Schrödinger Equation Solutions for the Central Field Power Potential Energy I. V(r) = V 0(r/a 0)2ν−2, ν ≥ 1

The solution of a generalized non-relativistic Schrödinger equation with radial potential energy V(r)=V ₀(r/a ₀)^2–2 is presented. After reviewing the general properties of the radial ordinary differential equation, power series solutions are developed. The Green's function is constructed, its trace and the trace of its first iteration are calculated, and the ability of the traces to provide upper and lower bounds for the ground eigenvalue is examined. In addition, WKB-like solutions for the eigenvalues and eigenfunctions are derived. The approximation method yields valid eigenvalues for large quantum numbers (Rydberg states).     

性别差异对研发投资决策影响的实验研究

基于Z-Tree实验平台利用研发投资自然博弈实验研究发现风险规避导致女性的研发投入强度低于男性,但女性在研发投资过程中愿意投入的研发努力程度高于男性。随着时间推移,男性的研发努力程度明显下降,但女性的研发努力程度反而上升。男性和女性在决策中均表现出心理账户效应,女性更加愿意通过努力来提高业绩。男性在投资中更多体现期权思想,而女性更多表现出长期倾向下的持续努力。文章在揭示性别因素影响研发投资机理基础上,为董事会性别多样性、分级董事会以及递延薪酬等治理机制发挥作用机理提供了新解释。研究结论可以为企业职位的性别配置、分级董事会以及激励契约设计提供借鉴。关键词:性别差异;研发投资;实验研究;心理账户;风险规避;长期倾向     

Estimation of a quantile in some nonstandard cases

A necessary condition for the asymptotic normality of the sample quantile estimator isf(Q(p))=F(Q(p))>0, whereQ(p) is thep-th quantile of the distribution functionF(x). In this paper, we estimate a quantile by a kernel quantile estimator when this condition is violated. We have shown that the kernel quantile estimator is asymptotically normal in some nonstandard cases. The optimal convergence rate of the mean squared error for the kernel estimator is obtained with respect to the asymptotically optimal bandwidth. A law of the iterated logarithm is also established.This research was partially supported by the new faculty award from the University of Oregon.     

Convolution semigroups and central limit theorem associated with a dual convolution structure

We consider hypergroups associated with Jacobi functions ⁽⁾ (x), (–1/2). We prove the existence of a dual convolution structure on [0,+[i(]0,s ₀]{{) =++1,s ₀=min(,–+1). Next we establish a Lévy-Khintchine type formula which permits to characterize the semigroup and the infinitely divisible probabilities associated with this dual convolution, finally we prove a central limit theorem.     

A central limit theorem for age- and density-dependent population processes

Consider a population consisting of one type of individual living in a fixed region with area A. In [8], we constructed a stochastic population model in which the death rate is affected by the age of the individual and the birth rate is affected by the population density P_A(t), i.e., the population size divided by the area A of the given region. In [8], we proposed a continuous deterministic model which in general is a nonlinear Volterra type integral equation and proved that under appropriate conditions the sequence P_A(t) would converge to the solution P(t) of our integral equation in the sense that

$\text{lim}\text{→∞}\text{P}\text{sup}\text{0?s?t}\text{|P}{}_{\mathrm{A}}\text{(s) ? P(s)|>ε}\text{=0}\text{for every}\text{ε > 0}$

.In this paper, we obtain a “central limit theorem” for the random element √A(P_A(t)?P(t)). We prove that under appropriate conditions √A(P_A(t)?P(t)) will converge to a Gaussian process. (See Theorem 3.4 for the explicit formula of this Gaussian process.)     

Evaluation of the use of consensus values in proficiency testing programmes

Proficiency testing (PT) is an essential tool used by laboratory accreditation bodies to assess the competency of laboratories. Because of limited resources of PT providers or for other reasons, the assigned reference value used in the calculation of z-score values has usually been derived from some sort of consensus value obtained by central tendency estimators such as the arithmetic mean or robust mean. However, if the assigned reference value deviates significantly from the ‘true value’ of the analyte in the test material, laboratories’ performance will be evaluated incorrectly. This paper evaluates the use of consensus values in proficiency testing programmes using the Monte Carlo simulation technique. The results indicated that the deviation of the assigned value from the true value could be as large as 40%, depending on the parameters of the proficiency testing programmes under investigation such as sample homogeneity, number of participant laboratories, concentration level, method precision and laboratory bias. To study how these parameters affect the degree of discrepancy between the consensus value and the true value, a fractional factorial design was also applied. The findings indicate that the number of participating laboratories and the distribution of laboratory bias were the prime two factors affecting the deviation of the consensus value from the true value.     

Anaerobic reduction of a sulfonated azo dye, Congo Red, by sulfate-reducing bacteria

The capacity for anaerobic decolorization of a sulfonated azo dye, Congo Red, by a strain of a sulfate-reducing bacterium was evaluated. After optimizing the growth rate of the bacteria on a simple carbon source and terminal electron acceptor pair, lactate and sulfate, respectively, the effect of the dye concentration on their growth rate was analyzed. The decolorization rate was affected by the dye concentration in the growth medium. The azo-bond cleavage mechanism of reductive decolorization with the formation of benzidine was consistent with the results, as this metabolite was identified by high-performance liquid chromatography. Several fractions of the culture medium, including lysed cell extracts, were examined for the capacity to reduce the azo dye. This reduction capacity was found in the culture medium in which the cells had previously grown. The results showed that the mechanism of reductive decolorization of this sulfonated azo dye was extracellular and nonenzymatic, consistent with the production of sulfide anion by the microorganisms while growing on lactate and sulfate. The sulfide anions were the cause of the reduction leading to the disappearance of color in the medium. To increase the rate of decolorization, the presence of ferrous ion was also necessary together with the lactate and sulfate substrates.     

A review ofL=λW and extensions

A fundamental principle of queueing theory isL=W (Little's law), which states that the time-average or expected time-stationary number of customers in a system is equal to the product of the arrival rate and the customer-average or expected customer-stationary time each customer spends in the system. This principle is now well known and frequently applied. However, in recent years there have been extensions, such as H=G and the continuous, distributional, ordinal and central-limit-theorem versions, which show that theL=W relation, when viewed properly, has much more power than was previously realized. Moreover, connections have been established between H=G and other fundamental relations, such as the rate conservation law and PASTA (Poisson arrivals see time averages), which show that there is a much greater unity in the overall theory than was previously realized. This paper provides a review.This paper is dedicated to the memory of our colleague Professor Peter Franken (1937–1989), who contributed greatly to the subject of this paper and to queueing theory more generally.     

Rates of convergence of α-stable random motions

The rate-of-convergence problem in the central limit theorem is considered for -stable distributions on the noncommutative group of motions in ^d. The method used here is based on the theory of probability metrics.     

Monte carlo calculations of the conformal charge

The conformal charge is an important quantity which characterizes the nature of the two-dimensional phase transition. We report a first attempt to use a new numerical method to calculate the conformal charge. In this paper, we apply our method to the 2-dimensional, ⁴, continuous-spin Ising model. By varying the parameters in the Hamiltonian, one can change continuously from the known Gaussian limit to the Ising limit. It is well known that the critical points for these two systems are not in the same universality class. We study this behavior for the Gaussian model, the single-well ⁴ model, the border model, and the double-well ⁴ model for a large lattice. Our results, while giving a good general picture, are not so far sufficient to differentiate whether the non-Gaussian cases studied belong to the Ising model universality class or not. Further studies of other lattice sizes should serve to improve greatly our conclusions.