玻璃光盘基片

本文对制备光盘的各种材料的性能做了比较,叙述了采用白冕平板玻璃研制直径为133.35mm可擦除光盘基片的工艺过程,以及产品达到的性能指标。     

Characterizations of an approximate minimum in optimal control

By using the classical variational methods based on geodesic coverings of a domain and on Hilbert's independent integral, further characterizations of an approximate solution in problems of control are described. The starting point is the Ekeland-type characterization, the variational principle. As consequences, sufficient conditions for optimality are obtained in a form similar to the Weierstrass conditions from the calculus of variations.The author is grateful to the referee for the valuable counterexamples to the first version of the paper.     

Lower bounds for coverings of pairs by large blocks

Letnkt be positive integers, andX—a set ofn elements. LetC(n, k, t) be the smallest integerm such that there existm k-tuples ofX B ₁ B ₂,...,B _m with the property that everyt-tuple ofX is contained in at least oneB _i . It is shown that in many cases the standard lower bound forC(n, k, 2) can be improved (k sufficiently large,n/k being fixed). Some exact values ofC(n, k, 2) are also obtained.     

Computational aspects of monotone dualization: A brief survey

Dualization of a monotone Boolean function represented by a conjunctive normal form (CNF) is a problem which, in different disguise, is ubiquitous in many areas including Computer Science, Artificial Intelligence, and Game Theory to mention some of them. It is also one of the few problems whose precise tractability status (in terms of polynomial-time solvability) is still unknown, and now open for more than 25 years. In this paper, we briefly survey computational results for this problem, where we focus on the famous paper by Fredman and Khachiyan [On the complexity of dualization of monotone disjunctive normal forms, J. Algorithms 21 (1996) 618-628], which showed that the problem is solvable in quasi-polynomial time (and thus most likely not co-NP-hard), as well as on follow-up works. We consider computational aspects including limited nondeterminism, probabilistic computation, parallel and learning-based algorithms, and implementations and experimental results from the literature. The paper closes with open issues for further research.     

The infinite sum of reciprocal Pell numbers

In this paper, we consider infinite sums derived from the reciprocals of the Pell numbers. Then applying the floor function to the reciprocals of this sums, we obtain a new and interesting identity involving the Pell numbers.     

All double coverings of cyclotomic fields

A double covering of a Galois extension K/F in the sense of [3] is an extension /K of degree ≤2 such that /F is Galois. In this paper we determine explicitly all double coverings of any cyclotomic extension over the rational number field in the complex number field. We get the results mainly by Galois theory and by using and modifying the results and the methods in [2] and [3]. Project 10571097 supported by NSFC     

Pentagons and cycle coverings

Let G be a graph of order n ≥ 5k + 2, where k is a positive integer. Suppose that the minimum degree of G is at least ?(n + k)/2?. We show that G contains k pentagons and a path such that they are vertex‐disjoint and cover all the vertices of G. Moreover, if n ≥ 5k + 7, then G contains k + 1 vertex‐disjoint cycles covering all the vertices of G such that k of them are pentagons. © 2006 Wiley Periodicals, Inc. J Graph Theory 54: 194–208, 2007     

Lifting of homeomorphisms to 4-sheeted branched coverings of a disk

Let p:X→D be a simple, possibly not connected, 4-sheeted branched covering of a closed 2-dimensional disk D with n branch values A₁,…,An. The isotopy classes of homeomorphisms of D which are fixed on the boundary of D and permute the branch values form a braid group Bn. Some of these homeomorphisms can be lifted to homeomorphisms of X. They form a subgroup L(p) of finite index in Bn. For each equivalence class of coverings we find a set of generators for L(p) which contains between n and n+4 elements, depending on the equivalence class of the covering, and the generators are powers of half-twists.     

Partial Degree Conditions and Cycle Coverings

Let G be a graph of order n. Let W be a subset of with , where k is a positive integer. We show that if for each , then G contains k vertex‐disjoint cycles covering W such that each of the k cycles contains at least three vertices of W.