Some results on the analysis of asymmetric stress

The structure of the principal directions of stress is characterized for the general case of asymmetric stress. It is shown that, for asymmetric stress, the principal directions can comprise either only one direction, or only two directions, or only three directions, or a plane of directions, or a plane of directions together with a direction external to this plane. The asymmetric stress obtained by combining a one-dimensional stress field with a skew-symmetric stress field is analyzed in detail.     

Probabilistic behavior of asymmetric level compressed tries

Level‐Compressed (in short LC) tries were introduced by Andersson and Nilsson in 1993. They are compacted versions of tries in which, from the top down, maximal height complete subtrees are level compressed. We show that when the input consists of n independent strings with independent Bernoulli (p) bits, p ≠ 1/2, then the expected depth of a typical node is in probability asymptotic to where H ? p log p ? (1 ? p) log (1 ? p) is the Shannon entropy of the source, and H_?∞ = log (1 / min(p, 1 ? p)). The height is in probability asymptotic to where H₂ = log(1/(p² + (1?p)²)). © 2005 Wiley Periodicals, Inc. Random Struct. Alg., 2005     

Some results on spanning trees

Some structures of spanning trees with many or less leaves in a connected graph are determined.We show（1） a connected graph G has a spanning tree T with minimum leaves such that T contains a longest path,and（2） a connected graph G on n vertices contains a spanning tree T with the maximum leaves such that Δ（G） =Δ（T） and the number of leaves of T is not greater than n D（G）＋1,where D（G） is the diameter of G.     

Some results on generalized exponents

A digraph G = (V, E) is primitive if, for some positive integer k, there is a u → v walk of length k for every pair u, v of vertices of V. The minimum such k is called the exponent of G, denoted exp(G). The exponent of a vertex u ∈ V, denoted exp(u), is the least integer k such that there is a u → v walk of length k for each v ∈ V. For a set X ⊆ V, exp(X) is the least integer k such that for each v ∈ V there is a X → v walk of length k, i.e., a u → v walk of length k for some u ∈ X. Let F(G, k) : = max{exp(X) : |X| = k} and F(n, k) : = max{F(G, k) : |V| = n}, where |X| and |V| denote the number of vertices in X and V, respectively. Recently, B. Liu and Q. Li proved F(n, k) = (n − k)(n − 1) + 1 for all 1 ≤ k ≤ n − 1. In this article, for each k, 1 ≤ k ≤ n − 1, we characterize the digraphs G such that F(G, k) = F(n, k), thereby answering a question of R. Brualdi and B. Liu. We also find some new upper bounds on the (ordinary) exponent of G in terms of the maximum outdegree of G, Δ⁺(G) = max{d⁺(u) : u ∈ V}, and thus obtain a new refinement of the Wielandt bound (n − 1)² + 1. © 1998 John Wiley & Sons, Inc. J. Graph Theory 28: 215–225, 1998     

关于n-可扩图的一些结果

本文讨论了关于m-可扩图的两个极值问题；并考查了下述图类的n-可扩性；正则偶图，单位区间图和分裂图。     

Some results on b-AM-compact operators

We show that for positive operator B : E → E on Banach lattices, if there exists a positive operator S : E → E such that:1.SB ≤ BS;2.S is quasinilpotent at some x₀ > 0; 3.S dominates a non-zero b-AM-compact operator, then B has a non-trivial closed invariant subspace. Also, we prove that for two commuting non-zero positive operators on Banach lattices, if one of them is quasinilpotent at a non-zero positive vector and the other dominates a non-zero b-AM-compact operator, then both of them have a common non-trivial closed invariant ideal. Then we introduce the class of b-AM-compact-friendly operators and show that a non-zero positive b-AM- compact-friendly operator which is quasinilpotent at some x₀ > 0 has a non-trivial closed invariant ideal.     

Some results on conversion matrices

In this paper we prove a theorem concerning weak lattice constants and hence three matricial equations for conversion matrices.Then we introduce a block-partition for conversion matrices and we write matricial equations for this block-partition; from these matricial equations we propose the calculus of conversion matrices by induction on their order.     

Some results on symplectic matrices

The principal results are that if A is an integral matrix such that AA^T is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.     

Some results on space-like self-shrinkers

We study space-like self-shrinkers of dimension n in pseudo-Euclidean space R_m^m+n with index m. We derive drift Laplacian of the basic geometric quantities and obtain their volume estimates in pseudo-distance function. Finally, we prove rigidity results under minor growth conditions in terms of the mean curvature or the image of Gauss maps.     

Some results on quasipolyhedral convexity

The objective of this paper is to analyze under what well-known operations the class of quasipolyhedral convex functions, which can be regarded as an extension of that of polyhedral convex functions, is closed. The operations that will be considered are those that preserve polyhedral convexity, such that the image and the inverse image under linear transformations, right scalar multiplication (including the case where λ=0⁺) and pointwise addition.      

Some results on sperner families

Let F be a Sperner family of subsets of {1,…,m}. Bollobás showed that if $\text{A ∈ F ?}\text{A}\text{= {1,…,m}βA ∈ F}$, and if the parameters of F are p₀,…,p_m then

$\text{∑}\text{i=0}\text{[}\text{m}\text{2}\text{Pi}\text{m?1}\text{i?1}\text{+}\text{∑}\text{i=[}\text{m}\text{2}\text{]+1}\text{m}\text{Pi}\text{m?1}\text{m?i?1}\text{? 2}$

Here we generalize this result and prove some analogues of it. A corollary of Bollobás' result is that $\text{|F| ? 2(}{}_{\mathrm{<mtext>[</mtext><mtext>m</mtext><mtext>2</mtext><mtext>]?1</mtext>}}{}^{\mathrm{m?1}}\text{)}$. Purdy proved that if $\text{A ∈ F ?}\text{A}\text{? F}$ then $\text{|F| ? (}{}_{\mathrm{<mtext>[</mtext><mtext>m</mtext><mtext>2</mtext><mtext>]+1</mtext>}}{}^{\mathrm{m}}\text{)}$, which implies Bollobás' corollary. We also show that Purdy's result may be deduced from Bollobás' by a short argument. Finally, we give a canonical form for Sperner families which are also pairwise intersecting.     

Some results on symplectic matrices

The principal results are that if A is an integral matrix such that AA^T is symplectic then A = CQ, where Q is a permutation matrix and C is symplectic; and that if A is a hermitian positive definite matrix which is symplectic, and B is the unique hermitian positive definite pth.root of A, where p is a positive integer, then B is also symplectic.     

Some results on fractionaln-factor-critical graphs

A simple graphG is said to be fractionaln-factor-critical if after deleting anyn vertices the remaining subgraph still has a fractional perfect matching. For fractionaln-factor-criticality, in this paper, one necessary and sufficient condition, and three sufficient conditions related to maximum matching, complete closure are given.