Multidimensional Permanents and an Upper Bound on the Number of Transversals in Latin Squares

The permanent of a multidimensional matrix is the sum of products of entries over all diagonals. A nonnegative matrix whose every 1‐dimensional plane sums to 1 is called polystochastic. A latin square of order n is an array of n symbols in which each symbol occurs exactly once in each row and each column. A transversal of such a square is a set of n entries such that no two entries share the same row, column, or symbol. Let T(n) be the maximum number of transversals over all latin squares of order n. Here, we prove that over the set of multidimensional polystochastic matrices of order n the permanent has a local extremum at the uniform matrix for whose every entry is equal to . Also, we obtain an asymptotic value of the maximal permanent for a certain set of nonnegative multidimensional matrices. In particular, we get that the maximal permanent of polystochastic matrices is asymptotically equal to the permanent of the uniform matrix, whence as a corollary we have an upper bound on the number of transversals in latin squares      

Existence of a-fold Perfect （v, {5, 8}, 1）-Mendelsohn Designs

Let v be a positive integer and let K be a set of positive integers. A （v, K, 1）-Mendelsohn design, which we denote briefly by （v, K, 1）-MD, is a pair （X, B） where X is a v-set （of points） and B is a collection of cyclically ordered subsets of X （called blocks） with sizes in the set K such that every ordered pair of points of X are consecutive in exactly one block of B. If for all t =1, 2,..., r, every ordered pair of points of X are t-apart in exactly one block of B, then the （v, K, 1）-MD is called an r-fold perfect design and denoted briefly by an r-fold perfect （v, K, 1）-MD. If K = {k） and r = k - 1, then an r-fold perfect （v, （k）, 1）-MD is essentially the more familiar （v, k, 1）-perfect Mendelsohn design, which is briefly denoted by （v, k, 1）-PMD. In this paper, we investigate the existence of 4-fold perfect （v, （5, 8}, 1）-Mendelsohn designs.     

Halving transversal designs

A factor H of a transversal design TD(k,n) = (V,𝒢, ℬ︁), where V is the set of points, 𝒢 the set of groups of size n and ℬ︁ the set of blocks of size k, is a triple (V,𝒢, 𝒟) such that 𝒟 is a subset of ℬ︁. A halving of a TD (k, n) is a pair of factors H_i = (V, 𝒢, 𝒟_i), i = 1,2 such that 𝒟₁ ∪ 𝒟₂ = ℬ︁, 𝒟₁ ∩ 𝒟₂ = ∅︁ and H₁ is isomorphic to H₂. A path of length q is a sequence x₀, x₁,…,x_q of points such that for each i = 1, 2,…, q the points x_i‐1 and x_i belong to a block B_i and no point appears more than once. The distance between points x and y in a factor H is the length of the shortest path from x to y. The diameter of a connected factor H is the maximum of the set of distances among all pairs of points of H. We prove that a TD (3, n) is halvable into isomorphic factors of diameter d only if d = 2,3,4, or ∞ and we completely determine for which values of n there exists such a halvable TD (3, n). We also show that if any group divisible design with block size at least 3 is decomposed into two factors with the same finite diameter d, then d≤ 4. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 83–99, 2000     

On simple and supersimple transversal designs

In this paper, the existence of a transversal design TD_λ (4, g) is proved for all indices λ satisfying 2 ≤ λ ≤ g such that any two of its blocks intersect in at most two elements. Similar results are obtained for transversal designs without repeated blocks. © 2000 John Wiley & Sons, Inc. J Combin Designs 8: 311–320, 2000     

New recursive methods for transversal covers

A transversal cover is a set of gk points in k disjoint groups of size g and a minimum collection of transversal subset s, called blocks, such that any pair of points not contained in the same group appear in at least one block. The case g = 2 was investigated and completely solved by Sperner, Renyi, Katona, Kleitman, and Spencer. For all g, asymptotic results are known, but little is understood for small values of k. Sloane and others have initiated the investigation of g = 3. The present article is concerned with constructive techniques for all g and k. One of the principal constructions generalizes Wilson's theorem for transversal designs. This article also discusses a simulated annealing algorithm for finding transversal covers and gives a table of the best known transversal covers at this time. © 1999 John Wiley & Sons, Inc. J Combin Designs 7: 185–203, 1999     

Stability of a Laser-diode-pumped Cr4+∶YAG Passively Q-switched Nd3+∶YAG Laser

In this paper, the stability of a laser-diode-pumped Cr4+∶YAG passively Nd3+∶YAG Q-switched laser and the influence of the transversal mode structure on the stability are investigated. With the laser operating in TEM00 mode, the pulse energy fluctuation and the repetition rate fluctuation as functions of the repetition rate are measured, and semi-quantitatively and qualitatively analyzed, respectively.     

舟山群岛峡道水动力及沉积特性

本文根据浙江舟山群岛地区金塘水道、册子水道、虾峙门水道和灌门水道的水文泥沙和沉积资料,以峡道水动力特性为出发点,分析了峡道环境中的水文泥沙特性和沉积特性,初步探讨了沉积峡道效应。文章指出,沉积峡道效应是指在峡道环境中,受峡道水动力特性及其影响下的泥沙运动作用,在峡道中所形成的沉积特点和变化过程,这种变化既不同于受单向水流作用的河流环境,也不同于受盐淡水共同作用的河口环境,也有别于开敞的海岸环境,峡道沉积的重要特点在于,在垂直于水流运动方向的横向上,存在显著的沉积相变。     

Stability for basic system of equations of atmospheric motion

The topological characteristics for the basic system of equations of atmospheric motion were analyzed with the help of method provided by stratification theory. It was proved that in the local rectangular coordinate system the basic system of equations of atmospheric motion is stable in infinitely differentiable function class. In the sense of local solution, the necessary and sufficient conditions by which the typical problem for determining solution is well posed were also given. Such problems as something about ＂speculating future from past＂ in atmospheric dynamics and how to amend the conditions for determining solution as well as the choice of underlying surface when involving the practical application were further discussed. It is also pointed out that under the usual conditions, three motion equations and continuity equation in the basic system of equations determine entirely the property of this system of equations.     

Vrănceanu connections and foliations with bundle-like metrics

We show that the Vrănceanu connection which was initially introduced on non-holonomic manifolds can be used to study the geometry of foliated manifolds. We prove that a foliation is totally geodesic with bundle-like metric if and only if this connection is a metric one. We introduce the notion of a foliated Riemannian manifold of constant transversal Vrănceanu curvature and the notion of a transversal Einstein foliated Riemannian manifold. The geometry of these two classes of manifolds is studied and the relationship between them is determined.     

An analytical approach to the design of electrodes in high-power, fast-transverse-flow lasers

Usually, the design of the electrodes of fast-transverse-flow CO₂ lasers relies more on experimental data than on theoretical analysis. Traditional systems sustain a stable, high-power discharge but the current distribution generates a peaked, non-uniform small-signal-gain pattern. We present a theoretical model suitable for the design of electrodes that achieves a quasi uniform gain distribution. The analysis, based on a two-temperature model and the gas-transport equations, computes the electron density that supports a flat gain pattern. Combining the desired electron density with the electron-balance equations, the model determines the required electric field. The results were used to design a new set of electrodes for a home-made fast transverse flow CO₂ laser. A stable, large volume discharge () with very good uniformity was obtained. The resulting gain distribution was registered in a bi-dimensional map. The peak gain rose from , attained with the old electrode set, to with the new one and the gain showed a homogeneous profile.