Nonlinear Matrix Research Articles

Decision making on the mappings' ideal solution of a fuzzy non-linear matrix system of Kannan-type

Since proving many fixed point theorems in a given space requires either growing the space itself or growing the self-mapping that works on it, both of these options are good. The operators' ideal generated by a weighted binomial matrix in the Nakano sequence space of extended s-fuzzy functions is constructed. Some structures for it based on geometry and topology are presented. It has been proven that the Kannan contraction operator has a unique fixed point in this class. Lastly, sufficient conditions such that a fuzzy non-linear matrix system of Kannan-type has a unique solution in this ideal class are investigated and a numerical example to explain our results are given.

Wave and scattering operators for the nonlinear matrix Schrödinger equation on the half-line with a potential

We consider the nonlinear matrix Schrödinger equation on the half-line with general selfadjoint boundary condition. We prove the existence of local solutions in the corresponding energy space. Moreover, we consider the scattering problem for this problem in the scattering-supercritical regime for both generic and exceptional potentials. We construct the wave, inverse wave and scattering operators in a certain neighborhood of the origin in the scattering space Σ=H1∩L2,1 for generic potentialsin L1,2+η, with η>1/2, and for exceptional potentials in L1,3 under further assumptions on the input data. In particular, thanks to the general boundary conditions, we are able to obtain a scattering result for the nonlinear matrix Schrödinger equation on the line with a potential and point interactions.

Versatile benchmark model reproducing snap-through, asymmetric, symmetric unstable/stable, and multiple bifurcation including hill-top branching in structural instability

This paper proposes a simple yet innovative two degrees of freedom (2-DoFs) benchmark model for structural instability. The proposed rigid-body-spring model is versatile in that it reproduces various kinds of instability phenomena. Its parameter combinations comprehensively realize snap-through, asymmetric, symmetric unstable/stable, and multiple bifurcation including hill-top branching in structural stability problems. Governing equations including the total potential energy, nonlinear equilibrium equations, stiffness matrix (Jacobian), and stability information are all explicitly formulated. For post-buckling analysis, path branching is visualized for intuitive diagnosis. The versatile benchmark model is expected to yield a qualitative understanding of the instability mechanism depending on the model parameters. The benchmark model also contributes to the phenomenological understanding of stability design.

Matlab-based Study of the Dynamic Performance of Flexible Jointed Robots

In the development and research of industrial robots, due to the defects of traditional mechanical structures in the drive system and sensors, control units and other aspects, so in order to solve these problems and improve the efficiency of flexible work, the author proposes a Matlab based core motion characteristics analysis method, that is mathematical modelling method (GUI) to improve the elastic joint robot dynamics The relationship between performance index evaluation and prediction model parameters. At present, most of the research on elastically articulated robots at home and abroad is focused on the development of one or several functionalised components. In this paper, the kinematic model is developed and its performance characteristics are analysed based on the Matlab c language platform. The kinetic equations are discretized to obtain the velocity and acceleration curves with time for each degree of freedom, so that the non-linear contact angle matrix and the driving torque can be established.

Multifactor Association Analysis Model of Preschool Education Based on Multivariate Nonlinear Matrix Theory

This paper uses the multivariate nonlinear matrix theory to conduct an in-depth study and analysis of the multifactor correlation in preschool education, and designs a multifactor correlation analysis model to be applied to the actual preschool education process. A nonlinear approach is used to define the dynamics of a planar segmented smooth system with visible folds on a switching surface, and conditions are given for the system to have a sliding periodic orbit through this visible fold. A curriculum relatedness analysis algorithm based on Euclidean distance is proposed, which takes distance as a perspective and calculates the curriculum relatedness by measuring the difference of the curriculum in different dimensions; a curriculum relatedness analysis algorithm based on cosine similarity is proposed, which takes a vector as a perspective and calculates the curriculum relatedness by measuring the cosine of the angle between different curriculum vectors; a curriculum relatedness analysis algorithm based on correlation coefficient is proposed. The course relevance analysis algorithm based on the correlation coefficient evaluates the relevance of different courses from a centralized perspective. After preprocessing the course results, two different types of datasets were selected for the experiments of the three algorithms. The results show that all three algorithms can calculate the correlation degree of the courses, which can provide scientific data support for the curriculum arrangement of the school so that the school can scientifically verify and optimize the curriculum arrangement. The three algorithms are compared from the calculation results and theories, among which the correlation coefficient correlation analysis algorithm has the best effect. This study proposes suggestions to promote students’ noncognitive ability development from the perspective of preschool education development; grasp the critical period of children’s noncognitive ability development and pay attention to preschool education investment; further develop inclusive preschool education to ensure the educational opportunities of disadvantaged groups; promote regional preschool education reform to ensure the quality of preschool education in rural areas; and guide parents’ participation and cooperate with schools to cultivate children’s noncognitive ability.

Nonlinear observability analysis of multi-robot cooperative localization

In this paper, the observability property of relative position-based cooperative localization is investigated by using spectral graph theory. First, a directed graph, called Extended Relative Measurement Graph (ERMG), is constructed in which all influencing factors of observability are integrated. Then, the rank equivalence is established between the nonlinear observability matrix and the configuration matrix associated with a reduced ERMG. Then, a necessary and sufficient observability condition is proved. Moreover, an analytic relation is derived by which the configuration matrix can be directly attained according to fundamental cycles of a reduced ERMG. Then, it is shown that the observability is uniquely determined by the number and placement of fundamental cycles in a reduced ERMG. The minimum number of fundamental cycles required to guarantee an n -robot system observable is given. Additionally, a sufficient condition is provided by which reduced ERMGs can be designed to enable an n -robot system observable. Finally, a graph-based observability checking algorithm is developed which greatly improves the computation efficiency. Some illustrative examples are presented to validate the theoretical results.

Deformation Mechanics of Fuel Cell Gas Diffusion Layer: Cyclic Response and Constitutive Model

The deformation mechanics of a typical gas diffusion layer using experimental and advanced modelling technique is reported. The experimental cyclic response is observed similar to pseudo-elastic materials with highly nonlinear loading/unloading. The cyclic compressive mechanical response of gas diffusion layer (GDL) is modelled to be the outcome of cumulative changes in deformation kinematics of matrix and fiber fractions. The individual mechanisms necessitating the energy dissipation, residual strain, and stress softening during cyclic mechanical response are related to nonlinear hyperelastic matrix with the damage function and inelastic activation function at the interface of constituents. The model predicts highly nonlinear elastic loading, residual strain, hysteresis, and damage quotient associated with stress softening as a function of several cycles. The significant takeaway from this study is in terms of quantifying strength, inelastic nature of individual constituents. The proposed model is simulated for low-level altering stresses of up to twenty cycles. The results show the build-up of residual strains and hysteresis as a function of fuel cell clamping pressure.

Matrix equations models for nonlinear dynamic analysis of two-dimensional and three-dimensional RC structures with lateral load resisting cantilever elements

In control engineering and structural dynamics, mathematical models such as the state-space representation, equation of motion, and the phase plane are matrix equations describing the system equilibrium. This paper develops novel matrix equations models for linear/nonlinear dynamic analysis of reinforced concrete (RC) buildings with cantilever elements lateral load resisting systems (e.g., RC shear wall, RC core). The models offer a new approach for introducing two-dimensional and three-dimensional cantilever structures to control the theory’s state-space representation and structural dynamics’ equation of motion. The development primarily addresses the stiffness and mass matrices. The proposed displacement-related stiffness matrix of cantilever elements satisfies the necessary conditions of symmetricity and elemental boundary conditions. The nonlinear matrix structural analysis employs a smooth hysteretic model for deteriorating inelastic structures, referring to the relation between the bending moment and the bending curvature through the bending stiffness. The parameters controlling the cyclic behavior regard a composite RC cross section subject to gravitational load and bending simultaneously. The paper includes four examples that exemplify the practical utilization of the matrix equations models in analyzing two-dimensional and three-dimensional structures of linearly elastic and inelastic properties. The four examples demonstrated the idealized applicability of the matrix equations models for modal analysis, pushover analysis, and inelastic earthquake response analysis.

Algebraic and differential-geometric constructions of set-theoretical solutions to the Zamolodchikov tetrahedron equation

We present several algebraic and differential-geometric constructions of tetrahedron maps, which are set-theoretical solutions to the Zamolodchikov tetrahedron equation. In particular, we obtain a family of new (nonlinear) polynomial tetrahedron maps on the space of square matrices of arbitrary size, using a matrix refactorisation equation, which does not coincide with the standard local Yang–Baxter equation. Liouville integrability is established for some of these maps. Also, we show how to derive linear tetrahedron maps as linear approximations of nonlinear ones, using Lax representations and the differentials of nonlinear tetrahedron maps on manifolds. We apply this construction to two nonlinear maps: a tetrahedron map obtained in Dimakis and Müller-Hoissen (2019 Lett. Math. Phys. 109 799–827) in a study of soliton solutions of vector Kadomtsev–Petviashvili equations and a tetrahedron map obtained in Konstantinou-Rizos (2020 Nucl. Phys. B 960 115207) in a study of a matrix trifactorisation problem related to a Darboux matrix associated with a Lax operator for the nonlinear Schrödinger equation. We derive parametric families of new linear tetrahedron maps (with nonlinear dependence on parameters), which are linear approximations for these nonlinear ones. Furthermore, we present (nonlinear) matrix generalisations of a tetrahedron map from Sergeev’s classification Sergeev (1998 Lett. Math. Phys. 45 113–9). These matrix generalisations can be regarded as tetrahedron maps in noncommutative variables. Besides, several tetrahedron maps on arbitrary groups are constructed.

Time-Optimal Control Problem of Two Non-Synchronous Oscillators

The time-optimal control problem for a system consisting of two non-synchronous oscillators is considered. Each oscillator is controlled with a shared limited scalar control. The objective of the control is to accelerate the oscillatory system to a given specific position, where the first oscillator must have non-zero phase coordinates, but the second one must remain motionless at the terminal moment. For an arbitrary number of unknown switching moments that determine the optimal relay control, the necessary extremum conditions in the form of nonlinear matrix equalities are proposed. The study of the necessary/sufficient conditions of the extremum made it possible to describe the reachability set in the phase space of the first oscillator, to find an analytical form of the curve corresponding to the two-switching control class, which also separates the reachability set of the three switching-control class. The corresponding theorems are proved and the dependence of the criteria on control constraints is shown. Matrix conditions for different classes of control switchings are found. All of the obtained analytical results are numerically validated and illustrated with mathematical modeling.

Finite-time guaranteed cost control for uncertain delayed switched nonlinear stochastic systems

This paper studies the finite-time guaranteed cost control problem for switched nonlinear stochastic systems with parameter uncertainties and time-varying delays. By choosing a model-dependent and delay-dependent Lyapunov-Krasovskii functional, applying the average dwell time approach and the Gronwall inequality, some novel sufficient conditions are derived to ensure that the switched nonlinear stochastic closed-loop system is finite-time stochastically stable and an upper bound is given on the performance index. The obtained nonlinear matrix is transformed into a linear matrix form, and then the feedback controller gains of the switched nonlinear stochastic systems with time-varying delay are obtained. Finally, two simulation examples are designed to verify the effectiveness of the suggested approach.

On solution and perturbation estimates for the nonlinear matrix equation X-A^{*}e^{X}A=I

This work incorporates an efficient inversion free iterative scheme into Newton’s method to solve Newton’s step regardless of the singularity of the Fr{acute{text {e}}}chet derivative. The proposed iterative scheme is constructed by extending the idea of the foundational form of the conjugate gradient method. Moreover, the resulting scheme is refined and employed to obtain a symmetric solution of the nonlinear matrix equation X-A^{*}e^{X}A=I. Furthermore, explicit expressions for the perturbation and residual bound estimates of the approximate positive definite solution are derived. Finally, five numerical case studies provided confirm both the preciseness of theoretical results and the effectiveness of the propounded iterative method.

A Study on the Application of Deep Learning Methods Based on Nonlinear Random Matrices in the Design of Intelligent Research Management System

The database of the intelligent science management system includes the exchange of data between the basic data of the school and the various business systems, with centralised storage and centralised management, constituting a unified data centre. The completed intelligent science management system will become a reliable source of data to integrate the system to be developed with existing systems, making the data of the whole system unified, thus providing reliable data for information enquiry and decision analysis within the university, and laying a good foundation for the construction of the digital campus. The NRM_DeepNet model is an application model of the deep learning method of the nonlinear random matrix in the intelligent research management system, mainly in data processing and analysis, using deep learning technology for data classification processing, and then for research data modal feature extraction, the process of modal extraction applies the nonlinear random matrix for feature weighting. In the process of modal extraction, a nonlinear random matrix is applied to weight the features, and finally, different research data features of different research projects can be classified as intelligent and regularised. In this paper, we combine the practice of scientific research management, introduce the nonlinear random matrix and deep learning technology into the intelligent scientific research management system, and update its data and system management so as to realize intelligent and humanised scientific research management, thus reducing the labour intensity and improving the efficiency of scientific research work.

A receptance based method for the calculation of nonlinear normal modes of large ordered structures with distributed localized nonlinearities

In recent years, the concept of nonlinear normal modes (NNMs) has gained interest for interpreting a broad range of nonlinear dynamic phenomena. Present study introduces a simple and efficient computational framework to compute NNMs of large order nonlinear structures overcoming difficulties associated with commonly used time-integration and discretization methods. In this paper, a Receptance Based Nonlinear Normal Mode Calculation Method (RBNM) is developed to determine the NNMs of large ordered realistic finite element models in frequency domain. Describing Function Method (DFM), which makes it possible to write the nonlinear internal forcing vector as a nonlinear displacement dependent stiffness matrix multiplied by displacement vector, is used to model the internal nonlinear forcing vector in frequency domain. A unique matrix manipulation based on dynamic stiffness and receptance concepts is used, as a result of which the number of governing nonlinear equations becomes independent from the total number of degrees of freedom of the model and it only depends on the number of degrees of freedom associated with the nonlinear elements. This brings a significant computational advantage over the classical reduced order modeling techniques. A simple 2-degree of freedom (DOF) system, the results of which are available in the literature, is used to verify the proposed method in determining NNMs. A 20-DOF lumped parameter system model and later a complex finite element model of a large structure are used to demonstrate the applicability of RBNM in determining NNMs of larger systems with several different types of nonlinear elements distributed among the degrees of freedom.

Full-Vector Mode Solver for Nonlinear Optical Waveguides Using Meshfree Finite Cloud Method

A full-vector mode solver based on the meshfree (MF) finite cloud method (FCM) is developed for analysis of nonlinear optical waveguide. Self-consistent solutions are approximated through a simple iterative scheme after the nonlinear eigenvalue matrix equation derived using the quadratic basis function as well as the MF shape function. The proposed mode solver can significantly reduce the computational requirements since major parts of formulations are tackled analytically. Results for two numerical examples are presented and compared with those from the well-known finite element method (FEM) and FEM-based imaginary-distance beam propagation method (FEM-IDBPM), respectively, showing the validity and utility of the proposed method.

Deep Learning Model for Stock Excess Return Prediction Based on Nonlinear Random Matrix and Esg Factor

Aiming at the problem that the traditional model has low accuracy in describing stock excess return, in order to further analyze the change law of stock excess, based on the nonlinear random matrix and esg factor theory, the traditional learning model is analyzed, and the corresponding optimized deep learning model is obtained by introducing the single ring theorem and statistical data. Through the analysis and research of related indexes, the change rules of different indexes are obtained, and the optimization model is used to calculate and forecast the excess return of stocks. The results show that the statistics and spectral radius show typical local linear variation with the increase of eigenvalue. The corresponding statistics show a trend of gradual increase. The corresponding spectral radius has a decreasing variation law, and the two curves have obvious symmetry at some eigenvalues. It can be seen from the change curves under different factors that the change trend of the yield curve is mainly affected by the investment factor, while the change rule of the specific value of the yield curve is controlled by the profit factor. This shows that the two factors have the same influence on the stock excess return. The influence of optimized deep learning model on stock excess index has typical linear characteristics, which can be divided into linear increase and linear decline according to different change rules. The basic type has the greatest influence, while the corresponding pattern analysis type has the least influence. Finally, the method of experimental verification is used to verify the stock excess data, and the results show that the optimized deep learning model can better characterize the experimental results. Therefore, the optimized deep learning model based on nonlinear random matrix and esg factor can carry out targeted analysis of different types of stock returns, thus improving research ideas and calculation methods for the application of deep learning model in different fields.

Development Model of College English Education Service Talents Based on Nonlinear Random Matrix

The development of artificial intelligence makes people’s life and work easier and more effective, and computer-based online exams and marking not only improve students’ learning efficiency but also reduce the pressure of teachers’ marking work. For objective questions, marking has gone from manual marking to cursor reader marking to computerized character matching, and the correct rate of marking has soared to 100%; for subjective questions, foreign systems such as PEG and E-rater have been used, and domestic systems such as those using English large corpus similarity matching and those based on natural language understanding using intelligent algorithms have been used for marking. Most of these systems are based on some shallow linguistic features such as rules and LSAs for marking, and there is no deep perception of English language sense. Although the current intelligent marking systems have made a lot of achievements, they do not fundamentally solve the problem of the rationality of intelligent marking of subjective questions. In this article, we propose a regularized discriminant analysis algorithm with good estimation of the mean, and a dimensionality reduction algorithm for high-dimensional missing data by using the relevant research results of random matrix theory to address the problems of traditional machine learning methods in high-dimensional data analysis. Although the linear discriminant analysis algorithm performs well in solving many practical problems, it works poorly in dealing with high-dimensional data. The specific analysis is as follows: in terms of age characteristics, the mobile population under the age of 35 has a significant preference for urban consumer comfort, and it increases with age, peaking at the stage of 30–35 years old and then decreasing rapidly. For this reason, a regularized discriminant analysis algorithm based on random matrix theory is proposed. First, a good estimate of the high-dimensional covariance matrix is made by the nonlinear shrinkage method or the eigenvalue interception method, respectively; then, the estimated high-dimensional covariance matrix is used to calculate the discriminant function values and perform the classification. The classification experiments conducted on simulated and real datasets show that the proposed algorithm is not only more widely applicable but also has a high correct classification rate.

A Neural Network Model for the Relationship between Hotel Marketing Strategies and Performance Based on Nonlinear Random Matrix Theory

The homogenization of hotel competition has become a key problem faced by the hotel enterprises. According to the marketing theory, one perfect marketing strategy will improve hotel performance. Based on nonlinear random matrix and neural network model, this paper empirically analyzes the relationship between different marketing strategies and hotel performance from the aspects of hotel demand collection, strategy support, performance matching, and strategy regulation. The results show that the hotel performance based on product marketing strategies is the best, which has improving hotel performance about 5%. In contrast, the hotel performance based on people-oriented marketing strategies is the worst. Therefore, we believe that the hotel enterprises should pay attention to the adjustment of product structure and the role of people in hotel operation and management. Adjusting product structure, strengthening hotel information construction, and paying attention to talent cultivation play an important role in improving hotel performance, in order to provide useful reference for the development of other hotels.

Efficient Biexciton State Preparation in a Semiconductor Quantum Dot Coupled to a Metal Nanoparticle with Linearly Chirped Gaussian Pulses.

We consider a hybrid nanostructure composed of a semiconductor quantum dot placed near a spherical metallic nanoparticle, and study the effect of the nanoparticle on the population transferral from the ground to the biexciton state of the quantum dot, when using linearly chirped Gaussian pulses. For various values of the system parameters (biexciton energy shift, pulse area and chirp, interparticle distance), we calculate the final population of the biexciton state by performing numerical simulations of the non-linear density matrix equations which describe the coupled system, as well as its interaction with the applied electromagnetic field. We find that for relatively large values of the biexciton energy shift and not very small interparticle distances, the presence of the nanoparticle improves the biexciton state preparation, since it effectively increases the area of the applied pulse. For smaller biexciton energy shifts and smaller distances between the quantum dot and the nanoparticle, the performance is, in general, degraded. However, even in these cases we can still find ranges of parameter values where the population transfer to the biexciton state is accomplished with high fidelity, when using linearly chirped Gaussian pulses. We anticipate that our results may be exploited for the implementation of novel nanoscale photonic devices or future quantum technologies.

Multi-harmonic Rutherford island theory

Rutherford island theory, which governs the nonlinear evolution of tearing modes in tokamak plasmas, is generalized to take into account situations in which the conventional one-harmonic approximation is not valid. The analysis incorporates non-inductive currents driven by radio frequency (RF) electromagnetic waves injected into the plasma. A multi-harmonic tearing mode dispersion relation is derived that takes the form of a nonlinear inhomogeneous matrix eigenvalue problem. The dispersion relation is solved in the so-called two-harmonic approximation, in which only the principal Fourier harmonic of the perturbed magnetic flux and its first overtone are included in the calculation. In the absence of RF current drive, the nonlinear behavior of a tearing mode predicted in the two-harmonic approximation does not differ substantially from that predicted in the one-harmonic approximation. On the other hand, RF current drive that is sufficiently localized in the vicinity of the O-points of the mode's magnetic island chain is capable of triggering bifurcations of the O-points (which is impossible in the one-harmonic approximation). However, the current drive is incapable of triggering bifurcations of the island X-points. This finding is significant because Bardóczi and Evans [Phys. Rev. Lett. 126, 085003 (2021)] recently observed bifurcations of magnetic island chain O-points in the presence of RF current drive in the DIII-D tokamak but did not observe bifurcations of the X-points. Finally, the changes in the topology of the magnetic island flux-surfaces induced by RF current drive are found to facilitate the stabilization of the tearing mode.