Gauge-invariant treatment of the exponential decay of an unstable atomic state

Summary By using a self-consistent projection-operator method, developed recently by the present author, a gauge-invariant Markovian treatment of the spontaneous decay of an unstable atomic state is presented. Without ignoring retardation and electron-spin effects a gauge-invariant result for the probability amplitude of finding the atom in the unstable state is obtained.     

On the Solvability of the Multidimensional Version of the First Darboux Problem for a Model Second-Order Degenerating Hyperbolic Equation

A multidimensional version of the first Darboux problem is considered for a model second-order degenerating hyperbolic equation. Using the technique of functional spaces with a negative norm, the correct formulation of this problem in the Sobolev weighted space is proved.     

Fluorescent humic substances–arsenic complex in well water in areas where blackfoot disease is endemic in Taiwan

The relationship between the four components, (1) fluorescence intensity, (2) arsenic concentration, (3) pH and (4) total dissolved solids, (TDS) measured in well waters from areas in Taiwan where blackfoot disease (BFD) is endemic was studied, as well as the relationships between the four degrees of BFD and each of the above four symptomatic components, in order to evaluate the etiological factors of BFD more progressively. The following 95% confidence intervals were obtained in well water samples (n = 1189): fluorescence intensity, 26.837–32.570; arsenic concentration, 0.103–0.127 mg dm^?3; pH, 7.466–7.519; and TDS 733.063–801.647 mg dm^?3. Fluorescence intensities of the four degrees of BFD were not all the same (F = 64.54, P < 0.001), and nor were arsenic concentrations (F = 72.03, P < 0.001), pH values (F = 7.30, P < 0.001), nor TDS values (F = 10.76, P < 0.001). In addition, multiple comparisons indicate that the higher the epidemical degree, the higher the fluorescence intensities, arsenic concentrations and pH values become; however, such a relationship is not found for TDS values. Moreover, the fluorescence intensities have positive linear correlations with arsenic concentrations (r = 0.49, P < 0.001), pH (r = 0.25, P < 0.001), and TDS (r = 0.18, P < 0.001), as do the arsenic concentrations with pH (r = 0.22, P < 0.001). Of the four epidemical degree groups, pairs are not significantly different from one another in correlation coefficients between fluorescence intensity and arsenic concentration, which implies a steady relationship between fluorescent compounds and arsenic. We conclude that fluorescent compounds in well water, as possible etiological factors of BFD, are closely related to arsenic along with pH and TDS values in the areas where BFD is endemic. In addition, we infer that a complex is formed by fluorescent compounds, arsenic and other metals.     

Classical limit of relativistic quantal system with attractive Coulomb interaction

Using the appropriate harmonic oscillator states and reasonable approximations, we construct coherent wavepackets corresponding to the solutions of the Klein-Gordon equation for the attractive potentialV(r)=−k/r, k>0, in two and three space dimensions. We deduce the corresponding classical limit in two dimension by requiring that the expectation value 〈r〉 of the radial variable is large. In the case of three dimensions, besides the condition of large 〈r〉, we make the uncertainty Δr=[〈r ²〉 − 〈r〉²]^1/2 a minimum with respect to certain parameter of the wavepacket. We then investigate the trajectory traversed by the wavepacket in the classical limit. We find that the classical limit of this relativistic quantal problem gives, in the leading order, the same expression for the rate of motion of the perihelion as that given by the solution of the corresponding special relativistic classical dynamical problem. We also briefly discuss some of the subtle aspects of the classical limit of the relativistic quantal system, in general.     

在相干理论的基础上分析光栅成像

本文在相干理论的基础上,推导了光栅成像中的光强分布和物像位置关系,并给出一组具有不同参数的实验照片及一组借助计算机绘制的强度分布图。     

Derived Brackets

We survey the many instances of derived bracket construction in differential geometry, Lie algebroid and Courant algebroid theories, and their properties. We recall and compare the constructions of Buttin and of Vinogradov, and we prove that the Vinogradov bracket is the skew-symmetrization of a derived bracket. Odd (resp., even) Poisson brackets on supermanifolds are derived brackets of canonical even (resp., odd) Poisson brackets on their cotangent bundle (resp., parity-reversed cotangent bundle). Lie algebras have analogous properties, and the theory of Lie algebroids unifies the results valid for manifolds on the one hand, and for Lie algebras on the other. We outline the role of derived brackets in the theory of Poisson structures with background'.     

Trefftz,collocation, and other boundary methods—A comparison

In this article we survey the Trefftz method (TM), the collocation method (CM), and the collocation Trefftz method (CTM). We also review the coupling techniques for the interzonal conditions, which include the indirect Trefftz method, the original Trefftz method, the penalty plus hybrid Trefftz method, and the direct Trefftz method. Other boundary methods are also briefly described. Key issues in these algorithms, including the error analysis, are addressed. New numerical results are reported. Comparisons among TMs and other numerical methods are made. It is concluded that the CTM is the simplest algorithm and provides the most accurate solution with the best numerical stability. © 2006 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2007     

Bivariate Free Knot Splines

We consider the least squares approximation of gridded 2D data by tensor product splines with free knots. The smoothing functional to be minimized—a generalization of the univariate Schoenberg functional—is chosen in such a way that the solution of the bivariate problem separates into the solution of a sequence of univariate problems in case of fixed knots. The resulting optimization problem is a constrained separable least squares problem with tensor product structure. Based on some ideas developed by the authors for the univariate case, an efficient method for solving the specially structured 2D problem is proposed, analyzed and tested on hand of some examples from the literature.     

Study of hot bands in the B2Σ+u-X2Σ+g system of C-2 anion

Optical heterodyne magnetic rotation enhanced velocity modulation spectroscopy was employed to observe the visible absorption spectra of the B^2Σ^+_u-X^2Σ^+_g electronic transition of C^-_2. Four hot bands (0,1), (1,2), (2,3) and (3,4) have been observed and the band (3,4) is measured directly for the first time, so far as we know, by absorption. A rotational analysis was carried out to obtain molecular constants. With the Franck－Condon principle and the vibrational Boltzmann distribution, we have estimated the vibrational temperature of C^-_2 to be about 3000K.