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.     

On the properties of solutions to the Goursat-Darboux problem with boundary and distributed controls

We consider a control system described by the Goursat-Darboux equation. The system is controlled by distributed and boundary controls. The controls are subject to the constraints given as multivalued mappings with closed, possibly nonconvex, values depending on the phase variable. Alongside the initial constraints, we consider the convexified constraints and the constraints whose values are the extreme points of the convexified constraints. We study the questions of existence of solutions and establish connections between the solutions under various constraints.     

Optimality and duality in multiobjective fractional programming involving ρ-semilocally type I-preinvex and related functions

Optimality conditions are obtained for a nonlinear fractional multiobjective programming problem involving η-semidifferentiable functions. Also, a general dual is formulated and a duality result is proved using concepts of generalized ρ-semilocally type I-preinvex functions.     

Generators of some Ramanujan formulas

In this paper we prove some Ramanujan type formulas for 1/π but without using the theory of modular forms. Instead we use the WZ—method created by H. Wilf and D. Zeilberger and find some hypergeometric functions in two variables which are second components of WZ—pairs than can be certified using Zeilberger's EKHAD package. These certificates have an additional property which allows us to get generalized Ramanujan's type series which are routinely proven by computer. We call these second hypergeometric components of the WZ—pairs generators. Finding generators seems a hard task but using a kind of experimental research (explained below), we have succeeded in finding some of them. Unfortunately we have not found yet generators for the most impressive Ramanujan's formulas. We also prove some interesting binomial sums for the constant 1/π². Finally we rewrite many of the obtained series using pochhammer symbols and study the rate of convergence. 2000 Mathematics Subject Classification Primary—33C20     

集值优化问题强有效解的Kuhn Tucker最优性条件

在局部凸空间中考虑集值优化问题（VP）在强有效解意义下的Kuhn-Tucker最优性条件．在近似锥．次类凸假设下利用择一性定理得到了（VP）取得强有效解的必要条件，利用基泛函的性质给出了（VP）取得强有效解的充分条件，最后给出了一种与（VP）等价的无约束规划。     

Linearisation and potential symmetries of certain systems of diffusion equations

We consider systems of two pure one-dimensional diffusion equations that have considerable interest in Soil Science and Mathematical Biology. We construct non-local symmetries for these systems. These are determined by expressing the equations in a partially and wholly conserved form, and then by performing a potential symmetry analysis on those systems that can be linearised. We give several examples of such systems, and in a specific case we show how linearising and hodograph-type mappings can lead to new solutions of the diffusion system.     

A web-based workforce management system for Sainsburys Supermarkets Ltd

Labour scheduling plays an important and critical role in Sainsburys day to day operations and profitability. The integrated workforce management solution for Sainsburys is an intelligent solution for the operational deployment of human resources, integrated business forecasting, generation and calculation of workload and staff scheduling. The workforce management technology has provided Sainsburys with a fully web-based and centralised system that produces optimised schedules for all employees satisfying the forecasted demand of work with granularity of fifteen minutes. The system is configured to respect all operational and business requirements as well as employee contractual constraints. This paper introduces the workforce management solution system designed, developed and implemented for Sainsburys. We concentrate on two modelling structures, namely the work pattern model and optimisation model, and briefly discuss the optimisation process.     

单调和Fuzzy单调变分不等式的补偿方法

在实自反Banach空间,用补偿方法研究单调和Fuzzy单调广义非线性变分不等式,得到了几个解的存在性定理.改进和推广了Browder,Pascali-Sburlan,Zhang等人的各种结果.     

Scattering from infinite rough tubular surfaces

We study the Helmholtz equation in the exterior of an infinite perturbed cylinder with a Dirichlet boundary condition. Existence and uniqueness of solutions are established using the variational technique introduced (SIAM J. Math. Anal. 2005; 37 (2):598–618). We also provide stability estimates with explicit dependence of the constants in terms of the frequency and the perturbed cylinder thickness. Copyright © 2006 John Wiley & Sons, Ltd.     

关于环网G（N;S1,S2）的直径

本文首先给出了环网G(N;s_1,s_2)存在有限直径的充要条件;用初等数论方法给出了其直径的简明计算公式;最后讨论了类似文献[3]的环网直径的对偶性问题.