The minimum forcing number of perfect matchings in the hypercube

Let $M$ be a perfect matching in a graph. A subset $S$ of $M$ is said to be a forcing set of $M$, if $M$ is the only perfect matching in the graph that contains $S$. The minimum size of a forcing set of $M$ is called the forcing number of $M$. Pachter and Kim (1998) conjectured that the forcing number of every perfect matching in the $n$-dimensional hypercube is $2^{n?2}$, for all $n\ge 2$. This was revised by Riddle (2002), who conjectured that it is at least $2^{n?2}$, and proved it for all even $n$. We show that the revised conjecture holds for all $n\ge 2$. The proof is based on simple linear algebra.     

A tripling construction for mutually orthogonal symmetric hamiltonian double Latin squares

We provide two new constructions for pairs of mutually orthogonal symmetric hamiltonian double Latin squares. The first is a tripling construction, and the second is derived from known constructions of hamilton cycle decompositions of when is prime.     

Perfect matchings extend to two or more hamiltonian cycles in hypercubes

Kreweras conjectured that every perfect matching of a hypercube $Q_n$ for $n\ge 2$ can be extended to a hamiltonian cycle of $Q_n$. Fink confirmed the conjecture to be true. It is more general to ask whether every perfect matching of $Q_n$ for $n\ge 2$ can be extended to two or more hamiltonian cycles of $Q_n$. In this paper, we prove that every perfect matching of $Q_n$ for $n\ge 4$ can be extended to at least $2^{2^{n?4}}$ different hamiltonian cycles of $Q_n$.     

On the threshold problem for Latin boxes

Let m ≤ n ≤ k. An m × n × k 0‐1 array is a Latin box if it contains exactly m n ones, and has at most one 1 in each line. As a special case, Latin boxes in which m = n = k are equivalent to Latin squares. Let be the distribution on m × n × k 0‐1 arrays where each entry is 1 with probability p, independently of the other entries. The threshold question for Latin squares asks when contains a Latin square with high probability. More generally, when does support a Latin box with high probability? Let ε > 0. We give an asymptotically tight answer to this question in the special cases where n = k and , and where n = m and . In both cases, the threshold probability is . This implies threshold results for Latin rectangles and proper edge‐colorings of K_n,n.     

一类正交空间填充设计的构造

空间填充设计在计算机试验中应用十分广泛,当拟合回归模型时,正交的空间填充设计保证了因子效应估计的独立性.基于广义正交设计,文章给出了构造二阶正交拉丁超立方体设计和列正交设计的方法,新构造的设计不仅满足任意两列之间相互正交,还能保证每一列与任一列元素平方组成的列以及任两列元素相乘组成的列都正交.当某些正交的空间填充设计不存在时,具有较小相关系数的近似正交设计可作为替代设计使用.设计构造的灵活性为计算机试验在实践中的广泛应用提供了必要的支持.     

The Development of Appropriate Upscaling Procedures

Permeability upscaling should be carried out with careful attention to the nature of rock heterogeneities. While there are many large-scale features which must be taken into account, there are also important heterogeneities at the small-scale. Many sedimentary structures contain laminae at the mm–cm scale, and beds at the m-scale, which give rise to strong contrasts in permeability. We use a 2D model derived from a photo-panel of an aeolian outcrop, along with permeability measurements from a North Sea oil field, to demonstrate the effects of small-scale heterogeneity. This model is similar in size to a typical cell of a reservoir geological model. We take imaginary probe and core plug measurements from the model, average them, and compare these with the effective permeability for the model computed from a finite difference flow calculation. Although this procedure is standard practice, we show that it can lead to biased estimates of the permeabilities used in flow simulation. As an alternative we suggest using models of representative beds, and performing flow simulation to calculate effective permeabilities for both single-phase and two-phase flow.     

The Order of Automorphisms of Quasigroups

We prove quadratic upper bounds on the order of any autotopism of a quasigroup or Latin square, and hence also on the order of any automorphism of a Steiner triple system or 1‐factorization of a complete graph. A corollary is that a permutation σ chosen uniformly at random from the symmetric group will almost surely not be an automorphism of a Steiner triple system of order n, a quasigroup of order n or a 1‐factorization of the complete graph . Nor will σ be one component of an autotopism for any Latin square of order n. For groups of order n it is known that automorphisms must have order less than n, but we show that quasigroups of order n can have automorphisms of order greater than n. The smallest such quasigroup has order 7034. We also show that quasigroups of prime order can possess autotopisms that consist of three permutations with different cycle structures. Our results answer three questions originally posed by D.  Stones.     

Classification of Graeco‐Latin Cubes

A q‐ary code of length n, size M, and minimum distance d is called an code. An code with is said to be maximum distance separable (MDS). Here one‐error‐correcting () MDS codes are classified for small alphabets. In particular, it is shown that there are unique (5, 5³, 3)₅ and (5, 7³, 3)₇ codes and equivalence classes of (5, 8³, 3)₈ codes. The codes are equivalent to certain pairs of mutually orthogonal Latin cubes of order q, called Graeco‐Latin cubes.     

沿海软粘土取土质量的对比分析

本文在简要介绍固定活塞式薄壁取土器取土技术基础上，详尽分析了在连云港、赤湾港和广深路所进行的软粘土取土技术对比试验结果。试验结果表明，土样扰动对于室内所测定的力学性指标影响极大，但对物理性指标影响不大。研究结果还有力地证明，只要操作正确，采用薄壁取土器能取得高质量土样，而采用敞口式厚壁取土器取出的土样扰动较大。