Eigenfunction expansion method of upper triangular operator matrixand application to two-dimensional elasticity problems based onstress formulation

This paper studies the eigenfunction expansion method to solve the two dimensional （2D） elasticity problems based on the stress formulation. The fundamental system of partial differential equations of the 2D problems is rewritten as an upper tri angular differential system based on the known results, and then the associated upper triangular operator matrix matrix is obtained. By further research, the two simpler com plete orthogonal systems of eigenfunctions in some space are obtained, which belong to the two block operators arising in the operator matrix. Then, a more simple and conve nient general solution to the 2D problem is given by the eigenfunction expansion method. Furthermore, the boundary conditions for the 2D problem, which can be solved by this method, are indicated. Finally, the validity of the obtained results is verified by a specific example.

A CHARACTERIZATION OF SCHECHTER'S ESSENTIAL SPECTRUM BY MEAN OF MEASURE OF NON-STRICT-SINGULARITY AND APPLICATION TO MATRIX OPERATOR

In this paper we use the notion of measure of non-strict-singularity to give some results on Fredholm operators and we establish a fine description of the Schechter essential spectrum of a closed densely defined linear operator.

Structure of continuous matrix product operator for transverse field Ising model:An analytic and numerical study

We study the structure of the continuous matrix product operator(cMPO)^([1]) for the transverse field Ising model(TFIM).We prove TFIM’s cMPO is solvable and has the form T=e^(-1/2H_(F)).H_(F) is a non-local free fermionic Hamiltonian on a ring with circumferenceβ,whose ground state is gapped and non-degenerate even at the critical point.The full spectrum of H_(F) is determined analytically.At the critical point,our results verify the state–operator-correspondence^([2]) in the conformal field theory(CFT).We also design a numerical algorithm based on Bloch state ansatz to calculate the lowlying excited states of general(Hermitian)cMPO.Our numerical calculations coincide with the analytic results of TFIM.In the end,we give a short discussion about the entanglement entropy of cMPO’s ground state.

Completeness of the system of eigenvectors of off-diagonal operator matrices and its applications in elasticity theory

This paper deals with off-diagonal operator matrices and their applications in elasticity theory. Two kinds of completeness of the system of eigenvectors are proven, in terms of those of the compositions of two block operators in the off-diagonal operator matrices. Using these results, the double eigenfunction expansion method for solving upper triangular matrix differential systems is proposed. Moreover, we apply the method to the two-dimensional elasticity problem and the problem of bending of rectangular thin plates on elastic foundation.

Completeness of eigenfunction systems for the product of two symmetric operator matrices and its application in elasticity

The completeness theorem of the eigenfunction systems for the product of two 2 × 2 symmetric operator matrices is proved. The result is applied to 4 × 4 infinite-dimensional Hamiltonian operators. A modified method of separation of variables is proposed for a separable Hamiltonian system. As an application of the theorem, the general solutions for the plate bending equation and the free vibration of rectangular thin plates are obtained. Finally, a numerical test is analysed to show the correctness of the results.

On the completeness of eigen and root vector systems for fourth-order operator matrices and their applications

In this paper, we consider the eigenvalue problem of a class of fourth-order operator matrices appearing in mechan- ics, including the geometric multiplicity, algebraic index, and algebraic multiplicity of the eigenvalue, the symplectic orthogonality, and completeness of eigen and root vector systems. The obtained results are applied to the plate bending problem.

Furi-Martelli-Vignoli spectrum and Feng spectrum of nonlinear block operator matrices

We investigate the Furi-Martelli-Vignoli spectrum and the Feng spectrum of continuous nonlinear block operator matrices,and mainly describe the relationship between the Furi-Martelli-Vignoli spectrum(compared to the Feng spectrum)of the whole operator matrix and that of its entries.In addition,the connection between the Furi-Martelli-Vignoli spectrum of the whole operator matrix and that of its Schur complement is presented by means of Schur decomposition.

Symplectic eigenvector expansion theorem of a class of operator matrices arising from elasticity theory

This paper deals with the completeness of the eigenvector system of a class of operator matrices arising from elasticity theory, i.e., symplectic eigenvector expansion theorem. Under certain conditions, the symplectic orthogonality of eigenvectors of the operator matrix is demonstrated. Based on this, a necessary and sufficient condition for the completeness of the eigenvector system of the operator matrix is given. Furthermore, the obtained results are tested for the free vibration of rectangular thin plates.

Functionalized Data Operator Model for System Analysis and Forecasting

A deconvolution data processing is developed for obtaining a Functionalized Data Operator (FDO) model that is trained to approximate past and present, input-output data relations. The FDO model is designed to predict future output features for deviated input vectors from any expected, feared of conceivable, future input for optimum control, forecast, or early-warning hazard evaluation. The linearized FDO provides fast analytical, input-output solution in matrix equation form. If the FDO is invertible, the necessary input for a desired output may be explicitly evaluated. A numerical example is presented for FDO model identification and hazard evaluation for methane inflow into the working face in an underground mine: First, a Physics-Based Operator (PBO) model to match monitored data. Second, FDO models are identified for matching the observed, short-term variations with time in the measured data of methane inflow, varying model parameters and simplifications following the parsimony concept of Occam’s Razor. The numerical coefficients of the PBO and FDO models are found to differ by two to three orders of magnitude for methane release as a function of short-time barometric pressure variations. As being data-driven, the significantly different results from an FDO versus PBO model is either an indication of methane release processes poorly understood and modeled in PBO, missing some physics for the pressure spikes;or of problems in the monitored data fluctuations, erroneously sampled with time;or of false correlation. Either way, the FDO model is originated from the functionalized form of the monitored data, and its result is considered experimentally significant within the specified RMS error of model matching.

Matrix Operations Design Tool for FPGA and VLSI Systems

Embedded systems used in real-time applications require low power, less area and high computation speed. For digital signal processing, image processing and communication applications, data are often received at a continuously high rate. The type of necessary arithmetic functions and matrix operations may vary greatly among different applications. The RTL-based design and verification of one or more of these functions could be time-consuming. Some High Level Synthesis tools reduce this design and verification time but may not be optimal or suitable for low power applications. The design tool proposed in this paper can improve the design time and reduce the verification process. The design tool offers a fast design and verification platform for important matrix operations. These operations range from simple addition to more complex matrix operations such as LU and QR factorizations. The proposed platform can improve design time by reducing verification cycle. This tool generates Verilog code and its testbench that can be realized in FPGA and VLSI systems. The designed system uses MATLAB-based verification and reporting.

A Numerical Investigation Based on Exponential Collocation Method for Nonlinear SITR Model of COVID-19

In this work,the exponential approximation is used for the numerical simulation of a nonlinear SITR model as a system of differential equations that shows the dynamics of the new coronavirus(COVID-19).The SITR mathematical model is divided into four classes using fractal parameters for COVID-19 dynamics,namely,susceptible(S),infected(I),treatment(T),and recovered(R).The main idea of the presented method is based on the matrix representations of the exponential functions and their derivatives using collocation points.To indicate the usefulness of this method,we employ it in some cases.For error analysis of the method,the residual of the solutions is reviewed.The reported examples show that the method is reasonably efficient and accurate.

Least square method based on Haar wavelet to solve multi-dimensional stochastic Ito-Volterra integral equations

This paper proposes a method combining blue the Haar wavelet and the least square to solve the multi-dimensional stochastic Ito-Volterra integral equation.This approach is to transform stochastic integral equations into a system of algebraic equations.Meanwhile,the error analysis is proven.Finally,the effectiveness of the approach is verified by two numerical examples.

Quantum algorithms for matrix operations and linear systems of equations

Fundamental matrix operations and solving linear systems of equations are ubiquitous in scientific investigations.Using the‘sender-receiver’model,we propose quantum algorithms for matrix operations such as matrix-vector product,matrix-matrix product,the sum of two matrices,and the calculation of determinant and inverse matrix.We encode the matrix entries into the probability amplitudes of the pure initial states of senders.After applying proper unitary transformation to the complete quantum system,the desired result can be found in certain blocks of the receiver’s density matrix.These quantum protocols can be used as subroutines in other quantum schemes.Furthermore,we present an alternative quantum algorithm for solving linear systems of equations.

Adaptive partition intuitionistic fuzzy time series forecasting model

To enhance the accuracy of intuitionistic fuzzy time series forecasting model, this paper analyses the influence of universe of discourse partition and compares with relevant literature. Traditional models usually partition the global universe of discourse, which is not appropriate for all objectives. For example, the universe of the secular trend model is continuously variational. In addition, most forecasting methods rely on prior information, i.e., fuzzy relationship groups (FRG). Numerous relationship groups lead to the explosive growth of relationship library in a linear model and increase the computational complexity. To overcome problems above and ascertain an appropriate order, an intuitionistic fuzzy time series forecasting model based on order decision and adaptive partition algorithm is proposed. By forecasting the vector operator matrix, the proposed model can adjust partitions and intervals adaptively. The proposed model is tested on student enrollments of Alabama dataset, typical seasonal dataset Taiwan Stock Exchange Capitalization Weighted Stock Index (TAIEX) and a secular trend dataset of total retail sales for social consumer goods in China. Experimental results illustrate the validity and applicability of the proposed method for different patterns of dataset.

A complete symplectic eigenfunction expansion for the elastic thin plate with simply supported edges

The eigenvalue problem for the Hamiltonian operator associated with the mathematical model for the deflection of a thin elastic plate is investigated.First,the problem for a rectangular plate with simply supported edges is solved directly.Then,the completeness of the eigenfunctions is proved,thereby demonstrating the feasibility of using separation of variables to solve the problem. Finally,the general solution is obtained by using the proved expansion theorem.

LEGENDRE SERIES SOLUTIONS FOR TIME-VARIATION DYNAMICS

In this topic, a new. approach to the analysis of time-variation dynamics is proposed by use of Legendre series expansion and Legendre integral operator matrix. The theoretical basis for effective solution of time-variation dynamics is therefore established, which is beneficial to further research of time-variation science.

The Decomposability for Operator Matrices and Perturbations

Let X and Y be Banach spaces.For A∈L(X),B∈L(Y),C∈L(Y,X),let MCbe the operator matrix defined on X⊕Y by M_(C)=(AC0B)∈L(X⊕Y).In this paper we investigate the decomposability for MC.We consider Bishop’s property(β),decomposition property(δ)and Dunford’s property(C)and obtain the relationship of these properties between M_(C) and its entries.We explore how σ_(*)(M_(C))shrinks from σ_(*)(A)∪σ_(*)(B),where σ_(*)denotes σ_(β),σ_(δ),σ_(C),σ_(dec).In particular,we develop some sufficient conditions for equality σ_(*)(MC)=σ_(*)(A)∪σ_(*)(B).Besides,we consider the perturbation of these properties for MCand show that in perturbing with certain operators C the properties for MCkeeps with A,B.Some examples are given to illustrate our results.Furthermore,we study the decomposability for(0AB0).Finally,we give applications of decomposability for operator matrices.

Property(R)for Upper Triangular Operator Matrices

Property(R)holds for an operator when the complement in the approximate point spectrum of the Browder essential approximate point spectrum coincides with the isolated points of the spectrum which are eigenvalues of finite multiplicity.Let A∈B(H)and B∈B(K),where H and K are complex infinite dimensional separable Hilbert spaces.We denote by M_(C)the operator acting on H⊕K of the form M_(C)=(AC0B).In this paper,we give a sufficient and necessary condition for M_(C)∈(R)for all C∈B(K,H).

Numerical Solutions of Fractional Differential Equations by Using Fractional Taylor Basis

In this paper, a new numerical method for solving fractional differential equations(FDEs) is presented. The method is based upon the fractional Taylor basis approximations. The operational matrix of the fractional integration for the fractional Taylor basis is introduced. This matrix is then utilized to reduce the solution of the fractional differential equations to a system of algebraic equations. Illustrative examples are included to demonstrate the validity and applicability of this technique.

Numerical Solution of Two-Dimensional Nonlinear Stochastic Ito-Volterra Integral Equations by Applying Block Pulse Functions

This paper investigates the numerical solution of two-dimensional nonlinear stochastic Itô-Volterra integral equations based on block pulse functions. The nonlinear stochastic integral equation is transformed into a set of algebraic equations by operational matrix of block pulse functions. Then, we give error analysis and prove that the rate of convergence of this method is efficient. Lastly, a numerical example is given to confirm the method.