On the structure of stationary sets

We isolate several classes of stationary sets of [k]ωand investigate implications among them. Under a large cardinal assumption, we prove a structure theorem for stationary sets.     

Optimisation of fault-tolerant fabric-cutting schedules using genetic algorithms and fuzzy set theory

In apparel industry, manufacturers developed standard allowed minutes (SAMs) databases on various manufacturing operations in order to facilitate better scheduling, while effective production schedules ensure smoothness of downstream operations. As apparel manufacturing environment is fuzzy and dynamic, rigid production schedules based on SAMs become futile in the presence of any uncertainty. In this paper, a fuzzification scheme is proposed to fuzzify the static standard time so as to incorporate some uncertainties, in terms of both job-specific and human related factors, into the fabric-cutting scheduling problem. A genetic optimisation procedure is also proposed to search for fault-tolerant schedules using genetic algorithms, such that makespan and scheduling uncertainties are minimised. Two sets of real production data were collected to validate the proposed method. Experimental results indicate that the genetically optimised fault-tolerant schedules not only improve the operation performance but also minimise the scheduling risks.     

Bases,spanning sets,and the axiom of choice

Two theorems are proved: First that the statement “there exists a field F such that for every vector space over F, every generating set contains a basis” implies the axiom of choice. This generalizes theorems of Halpern, Blass, and Keremedis. Secondly, we prove that the assertion that every vector space over ?₂ has a basis implies that every well‐ordered collection of two‐element sets has a choice function. (© 2007 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

An algorithm for decomposing a polynomial system into normal ascending sets

We present an algorithm to decompose a polynomial system into a finite set of normal ascending sets such that the set of the zeros of the polynomial system is the union of the sets of the regular zeros of the normal ascending sets.If the polynomial system is zero dimensional,the set of the zeros of the polynomials is the union of the sets of the zeros of the normal ascending sets.     

解约束优化问题的QP-free可行域方法

本文利用一个新的分片线性NCP函数提出一个新的可行的QP-free方法解非线性不等式约束优化问题.不同于其他的QP-free方法,这个方法只考虑在工作集中的约束函数,工作集是积极集的一个估计,因此子问题的维数不是满秩的.这个方法可行的并且不需假定严格互补条件、聚点的孤立性得到算法的全局收敛性,并且积极约束函数的梯度不要求线性独立的,其中由拟牛顿法得到的子矩阵不需要求一致正定性.     

混合方次的Waring——Goldbach问题

研究混合方次的Waring--Goldbach     

THE DIMENSION FOR RANDOM SUB-SELF-SIMILAR SET

In this article,the Hausdorff dimension and exact Hausdorff measure function of any random sub-self-similar set are obtained under some reasonable conditions.Several examples are given at the end.     

On then-cutset property

For a partially ordered setP and an elementx ofP, a subsetS ofP is called a cutset forx inP if every element ofS is noncomparable tox and every maximal chain ofP meets {x}∪S. We letc(P) denote the smallest integerk such that every elementx ofP has a cutsetS with ‖S‖?k: Ifc(P)?n we say thatP has then-cutset property. Our results bear on the following question: givenP, what is the smallestn such thatP can be embedded in a partially ordered set having then-cutset property? As usual, 2 ⁿ denotes the Boolean lattice of all subsets of ann-element set, andB _n denotes the set of atoms and co-atoms of 2 ⁿ . We establish the following results: (i) a characterization, by means of forbidden configurations, of whichP can be embedded in a partially ordered set having the 1-cutset property; (ii) ifP contains a copy of 2 ⁿ , thenc(P)?2^[n/2]?1; (iii) for everyn>3 there is a partially ordered setP containing 2 ⁿ such thatc(P)<c(2 ⁿ ); (iv) for every positive integern there is a positive integerN such that, ifB _m is contained in a partially ordered set having then-cutset property, thenm?N.     

A family of skew Hadamard difference sets

In 1933 a family of skew Hadamard difference sets was described by Paley using matrix language and was called the Paley-Hadamard difference sets in the literature. During the last 70 years, no new skew Hadamard difference sets were found. It was conjectured that there are no further examples of skew Hadamard difference sets. This conjecture was proved to be true for the cyclic case in 1954, and further progress in favor of this conjecture was made in the past 50 years. However, the conjecture remains open until today. In this paper, we present a family of new perfect nonlinear (also called planar) functions, and construct a family of skew Hadamard difference sets using these perfect nonlinear functions. We show that some of the skew Hadamard difference sets presented in this paper are inequivalent to the Paley-Hadamard difference sets. These new examples of skew Hadamard difference sets discovered 70 years after the Paley construction disprove the longstanding conjecture on skew Hadamard difference sets. The class of new perfect nonlinear functions has applications in cryptography, coding theory, and combinatorics.     

On a Polynomial Fractional Formulation for Independence Number of a Graph

In this paper we characterize the local maxima of a continuous global optimization formulation for finding the independence number of a graph. Classical Karush-Kuhn-Tucker conditions and simple combinatorial arguments are found sufficient to deduce several interesting properties of the local and global maxima. These properties can be utilized in developing new approaches to the maximum independent set problem.