Critical sets in parametric optimization

We deal with one-parameter families of optimization problems in finite dimensions. The constraints are both of equality and inequality type. The concept of a generalized critical point (g.c. point) is introduced. In particular, every local minimum, Kuhn-Tucker point, and point of Fritz John type is a g.c. point. Under fairly weak (even generic) conditions we study the set consisting of all g.c. points. Due to the parameter, the set is pieced together from one-dimensional manifolds. The points of can be divided into five (characteristic) types. The subset of nondegenerate critical points (first type) is open and dense in (nondegenerate means: strict complementarity, nondegeneracy of the corresponding quadratic form and linear independence of the gradients of binding constraints). A nondegenerate critical point is completely characterized by means of four indices. The change of these indices along is presented. Finally, the Kuhn-Tucker subset of is studied in more detail, in particular in connection with the (failure of the) Mangasarian-Fromowitz constraint qualification.     

Polynomial Forms of Typical Interatomic Potential Functions

The use of polynomial functionals for describing two-body interactions in computational chemistry softwares has been surveyed and found to be prevalent. In this paper, Binomial and Maclaurin series expansions are used for expressing typical interatomic potential functions – such as Lennard-Jones, Morse, Rydberg and Buckingham potential – in a generic polynomial function, with the coefficients presented in a tabular format. Theoretical plots of these potential functions and their corresponding polynomial forms show increasing correlation with the order of polynomial, thereby validating the obtained polynomial’s coefficients. Conversely, a polynomial functional obtained by curve-fitting of experimental data can be converted into Morse, Rydberg and Buckingham potentials by using the generated table.     

Primal-relaxed dual global optimization approach

A deterministic global optimization approach is proposed for nonconvex constrained nonlinear programming problems. Partitioning of the variables, along with the introduction of transformation variables, if necessary, converts the original problem into primal and relaxed dual subproblems that provide valid upper and lower bounds respectively on the global optimum. Theoretical properties are presented which allow for a rigorous solution of the relaxed dual problem. Proofs of -finite convergence and -global optimality are provided. The approach is shown to be particularly suited to (a) quadratic programming problems, (b) quadratically constrained problems, and (c) unconstrained and constrained optimization of polynomial and rational polynomial functions. The theoretical approach is illustrated through a few example problems. Finally, some further developments in the approach are briefly discussed.The authors gratefully acknowledge financial support from National Science Foundation Presidential Young Investigator Award CBT-88-57013. The authors are also grateful to Drs. F. A. Al-Khayyal, B. Jaumard, P. M. Pardalos, and H. D. Sherali for helpful comments on an earlier draft of this paper.     

Signature classes of transportation polytopes

Signature algorithms solve certain classes of transportation problems in a number of steps bounded by the diameter of the dual polyhedron. The class of problems to which signature algorithms may be applied—the signature classes of the title—are characterized, and the monotonic Hirsch conjecture is shown to hold for them. In addition, certain more precise results are given for different cases. Finally, it is explained why a supposed proof of the Hirsch conjecture for all transportation polytopes is incomplete and apparently irremedial.Dedicated with affection to Philip Wolfe on the occasion of his 65th birthday.     

Some Combinatorial Properties of Schubert Polynomials

Schubert polynomials were introduced by Bernstein et al. and Demazure, and were extensively developed by Lascoux, Schützenberger, Macdonald, and others. We give an explicit combinatorial interpretation of the Schubert polynomial in terms of the reduced decompositions of the permutation w. Using this result, a variation of Schensted's correspondence due to Edelman and Greene allows one to associate in a natural way a certain set of tableaux with w, each tableau contributing a single term to . This correspondence leads to many problems and conjectures, whose interrelation is investigated. In Section 2 we consider permutations with no decreasing subsequence of length three (or 321-avoiding permutations). We show for such permutations that is a flag skew Schur function. In Section 3 we use this result to obtain some interesting properties of the rational function , where denotes a skew Schur function.Sara C. Billey: Supported by the National Physical Science Consortium. William Jockusch: Supported by an NSF Graduate Fellowship. Richard P. Stanley: Partially supported by NSF grants DMS-8901834 and DMS-9206374     

Interpreting divergences and Madelung-type sums via extrapolations of sequence generating polynomials: Method of intersection and area rule

Based on explorations in estimating certain Madelung constants, we put forward here two separate strategies to understand the meaning of two distinct classes of divergent non-power-series expansions. One class refers to alternating series representations, the other to monotonic ones. They chiefly rest on precise and approximate polynomial extrapolations, depending on situations. In case of sawtooth sequences, e.g., the partial-sums obtainable from Dirichlet eta or beta function at negative integer arguments, exact sequence-generating polynomials are found. Extrapolations yield a graphical meaning to anti-limit here, along with the exact answer. For staircase sequences, like the ones obtained from partial-sums of series representations for lambda and zeta functions, again at negative integer arguments, anti-limits do not exist. But, correct sequence-generating polynomials are obtained. There, our recipe relies on estimation of specific, finite areas embedded by such polynomials. The schemes put forward here are direct, independent and conceptually appealing. A subsequent extension of the latter strategy to alternating series also lends extra credence. Two new interpretations of summability are gained. Pilot calculations on several types of lattice sums reveal the worth of our endeavor with approximate extrapolations as well.     

证明微分中值问题的辅助多项式法

通过几个典型实例,介绍辅助多项式函数在微分中值证明问题中的应用.     

一类参数二次规划参数延拓的作用集法和稳定性分析

本文用作用集法考虑一类参数二次规划的参数延拓问题，并研究稳定性问题。     

随机扰动下的增长模型

以确定性的Solow增长模型为背景,文章将随机扰动引入其中.因此模型中的要素方程从ODE过渡到It SDE,并对其作某些定性的分析.进而运用二次Liapunov函数等方法对随机Solow模型的零解的稳定性及其平稳分布进行了有益的探索,得出了一些有意义的结果.文章的主要结论是:随机干扰将实质性地改变由确定性增长模型所描述的经济增长路径.     

Binary Bell polynomial application in generalized （2＋1）-dimensional KdV equation with variable coefficients

In this paper, we apply the binary Bell polynomial approach to high-dimensional variable-coefficient nonlinear evolution equations. Taking the generalized (2+1)-dimensional KdV equation with variable coefficients as an illustrative example, the bilinear formulism, the bilinear Bäcklund transformation and the Lax pair are obtained in a quick and natural manner. Moreover, the infinite conservation laws are also derived.