Geometrical study of paraxial light beam transmission in free space

By introducing an imaginary space transform curvature ρ_s, a complex space called Riemannian space is constructed, in which the light propagating in free space has the trajectory of straight line while propagating. Moreover, this curvature couples with that of the wave front of the paraxial beam ρ_w, and therefore a complex curvature ρ_c is constructed, which can be employed to investigate the behavior of the light transmission and to generalize the ABCD law. Project supported by the National Hi-Tech Inertial Confinement Fusion Committee, the Guangdong Natural Science Foundation the Postdoctoral Foundation of Guangdong and National Postdoctoral Foundation of China.     

How to decrease the levels of modular integrals

A method to decrease the level of a modular integral with a given level is advanced. Project supported by the National Natural Science Foundation of China (Grant No. 19501009) and the Natural Science Foundation of Guangdong Province.     

Hausdorff measure of sets of finite type of one-sided symbolic space

Some estimation formulae and a computation formula for the Hausdorff measure of the sets of finite type of the one-sided symbolic space are given. Project partially supported by the Natural Science Foundation of Guangdong Province and the Foundation of Advanced Research of Zhongshan University.     

Ideals in Morita Rings and Morita Semigroups

We characterize the lattice of all ideals of a Morita ring （semigroup） when the corresponding pair of rings （semigroups） in the Morita context are Morita equivalent s-unital （like-unitv） rings （semigroups）.     

The Hausdorff measure of a Sierpinski carpet

The exact value of the Hausdorff measure of a Sierpinski carpet has been obtained. Project partially supported by the Science Foundation of Guangdong Province.     

General A-P iterative algorithm in shift-invariant spaces

A general A-P iterative algorithm in a shift-invariant space is presented. We use the algorithm to show reconstruction of signals from weighted samples and also show that the general improved algorithm has better convergence rate than the existing one. An explicit estimate for a guaranteed rate of convergence is given.     

Existence of positive solutions for p -laplacian singular boundary value problems of functional differential equation

In this paper, the boundary value problems of p-Laplacian functional differential equation are studied. By using a fixed point theorem in cones, some criteria for the existence of positive solutions are given. Partially supported by the Natural Science Foundation of Guangdong Province (011471).     

A Schr(o)idinger formulation research for light beam propagation

The wave equation of light beam propagation was written in the form of an axial-coordinate-dependent Schrdinger equation, and the expectation value of a dynamical variable, the trial function of variational approach and the ABCD law were discussed by use of quantum mechanics approach. In view of the evolution equations of expectation values of dynamical variables in the framework of quantum mechanics, the definition of a potential function representing the beam propagation stability and its universal formula with the quality factor, the universal formula of beam width and curvature radius for a paraxial beam and cylindrically symmetric non-paraxial beam, the general formula of second derivative of beam width with respect to the axial coordinate of beam for a paraxial beam, and the general criteria of the conservation of beam quality factor and the existence of a potential well of a potential function for a paraxial beam, were given or derived, respectively. Starting with the same trial function, the comparative research of our formulation with variational approach was done, which gave some further insight into the physical nature of a beam propagation parameters. The ABCD law of non-paraxial beam was discussed in terms of the definition of the non-paraxial expectation value of a dynamical variable for the first time. The applications to the media of constant second derivative of beam width with respect to the axial coordinate of a beam, square law media and the media of constant refractive index in the momentum representation were discussed, respectively.     

A Schr?dinger formulation research for light beam propagation

The wave equation of light beam propagation was written in the form of an axial-coordinate-dependent Schr?dinger equation, and the expectation value of a dynamical variable, the trial function of variational approach and the ABCD law were discussed by use of quantum mechanics approach. In view of the evolution equations of expectation values of dynamical variables in the framework of quantum mechanics, the definition of a potential function representing the beam propagation stability and its universal formula with the quality factor, the universal formula of beam width and curvature radius for a paraxial beam and cylindrically symmetric non-paraxial beam, the general formula of second derivative of beam width with respect to the axial coordinate of beam for a paraxial beam, and the general criteria of the conservation of beam quality factor and the existence of a potential well of a potential function for a paraxial beam, were given or derived, respectively. Starting with the same trial function, the comparative research of our formulation with variational approach was done, which gave some further insight into the physical nature of a beam propagation parameters. The ABCD law of nonparaxial beam was discussed in terms of the definition of the non-paraxial expectation value of a dynamical variable for the first time. The applications to the media of constant second derivative of beam width with respect to the axial coordinate of a beam, square law media and the media of constant refractive index in the momentum representation were discussed, respectively.     

<Emphasis Type="Italic">m</Emphasis> Components-admissible solutions of systems of higher-order partial differential equations on <Emphasis Type="Italic">C</Emphasis><Superscript><Emphasis Type="Italic">n</Emphasis></Superscript>

Using value distribution theory and techniques in several complex variables,we investigate the problem of existence of m components-admissible solutions of a class of systems of higher-order partial differential equations in several complex variables and estimate the number of admissible components of solutions.Some related results will also be obtained.     

Asymptotic Speed of Wave Propagation for A Discrete Reaction-Diffusion Equation

We deal with asymptotic speed of wave propagation for a discrete reactlon-diffusion equation. We find the minimal wave speed c★ from the characteristic equation and show that c★ is just the asymptotic speed of wave propagation. The isotropic property and the existence of solution of the initial value problem for the given equation are also discussed.     

Moment inequalities and weak convergence for negatively associated sequences

A probability inequality for S_n and somepth moment (p⩾2) inequalities for |S_n| and max 1⩽k⩽n | S_k| are established. Here S_n is the partial sum of a negatively associated sequence. Based on these inequalities, a weak invariance principle for strictly stationary negatively associated sequences is proved under some general conditions. Project supported by the National Natural Science Foundation of China, the Doctoral Program Foundation of the State Education Commission of China and the High Eductional Natural Science Foundation of Guangdong Province.     

Asymptotic Smoothing Effect of Solutions to Davey-Stewartson Systems on the Whole Plane

This paper discusses the Cauchy problem of elliptic-elliptic-type Davey-Stewartson systems with zero-order dissipation on R^2. Making use of the Fourier spectral projector, together with a long time comparison between the solutions to the Davey-Stewartson systems and to an auxiliary problem, we prove that the global attractor in H^1 （R^2） for the addressed Davey-Stewartson systems is a compact subset of H^2（R^2）, and thus reveal the asymptotic smoothing effect of the solutions for the systems.     

On the Growth of Components of Meromorphic Solutions of Systems of Complex Differential Equations

This paper investigates the problem of the growth of the components of meromorphic solutions of a class of a system of complex algebraic differential equations, and generalized some of N. Toda＇s results concerning the growth of differential equations to the case of systems of differential equations. The paper considers the existence of admissible solutions of the system of differential equations.     

Separate Necessary and Sufficient Conditions for the Local Minimum of a Function

This paper develops two necessary conditions for a local minimum of an arbitrary extended real-valued function (Theorem 2.2) and, quite separately, two conditions, each sufficient for such a function to have a strict local minimum (Theorem 2.3).This work was supported by the National Science Foundation of China, Grant 10171034, and the Natural Science Foundation of the Department of Education of Guangdong Province, Grant 820138.The author thanks the referees for helpful comments.     

On perturbation bounds of generalized eigenvalues for diagonalizable pairs

The purpose of this paper is to study the perturbation of generalized eigenvalues. Two perturbation bounds of the diagonalizable pairs are given. These results extend the corresponding ones given by Sun (Math Numer Sinica 4:23–29, 1982). This work is supported by the Natural Science Foundation of Guangdong Province (No. 06025061) and by the National Natural Science Foundations of China (No. 10671077 and 10626021).     

Iterated processes and their applications to higher order differential equations

In this paper we construct models obtained by suitably combining Brownian motions and telegraphs in such a way that their transition functions satisfy higher-order parabolic or hyperbolic equations of different types. Equations with time-varying coefficients are also derived by considering processes endowed either with drift or with suitable modifications of their structure. Finally the distribution of the maximum of the iterated Brownian motion (along with some other properties) is presented. This work is partially supported by the Natural Science Foundation of Guangdong Province, National Natural Science Foundation of China grant No. 19501026 and the Alexander von Humbodlt Foundation     

Further generalization of Fermat’s principle

From Maxwell’ s equations for electromagnetic fields, time-averaged energy flow density vector of stable monochromatic linearly polarized light in an isotropic insulative nonmagnetic medium is deduced. By the introduction of time-averaged energy flow density rays and the definition of new generalized refractive indexn _G1, Fermat’s principle of geometric optics is further generalized and its application conditions are discussed. The generalized Fermat' s principle can be used to describe stable transmission of light in a medium with variable refractive index. The necessary and sufficient conditions of a nondivergent and nonfocusing light beam are derived from this Fermat’s principle. Project supported by the National Natural Science Foundation of China (Grant No. 69789801) and Guangdong Provincial Natural Science Foundation of China.     

The Essential Component of the Solution Set for Vector Equilibrium Problems

In the paper, by using Ky Fan’s section theorem, we obtain an existence theorem for vector equilibrium problems. Motivated by the ideas of Kinoshita and McLennan, we introduce the concept of the essential component of the solution set for vector equilibrium problems, and we prove that there exists at least one essential component of the solution set for every vector equilibrium problem satisfying some conditions. Research was partially supported by the Natural Science Foundation of Guangdong Province, P. R. China.     

Electric field effect on properties of PT/epoxy nanocomposites

The structure and optic properties of PbTiO₃ nanocrystalline powder/epoxy composites are studied under different electric fields during solidification. X-ray diffraction (XRD), scattered light, ultraviolet optic absorption and second harmonic generation (SHG) of the composites are measured. Results show that the electric field has a great effect on the properties of nanocomposites. The mechanism of electric field production is discussed. Project supported by the National Advanced Materials Research Project and the Natural Science Foundation of Guangdong Province of China.