Support-type properties of generalized convex functions

Chebyshev systems induce in a natural way a concept of convexity. The functions convex in this sense behave in many aspects similarly to ordinary convex functions. In this paper support-type properties are investigated. Using osculatory interpolation, the existence of support-like functions is established for functions convex with respect to Chebyshev systems. Unique supports are determined. A characterization of the generalized convexity via support properties is presented.     

A second-order pseudo-transient method for steady-state problems

This article gives a second pseudo-transient method for a special system of nonlinear equations, which arises from chemical reaction rate equations. This method uses a special second-order Rosenbrock method as the discrete difference scheme, which satisfies a linear conservation law. Moreover, it adaptively adjusts the time step in inverse proportion to an arithmetic mean of the current residual and the previous residual at every iteration step. For a singular system of nonlinear equations, under some standard assumptions, local convergence of the new method is addressed. Finally, some promise numerical results are also reported.     

Bifurcations in parametric nonlinear programming

Bifurcation and continuation techniques are introduced as a class of methods for investigating the parametric nonlinear programming problem. Motivated by the Fritz John first-order necessary conditions, the parametric programming problem is first reformulated as a closed system of nonlinear equations which contains all Karush-Kuhn-Tucker and Fritz John points, both feasible and infeasible solutions, and relative minima, maxima, and saddle points. Since changes in the structure of the solution set and critical point type can occur only at singularities, necessary and sufficient conditions for the existence of a singularity are developed in terms of the loss of a complementarity condition, the linear dependence constraint qualification, and the singularity of the Hessian of the Lagrangian on a tangent space. After a brief introduction to elementary bifurcation theory, some simple singularities in this parametric problem are analyzed for both branching and persistence of local minima. Finally, a brief introduction to numerical continuation and bifurcation procedures is given to indicate how these facts can be used in a numerical investigation of the problem.This research was supported by the Air force Office of Scientific Research through grant number AFOSR-88-0059.     

Solving the radiation diffusion and energy balance equations using pseudo-transient continuation

We develop a scheme for the system coupling the radiation diffusion and matter energy balance equations. The method is based on fully implicit, first-order, backward Euler differencing; Picard-Newton iterations solve the nonlinear system. We show that iterating on the radiation energy density and the emission source is more robust. Since the Picard-Newton scheme may not converge for all initial conditions and time steps, pseudo-transient continuation (Ψtc) is introduced. The combined Ψtc-Picard-Newton scheme is analyzed. We derive conditions on the Ψtc parameter that guarantee physically meaningful iterates, e.g., positive energies. Successive Ψtc iterates are bounded and the radiation energy density and emission source tend to equilibrate. The scheme is incorporated into a multiply dimensioned, massively parallel, Eulerian, radiation-hydrodynamic computer program with automatic mesh refinement (AMR). Three examples are presented that exemplify the scheme's performance. (1) The Pomraning test problem that models radiation flow into cold matter. (2) A similar, but more realistic problem simulating the propagation of an ionization front into tenuous hydrogen gas with a Saha model for the equation-of-state. (3) A 2D axisymmetric (R,Z) simulation with real materials featuring jetting, radiatively driven, interacting shocks.     

On the asymmetric clocked buffered switch

A 2 × 2 clocked buffered switch is a device used in data-processing networks for routing messages from one node to another. The message handling process of this switch can be modelled as a two-server, time slotted, queueing process with state space the number of messages (x_n , y_n) present at the servers at the end of a time slot. The x_n , y_n-process is a two-dimensional nearest-neighbour random walk. In the present study the bivariate generating function Φ(p,q) of the stationary distribution of this random walk is determined, assuming that this distribution exists. Φ(p,q) is known, whenever Φ(p,0) and Φ(0,q) are known. The essential points of the present study are the construction of these two functions from the knowledge of their poles and zeros and the simple determination of these poles and zeros. This revised version was published online in June 2006 with corrections to the Cover Date.     

一类抽象算子方程组的迭代解及其应用

在Banach空间中, 利用半序方法讨论了一类抽象算子方程组解的存在唯一性, 推广和统一了以前的一些结果. 然后应用到 Banach 空间非线性积分方程组, 得到了方程组的唯一解, 构造了收敛于方程组唯一解的迭代序列并给出了相应的误差估计.     

Analytic continuation of the exponential transform from convex cavities

The analytic continuation of the exponential transform of a domain in Rn is proved under some global geometric assumptions on the boundary. Two approximation schemes of the continued transform (one based on a Taylor series truncation, the other on a global eigenfunction expansion) are also discussed.     

包含极值点和分枝点的解曲线跟踪算法的摄动解法

本文提出了采用摄动格式求解非线性方程组的解曲线跟踪算法的计算格式.文中着重讨论了解曲线上非正则点的搜索,以及从这些非正则点——转向点或分枝点——继续跟踪超临界平衡路径的计算方法.文中把这一算法应用于弹性薄壳的屈曲分析。通过柱壳和环壳的算例得到它们的整个屈曲过程的平衡路径和变形形态.     

一类多偏差变元的二阶微分方程周期解

本文利用重合度理论研究了一类二阶多偏差变元的微分方程 x＂（t）＋f（t,x（t）,x（t-τ0（t））,x＇（t））＋∑nj=1g（x（t-τj（t）））=p（t） 的周期解问题，得到了存在周期解的新的结果．