ON THE PRODUCT OF GTEAUX DIFFERENTIABILITY LOCALLY CONVEX SPACES

A locally convex space is said to be a Gateaux differentiability space (GDS) provided every continuous convex function defined on a nonempty convex open subset D of the space is densely Gateaux differentiable in .D.This paper shows that the product of a GDS and a family of separable Prechet spaces is a GDS,and that the product of a GDS and an arbitrary locally convex space endowed with the weak topology is a GDS.     

Curvature-compensated convective transport: SMART,A new boundedness- preserving transport algorithm

The paper describes a new approach to approximating the convection term found in typical steady-state transport equations. A polynomial-based discretization scheme is constructed around a technique called ‘curvature compensation’; the resultant curvature-compensated convective transport approximation is essentially third-order accurate in regions of the solution domain where the concept of order is meaningful. In addition, in linear scalar transport problems it preserves the boundedness of solutions. Sharp changes in gradient in the dependent variable are handled particularly well. But above all, the scheme, when used in conjunction with an ADI pentadiagonal solver, is easy to implement with relatively low computational cost, representing an effective algorithm for the simulation of multi-dimensional fluid flows. Two linear test problems, for the case of transport by pure convection, are employed in order to assess the merit of the method.     

关于n阶微分方程边值问题（Ⅲ）——奇摄动的渐近展开

本文研究n阶非线性过值问题(NB)的奇异摄动。在较一般的条件下,应用高阶微分不等式理论证明了摄动解的存在性,并给出了摄动解直到n阶导函数的一致有效渐近展开式,推广和改进了已有的结果。     

正则局部Lipschitz函数可微性的几何条件

Ｂａｎａｃｈ空间的范数可微性可用它的单位球面来刻画．本文把这些范数可微性的几何条件推广到正则局部Ｌｉｐｓｃｈｉｔｚ函数情形．     

On the Optimal Value Function of a Linearly Perturbed Quadratic Program

The optimal value function of the quadratic program , where is a given symmetric matrix, a given matrix, and are the linear perturbations, is considered. It is proved that is directionally differentiable at any point in its effective domain . Formulae for computing the directional derivative of at in a direction are obtained. We also present an example showing that, in general, is not piecewise linear-quadratic on W. The preceding (unpublished) example of Klatte is also discussed.     

On the relationship between differentiability conditions and existence of a strong gradient

It is proved that for any n 2 there exists continuous function f : ⁿ which is differentiable almost everywhere, but has no strong gradient almost everywhere.Translated from Matematicheskie Zametki, vol. 77, no. 1, 2005, pp. 93–98.Original Russian Text Copyright © 2005 by G. G. Oniani.This revised version was published online in April 2005 with a corrected issue number.     

Fréchet differentiability of the solutions of a semilinear abstract Cauchy problem

Sufficient conditions are found under which the solutions z(t;q) of a semilinear abstract Cauchy problem of the form are Fréchet differentiable with respect to the parameter q. An explicit form is provided for the sensitivity equation satisfied by the Fréchet derivative Dqz(t;q).     

A critical evaluation of seven discretization schemes for convection–diffusion equations

A comparative study of seven discretization schemes for the equations describing convection-diffusion transport phenomena is presented. The (differencing) schemes considered are the conventional central- and upwind-difference schemes, together with the Leonard,¹ Leonard upwind¹ and Leonard super upwind difference¹ schemes. Also tested are the so called locally exact difference scheme² and the quadratic-upstream difference scheme.^3,4 In multidimensional problems errors arise from ‘false-diffusion’ and function approximations. It is asserted that false diffusion is essentially a multidimensional source of error. No mesh constraints are associated with errors in function approximation and discretization. Hence errors associated with discretization only may be investigated via one-dimensional problems. Thus, although the above schemes have been tested for one- and two-dimensional flows with sources, only the former are presented here. For 1D flows, the Leonard super upwind difference scheme and the locally exact scheme are shown to be far superior in accuracy to the others at all Peclet numbers and for most source distributions, for the test cases considered. Furthermore, the latter is shown to be considerably cheaper in computational terms than the former. The stability of the schemes and their CPU time requirements are also discussed.