Model-theory of vector-spaces over unspecified fields

Vector spaces over unspecified fields can be axiomatized as one-sorted structures, namely, abelian groups with the relation of parallelism. Parallelism is binary linear dependence. When equipped with the n-ary relation of linear dependence for some positive integer n, a vector-space is existentially closed if and only if it is n-dimensional over an algebraically closed field. In the signature with an n-ary predicate for linear dependence for each positive integer n, the theory of infinite-dimensional vector spaces over algebraically closed fields is the model-completion of the theory of vector spaces.      

基于主成分分析的湖南竞争力评价研究

介绍了主成分分析方法,提出了运用主成分评价模型必须满足的条件。并且借鉴中国科学院可持续发展研究组构建的可持续发展指标体系,运用主成分分析构建了区域竞争力的综合评价指标,将湖南的经济发展与其他地区比较,并提出相应的策略。     

Random sorting networks

A sorting network is a shortest path from 12?n to n?21 in the Cayley graph of Sn generated by nearest-neighbour swaps. We prove that for a uniform random sorting network, as n→∞ the space-time process of swaps converges to the product of semicircle law and Lebesgue measure. We conjecture that the trajectories of individual particles converge to random sine curves, while the permutation matrix at half-time converges to the projected surface measure of the 2-sphere. We prove that, in the limit, the trajectories are Hölder-1/2 continuous, while the support of the permutation matrix lies within a certain octagon. A key tool is a connection with random Young tableaux.     

Symbolic topological sorting with OBDDs

We present a symbolic OBDD algorithm for topological sorting which requires O(log²|V|) OBDD operations. Then we analyze its true runtime for the directed grid graph and show an upper bound of O(log⁴|V|loglog|V|). This is the first true runtime analysis of a symbolic OBDD algorithm for a fundamental graph problem, and it demonstrates that one can hope that the algorithm behaves well for sufficiently structured inputs.     

Parallel complexity of sorting problems

The paper discusses three basic sorting problems using a parallel model, the selection problem, the merging problem and the full sorting problem. The merging problem is completely solved, for the other two upper and lower bounds for the cost functions are derived.     

On sorting by 3-bounded transpositions

Heath and Vergara [Sorting by short block moves, Algorithmica 28 (2000) 323-352] proved the equivalence between sorting by 3-bounded transpositions and sorting by correcting skips and correcting hops. This paper explores various algorithmic as well as combinatorial aspects of correcting skips/hops, with the aim of understanding 3-bounded transpositions better.We show that to sort any permutation via correcting hops and skips, ⌊n/2⌋ correcting skips suffice. We also present a tighter analysis of the approximation algorithm of Heath and Vergara, and a possible simplification. Along the way, we study the class H_n of those permutations of S_n which can be sorted using correcting hops alone, and characterize large subsets of this class. We obtain a combinatorial characterization of the set G_n⊆S_n of all correcting-hop-free permutations, and describe a linear-time algorithm to optimally sort such permutations. We also show how to efficiently sort a permutation with a minimum number of correcting moves.     

Inverse sorting problem by minimizing the total weighted number of changes and partial inverse sorting problems

In this paper, we consider two types of inverse sorting problems. The first type is an inverse sorting problem by minimizing the total weighted number of changes with bound constraints. We present an O(n ²) time algorithm to solve the problem. The second type is a partial inverse sorting problem and a variant of the partial inverse sorting problem. We show that both the partial inverse sorting problem and the variant can be solved by a combination of a sorting problem and an inverse sorting problem. Supported by the Hong Kong Universities Grant Council (CERG CITYU 103105) and the National Key Research and Development Program of China (2002CB312004) and the National Natural Science Foundation of China (700221001, 70425004).     

Bounds for sorting by prefix reversal

For a permutation σ of the integers from 1 to n, let ?(σ) be the smallest number of prefix reversals that will transform σ to the identity permutation, and let ?(n) be the largest such ?(σ) for all σ in (the symmetric group) S_n. We show that $\text{?(n)?}\text{(5n+5)}\text{3}$, and that $\text{?(n)?}\text{17n}\text{16}$ for n a multiple of 16. If, furthermore, each integer is required to participate in an even number of reversed prefixes, the corresponding function g(n) is shown to obey $\text{3n}\text{2?1}\text{?g(n)?2n+3}$.     

Exploiting locality: approximating sorting buffers

The sorting buffers problem is motivated by many applications in manufacturing processes and computer science, among them car-painting and file servers architecture. The input is a sequence of items of various types. All the items must be processed, one by one, by a service station. We are given a random-access sorting buffer with a limited capacity. Whenever a new item arrives it may be moved directly to the service station or stored in the buffer. Also, at any time items can be removed from the buffer and assigned to the service station. Our goal is to give the service station a sequence of items with minimum type transitions. We generalize the problem to allow items with different sizes and type transitions with different costs. We give a polynomial-time 9-approximation algorithm for the maximization variant of this problem, which improves the best previously known 20-approximation algorithm.     

Minimal spanning trees and partial sorting

Two partial sorting algorithms used in conjunction with Kruskal's algorithm to find minimal spanning trees, are tested. The superior method can be used in the computation of the Held-Karp bound for the traveling salesman problem and other sort-based greedy algorithms.     

A select and insert sorting algorithm

A general sorting algorithm, having the well knownO(n ²) algorithmsStraight Insertion Sort andSelection Sort as special cases, is described. This algorithm is analyzed in the case that certain choices in the algorithm are done randomly, and this yields an algorithm that has an average complexity ofO(n ^1.5) and a worst case complexity ofO(n ²). However, making random choices (by some random number generator) is time consuming, and a simple quasi-random scheme is therefore implemented. Test runs indicate that also this version has average complexity ofO(n ^1.5), and it even seems to perform better than the version using real random choices (even if we disregard the time used for the random choices). This version also needs very little administrative overhead, and it therefore compares favourably to many other sorting algorithms for small and medium sized arrays.     

On parallel hashing and integer sorting

The problem of sorting n integers from a restricted range [1…m], where m is a superpolynomial in n, is considered. An o(n log n) randomized algorithm is given. Our algorithm takes O(n log log m) expected time and O(n) space. (Thus, for m = n^polylog(n) we have an O(n log log n) algorithm.) The algorithm is parallelizable. The resulting parallel algorithm achieves optimal speedup. Some features of the algorithm make us believe that it is relevant for practical applications. A result of independent interest is a parallel hashing technique. The expected construction time is logarithmic using an optimal number of processors, and searching for a value takes O(1) time in the worst case. This technique enables drastic reduction of space requirements for the price of using randomness. Applicability of the technique is demonstrated for the parallel sorting algorithm and for some parallel string matching algorithms. The parallel sorting algorithm is designed for a strong and nonstandard model of parallel computation. Efficient simulations of the strong model on a CRCW PRAM are introduced. One of the simulations even achieves optimal speedup. This is probably the first optimal speedup simulation of a certain kind.     

Friction and rheological properties of resin-impregnated glass tape

The dependence of the dynamic coefficient of friction of resin-impregnated glass tape on the sliding speed over the mandrel surface and the viscosity of the resin is derived. The formulas obtained are confirmed by the results of experiments which indicate a linear dependence of the coefficient of friction on the sliding speed of the tape over a metal mandrel. Graphically, this dependence is expressed by straight lines with a common origin and a slope that increases with the viscosity of the resin. The coefficient of friction of glass tape sliding over a prewound layer at low speeds also has a linear dependence. At high speeds the dependence takes the form of a damped curve.Translated from Mekhanika Polimoerov, No. 1, pp. 166–170, January–February, 1976.