Representation Of Complex Numbers Research Articles

On logic gates with complex numbers

Logic gates can be written in terms of complex differential operators, where the inputs and outputs are holomorphic functions with several variables. Using the polar representation of complex numbers, we arrive at an immediate connection between the oscillatory behavior of the system and logic gates. We discuss the universality of this formalism in a variety of computing systems.

The imaginary unit described as an algebraic ambiguity

A complex number is formed by a real part and an imaginary part, the latter being composed of a real number multiplied by the imaginary unit “i,” defined as the square root of . For this reason, imaginary numbers cannot be placed on the real number line. The geometric representation of complex numbers is done in the so-called complex plane, composed of a real axis and an imaginary axis, orthogonal to the real line. However, from a philosophical point of view, the fact that the imaginary axis is not identified with any axis in three-dimensional Euclidean space raises ontological questions that transcend the well-established mathematical formalism of the set of complex numbers: are imaginary numbers just a sophism, an arbitrary invention, or a true mathematical discovery? Are imaginary numbers a glimpse of a parallel reality that is beyond our perception? In this article, I seek to demonstrate that the imaginary unit can be interpreted as an algebraic ambiguity that arises when representing certain vectors in the real plane as one-dimensional variables. Consequently, we will see how it is possible to express complex numbers in terms of real numbers.

Inter-robot management via neighboring robot sensing and measurement using a zeroing neural dynamics approach

This paper proposes a complex number representation method for dynamically recording robot positions and develops an optimization strategy for measuring and minimizing inter-robot distances in real-time. Based on this, the optimization strategy is transformed into finding the theoretical solution of the complex time-varying matrix equation by combining the leader–follower frameworks, and a complex zeroing neural dynamics (CZND) model is proposed for this objective. Meanwhile, the effectiveness of the proposed CZND model and the ideal trajectories of the follower robots are ensured and given through rigorous theoretical analyses. Subsequently, these theoretical findings are validated by two numerical experiments (four cases). In addition, the superiority of the CZND model for inter-robot management tasks is comparatively analyzed from three perspectives: initial positions, number of follower robots, and design parameters. Compared to conventional gradient-based neural network (CGNN) methods, the proposed CZND model, based on a dynamic error function (which is not required to be non-negative as in CGNN), demonstrates improved handling of the dynamic behavior of the system by utilizing the first-order time derivatives of matrices, resulting in lower residual errors in the implementation of inter-robot management. Specifically, for the same inter-robot management task, the steady-state error implemented by CZND is on the order of 10−4, whereas for CGNN, it is on the order of 10−1, indicating a significantly larger lagging error.

On the Vector Representation of Characteristic Functions

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.

Waving arms around to teach quantum mechanics

Kinesthetic (or embodied) representations help students build intuition and deep understanding of concepts. This paper presents a series of kinesthetic activities for a spins-first undergraduate quantum mechanics course that supports students in reasoning and developing intuition about the complex-valued vectors of spin states. The arms representation, used in these activities, was developed as a tangible representation of complex numbers: Students act as an Argand diagram, using their left arm to represent numbers in the complex plane. The arms representation is versatile and can be expanded to depict complex-valued vectors with groups of students. This expansion enables groups of students to represent quantum mechanical state vectors with their arms. We have developed activities using the arms representation that parallel the progression of a spins-first approach by starting with complex numbers, then representing two- and three-state systems, considering time-dependence, and, eventually, extending to approximate wavefunctions. Each activity illustrates the complex nature of quantum states and provides a tangible manipulative from which students can build intuition about quantum phenomena.

Nonlinear Identification with Constraints in Frequency Domain of Electric Direct Drive with Multi-Resonant Mechanical Part

Knowledge of a direct-drive model with a complex mechanical part is important in the synthesis of control algorithms and in the predictive maintenance of digital twins. The identification of two-mass drive systems with one low mechanical resonance frequency is often described in the literature. This paper presents an identification workflow of a multi-resonant mechanical part in direct drive with up to three high-frequency mechanical resonances. In many methods, the identification of a discrete time (DT) model is applied, and its results are transformed into a continuous-time (CT) representation. The transformation from a DT model to a CT model has limitations due to nonlinear mapping of discrete to continuous frequencies. This problem may be overcome by identification of CT models in the frequency domain. This requires usage of a discrete Fourier transform to obtain frequency response data as complex numbers. The main work presented in this paper is the appropriate fitting of a CT model of a direct-drive mechanical part to complex number datasets. Fitting to frequency response data is problematic due to the attraction of unexcited high frequency ranges, which lead to wrong identification results of multi-mass (high order) drive systems. Firstly, a CT fitting problem is a nonlinear optimization problem, and, secondly, complex numbers may be presented in several representations, which leads to changes in the formulation of the optimization problem. In this paper, several complex number representations are discussed, and their influence on the optimization process by simulation evaluation is presented. One of the best representations is then evaluated using a laboratory setup of direct drive with unknown parameters of three high mechanical resonance frequencies. The mechanical part of the examined direct drive is described by three mechanical resonances and antiresonances, which are characteristic of a four-mass drive system. The main finding is the addition of frequency boundaries in the identification procedure, which are the same as those in the frequency range of the excitation signal. Neither a linear least-square algorithm nor a nonlinear least-square algorithm is suitable for this approach. The usage of nonlinear least-square algorithm with constraints as a fitting algorithm allows one to solve the issue of modeling multi-mass direct-drive systems in the frequency domain. The second finding of this paper is a comparison of different cost functions evaluated to choose the best complex number representation for the identification of multi-mass direct-drive systems.

Filtration network: A frame sampling strategy via deep reinforcement learning for video captioning

Recently many methods use encoder-decoder framework for video captioning, aiming to translate short videos into natural language. These methods usually use equal interval frame sampling. However, lacking a good efficiency in sampling, it has a high temporal and spatial redundancy, resulting in unnecessary computation cost. In addition, the existing approaches simply splice different visual features on the fully connection layer. Therefore, features cannot be effectively utilized. In order to solve the defects, we proposed filtration network (FN) to select key frames, which is trained by deep reinforcement learning algorithm actor-double-critic. According to behavior psychology, the core idea of actor-double-critic is that the behavior of agent is determined by both the external environment and the internal personality. It avoids the phenomenon of unclear reward and sparse feedback in training because it gives steady feedback after each action. The key frames are sent to combine codec network (CCN) to generate sentences. The operation of feature combination in CCN make fusion of visual features by complex number representation to make good semantic modeling. Experiments and comparisons with other methods on two datasets (MSVD/MSR-VTT) show that our approach achieves better performance in terms of four metrics, BLEU-4, METEOR, ROUGE-L and CIDEr.

Fuzzy complex numbers: Representations, operations, and their analysis

As a key theoretical basis of fuzzy complex analysis, a new representation of fuzzy complex numbers is proposed in this paper which unifies the exponential form, trigonometric form and algebraic form of fuzzy complex numbers, and some arithmetic operations are investigated. The concepts of strong sum and strong difference are defined in order to simplify the addition and subtraction operations of complex fuzzy numbers and establish the calculus theory of fuzzy complex valued functions. The metric between two fuzzy complex numbers, the derivative, differentiability and analyticity of fuzzy complex-valued functions are defined and characterized. The necessary and sufficient conditions of the analyticity of fuzzy complex-valued functions are investigated. Furthermore, the integral of fuzzy complex-valued functions and Cauchy integral theorem are given and discussed.

Representations for complex numbers with integer digits

We present the zeta-expansion as a complex version of the well-known beta-expansion. It allows us to expand complex numbers with respect to a complex base by using integer digits. Our concepts fits into the framework of the recently published rotational beta-expansions. But we also establish relations with piecewise affine maps of the torus and with shift radix systems.

CPL-SLAM: Efficient and Certifiably Correct Planar Graph-Based SLAM Using the Complex Number Representation

In this article, we consider the problem of planar graph-based simultaneous localization and mapping (SLAM) that involves both poses of the autonomous agent and positions of observed landmarks. We present complex (CPL)-SLAM, an efficient and certifiably correct algorithm to solve planar graph-based SLAM using the complex number representation. We formulate and simplify planar graph-based SLAM as the maximum likelihood estimation on the product of unit complex numbers, and relax this nonconvex quadratic complex optimization problem to convex complex semidefinite programming (SDP). Furthermore, we simplify the corresponding complex SDP to Riemannian staircase optimization (RSO) on the complex oblique manifold that can be solved with the Riemannian trust region method. In addition, we prove that the SDP relaxation and RSO simplification are tight as long as the noise magnitude is below a certain threshold. The efficacy of this work is validated through applications of CPL-SLAM and comparisons with existing state-of-the-art methods on planar graph-based SLAM, which indicates that our proposed algorithm is capable of solving planar graph-based SLAM certifiably, and is more efficient in numerical computation and more robust to measurement noise than existing state-of-the-art methods. The C++ code for CPL-SLAM is available at https://github.com/MurpheyLab/CPL-SLAM.

On lp-Complex Numbers

Dispensing with the common property of distributivity and replacing classical trigonometric functions with their l p -counterparts in Euler’s trigonometric representation of complex numbers, classes of l p -complex numbers are introduced and some of their basic properties are proved. The collection of all points that leave the l p -absolute value of each l p -complex number invariant under l p -complex numbers multiplication is shown to be a group of elements that have l p -absolute value one but not the symmetry group.

Complex and Hypercomplex-Valued Support Vector Machines: A Survey

In recent years, the field of complex, hypercomplex-valued and geometric Support Vector Machines (SVM) has undergone immense progress due to the compatibility of complex and hypercomplex number representations with analytic signals, as well as the power of description that geometric entities provide to object descriptors. Thus, several interesting applications can be developed using these types of data and algorithms, such as signal processing, pattern recognition, classification of electromagnetic signals, light, sonic/ultrasonic and quantum waves, chaos in the complex domain, phase and phase-sensitive signal processing and nonlinear filtering, frequency, time-frequency and spatiotemporal domain processing, quantum computation, robotics, control, time series prediction, and visual servoing, among others. This paper presents and discusses the importance, recent progress, prospective applications, and future directions of complex, hypercomplex-valued and geometric Support Vector Machines.

A Polar Representation for Complex Interval Numbers

The present work defines the basic elements for the introduction to the Study of Complex variables under the mathematical interval context with the goal of using it as a foundation for the understanding of pure mathematical problems, associating the mathematical interval to support the results. The present article contributes to the complex interval theory taking into consideration Euler’s Identity and redefining the polar representation of interval complex numbers. In engineering, the present article could be used as a subsidy for many applications where complex variable theory is applicable and requires accurate results.

복소수 개념에 관한 학생 오류 해석에 필요한 교사지식

In this study, the teacher’s knowledge and its features necessary to analyze student errors about the concept of complex numbers are revealed. The instruments used for the diagnosis of and correcting student’s responses were administered to 36 pre-service teachers. The results and analysis of this study showed that. First, a teacher’s knowledge in the step of error judgement is the key to determine the direction and accuracy of diagnosis of and correcting student’s errors. Secondly, it was confirmed that there is the need for the precise knowledge about the representation of complex numbers and the knowledge about the essential prerequisites for the harmony of the complex number system and real number system. Thirdly, pre-service teachers need to have a critical approach to the definition and operations of complex numbers. Lastly, the knowledge about the internal and external value and the usefulness of complex numbers was proposed as the teacher's knowledge necessary to analyze student's errors about the concept of complex numbers.

Una secuencia didáctica para favorecer la conversión de representaciones semióticas de curvas y regiones del plano complejo

En este trabajo se presentan la descripción, y los resultados de la implementación de una secuencia didáctica orientada a favorecer la habilidad de efectuar conversiones de representaciones de curvas y regiones del plano complejo, desde el registro gráfico al registro algebraico. Se detallan las variables de conversión que regulan los niveles de complejidad en estas actividades cognitivas, que determinaron la estructura de la secuencia didáctica presentada. Esta fue implementada con un grupo de estudiantes de la asignatura Álgebra de primer año de la Facultad de Ingeniería de la Universidad Nacional de Mar del Plata, Argentina. Se evaluó el impacto de la intervención didáctica realizada y se obtuvieron resultados satisfactorios, no solo sobre las conversiones a las que buscó específicamente contribuir, sino también en prácticas matemáticas que requieren otras actividades semióticas vinculadas a representaciones de números complejos en los dos registros considerados.

The Method of Interacting Configurations in Complex Number Representations: From Helium to the Complex Atoms

The transition from the He atom to the complex atoms description in the method of interacting configurations in the complex number representation has been presented. As a first step the problem of ionization of H- and Li+ ions is considered. The spectroscopic characteristics of the Be, Mg and Ca atoms in the problem of the electron-impact ionization of these atoms are investigated. Few results in the photoionization problem on the 1P autoionizing states above the n=2 threshold of helium-like Be++ ion are presented. The energies and the widths of the lowest 1S; 1P; 1D; and 1F autoionizing states of the Be, Mg atoms, and the lowest (1P) autoionizing states of Ca atom, are calculated. We consider briefly both a review of our previous results (together with results of other authors) and new calculations of our group. A brief review of the methods of the quasi-stationary states calculation is given.

Pythagorean Triples, Complex Numbers, and Perplex Numbers

SummaryAfter reviewing the use of Gaussian integers to generate Pythagorean triples, we consider the history of the perplex numbers and use them in an analogous way to generate Pythagorean triples. We also show that primitive Pythagorean triples form a group under the perplex number system, different from the group determined by the complex number representations.

Genre-based Link Prediction in Bipartite Graph for Music Recommendation

The bipartite graph method of link prediction can apply in many fields of recommendations, with the nodes (users and items) and links (interactions between users and items). However, that links cannot represent the users’ dual preferences (like and dislike). Some researchers improved that limits by complex number representations, but still not consider the influence of users’ similarity recommendation performance. Here, we proposed an improved method to cope with this deficiency, build the relational dualities by complex number representations and computing the users’ similarity by genres weight relations. In experiments with the Xiami.com music dataset, the proposed music genre weight-based music recommendation model (MGW) performances better than the CORLP method.

Застосування методу взаємодіючих конфігурацій у зображенні комплексних чисел до розрахунків спектроскопічних характеристик автоіонізаційних станів атомів Ве, Мg, Ca

Застосування методу взаємодіючих конфігурацій у зображенні комплексних чисел до розрахунків спектроскопічних характеристик автоіонізаційних станів атомів Ве, Мg, Ca

Uma proposta pedagógica: usando o software Geogebra na rotação vectores complexos

Este artigo apresenta um relato de experiência de uma proposta pedagógica posta em prática em uma turma do curso de Licenciatura em Matemática, na área de variáveis complexas, lidando com o conteúdo dos números complexos e sua representação geométrica no plano de Argand-Gauss, mais especificamente, o vector de rotação no plano. A proposta apresenta uma análise de livros didáticos e dissertações a respeito desse conteúdo de ensino, de onde pode se justificar a necessidade de trabalhar com diferentes representações de números complexos para que os alunos alcancem um bom aprendizado. O software Geogebra foi usado com um recurso de tecnologia de informação para a realização das aulas.