Complex Linear Equations Research Articles

AN ALGEBRAIC SOLUTION OF DUAL FUZZY COMPLEX LINEAR SYSTEMS

The main aims of this research is to solve the dual fuzzy complex linear system AX +C =BX +D, where A,B are crisp coefficient matrices and C ,D are fuzzy number matrices. The research findings indicate that the system of two fuzzy complex linear equations can be solved under the conditional that (A-B)^(-1) and (|B|-|A|)^(-1) exists.

Growth of meromorphic solution related to linear difference equations and value sharing

In this paper, we consider the growth of meromorphic solutions of complex linear difference equations of certain types and derive some results, which improve the results of Chiang and Feng [On the Nevanlinna characteristic of f ( z + η ) and difference equations in the complex plane. Ramanujan J. 2008;16(1):105–129; On the growth of logarithmic differences, difference quotients and logarithmic derivatives of meromorphic functions. Trans Am Math Soc. 2009;361:3767–3791] and Chen [Growth and zeros of meromorphic solution of some linear difference equations. J Math Anal Appl. 2011;373:235–241]. Also in this paper, we consider the sharing value problems of entire functions with their difference operators and obtain a result which improves the result of Li, Yang and Yi [Entire functions sharing an entire function of smaller order with their shifts. Proc Jpn Acad. 89 (A) (2013), 34–39]. Moreover, we show by illustrating a number of examples that our results are best possible in certain senses.

An Effective Algorithm for Solving Weak Fuzzy Complex Diophantine Equations in Two Variables

Weak fuzzy complex numbers are defined as , with as an extension of real numbers with . This paper is dedicated to studying weak fuzzy complex linear Diophantine equations in two weak fuzzy complex variables, by transforming the weak fuzzy complex Diophantine equation to a classical equivalent Diophantine system and going directly from the solutions of classical system into the desired equation. Algorithms for generating solutions of the previous equation will be presented in terms of theorems with many related examples that clarify the validity of our work.

Finite Logarithmic Order Meromorphic Solutions of Complex Linear Delay-Differential Equations

Finite Logarithmic Order Meromorphic Solutions of Complex Linear Delay-Differential Equations

Finite Element–Boundary Element Acoustic Backscattering with Model Reduction of Surface Pressure Based on Coherent Clusters

Computing backscattering of harmonic acoustic waves from underwater elastic targets of arbitrary shape is a problem of considerable practical significance. The finite element method is commonly applied to the discretization of the target; on the other hand, the boundary element method naturally incorporates the radiation boundary condition at infinity. The coupled model tends to be expensive, primarily due to the need to manipulate large, dense, and complex matrices and to repeatedly solve systems of complex linear algebraic equations of significant size for each frequency of interest. In this article, we develop a model reduction transformation based on the notion of coherence applied to the surface pressures, which considerably reduces the size of the systems to be solved. We found that the proposed model reduction approach delivers acceptably accurate results at a fraction of the cost of the full model. A typical speedup of an order of magnitude was realized in our numerical experiments. Our approach enables backscattering computations with considerably larger models than have been feasible to date.

Performance analysis of compressive sensing and reconstruction by LASSO and OMP for audio signal processing applications

Audio signal processing is used in acoustic IoT sensor nodes which have limitations in data storage, computation speed, hardware size and power. In most audio signal processing systems, the recovered data constitutes far less fraction of the sampled data providing scope for compressive sensing (CS) as an efficient way for sampling and signal recovery. Compressive sensing is a signal processing technique in which a sparse approximated signal is reconstructed at the receiving node by a signal recovery algorithm, using fewer samples compared to traditional sampling methods. It has two main stages: sparse approximation to convert the signal into a sparse domain and reconstruction through sparse signal recovery algorithms. Recovery algorithms involve complex matrix multiplication and linear equations in sampling and reconstruction, increasing the computational complexity and leading to highly resourceful hardware implementations. This work reconstructs the sparse audio signal using LASSO and orthogonal matching pursuit (OMP) algorithm. OMP is an iterative greedy algorithm involving least square method that takes a compressed signal as input and recovers it from the sparse approximation, while LASSO is L1 norm based with a controlled L2 penalty. The paper reviews the reconstruction and study of sparsity and error obtained for reconstructing an audio signal by OMP and LASSO.

A fixed-time distributed extended state observer for uncertain second-order nonlinear system

This paper investigates the fixed-time distributed estimation problem for a class of second-order nonlinear systems with uncertain input, unknown nonlinearity and matched perturbation. A fixed-time distributed extended state observer (FxTDESO) consisting of a group of local observer nodes under directed communication topology is proposed, and each node can reconstruct both the full state and unknown dynamics of the system. To achieve fixed-time stability, a Lyapunov function is elaborated, and based on this, sufficient conditions for the existence of the FxTDESO are established. Under time-invariant and time-varying disturbance, the observation errors can converge to the origin and a small region of the origin within a fixed time, respectively, where the upper bound of the settling time (UBST) is irrelevant to the initial conditions. Compared to the existing fixed-time distributed observers, the proposed observer can reconstruct both the unknown states and uncertain dynamics, and only the output of the leader and 1-dimensional output estimates from the neighboring nodes are needed in the observer design which effectively reduces the communication load. The paper also extends previous finite-time distributed extended state observer to the case of time-variant disturbance and eliminates the complex linear matrix equation assumption that guarantees the finite-time stability. Furthermore, the FxTDESO design for a class of high-order nonlinear systems is also discussed. Finally, simulation examples are conducted to demonstrate the effectiveness of the proposed observer.

A Geometric Multigrid Method for 3D Magnetotelluric Forward Modeling Using Finite-Element Method

The traditional three-dimensional (3D) magnetotelluric (MT) forward modeling using Krylov subspace algorithms has the problem of low modeling efficiency. To improve the computational efficiency of 3D MT forward modeling, we present a novel geometric multigrid algorithm for the finite element method. We use the vector finite element to discretize Maxwell’s equations in the frequency domain and apply the Dirichlet boundary conditions to obtain large sparse complex linear equations for the solution of EM responses. To improve the convergence of the solution at low frequencies we use the divergence correction to correct the electric field. Then, we develop a V-cycle geometric multigrid algorithm to solve the linear equations system. To demonstrate the efficiency and effectiveness of our geometric multigrid method, we take three synthetic models (COMMEMI 3D-2 model, Dublin test model 1, modified SEG/EAEG salt dome model) and compare our results with the published ones. Numerical results show that the geometric multigrid algorithm proposed in this paper is much better than the commonly used Krylov subspace algorithms (such as SOR-GMRES, ILU-BICGSTAB, SOR-BICGSTAB) in terms of the iteration number, the solution time, and the stability, and thus is more suitable for large-scale 3D MT forward modeling.

Efficient and Explicit Fourier Modal Method for Ultrathin Metallic Gratings

The solution of large matrix eigenvalues and complex linear equations limits the Fourier modal method (FMM) application in ultrathin metallic gratings (UMG) analysis. This paper proposes an efficient and explicit FMM method for analyzing UMG. The proposed method avoids solving complex linear equations and eigenvalues of eigenmatrix in the conventional method by simplifying the implementation equations. Two numerical examples then verify the reliability of the proposed method compared with CST simulations and the conventional method. The proposed method is proven efficient in decreasing the CPU time by over 80% and demanding significantly less memory.

The Multivariate Statistical Analysis – Multiple Linear Regression

Background. When processing the results of observational studies we ave to usemultivariate statistical methods that will examine the simultaneous influence of both independent and confounding variables on the outcome, Objective: The aim of this paper is to further explain how a researcher could decide whether multiple linear regression is suitable statistical option for processing his (her) data, and then how to implement it properly. Methods: This article is a narrative review of literature about logic, assumptions, quality check and interpretation of multiple linear regression. Results: Multiple linear regression is a complex linear equation (model) in which on one side of the equal sign is the absolute value of the dependent variable (i.e., the outcome), and on the other is a sum of additions, of which only one is a constant, and all others are the product of an independent or confounding variable and their coefficients. After checking assumptions and quality of the model, we may decide whether a predictor has significant influence on outcome, or not, and calculate size of this influence. Conclusions: Multiple linear regression is an extremely useful statistical model for explaining the influence of multiple predictors simultaneously on a continuous type dependent variable, but it requires the fulfillment of fairly strict assumptions in order to be used. That's why multiple linear regression should be used only when the conditions are met, otherwise other types of linear and non-linear models whose assumptions are far more lenient should be resorted to. .

Investigation of Flow of the Disc-and-Doughnut Baffles and 40% Cut Segmental Baffles

Heat exchanger is usually used in manufacturing process. At present, many researchers have efforts to increase the performance of the heat exchanger with less of the cost. This research discussed about the performance of heat exchanger using 40% cut segmental baffles compared with modified double segmental baffles disc-and-doughnut type. In this study, the investigation of the computational results consisted of heat flux, velocity profile along the heat exchanger, pressure distribution and, theoretical heat transfer coefficient and heat exchanger effectiveness. The model was calculated using finite difference method forward modeling with Multiphysics Software and focuses on the performance evaluation of the small shell-and-tube heat exchanger (STHE) – laboratory type. The tubes are composed of 14 tubes with 0.583 m length, triangular 30° rotated pitch. The pipe radius of shell and tube are 0.055 m and 0.00635 m, respectively. The baffle radius of disc and doughnut are 0.055 m and 0.025 m, respectively, and the baffle radius of 40% cut segmental baffles are 0.055 m. Both types of the baffle have a distance of 0.116 m which is evenly distributed along the shell. The generalized minimal residual (GMRES) method for the fluid flow case used as an iterative method for solving some of the complex linear equations shows good performance as in reliability and validity. For the 40% cut segmental baffles, fluid flow makes a zigzag pattern, an Eddy or swirling of a fluid, and there was some back mixing of fluid stream which caused several dead zones along the shell. The occurrence of the dead zones caused the heat transfer to be ineffective and gave lower value compared to the double segmental disc-and-doughnut baffles. The 40% cut segmental baffles was also seen to have a higher pressure at the outlet region than the double segmental disc-and-doughnut baffles. The disc-and-doughnut baffles leads to a turbulent fluid flow which causes an increase in heat transfer characteristics and also lower pressure drop than the 40% cut segmental baffles. Based on the theoretical, both types of disc-and-doughnut baffles and the 40% cut segmental baffles of heat exchanger investigated have highest effectiveness at the lowest mass flow rate of the hot fluids (tube).

Growth of solutions of linear differential-difference equations with coefficients having the same logarithmic order

In this paper, we investigate the relations between the growth of meromorphic coefficients and that of meromorphic solutions of complex linear differential-difference equations with meromorphic coefficients of finite logarithmic order. Our results can be viewed as the generalization for both the cases of complex linear differential equations and complex linear difference equations.

Double parameter splitting (DPS) iteration method for solving complex symmetric linear systems

Complex linear equation systems appear in many branches of mathematical sciences and have many applications in science and engineering. Therefore, there have been developed many methods to solve them. Here a new method based on decomposition of coefficient matrix for the solution of a positive symmetric linear system is studied. Our method uses two parameters. We obtain a domain on which this new method converges without any condition. We will also discuss how to calculate optimal parameters. Moreover we will prove that there exists a region in R2 such that DPS method is faster than CRI method (Wang et al. (2017) [48]). Finally some test problems will be given and simulation results will be reported to support the theoretical results.

Growth Properties of Solutions of Complex Linear Differential-difference Equations with Coefficients having the Same Logarithmic Order in the Unit Disc

In this paper, we investigate the relations between the growth of meromorphic coefficients and that of meromorphic solutions of complex linear differential-difference equations with meromorphic cofficients of finite logarithmic order in the unit disc. Our results can be viewed as the generalization for both the cases of complex linear differential equations and complex linear difference equations.

Learning linear equations: capitalizing on cognitive load theory and learning by analogy

ABSTRACT Capitalizing on cognitive load theory and learning by analogy, we propose two instructional methods to learn a complex linear equation (e.g. two-step equation) by building on prior knowledge of a simpler linear equation (e.g. one-step equation). We will examine the proposal theoretically in this paper. In line with the design principles of cognitive load theory, we propose to strengthen students’ prior knowledge of simpler linear equations before they learn complex linear equations with the aid of worked examples. Because a subset of the complex linear equation shares the same schema as the simpler linear equation, students can draw on their schema for the simpler linear equation to understand the complex linear equation, thus alleviating the limitation on working memory load. Based on the principles of learning by analogy, we place a simpler linear equation and a complex linear equation side-by-side and label the solution procedure of both linear equations to encourage active analogical comparison between these two equations. Making both the simpler linear equation and the complex linear equation visible to learners may help to reduce cognitive load demands in retrieving the simpler linear equation in order to facilitate the learning of the complex linear equation.

An efficient block Gauss–Seidel iteration method for the space fractional coupled nonlinear Schrödinger equations

In this paper, we propose an efficient block Gauss–Seidel iteration method for solving the complex linear equations arising from the space fractional coupled nonlinear Schrödinger (CNLS) equations. The proposed method avoids the inverse of coefficient matrix when solving linear equations, which can greatly reduce computation load and storage space. Furthermore, the convergence of the iteration method is proved theoretically and the number of iteration steps is estimated. Numerical results are presented to show the block Gauss–Seidel iteration method can be quite competitive when compared with others.

Exciting Fixed Point Results on a Novel Space with Supportive Applications

The goal of this paper is to present a new space, a complex valued controlled rectangular b -metric space (for short, υ ℂ -metric space). Some examples and topological properties of υ ℂ -metric spaces are given. Also, some related common fixed point results are discussed. Our results generalize a lot of works in this direction. Moreover, we apply the theoretical results to find a unique solution of a complex valued Atangana-Baleanu fractional integral operator and a system of complex linear equations. Finally, a numerical example to find the current that passes through the RLC circuit is illustrated.

Comprehensive study on complex-valued ZNN models activated by novel nonlinear functions for dynamic complex linear equations

This paper proposes two complex-valued zeroing neural network (Cv-ZNN) models for solving dynamic complex time-variant linear equations. The models involve two complex-valued nonlinear processing methods and adapt two real-valued activation functions. The convergence and robustness of the two Cv-ZNN models are analyzed comprehensively. Firstly, the convergence discussion shows that the corresponding stable error can rapidly converge to zero in finite time, with the upper limit of constriction time calculated. The upper limit of stable error is successfully obtained when the model implementation error is injected into the Cv-ZNN models, which displays their superior robustness. Besides, the results of numerical simulations reveal that the Cv-ZNN models effectively address the time-variant linear system of equations over complex field, even in noisy environments. The simulation results also demonstrate that two novel nonlinear activation functions perform much better than the Linear, Power, Bipolar Sigmoid Power-Sigmoid and Sign-bi-power activation functions.

A Parameter-Changing and Complex-Valued Zeroing Neural-Network for Finding Solution of Time-Varying Complex Linear Matrix Equations in Finite Time

For solving complex-valued linear matrix equations with time-varying coefficients (CV-LME-TVC) in the complex field, this article proposes a parameter-changing and complex-valued zeroing neural network (PC-CVZNN) model through integrating a new parameter-changing function. As compared to previous complex-valued zeroing neural networks (CVZNNs) with fixed parameters and existing parameter-changing functions, the PC-CVZNN model can achieve superior performance due to the accelerated role of the new parameter-changing function. In parts of theoretical analysis, we take advantage of Lyapunov methodology to prove that the proposed PC-CVZNN model can acquire the global and super-exponential convergence when the linear activation function is adopted, and even acquire super finite-time convergence when the new sign-bi-power activation function and its modified one are used. In parts of numerical comparison experiments, it is shown that the PC-CVZNN model possesses faster convergence rate than fixed-parameter CVZNN models and other analogy neural networks with parameter-changing function, when applied to finding the solution of CV-LME-TVC. Importantly, an application of the proposed method to the mobile manipulator control provides the potential practical value of the PC-CVZNN model in the industrial field.

Core-valence-separated coupled-cluster-singles-and-doubles complex-polarization-propagator approach to X-ray spectroscopies.

The iterative subspace algorithm to solve the complex linear response equation of damped coupled cluster response theory presented, up to CCSD level, by Kauczor et al., J. Chem. Phys., 2013, 139, 211102, and recently extended to the solution of the complex left response multipliers by Faber and Coriani, J. Chem. Theory Comput., 2019, 15, 520, has been modified to include a core-valence separation projection step in the iterative procedure. This allows one to overcome serious convergence issues that specifically manifest themselves at the CCSD level when addressing core-related spectroscopic effects using large basis sets. The spectra, obtained adopting the new scheme for X-ray absorption and circular dichroism, as well as resonant inelastic-X-ray scattering, are presented and discussed. Core-valence separated results for non-resonant X-ray emission are also reported.