Form Of Complex Number Research Articles

Kinematic Analysis of the Draglink Mechanism Using the Newton Rhapson Iteration Method

Position, speed, and acceleration analysis are all included in kinematic analysis. Vectors of location, velocity, and acceleration can be expressed analytically in the form of complex numbers using the Newton Rhapson Iteration Method. The location vector of the point to be investigated must be found first, and then the acceleration and velocity vectors must be found by differentiating against time. This study analyzes the relationship between the θ1 value and the θ2, θ3, θ4, θ5, and x2 values in a drag-link mechanism. The Newton-Raphson iteration method is employed for kinematic analysis to solve nonlinear equations. In this research, the analysis results indicate a correlation between the θ1 value and the θ2, θ3, θ4, θ5, and x2 values. As θ1 increases, the θ2, θ3, θ4, and θ5 values also proportionally increase. The x2 value is also influenced by changes in θ1. This study combines the Newton-Raphson iteration method with graphical analysis using CAD software. The conclusion of this research highlights a significant relationship between the θ1 value and the θ2, θ3, θ4, θ5, and x2 values in the drag-link mechanism. The analysis provides valuable insights for modeling, designing, or controlling systems involving these variables.

Empowering complex-valued data classification with the variational quantum classifier

Empowering complex-valued data classification with the variational quantum classifier

Closed Loop Vector Formulation in Eulers Complex Numbers for Multi Loop Planar Mechanisms With N bars A Novel Modeling Approach and Algorithm

This paper presents a novel iterative algorithm incorporated in a user-friendly GUI for modeling the kinematics of multiple looped N-bar closed-loop mechanisms. Past research works have used custom coding or expensive commercial software to analyze the mechanisms of specific applications. The proposed algorithm focuses on kinematics and offers a quick, easy-to-use, cost-effective solution to analyze a wide range of generic mechanisms, reducing the need for custom coding and lowering computational costs. The algorithm employs algebraic equations, such as solving complex closed-loop vector equations using the Euler form of complex numbers, to simulate and derive the unknowns necessary to characterise any generic closed-loop mechanism. The Python code implemented in the algorithm adapts to various scenarios by utilising available information on the position, velocity, and acceleration variables of the mechanisms. The simulation tool can display real-time color contour plots (RGB color scale) for linear and angular velocities and accelerations, simulate mechanisms with multiple loops and switch configurations, and find inverse mechanisms. The approach for solving multiple loop problems and the algorithm utilized to solve the configurations, methods, equations used and GUI features implementation are all described in this study. The case study considered for a four-bar mechanism indicates a strong agreement between the results obtained from the proposed kinematics-based simulator and ANSYS software. These results demonstrate the simulator’s effectiveness in providing low-cost and user-friendly simulation results for various generic mechanisms involving multiple interconnected loops.

Exploring Prospective Teachers’ Development of the Cartesian Form of Complex Numbers

In this study, we investigated prospective secondary mathematics teachers’ development of a meaning for the Cartesian form of complex numbers by examining the roots of quadratic equations through quantitative reasoning. Data included transcripts of the two sessions of classroom teaching experiments prospective teachers participated in, written artifacts from these teaching sessions, and their answers to pre-and-post written assessment questions. Results point toward prospective teachers’ improved meanings regarding the definition of complex numbers and the algebraic and geometrical meanings of the Cartesian form of complex numbers. Implications for mathematics teacher education include providing specific tasks and strategies for strengthening the knowledge of prospective and in-service teachers.

Analisis Kinematika pada Quick Return Mechanism dengan Menggunakan Metode Bilangan Kompleks

Kinematics analysis includes analysis of position, velocity and acceleration of the rods of a mechanism. The complex number method is a kinematic analytical way to express position, velocity and acceleration vectors in the form of complex numbers. The first step that needs to be done is to determine the position vector of the point to be analyzed, then determine the velocity and acceleration vectors by differentiating with time. After that is done, then the algorithm can be made. The mechanism used for this study is the quick return mechanism. The assumed data for this analysis is the angular velocity of rod 2 is 19.0986 rpm and the acceleration of rod 2 is 0 rad/s2. The results of the position analysis show that if rod 2 rotates one full rotation (360o), then rod 4 rotates about 1/6 circle (232.72o - 294.12o) and the slider moves along the X axis (406.939 - 504.111). The maximum speed behind the slider is 121.45 mm/s while the maximum speed forward for the slider is 101.09 mm/s. A large change in velocity (acceleration) is needed when bar 4 and the slider change direction (from CW to CCW or vice versa), this is indicated by a decrease and increase at the beginning and end of the graphic with significant acceleration

Temporal Neural Network Framework Adaptation in Reconfigurable Intelligent Surface-Assisted Wireless Communication

A reconfigurable intelligent surface (RIS) has potential for enhancing the performance of wireless communication. A RIS includes cheap passive elements, and the reflecting of signals can be controlled to a specific location of users. In addition, machine learning (ML) techniques are efficient in solving complex problems without explicit programming. Data-driven approaches are efficient in predicting the nature of any problem and can provide a desirable solution. In this paper, we propose a temporal convolutional network (TCN)-based model for RIS-based wireless communication. The proposed model consists of four TCN layers, one fully connected layer, one ReLU layer, and lastly a classification layer. In the input, we provide data in the form of complex numbers to map a specified label under QPSK and BPSK modulation. We consider 2×2 and 4×4 MIMO communication using one base station and two single-antenna users. We have considered three types of optimizers to evaluate the TCN model. For benchmarking, long short-term memory (LSTM) and without ML are compared. The simulation results are conducted in terms of the bit error rate and symbol error rate which show the effectiveness of the proposed TCN model.

A complex Jensen–Shannon divergence in complex evidence theory with its application in multi-source information fusion

Multi-source information fusion has attracted considerable attention in the few past years and plays a great role in real applications. However, the uncertainty or conflict will make the fusion results unreasonable. Furthermore, the information may be collected in the form of complex number that cannot be processed by existing methods. In this article, to handle the above issues, the complex evidence theory (CET) is exploited. CET is the generalization of Dempster–Shafer evidence theory, where the mass function is modeled by complex number, called complex mass function (CMF). In order to deal with multi-source information fusion from the perspective of the complex plane, a new Complex Jensen–Shannon divergence (CJS divergence) is presented in this article. The proposed CJS divergence can effectively measure the conflict between two CMFs, and it satisfies the properties of boundedness, symmetry and nondegeneracy. In addition, for a better combination result, we have adjusted the complex Dempster’s rule of combination, which is called the reinforced complex evidence combination rule (RCECR). Then an algorithm for multi-source information fusion is proposed based on the CJS divergence and the RCECR. Some numerical examples and two applications in target identification and medical diagnosis illustrate the effectiveness of the new approach.

Новые задачи начертательной геометрии. Продолжение

Complex geometry is a synthesis of Euclidean E-geometry (circle geometry) and pseudo-Euclidean M-geometry (hyperbola geometry). Each of them individually defines a non-closed system in which a correctly posed problem may not give a solution. Analytical geometry represents a closed system. In it, a correctly posed problem always gives solutions in the form of complex numbers, for each of which, one of the parts may be equal to zero. Finding imaginary solutions and imaginary figures formed by a set of such solutions is a new problem in descriptive geometry. Degenerated conics and quadrics, or curves and surfaces of higher orders, constitute a new class of figures and a new class of problems in descriptive geometry. For example, null-circle, null-sphere, null-cylinder, null-torus. In this paper the problem for studying the shape of second (conics, quadrics), third (conoid), and fourth (torus) order figures is posed. The latest suggest a meeting with new geometric properties of figures. Geometric operations are still immersed in the complex space E + M or real - imaginary. The examples under consideration continue a series of degenerated figures in which the null-circle splits into isotropic lines. Isotropic lines are taken as generators of surfaces. They form a cone of revolution and a hyperbolic paraboloid (an oblique plane).

Translation-Based Embeddings with Octonion for Knowledge Graph Completion

Knowledge representation learning achieves the automatic completion of knowledge graphs (KGs) by embedding entities into continuous low-dimensional vector space. In knowledge graph completion (KGC) tasks, the inter-dependencies and hierarchical information in KGs have gained attention. Existing methods do not well capture the latent dependencies between all components of entities and relations. To address this, we introduce the mathematical theories of octonion, a more expressive generalized form of complex number and quaternion, and propose a translation-based KGC model with octonion (TransO). TransO models entities as octonion coordinate vectors, relations as the combination of octonion component matrices and coordinate vectors, and uses specific grouping calculation rules to interact between entities and relations. In addition, since hyperbolic Poincaré space in non-Euclidean mathematics can represent hierarchical data more accurately and effectively than traditional Euclidean space, we propose a Poincaré-extended TransO model (PTransO). PTransO transforms octonion coordinate vectors into hyperbolic embeddings by exponential mapping, and integrates the Euclidean-based calculations into hyperbolic space by operations such as Möbius addition and hyperbolic distance. The experimental results of link prediction indicate that TransO outperforms other translation-based models on the WN18 benchmark, and PTransO further achieves state-of-the-art performance in low-dimensional space on the well-established WN18RR and FB15k-237 benchmarks.

CSI Feedback Based on Complex Neural Network for Massive MIMO Systems

In order to solve the problem of large channel state information (CSI) feedback overhead and low feedback accuracy in massive multiple-input multiple-output (MIMO) systems. We propose a CSI feedback method based on complex-valued convolutional neural networks to improve the representation capability of the network. In this method, a complex-valued encoder-decoder structure is constructed considering the fact that CSI exists in the form of complex numbers. We use complex convolutional downsampling (CCD) to extract CSI features in the encoder, and reconstruct the compressed CSI with high accuracy in the decoder by using a complex dense block (CDBlock). Simulation results show that the average accuracy is improved by 17.5% compared with several classical deep learning CSI feedback methods. Our proposed CSI feedback method has higher feedback accuracy and better system performance in massive MIMO systems.

Modulation Recognition Method Based on Complex Neural Network

With the increasingly intensive communication environment and the indefinite appearance of signal modulation, it is more and more difficult to identify signal modulation. Modulation recognition technology is very important in both civil and military fields. In order to solve the problems of multiple signal styles and low signal-to-noise ratio, a modulation recognition method based on complex neural network is proposed, which involves the field of wireless communication technology. For the complex signal widely existing in the field of communication, this method proposes a new method, which uses the complex signal received in time domain. It does not need to extract any parameters from the received signal. It only needs to input the complex data into the complex neural network for training, and fully learn the characteristics of the real part and the imaginary part of the data. It can obtain higher accuracy than the traditional high-order accumulation method, the complex number has more abundant expression ability. The complex number neural network can learn the real part and imaginary part of the complex number, which is more suitable for most communication signals in the form of complex number. This method does not need to manually calculate and observe the differences between various signals or eigenvalues, and does not need to manually set the threshold to distinguish the modulation types of signals, so the neural network has the function of classifier.

Some Novel Cosine Similarity Measures Based on Complex Hesitant Fuzzy Sets and Their Applications

The theory of complex hesitant fuzzy set (CHFS) is a modification technique of the complex fuzzy set (CFS) to cope with awkward and unreliable information’s in daily life issues. CHFS contains the grade of truth in the form of complex number, whose real and imaginary parts are in the form of the finite subset of the unit interval. CHFS is the mixture of hesitant fuzzy set (HFS) and CFS, which handles the complex and uncertain information in real-world issues which is compared with fuzzy sets and complex fuzzy sets. The positive membership in CHFS is in the form a polar coordinate belonging to unit disc in the complex plane. The aims of this manuscript are to explore some similarity measures (SMs), weighted SMs (WSMs) such as cosine SMs, weighted cosine SMs, SMs based on cosine function, WSMs based on cosine function, SMs based on tangent function, and WSMs based on tangent function of CHFS. Some special cases of the presented measures are discussed in detail. Moreover, we use our described SMs and weighted SMs of CHFS in the environment of medical diagnosis and pattern recognition to assess the practicality and competence of the described SMs. Finally, to find the validity and proficiency of the investigated measures based on CHFSs, the comparison between explored measures with some already defined measures and their graphical representations are also discussed in detail.

The base matrix of hermitian operator order n

The Eigen problem is a problem that exists when there is an operator that is affected by the function, then that function does not change except being multipled by the constant in the form of an eigenvalue. In general, the eigenvalue problem uses the Hermitian operators by using observable variable in the form of mathematical operations. Hermitian operator is an operator that happens when the Hermit conjugation is performed, then the operator will return to its original condition. This research aims to determine the base matrix of Hermitian operators with order n ≤ 4 (size 2 × 2, 3 × 3 and 4 × 4). Entry element of the hermitian matrix operator uses complex numbers of which each order consists of few Hermitian operators. The method used is an analytical method by inserting problems into the eigen equation then normalized to obtain a diagonal matrix, so as to obtain a complete solution of the eigen problem on form of a matrix (including eigenvalues, normalized eigenvectors, and base matrix). Base matrix that has been obtained will be a diagonal matrix with diagonal elements of the eigenvalue. The results of this research indicate that the base matrix obtained in each order forms a square matrix that has entry elements in the form of complex numbers.

Complex fuzzy ordered weighted quadratic averaging operators

A complex fuzzy set, the generalization of fuzzy set provides a powerful mathematical framework whose membership degrees are in the form of complex numbers in the unit disc. The averaging operators consisting of the properties of both t-norm and t-conorm are of great importance in complex fuzzy environment. In this paper, we present certain quadratic averaging operators, including complex fuzzy weighted quadratic averaging, complex fuzzy ordered weighted quadratic averaging, complex fuzzy Einstein weighted quadratic averaging and complex fuzzy Einstein ordered weighted quadratic averaging operators. These operators are used to study many different issues of periodic nature. We apply these models to the multi-attribute decision-making problems and wireless detection of target location. Conclusively, we can choose the best opinion by the ranking of the aggregated outputs and detect the position and direction of a target. Moreover, we describe these models through numerical examples to check their validity and importance in real life problems. To explain the consistency and authenticity of our model, we examine a comparative analysis with existing aggregation techniques.

The Polar form of a Neutrosophic Complex Number

In this paper, we will define the exponential form of a neutrosophic complex number. We have proven some characteristics and theories, including the conjugate of the exponential form of a neutrosophic complex number, division of the exponential form of a neutrosophic complex numbers, multiplication of the exponential form of a neutrosophic complex numbers. In addition, we have given the method of changing from the exponential to the algebraic form of a complex number.

Group Decision-Making Using Complex q-Rung Orthopair Fuzzy Bonferroni Mean

Complex q-rung orthopair fuzzy set (CQROFS), as a modified notion of complex fuzzy set (CFS), is an important tool to cope with awkward and complicated information. CQROFS contains two functions which are called truth grade and falsity grade by the form of complex numbers belonging to unit disc in a complex plane. The condition of CQROFS is that the sum of q-powers of the real part (Also for imaginary part) of the truth grade and real part (Also for imaginary part) of the falsity grade is limited to the unit interval. Bonferroni mean (BM) operator is an important and meaningful concept to examine the interrelationships between the different attributes. Keeping the advantages of the CQROFS and BM operator, in this manuscript, the complex q-rung orthopair fuzzy BM (CQROFBM) operator, complex q-rung orthopair fuzzy weighted BM (CQROFWBM) operator, complex q-rung orthopair fuzzy geometric BM (CQROFGBM) operator, and complex q-rung orthopair fuzzy weighted geometric BM (CQROFWGBM) operator are proposed, and some properties are discussed, further, based on the CQROFWGBM operator, a multi-attribute group decision-making (MAGDM) method is developed, and the ranking results are examined by score function. Finally, we give some numerical examples to verify the rationality of the established method, and show its advantages by comparative analysis with some existing methods.

Secondary teachers’ conception of various forms of complex numbers

This study explores in-service high school mathematics teachers’ conception of various forms of complex numbers and ways in which they transition between different representations of these forms. One 90-min interview was conducted with three high school mathematics teachers after they completed three professional development sessions, each 4 h, on complex numbers. Results indicate that, in general, these teachers did not necessarily have a dual conception of complex numbers. However, they demonstrated varying conceptions with different forms of complex numbers. Teachers worked at an operational level with the exponential form of complex numbers, but there was no evidence to indicate that they had a structural conception of this form. On the other hand, two teachers were very comfortable with the Cartesian form and exhibited a process/object duality by translating between different representations of this form. These results indicate that high school teachers need more opportunities to help them develop a dual conception of each form (multiple duals), which in turn can result in developing a dual conception of complex numbers. An interesting phenomenon that we found was that teachers who taught courses such as geometry and international baccalaureate were able to draw from their teaching experiences as they attempted the interview tasks. This particular observation may suggest that teachers’ teaching assignments coupled with appropriate professional development activities could facilitate their understanding of these concepts.

Euler–Savary formula for the homothetic motion in the complex plane [formula omitted

This brief note expanding on one aspect of paper [1], which deals with the complex number form of Euler–Savary formula. This formula is the relationship between the radius of the curvature and the curvature center of a path drawn by a point in the coplanar rolling of two conjugate curves on one another.We study complex number form of the Euler–Savary formula for one parameter homothetic planar motion and the case of the homothetic scale h is equal to 1, given in [1], will appear as a special case of this study.

Probabilistic Interpretation of Complex Fuzzy Set

The innovative concept of Complex Fuzzy Set is introduced. The objective of the this paper to investigate the concept of Complex Fuzzy set in constraint to a traditional Fuzzy set , where the membership function ranges from [0, 1], but in the Complex fuzzy set extended to a unit circle in a complex plane, where the member ship function in the form of complex number. The Compressive study of mathematical operation of Complex Fuzzy set is presented. The basic operation like addition, subtraction, multiplication and division are described here. The Novel idea of this paper to measure the similarity between two fuzzy relations by evaluating δ -equality. Here also we introduce the probabilistic interpretation of the complex fuzzy set where we attempted to clarify the distinction between Fuzzy logic and probability. KEY WORD Complex fuzzy set, Complex fuzzy relation, complement of complex fuzzy relation, δ -equality, probability.

A distribution for a pair of unit vectors generated by Brownian motion

We propose a bivariate model for a pair of dependent unit vectors which is generated by Brownian motion. Both marginals have uniform distributions on the sphere, while the conditionals follow so-called “exit” distributions. Some properties of the proposed model, including parameter estimation and a pivotal statistic, are investigated. Further study is undertaken for the bivariate circular case by transforming variables and parameters into the form of complex numbers. Some desirable properties, such as a multiplicative property and infinite divisibility, hold for this submodel. Two estimators for the parameter of the submodel are studied and a simulation study is carried out to investigate the finite sample performance of the estimators. In an attempt to produce more flexible models, some methods to generalize the proposed model are discussed. One of the generalized models is applied to wind direction data. Finally, we show how it is possible to construct distributions in the plane and on the cylinder by applying bilinear fractional transformations to the proposed bivariate circular model.