On Maximal, Discrete, and Area Operators

In this paper, we study the boundednesses of maximal operator g., the discrete operator gd, and the area operator A on Bergman spaces.

Symplectic partitioned Runge-Kutta method based onthe eighth-order nearly analytic discrete operator and its wavefield simulations

We propose a symplectic partitioned Runge-Kutta （SPRK） method with eighth-order spatial accuracy based on the extended Hamiltonian system of the acoustic waveequation. Known as the eighth-order NSPRK method, this technique uses an eighth-orderaccurate nearly analytic discrete （NAD） operator to discretize high-order spatial differentialoperators and employs a second-order SPRK method to discretize temporal derivatives.The stability criteria and numerical dispersion relations of the eighth-order NSPRK methodare given by a semi-analytical method and are tested by numerical experiments. We alsoshow the differences of the numerical dispersions between the eighth-order NSPRK methodand conventional numerical methods such as the fourth-order NSPRK method, the eighth-order Lax-Wendroff correction （LWC） method and the eighth-order staggered-grid （SG）method. The result shows that the ability of the eighth-order NSPRK method to suppress thenumerical dispersion is obviously superior to that of the conventional numerical methods. Inthe same computational environment, to eliminate visible numerical dispersions, the eighth-order NSPRK is approximately 2.5 times faster than the fourth-order NSPRK and 3.4 timesfaster than the fourth-order SPRK, and the memory requirement is only approximately47.17% of the fourth-order NSPRK method and 49.41% of the fourth-order SPRK method,which indicates the highest computational efficiency. Modeling examples for the two-layermodels such as the heterogeneous and Marmousi models show that the wavefields generatedby the eighth-order NSPRK method are very clear with no visible numerical dispersion.These numerical experiments illustrate that the eighth-order NSPRK method can effectivelysuppress numerical dispersion when coarse grids are adopted. Therefore, this methodcan greatly decrease computer memory requirement and accelerate the forward modelingproductivity. In general, the eighth-order NSPRK method has tremendous potential value forseismic exploration and seismology research.

ELEMENT FUNCTIONS OF DISCRETE OPERATOR DIFFERENCE METHOD

The discrete scheme called discrete operator difference for differential equations was given. Several difference elements for plate bending problems and plane problems were given. By investigating these elements, the ability of the discrete forms expressing to the element functions was talked about. In discrete operator difference method, the displacements of the elements can be reproduced exactly in the discrete forms whether the displacements are conforming or not. According to this point, discrete operator difference method is a method with good performance.

Controlled Implementation of Non-local CNOT Operation Using Three-Qubit Entanglement

We investigate the controlled implementation of a non-local CNOT operation using a three-qubit entangled state. Firstly, we show how the non-local CNOT operation can be implemented with unit fidelity and unit probability by using a maximally entangled GHZ state as controlled quantum channel. Then, we put forward two schemes for conclusively implementing the non-local operation with unit fidelity by employing a partially entangled pure GHZ state as quantum channel. The feature of these schemes is that a third side is included, who may participate the process of quantum non-local implementation as a supervisor. Furthermore, when the quantum channel is partially entangled,the third one can rectify the state distorted by imperfect quantum channel. In addition to the GHZ class state, the W class state can also be used to implement the same non-local operation probabilistically. The probability of successful implementation using the W class state is always less than that using the GHZ class state.

A DISCRETE CHARACTERIZATION OF HERZ-TYPE TRIEBEL-LIZORKIN SPACES AND ITS APPLICATIONS

In this paper, the author establishes a discrete characterization of the Herz-type Triebel-Lizorkin spaces which is used to prove the boundedness of pseudo-differential operators on these function spaces.

Discrete Fractional Lagrange Equations of Nonconservative Systems

In order to study discrete nonconservative system,Hamilton's principle within fractional difference operators of Riemann-Liouville type is given. Discrete Lagrange equations of the nonconservative system as well as the nonconservative system with dynamic constraint are established within fractional difference operators of Riemann-Liouville type from the view of time scales. Firstly,time scale calculus and fractional calculus are reviewed.Secondly,with the help of the properties of time scale calculus,discrete Lagrange equation of the nonconservative system within fractional difference operators of Riemann-Liouville type is presented. Thirdly,using the Lagrange multipliers,discrete Lagrange equation of the nonconservative system with dynamic constraint is also established.Then two special cases are discussed. Finally,two examples are devoted to illustrate the results.

SINGULAR INTEGRAL OPERATORS AND SINGULAR QUADRATURE OPERATORS ASSOCIATED WITH SINGULAR INTEGRAL EQUATIONS OF THE FIRST KIND AND THEIR APPLICATIONS

In this paper, we discuss some singulal integral operators, singular quadrature operators and discrethation matrices associated with singular integral equations of the first kind, and obtain some useful Properties for them. Using these operators we give a unified framework for various collocation methods of numerical solutions of singular integral equations of the fine kind, which appears very simple.

Generalized Parseval’s Theorem on Fractional Fourier Transform for Discrete Signals and Filtering of LFM Signals

This paper investigates the generalized Parseval’s theorem of fractional Fourier transform (FRFT) for concentrated data. Also, in the framework of multiple FRFT domains, Parseval’s theorem reduces to an inequality with lower and upper bounds associated with FRFT parameters, named as generalized Parseval’s theorem by us. These results theoretically provide potential valuable applications in filtering, and examples of filtering for LFM signals in FRFT domains are demonstrated to support the derived conclusions.

BASISITY PROBLEM AND WEIGHTED SHIFT OPERATORS

We investigate a basisity problem in the space =lpA（D）and in its invariant sub-spaces. Namely, let W denote a unilateral weighted shift operator acting in the spacelpA（D）, 1≤p《∞, by W zn=λnzn＋1, n≥0, with respect to the standard basis λzn＋1 n≥0. Applying the so-called ＂discrete Duhamel product＂ technique, it is proven that for any integer k ≥1 the sequence {（wi＋nk）-1｜W ｜Ei）knf}n≥0 is a basic sequence in Ei ：=span{zi＋n ：n≥0} equivalent to the basis {zi＋n}n≥0 if and only if fb（i） 6= 0. We also investigate a Banach algebra structure for the subspaces Ei, i≥0.

Discrete (J,J′)-Lossless Factorization

Discrete (J,J′) lossless factorization is established by using conjugation.For stable case ,the existence of such factorization is equivalent to the existence of a positive solution of a Riccati equation. For unstable case ,the existence conditions can be reduced to the existence of two positive solution of two Riccati equations.

A Principal Theorem of Normal Discretization Schemes for Operator Equations of the First Kind

The authors announce a newly-proved theorem of theirs. This theorem is of principal significance to numerical computation of operator equations of the first kind.

Discrete Heat Equation Model with Shift Values

We investigate the generalized partial difference operator and propose a model of it in discrete heat equation in this paper. The diffusion of heat is studied by the application of Newton’s law of cooling in dimensions up to three and several solutions are postulated for the same. Through numerical simulations using MATLAB, solutions are validated and applications are derived.

The Discrete Horizontal Complex on Lattice Space

We define discrete total differential forms on lattice space by. changing coefficients of discrete differential forms from functions only of n to functions also of dependent variables un and their partial differences. And the discrete exterior derivative extends to be discrete total differential map which is also nilpotent. Then a discrete horizontal complex can be derived and be proved to be exact by constructing homotopy operators.

SINGULAR INTEGRAL OPERATORS ANDSINGULAR QUADRATURE OPERATORSASSOCIATED WITH SINGULAR INTEGRAL EQUATIONS

In this paper, the author discusses some singular integral operators, singular quadrature operators and discretization matrices associated with singular integral equations with Cauchy kernels, and obtain some useful properties for them. These results improve both the classical theory of singular integral equation and the classical theory of singular quadrature.

The Optimal Matching Parameter of Half Discrete Hilbert Type Multiple Integral Inequalities with Non-Homogeneous Kernels and Applications

By using the weight function method,the matching parameters of the half discrete Hilbert type multiple integral inequality with a non-homogeneous kernel K(n,||x||ρ,m)=G(nλ1||x||ρmλ,2)are discussed,some equivalent conditions of the optimal matching parameter are established,and the expression of the optimal constant factor is obtained.Finally,their applications in operator theory are considered.

Uniqueness Theorem for the Non-Local Ionization Source in Glow Discharge and Hollow Cathode

The paper is devoted to the proof of the uniqueness theorem for solution of the equation for the non-local ionization source in a glow discharge and a hollow cathode in general 3D geometry. The theorem is applied to wide class of electric field configurations, and to the walls of discharge volume, which have a property of incomplete absorption of the electrons. Cathode is regarded as interior singular source, which is placed arbitrarily close to the wall. The existence of solution is considered also. During the proof of the theorem many of useful structure formulae are obtained. Elements of the proof structure, which have arisen, are found to have physical sense. It makes clear physical construction of non-local electron avalanche, which builds a source of ionization in glow discharge at low pressures. Last has decisive significance to understand the hollow cathode discharge configuration and the hollow cathode effect.

An operator methodology for the global dynamic analysis of stochastic nonlinear systems

In a global dynamic analysis,the coexisting attractors and their basins are the main tools to understand the system behavior and safety.However,both basins and attractors can be drastically influenced by uncertainties.The aim of this work is to illustrate a methodology for the global dynamic analysis of nondeterministic dynamical systems with competing attractors.Accordingly,analytical and numerical tools for calculation of nondeterministic global structures,namely attractors and basins,are proposed.First,based on the definition of the Perron-Frobenius,Koopman and Foias linear operators,a global dynamic description through phase-space operators is presented for both deterministic and nondeterministic cases.In this context,the stochastic basins of attraction and attractors’distributions replace the usual basin and attractor concepts.Then,numerical implementation of these concepts is accomplished via an adaptative phase-space discretization strategy based on the classical Ulam method.Sample results of the methodology are presented for a canonical dynamical system.

基于神经算子与类物理信息神经网络智能求解新进展

深度学习通过多层神经网络对数据进行学习,不仅能揭示潜藏信息,还能很好地解决复杂非线性问题.偏微分方程(PDE)是描述自然界中许多物理现象的基本数学模型.两者的碰撞与融合,产生了基于深度学习的PDE智能求解方法,它具有高效、灵活和通用等优点.文章聚焦PDE智能求解方法,以是否求解单一问题为判定依据,把求解方法分为两类:神经算子方法和类物理信息神经网络(PINN)方法,其中神经算子方法用于求解一类具有相同数学特征的PDE问题,类PINN方法用于求解单一问题.对于神经算子方法,从数据驱动和物理约束两个方面展开介绍,分析研究现状并指出现有方法的不足.对于类PINN方法,首先介绍了基础PINN的3种改进方法 (基于数据优化、基于模型优化和基于领域知识优化),然后详细介绍了基于物理驱动的两类解决方案:基于传统PDE离散方程的智能求解方案和无网格的非离散求解方案.最后总结技术路线,探讨现有研究存在的不足,给出可行的研究方案.最后,简要介绍智能求解程序发展现状,并对未来研究方向给出建议.

梯级水库优化调度的改进ODDDP算法研究

“维数灾”问题和易陷入局部最优问题是梯级水库优化调度模型求解中的热点和难点。正交离散微分动态规划算法依据正交试验原理,解决了离散微分动态规划算法在求解水库群联合优化调度问题中的维数灾问题,但并未克服其依赖初始解质量和易陷入局部最优的缺陷。对此,借鉴入侵杂草算法空间扩散机制,引入高斯随机变量,并基于三角余弦函数的周期性质,设计提出了M-IWO-ODDDP算法,从算法全局搜索能力和搜索深度两方面,提高算法全局收敛能力。采用Schaffer、Shubert两种测试函数,分析测试了M-IWOODDDP算法的收敛性和鲁棒性。并以乌江流域11级梯级水库为例开展实例应用研究,结果表明,M-IWOODDDP寻优效果显著且对初始解质量无明显依赖性,算法收敛能力强、鲁棒性好。

基于离散粒子群算法的电力芯片电路运行参数优化

电力芯片电路运行中受到不确定因素的干扰,信噪比较低。为此基于离散粒子群算法提出了一种新的电力芯片电路运行参数优化方法,分析输出功率和发电功率,根据计算结果得到目标函数,确定离散粒子群的分布状态,平行转换最优粒子,建立参数序列,构建电力芯片电路运行参数优化模型,提取运行参数特征,二次优化调试数据,剔除异常数据,建立离散阵列模型,通过数据扫描实现参数优化。实验结果表明,所设计方法在面对连续运行电路数据和随机电路数据时,具有极好的优化效果,平均绝对值误差低于0.10,均方根误差低于0.20,信噪比较高,具有较好的效果。