The Schrodinger-like Equation for a Nonrelativistic Electron in a Photon Field of Arbitrary Intensity

The ordinary Schrodinger equation with minimal coupling for a nonrelativistic electron interacting with a single-mode photon field is not satisfied by the nonrelativistic limit of the exact solutions to the corresponding Dirac equation. A Schrodinger-like equation valid for arbitrary photon intensity is derived from the Dirac equation without the weak-field assumption. The "eigenvalue" in the new equation is an operator in a Cartan subalgebra. An approximation consistent with the nonrelativistic energy level derived from its relativistic value replaces the "eigenvalue" operator by an ordinary number, recovering the ordinary Schrodinger eigenvalue equation used in the formal scattering formalism. The Schrodinger-like equation for the multimode case is also presented.Comment: Tex file, 13 pages, no figur

Spatio-temporal generalised frequency response functions

The concept of generalised frequency response functions (GFRFs), which were developed for nonlinear system identification and analysis, is extended to continuous spatio-temporal dynamical systems normally described by partial differential equations (PDEs). The paper provides the definitions and interpretation of spatio-temporal generalised frequency response functions for linear and nonlinear spatio-temporal systems based on an impulse response procedure. A new probing method is also developed to calculate the GFRFs. Both the Diffusion equation and Fisher’s equation are analysed to illustrate the new frequency domain methods

Photoproduction of K∗+ΛK^{*+}\Lambda and K+Σ(1385)K^+\Sigma(1385) in the reaction \gamma \lowercase{p} \to K^+ \Lambda \pi^0 at Jefferson Lab

The search for missing nucleon resonances using coupled channel analysis has mostly been concentrated on $N\pi$ and $KY$ channels, while the contributions of $K^*Y$ and $KY^*$ channels have not been investigated thoroughly mostly due to the lack of data. With an integrated luminosity of about 75 $pb^{-1}$, the photoproduction data using a proton target recently collected by the CLAS Collaboration at Jefferson Lab with a photon energy range of 1.5-3.8 GeV provided large statistics for the study of light hyperon photoproduction through exclusive reactions. The reaction $\gamma p \to K^+ \Lambda \pi^0$ has been investigated. Preliminary results of the $K^{*+}\Lambda$ and $K^+\Sigma(1385)$ cross sections are not negligible compared with the $KY$ channels. The $\Lambda \pi^0$ invariant mass spectrum is dominated by the $\Sigma(1385)$ signal and no significant structure was found around the $\Sigma(1480)$ region.Comment: 4 pages, 3 figures, to be publised on the NSTAR05 proceeding

Spatio-temporal generalised frequency response functions over unbounded spatial domains

The concept of generalised frequency response functions (GFRFs), which were developed for nonlinear system identification and analysis, is extended to continuous spatio-temporal dynamical systems normally described by partial differential equations (PDEs). The paper provides the definitions and interpretation of spatio-temporal generalised frequency response functions for linear and nonlinear spatio-temporal systems, defined over unbounded spatial domains, based on an impulse response procedure. A new probing method is also developed to calculate the GFRFs. Both the Diffusion equation and Fisher’s equation are analysed to illustrate the new frequency domain methods

Identification of partial differential equation models for a class of multiscale spatio-temporal dynamical systems

In this paper, the identification of a class of multiscale spatio-temporal dynamical sys-tems, which incorporate multiple spatial scales, from observations is studied. The proposed approach is a combination of Adams integration and an orthogonal least squares algorithm, in which the multiscale operators are expanded, using polynomials as basis functions, and the spatial derivatives are estimated by finite difference methods. The coefficients of the polynomials can vary with respect to the space domain to represent the feature of multiple scales involved in the system dynamics and are approximated using a B-spline wavelet multi-resolution analysis (MRA). The resulting identified models of the spatio-temporal evolution form a system of partial differential equations with different spatial scales. Examples are provided to demonstrate the efficiency of the proposed method

On the least common multiple of qq-binomial coefficients

In this paper, we prove the following identity \lcm({n\brack 0}_q,{n\brack 1}_q,...,{n\brack n}_q) =\frac{\lcm([1]_q,[2]_q,...,[n+1]_q)}{[n+1]_q}, where ${n\brack k}_q$ denotes the $q$-binomial coefficient and $[n]_q=\frac{1-q^n}{1-q}$. This result is a $q$-analogue of an identity of Farhi [Amer. Math. Monthly, November (2009)].Comment: 5 page

Consistent parameter identification of partial differential equation models from noisy observations

This paper introduces a new residual-based recursive parameter estimation algorithm for linear partial differential equations. The main idea is to replace unmeasurable noise variables by noise estimates and to compute recursively both the model parameter and noise estimates. It is proven that under some mild assumptions the estimated parameters converge to the true values with probability one. Numerical examples that demonstrate the effectiveness of the proposed approach are also provided

Identification of N-state spatio-temporal dynamical systems using a polynomial model

A multivariable polynomial model is introduced to describe n-state spatio-temporal systems. Based on this model, a new neighbourhood detection and transition rules determination method is proposed. Simulation results illustrate that the new method performs well even when the patterns are corrupted by static and dynamical noise