, called Random Forest. The problem of the current diversity prediction theory is detailed in this paper. • Diversity is supposed to create better groups and societies but sometimes fails. • Diversity in real numbers ignores individual abilities. • Diversity of population size plays a key role in collective intelligence. • Diversity in complex numbers allows us to express individual abilities. • Diversity in complex numbers always produces better groups and societies. [ABSTRACT FROM AUTHOR]
- Published
- 2023
- Full Text
- View/download PDF
30. Implementation and Performance Evaluation of the Frequency-Domain-Based Bit Flipping Controller for Stabilizing the Single-Bit High-Order Interpolative Sigma Delta Modulators
- Author
-
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
31. Resolution of singularities via deep complex‐valued neural networks.
- Author
-
Nitta, Tohru
- Subjects
- *
MATHEMATICAL singularities , *ARTIFICIAL neural networks , *MACHINE learning , *CRITICAL point (Thermodynamics) , *ERROR functions - Abstract
It has been reported that training deep neural networks is more difficult than training shallow neural networks. Hinton et al. proposed deep belief networks with a learning algorithm that trains one layer at a time. A much better generalization can be achieved when pre‐training each layer with an unsupervised learning algorithm. Since then, deep neural networks have been extensively studied. On the other hand, it has been revealed that singular points affect the training dynamics of the learning models such as neural networks and cause a standstill of training. Naturally, training deep neural networks suffer singular points. As described in this paper, we present a deep neural network model that has fewer singular points than the usual one. First, we demonstrate that some singular points in the deep real‐valued neural network, which is equivalent to a deep complex‐valued neural network, have been resolved as its inherent property. Such deep neural networks are less likely to become trapped in local minima or plateaus caused by critical points. Results of experiments on the two spirals problem, which has an extreme nonlinearity, support our theory. Copyright © 2017 John Wiley & Sons, Ltd. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
32. Dual-numbered Hopfield neural networks.
- Author
-
Kobayashi, Masaki
- Subjects
- *
ARTIFICIAL neural networks , *HOPFIELD network stability , *ALGEBRA , *GEOMETRIC analysis , *CLIFFORD algebras - Abstract
In recent years, Hopfield neural networks using Clifford algebra have been studied. Clifford algebra is also referred to as geometric algebra, and is useful to deal with geometric objects. There are three kinds of Clifford algebra with degree 2; complex, hyperbolic, and dual-numbered. Complex-valued Hopfield neural networks have been studied by many researchers. Several models of hyperbolic Hopfield neural networks have also been proposed. It has been difficult to construct dual-numbered Hopfield neural networks. In this work, we propose dual-numbered Hopfield neural networks by modification of hyperbolic Hopfield neural networks with the split activation function. The stability condition and Hebbian learning rule are also provided. © 2017 Institute of Electrical Engineers of Japan. Published by John Wiley & Sons, Inc. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
33. Fixed points of symmetric complex-valued Hopfield neural networks.
- Author
-
Kobayashi, Masaki
- Subjects
- *
FIXED point theory , *HOPFIELD networks , *INFORMATION retrieval , *MAXIMA & minima , *ATTRACTORS (Mathematics) - Abstract
A complex-valued Hopfield neural network (CHNN) is a model of a multistate Hopfield neural network, and has been applied to the storage of multilevel data. Weak noise tolerance, however, is a disadvantage of CHNNs. Symmetric CHNNs (SCHNNs), modified CHNNs, improve the noise tolerance of CHNNs. In the present work, we study the global and local minima of SCHNNs with one training pattern. In CHNNs, the global minima are the training and rotated patterns, and there are no local minima. In SCHNNs, it has been hard to determine all the global and local minima. It is thought that the global minima are the training and reversed patterns, and that there are no local minima in most cases. In the present work, however, we find many local minima, and show that they are very weak attractors, which reduce noise tolerance very little. In addition, we determine all the global minima of SCHNNs. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
34. Incidencia del discurso metafórico del profesor en la enseñanza del concepto de número complejo.
- Author
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Rojas, C. V. and Fernández, O.
- Abstract
In this paper, it is presented two math teachers' discourse analysis results, to identify the possible incidence of the metaphorical language used by them, in the learning of the concept of a complex number with ninth grade students. The analysis of the metaphorical language of the teacher is done in the light of the Cognitive Theory of Mathematics. It seeks the understanding of mathematics in the metaphorical thinking of people, and not in the rigorous demonstrations of books, emphasizing the connection that has the body with the cognitive processes, inducing the main key for the reform of teaching and learning processes focused on mathematical communication. [ABSTRACT FROM AUTHOR]
- Published
- 2018
- Full Text
- View/download PDF
35. Karmaşık Sayılar Konusundaki Kavram Yanılgıları ve Hataları Belirlemek için Alternatif Bir Araç: Tanılayıcı Dallanmış Ağaç.
- Author
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KARAASLAN, Gökhan and TURANLI, Necla
- Abstract
Copyright of Hacettepe University Journal of Education is the property of Hacettepe University Journal of Education 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
- 2018
- Full Text
- View/download PDF
36. ТРИ ИНВАРИАНТЫ В ОДНУ ЗАДАЧУ.
- Author
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Горская, Ксения, Коптева, Дарья, Ермекбаев, Асхат, Жетиру, Арман, Бермухамедов, Азат, Кошер, Салтанат, Стефанова, Лили, Христова, Ирина, and Йовкова, Александра
- Abstract
The paper presents results of the work of an international group of students in the frames of the realization of the research net project “Mathematical mosaic”. The results are obtained during investigations applying software products GeoGebra, Geometer's Sketchpad and Maple. The method of the complex numbers is used for the proofs of the assertions. [ABSTRACT FROM AUTHOR]
- Published
- 2017
37. An improved approach to evaluating pile length using complex continuous wavelet transform analysis.
- Author
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Sheng-Huoo Ni, Ji-Lung Li, Yu-Zhang Yang, and Zih-Tong Yang
- Subjects
- *
WAVELET transforms , *PILES & pile driving , *NONDESTRUCTIVE testing , *STRESS waves , *TIME-frequency analysis - Abstract
Non-destructive testing has been used widely for the integrity testing of piles, especially low-strain testing methods, for example the sonic echo test, the impulse response method, etc. The length of a long pile is often difficult to evaluate correctly due to the fact that the energy of a reflected stress wave generally fades with the wave travel path during testing. The purpose of this paper is to use the three-dimensional amplitude and phase angle message of a complex continuous wavelet transform to determine the length of piles by analysing a time-frequency phase angle diagram. Five piles of different lengths were tested to verify the new approach used in this study. The testing results show that a complex continuous wavelet transform is not only able to provide high-resolution results in different frequency bands but can also simplify the identification of the refection point using the three-dimensional phase angle spectrum. The pile length and pile tip condition can be easily determined using a phase angle diagram under the corresponding frequency of maximum amplitude in the three-dimensional amplitude spectrum. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
38. Dialectical Multivalued Logic and Probabilistic Theory.
- Author
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Doménech, José Luis Usó, Nescolarde-Selva, Josué Antonio, and Segura-Abad, Lorena
- Subjects
- *
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]
- Published
- 2017
- Full Text
- View/download PDF
39. THE CORIOLIS FORCE AND THE CONCEPT OF THE CORIOLIS POWER PLANT.
- Author
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Janaszak, Tadeusz
- Subjects
CORIOLIS force ,CORIOLIS acceleration ,COMPLEX numbers ,ROTATION of the earth ,MATHEMATICAL models - Abstract
This paper presents analyses of the fictitious forces which are appearing where the frame of reference is moving simultaneously on the linear and rotational path. Finally, it gives a concept of the construction of a power plant. The power plant used the energy which derived from the rotation of the Earth. [ABSTRACT FROM AUTHOR]
- Published
- 2017
- Full Text
- View/download PDF
40. Directed partial orders on complex numbers and quaternions II.
- Author
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Ma, Jingjing
- Subjects
PARTIAL differential equations ,COMPLEX numbers ,QUATERNIONS ,SIGNED numbers ,MATHEMATICAL functions - Abstract
Suppose that F is a partially ordered field with a directed partial order and K is a non-archemedean totally ordered subfield of F with $$K^{+} = F^{+} \cap K$$ . In this note, directed partial orders are constructed for complex numbers and quaternions over F. It is also shown that real quaternions cannot be made into a directed algebra over the real field with the total order. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
41. Neighborhoods for certain analytic functions.
- Author
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Talafha, Y., Frasin, B. A., Al-Hawary, Tariq, and Bani Ata, R.
- Abstract
In this paper, we introduce the neighborhood of analytic functions and defined in the open unit disc. Furthermore, we derive some sufficient and necessary conditions to be in . In addition, we see some applications of Jack's lemma. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
42. COMPARATIVE ANALYSIS OF THE STUDENTS' SCORES IN SOLVING CONSTRUCTIVE TASKS WHEN GEOMETRY IS STUDIED IN A STANDARD WAY AND WITH THE USE OF COMPLEX NUMBERS.
- Author
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Anevska, Katerina, Glavche, Metodi, and Malceski, Risto
- Subjects
- *
EUCLIDEAN geometry , *COMPLEX numbers , *COMPARATIVE studies , *STUDENTS , *MATHEMATICAL transformations - Abstract
The students learn about complex numbers and Euclidean plane geometry in high school. Nevertheless, this material is separated into units instead of being integrated. (Anevska, 2014) presents methodological aspects about the study of exponential presentation of the complex numbers and (Anevska & al.2015-1) presents the possibility for inter-subject integration of the mathematics instruction in the study of the mentioned topics. Further on, (Anevska & al., 2015-2) and (Anevska & al., 2016) present the results of the comparative analyses of the scores of the students regarding the transformations in the Euclidean plane and the metric characteristics of the geometric figures and their use when studied in the standard way (see (Mitrović & al., 1998)), and when studied with the use of complex numbers (see (Malcheski & al., 2015)). This paper offers analogous analyses regarding the acquired knowledge in these two ways when solving constructive tasks. [ABSTRACT FROM AUTHOR]
- Published
- 2016
43. COMPARATIVE ANALYSIS REGARDING THE USE OF COMPLEX NUMBERS IN SECONDARY SCHOOL.
- Author
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Anevska, Katerina, Grozdev, Sava, and Malčeski, Risto
- Subjects
- *
COMPLEX numbers , *SECONDARY schools , *EDUCATION - Abstract
The study of Euclidean plane geometry is compulsory for high school students. Practice shows that even the most advanced students face diffi culties when learning this material, and especially the content related to the metric characteristics of the geometric fi gures and their use. On the other hand, the high school syllabus also includes complex numbers, characterized by an analytical apparatus, which is adequate for learning Euclidean plane geometry. This paper presents the results of a comparative analysis of the scores of the students related to the metric characteristics of the geometric fi gures in the Euclidean plane and their use, when studied with the use of complex numbers (Malcheski, Grozdev & Anevska, 2015), and when studied in a standard way (Mitrović & al., 1998). [ABSTRACT FROM AUTHOR]
- Published
- 2016
44. VORTICES ON A PINCHED SPHERE.
- Author
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Maciuk, Arkadiusz and Smoluk, Antoni
- Subjects
VORTEX motion ,STOCK exchanges ,ECONOMICS ,EQUATIONS of motion ,SOLID geometry - Abstract
The vortex motion is a sign of the desire to achieve balance. In the study, vortices described by complex numbers are moved from a sphere into a "squeezed" sphere - where the pair of opposite poles become one - called a pinched sphere or concave disk. Vortices on the pinched sphere reflect what is commonly observed in nature. A family of the pinched spheres very well represents the spatial vortices observed daily in gusts of wind. Stock market zigzags constitute an economic vortex - a spiral on the cone whose equivalent is the pinched sphere. [ABSTRACT FROM AUTHOR]
- Published
- 2016
- Full Text
- View/download PDF
45. ИСПОЛЬЗОВАНИЕ ТЕОРИИ КОМПЛЕКСНЫХ ЧИСЕЛ В КУРСОВОМ ПРОЕКТИРОВАНИИ ПО ТЕОРИИ МЕХАНИЗМОВ И МАШИН
- Author
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Пуголовкина, А. Г. and Кутало, Н. В.
- Abstract
Copyright of International Scientific Conference: Contemporary Innovation Technique of the Engineering Personnel Training for the Mining & Transport Industry is the property of National Mining University, State Higher Educational Institution 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
- 2016
46. СБОРКИ ТРЕХПОВОДКОВОЙ ГРУППЫ ТРЕТЬЕГО КЛАССА С ДВУМЯ ВНЕШНИМИ ПОСТУПАТЕЛЬНЫМИ КИНЕМАТИЧЕСКИМИ ПАРАМИ
- Author
-
Мацюк, И. Н., Шляхов, Э. М., and Зима, Н. В.
- Abstract
Copyright of International Scientific Conference: Contemporary Innovation Technique of the Engineering Personnel Training for the Mining & Transport Industry is the property of National Mining University, State Higher Educational Institution 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
- 2016
47. Dialectical Multivalued Logic and Probabilistic Theory
- Author
-
José Luis Usó Doménech, Josué Antonio Nescolarde-Selva, and Lorena Segura-Abad
- Subjects
complex number ,fortuity ,paraconsistency ,probability ,quantum mechanics ,truth value ,Mathematics ,QA1-939 - 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.
- Published
- 2017
- Full Text
- View/download PDF
48. [formula omitted]-loss: A symmetric loss function for magnetic resonance imaging reconstruction and image registration with deep learning.
- Author
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Terpstra, Maarten L., Maspero, Matteo, Sbrizzi, Alessandro, and van den Berg, Cornelis A.T.
- Subjects
- *
DEEP learning , *MAGNETIC resonance imaging , *IMAGE registration , *IMAGE reconstruction , *SYMMETRIC functions , *MAGNETIC flux leakage - Abstract
• The impact of loss functions on complex MRI reconstruction with CNNs was analyzed. • The mean squared loss is biased towards magnitude underestimations with phase error. • A newly-proposed unbiased loss function called ⊥ -loss yields improved performance. • ⊥ -loss can be applied to any high-dimensional problem defined in R 2. • ⊥ -loss was applied to image registration tasks, showing superior performance. Convolutional neural networks (CNNs) are increasingly adopted in medical imaging, e.g., to reconstruct high-quality images from undersampled magnetic resonance imaging (MRI) acquisitions or estimate subject motion during an examination. MRI is naturally acquired in the complex domain C , obtaining magnitude and phase information in k-space. However, CNNs in complex regression tasks are almost exclusively trained to minimize the L2 loss or maximizing the magnitude structural similarity (SSIM), which are possibly not optimal as they do not take full advantage of the magnitude and phase information present in the complex domain. This work identifies that minimizing the L2 loss in the complex field has an asymmetry in the magnitude/phase loss landscape and is biased, underestimating the reconstructed magnitude. To resolve this, we propose a new loss function for regression in the complex domain called ⊥ -loss, which adds a novel phase term to established magnitude loss functions, e.g., L2 or SSIM. We show ⊥ -loss is symmetric in the magnitude/phase domain and has favourable properties when applied to regression in the complex domain. Specifically, we evaluate the ⊥ + ℓ 2 -loss and ⊥ +SSIM-loss for complex undersampled MR image reconstruction tasks and MR image registration tasks. We show that training a model to minimize the ⊥ + ℓ 2 -loss outperforms models trained to minimize the L2 loss and results in similar performance compared to models trained to maximize the magnitude SSIM while offering high-quality phase reconstruction. Moreover, ⊥ -loss is defined in R n , and we apply the loss function to the R 2 domain by learning 2D deformation vector fields for image registration. We show that a model trained to minimize the ⊥ + ℓ 2 -loss outperforms models trained to minimize the end-point error loss. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
49. Estudo dos Polígonos no plano de Argand-Gauss.
- Author
-
Silva Costa, Felix and Batista Pinheiro, Robert
- Abstract
In this work, we present the historical part of the complex numbers, showing its emergence and its evolution over time. We classify the triangles in the Argand-Gauss plane, considering its vertices and we discut some results involving the calculus of area to convex polygons, in particular, a formula to determine the area of polygons whose vertices correspond to the roots of a complex number. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
50. Estudo dos Polígonos no plano de Argand-Gauss.
- Author
-
Costa, Felix Silva
- Abstract
In this work, we present the historical part of the complex numbers, showing its emergence and its evolution over time. We classify the triangles in the Argand-Gauss plane, considering its vertices and we discut some results involving the calculus of area to convex polygons, in particular, a formula to determine the area of polygons whose vertices correspond to the roots of a complex number. [ABSTRACT FROM AUTHOR]
- Published
- 2015
- Full Text
- View/download PDF
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