具有稠密临界轨道和预不动临界轨道的单峰映射的丰富性

证明了对于实二次族在参数空间存在正Ｌｅｂｅｓｇｕｅ测度集合Ｅ,使得Ｅ中几乎所有的参数,相应的映射在不变测度的支集上具有稠密的临界轨道;还证明了Ｅ中存在稠密集合使得相应映射的临界轨道进入它的反向不动点。     

齐次完全集的拟对称极小性

作为Cantor型集的推广,文志英和吴军引入了齐次完全集的概念,并基于齐次完全集的基本区间的长度以及基本区间之间的间隔的长度,得到了齐次完全集的Hausdorff维数.本文研究齐次完全集的拟对称极小性,证明在某些条件下Hausdorff维数为1的齐次完全集是1维拟对称极小的.     

拟度量空间中弱拟对称映射的一些特征

该文研究了弱拟对称映射在拟度量空间中的相关性质.引入了环与环性质的概念,并用环的性质来刻画了弱拟对称映射在拟度量空间中的一些特征.     

On global quadratic growth condition for min-max optimization problems with quadratic functions

Second-order sufficient condition and quadratic growth condition play important roles both in sensitivity and stability analysis and in numerical analysis for optimization problems. In this article, we concentrate on the global quadratic growth condition and study its relations with global second-order sufficient conditions for min-max optimization problems with quadratic functions. In general, the global second-order sufficient condition implies the global quadratic growth condition. In the case of two quadratic functions involved, we have the equivalence of the two conditions.     

Low-temperature phase transitions in the quadratic family

We give the first example of a quadratic map having a phase transition after the first zero of the geometric pressure function. This implies that several dimension spectra and large deviation rate functions associated to this map are not (expected to be) real analytic, in contrast to the uniformly hyperbolic case. The quadratic map we study has a non-recurrent critical point, so it is non-uniformly hyperbolic in a strong sense.     

Topological conditions for the existence of absorbing Cantor sets

This paper deals with strange attractors of S-unimodal maps . It generalizes earlier results in the sense that very general topological conditions are given that either

i)

guarantee the existence of an absorbing Cantor set provided the critical point of is sufficiently degenerate, or

ii)

prohibit the existence of an absorbing Cantor set altogether.

As a by-product we obtain very weak topological conditions that imply the existence of an absolutely continuous invariant probability measure for .

     

Quasisymmetric and Schur expansions of cycle index polynomials

Given a subgroup $G$ of the symmetric group $S_n$, the cycle index polynomial ${\mathrm{cyc}}_G$ is the average of the power-sum symmetric polynomials indexed by the cycle types of permutations in $G$. By Pólya’s Theorem, the monomial expansion of ${\mathrm{cyc}}_G$ is the generating function for weighted colorings of $n$ objects, where we identify colorings related by one of the symmetries in $G$. This paper develops combinatorial formulas for the fundamental quasisymmetric expansions and Schur expansions of certain cycle index polynomials. We give explicit bijective proofs based on standardization algorithms applied to equivalence classes of colorings. Subgroups studied here include Young subgroups of $S_n$, the alternating groups $A_n$, direct products, conjugate subgroups, and certain cyclic subgroups of $S_n$ generated by $(1,2,\dots ,k)$. The analysis of these cyclic subgroups when $k$ is prime reveals an unexpected connection to perfect matchings on a hypercube with certain vertices identified.     

Non-trivial quadratic approximations to zero of a family of cubic Pisot numbers

This paper gives exact rates of quadratic approximations to an infinite class of cubic Pisot numbers. We show that for any cubic Pisot number , with minimal polynomial , such that , and where has only one real root, then there exists a , explicitly given here, such that:

(1)

For all 0$">, all but finitely many integer quadratics satisfy

where is the height function.

(2)

For all 0$"> there exists a sequence of integer quadratics such that

Furthermore, for all in this class of cubic Pisot numbers. What is surprising about this result is how precise it is, giving exact upper and lower bounds for these approximations.

     

On the average speed of Lemke's algorithm for quadratic programming

In this paper, we show that the average number of steps of the Lemke algorithm for the quadratic programming problems grows at most linearly in the number of variables while fixing the number of constraints. The result and method were motivated by Smale's result on linear programming problems [cf. 4]. We also give the probability that a quadratic programming problem indeed possesses a finite optimal solution.     

面积定理推广

本文把单位国外ｍ叶半纯函数的面积定理，推广到只假定函数在无穷有ｍ级极点情形．并给出面积大于零的充要条件．     

The quadratic form in the Lévy-Khinchin formula on semigroups

In this paper we obtain the quadratic form in the Lévy-Khinchin formula on a commutative involutive semigroup, with a neutral element, as a sum of two simpler quadratic forms.

     

Reducing the number of variables in integer quadratic programming problem

In this paper, a new variable reduction technique is presented for general integer quadratic programming problem (GP), under which some variables of (GP) can be fixed at zero without sacrificing optimality. A sufficient condition and a necessary condition for the identification of dominated terms are provided. By comparing the given data of the problem and the upper bound of the variables, if they meet certain conditions, some variables can be fixed at zero. We report a computational study to demonstrate the efficacy of the proposed technique in solving general integer quadratic programming problems. Furthermore, we discuss separable integer quadratic programming problems in a simpler and clearer form.     

Necessary and sufficient condition for local minima of a class of nonconvex quadratic programs

The author (1992, 1993) earlier studied the equivalence of a class of 0–1 quadratic programs and their relaxed problems. Thus, a class of combinatorial optimization problems can be solved by solving a class of nonconvex quadratic programs. In this paper, a necessary and sufficient condition for local minima of this class of nonconvex quadratic programs is given; this will be the foundation for study of algorithms.Research supported by Huo Yingdong Educational Foundation '93.     

On Urysohn-Volterra fractional quadratic integral equations

In this paper the authors study a fractional quadratic integral equation of Urysohn-Volterra type. They show that the integral equation has at least one monotonic solution in the Banach space of all real functions defined and continuous on the interval $[0,1]$. The main tools in the proof are a fixed point theorem due to Darbo and a monotonicity measure of noncompactness.     

On the continuity of the minimum in parametric quadratic programs

This paper establishes results on the lower semicontinuity and continuity of the optimal objective value of a parametric quadratic program with continuous dependence of the coefficients on parameters. The results established herein generalize directly well-known results on parametric linear programs.This research was supported by the Natural Sciences and Engineering Research Council of Canada under Grant No. A8189.     

Note on a sufficient condition for a local minimum of a class of nonconvex quadratic programs

A counterexample is given to show that a previously proposed sufficient condition for a local minimum of a class of nonconvex quadratic programs is not correct. This class of problems arises in combinatorial optimization. The problem with the original proof is pointed out. © 1998 The Mathematical Programming Society, Inc. Published by Elsevier Science B.V.