32 results on '"Complex number"'
Search Results
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. 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
6. 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
7. Essential requirement of complex number for oscillatory phenomenon in intracellular trafficking process
- Author
-
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
8. 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
9. 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
10. De-Moivre and Euler Formulae for Dual-Complex Numbers
- Author
-
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
11. 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
12. 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
13. 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
14. 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
15. 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
16. 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
17. 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
18. 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
19. 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
20. 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
21. 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
22. Dialectical Multivalued Logic and Probabilistic Theory.
- Author
-
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
23. 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
24. Asset Pricing Model Based on Fractional Brownian Motion.
- Author
-
Yan, Yu and Wang, Yiming
- Subjects
- *
BROWNIAN motion , *COMPLEX numbers , *REAL numbers , *MERTON Model , *DECISION making - Abstract
This paper introduces one unique price motion process with fractional Brownian motion. We introduce the imaginary number into the agent's subjective probability for the reason of convergence; further, the result similar to Ito Lemma is proved. As an application, this result is applied to Merton's dynamic asset pricing framework. We find that the four order moment of fractional Brownian motion is entered into the agent's decision-making. The decomposition of variance of economic indexes supports the possibility of the complex number in price movement. [ABSTRACT FROM AUTHOR]
- Published
- 2022
- Full Text
- View/download PDF
25. Free in-plane vibration analysis of a curved beam (arch) with arbitrary various concentrated elements.
- Author
-
Wu, J.S., Lin, F.T., and Shaw, H.J.
- Subjects
- *
FREE vibration , *CURVED beams , *INFORMATION theory , *REAL numbers , *FINITE element method , *PARAMETER estimation - Abstract
Abstract: In the existing literature, the information regarding the exact solutions for free in-plane vibrations of the curved beams (or arches) carrying various concentrated elements is rare, particularly for the case with multiple attachments including eccentricities and mass moments of inertias. For this reason, this paper aims at presenting an effective approach to tackle the title problem. First of all, the un-coupled equation of motion for the circumferential displacement of an arch segment is derived. Next, based on the value of the discriminate parameter for a cubic equation, the exact solutions for the three types of roots of the un-coupled equation are determined and, corresponding to each type of roots, all displacement functions for the arch segment in terms of the real numbers (instead of the complex ones) are obtained. Finally, use of the compatible equations for the displacements and slopes together with the equilibrium equations for the forces and moments at each intermediate node and two ends of the entire curved beam, a frequency equation of the form ∣H(ω)∣=0 is obtained. It is found that the conventional approach by using the condition “∣H(ω t )∣⩽ ε” to search for the approximate value of ω t is difficult even if the convergence tolerance ε is greater than 10+3 (i.e., ε >10+3) instead of less than 10−3 (i.e., ε <10−3), however, the half-interval method is one of the effective tools for solving the problem if all coefficients of the determinant ∣H(ω)∣ are the real numbers. In addition to comparing with the existing literature, most of the numerical results obtained from the presented method are compared with those obtained from the conventional finite element method (FEM) and good agreement is achieved. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
26. Neural Networks with Comparatively Few Critical Points.
- Author
-
Nitta, Tohru
- Subjects
ERROR functions ,ARTIFICIAL neural networks ,COMPARATIVE studies ,CRITICAL point theory ,MATHEMATICAL analysis ,SADDLEPOINT approximations - Abstract
Abstract: A critical point is a point on which the derivatives of an error function are all zero. It has been shown in the literatures that the critical points caused by the hierarchical structure of the real-valued neural network could be local minima or saddle points, whereas most of the critical points caused by the hierarchical structure are saddle points in the case of complex- valued neural networks. Several studies have demonstrated that that kind of singularity has a negative effect on learning dynamics in neural networks. In this paper, we will demonstrate via some examples that the decomposition of high- dimensional NNs into real-valued NNs equivalent to the original NNs yields the NNs that do not have critical points based on the hierarchical structure. [Copyright &y& Elsevier]
- Published
- 2013
- Full Text
- View/download PDF
27. THE UNIQUENESS THEOREM FOR COMPLEX-VALUED NEURAL NETWORKS WITH THRESHOLD PARAMETERS AND THE REDUNDANCY OF THE PARAMETERS.
- Author
-
NITTA, TOHRU
- Subjects
- *
ARTIFICIAL neural networks , *NEURAL circuitry , *ARTIFICIAL intelligence , *DIGITAL computer simulation , *COMPUTER software - Abstract
This paper will prove the uniqueness theorem for 3-layered complex-valued neural networks where the threshold parameters of the hidden neurons can take non-zeros. That is, if a 3-layered complex-valued neural network is irreducible, the 3-layered complex-valued neural network that approximates a given complex-valued function is uniquely determined up to a finite group on the transformations of the learnable parameters of the complex-valued neural network. [ABSTRACT FROM AUTHOR]
- Published
- 2008
- Full Text
- View/download PDF
28. Iterants, Majorana Fermions and the Majorana-Dirac Equation.
- Author
-
Kauffman, Louis H.
- Subjects
- *
DIRAC equation , *CLIFFORD algebras , *MATRICES (Mathematics) , *COMPLEX numbers , *EQUATIONS , *MAJORANA fermions , *SCHRODINGER equation - Abstract
This paper explains a method of constructing algebras, starting with the properties of discrimination in elementary discrete systems. We show how to use points of view about these systems to construct what we call iterant algebras and how these algebras naturally give rise to the complex numbers, Clifford algebras and matrix algebras. The paper discusses the structure of the Schrödinger equation, the Dirac equation and the Majorana Dirac equations, finding solutions via the nilpotent method initiated by Peter Rowlands. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
29. Formulation of Strain Fatigue Criterion Based on Complex Numbers.
- Author
-
Łagoda, Tadeusz, Głowacka, Karolina, Kurek, Marta, Skibicki, Dariusz, Maletta, Carmine, and Marsavina, Liviu
- Subjects
- *
COMPLEX numbers , *SHEAR strain , *PHYSICAL constants , *TORSION , *NUMBER systems - Abstract
In the case of many low-cycle multiaxial fatigue criteria, we encounter a mathematical problem of adding vectors of normal and shear strains. Typically, the problem of defining an equivalent strain is solved by weighting factors. Unfortunately, this ignores the fact that these vectors represent other physical quantities: the normal strain is a longitudinal strain, and the shear strain is a rotation angle. Therefore, the goal of the present work was to propose a method of combining different types of strains by adopting a system of complex numbers. The normal strain was defined as the real part and the shear strain was defined as the imaginary part. Using this approach, simple load states, such as pure bending and pure torsion, have been transformed into an expression for equivalent strain identical to the previously proposed criteria defined by Macha. [ABSTRACT FROM AUTHOR]
- Published
- 2021
- Full Text
- View/download PDF
30. Essential requirement of complex number for oscillatory phenomenon in intracellular trafficking process.
- Author
-
Marunaka Y and Yagi K
- 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) ' k I ' is the insertion rate from an insertion state trafficking to the plasma membrane state; 2) ' k E ', the endocytotic rate from the plasma membrane state trafficking to a recycling state; 3) ' k R ', the recycling rate from the recycling state trafficking to the insertion state. Amounts of proteins in three states are expressed as α e lt + β e mt + γ with α , β , γ = constant and l and m are roots of a quadratic equation, r 2 + k I + k E + k R r + k I k E + k I k R + k E k R = 0 . When l and m are real k I 2 + k E 2 + k R 2 > 2 k I k E + k E k R + k R k I , 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 k I 2 + k E 2 + k R 2 < 2 k I k E + k E k R + k R k I , amounts of proteins in three states are expressed as α e lt + β e mt + γ = 2 g 2 + h 2 sin b t + σ e at + γ ( α , β , l , m = complex and γ , a , b , g , h , σ = real: α , β = conjugate each other; l , m = conjugate each other), showing an oscillatory change with time. The frequency of oscillatory change appearance is evaluated to be 60% at random combinations of three trafficking rates, k I , k E and k R . The present study indicates that complex numbers have an essentially important meaning in appearance of oscillatory phenomena in bodily and cellular function., Competing Interests: The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper., (© 2021 The Author(s).)
- Published
- 2021
- Full Text
- View/download PDF
31. Are all matrices diagonalisable?
- Author
-
Paranjape, Kapil H.
- Published
- 1997
- Full Text
- View/download PDF
32. Note on Wiener’s prediction theory
- Author
-
Takano, Kinsaku
- Published
- 1953
- Full Text
- View/download PDF
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