含故障点的加强超立方体中路和圈的嵌入（英文）

本文研究了含故障点的n-维加强超立方体Q_(n,k)中的路和圈嵌入的问题.充分分析了加强超立方体网络的潜在特性,利用了构造的方法.得到了含2n-4个故障点的加强超立方体Q_(n,k)中含长为2~n-2f的容错圈的结论,推广了折叠超立方体网络中1-点容错圈嵌入的结果.其中折叠超立方体网络为加强超立方体网络的一种特殊情况.     

故障折叠超立方体中的路和圈

本文研究了含故障点的n-维折叠超立方体FQn中的路和圈嵌入的问题,分析了折叠超立方体网络的潜在特性.利用了构造的方法,得到了含2n-3个故障点的折叠超立方体FQn中含长为2n-2f的圈的结论,推广了折叠超立方体网络中1-点容错圈嵌入的结果.     

故障加强超立方体中的路和圈

本论文研究了含故障点的加强超立方体中路和圈的嵌入问题.利用数学归纳法,获得了故障加强超立方体中的路和圈,推广了超立方体中点容错路和圈嵌入的结果.     

不动点集为常余维数6的(Z2)2作用

记J_n,k^r为具有如下性质的n维未定向上协边类α构成的集合:存在α的一个代表元Mⁿ及(Z₂)^k在Mⁿ上的作用,其不动点集为常余维数r .记J_n,k^r=∑_nJ_n,k^r,则J_n,k^r为未定向上协边环MO_n=     

对称凸集上初等对称函数为Schur-凹的充要条件

设x,y∈Rⁿ,x被y所控制记作x(?)y。又w∈Rⁿ,令S_k(x)为第k个初等对称函数。Q_m,n为前n个自然数取m个的严格增序列的集合。对于β∈Q_m,n写w_β=(W_{^β(1),…,WB(m)}∈R^m。本文主要证明了下面的结论:(1)S_k(x)在(?)_w上是Schur-凹的充要条件是(2)S_k(x)≥0,(?)x∈(?)_w的充要条件是S_k(w)≥0且S_k(x)在(?)_w上是Schur-凹的(3)S_k(x)≥0,(?)x∈(?)的充要条件是     

关于多元函数最佳逼近精确阶的Timan问题

关于找一个充分必要条件使Ω^k(f,1/σ)L_p(Rⁿ)=O(A_σ(f)L_p(Rⁿ)),σ→∞,成立的Timan问题被解决.这个条件是Q^k(f,δ)L_p(Rⁿ)=O(Ω^k+1(f,δ)L_p(Rⁿ)),δ→0.     

交换p-群的子群格的强Sperner性质

设n和k是任意正整数 ,p是素数 ,L_(kⁿ) (p)是交换p 群 (Z/p^kZ)ⁿ 的子群格 ,则存在正整数N(n ,k) ,使得当p >N(n ,k)时 ,L(_{kⁿ) (p)}具有强Sperner性质 .     

故障加强超立方体中的路和圈（英文）

本论文研究了含故障点的加强超立方体中路和圈的嵌入问题.利用数学归纳法,获得了故障加强超立方体中的路和圈,推广了超立方体中点容错路和圈嵌入的结果.     

QK空间的插值序列

本文研究了Q_K空间的插值问题.利用复分析和调和分析的方法,获得了单位圆盘上的一个序列{z_n}是Q_K∩H^∞空间的插值序列的一个充分必要条件,推广了Q_p空间的部分结果.     

部分和乘积的几乎处处中心极限定理

设X_n, n≥1是独立同分布正的随机变量序列, E(X₁)=u >0, Var(X₁)=σ², E|X₁|³<∞, 记S_n==∑^N_k=1X_k, 变异系数γ=σ/u.g是满足一定条件的无界可测函数, 证明了 lim_N→∞1/logN∑^N_n=11/n g((∏ⁿ_k=1S_k/n!uⁿ )^1/γ√n )=∫^∞₀g(x)dF(x),a.s., 其中 F(•) 是随机变量e^√2ξ 的分布函数, ξ 是服从标准正态分布的随机变量.     

VERTEX-FAULT-TOLERANT CYCLES EMBEDDING ON ENHANCED HYPERCUBE NETWORKS

In this paper, we study the enhanced hypercube, an attractive variant of the hypercube and obtained by adding some complementary edges from a hypercube, and focus on cycles embedding on the enhanced hypercube with faulty vertices. Let Fu be the set of faulty vertices in the n-dimensional enhanced hypercube Qn,k （n ≥ 3, 1 ≤ k 〈≤n - 1）. When IFvl = 2, we showed that Qn,k - Fv contains a fault-free cycle of every even length from 4 to 2n - 4 where n （n ≥ 3） and k have the same parity; and contains a fault-free cycle of every even length from 4 to 2n - 4, simultaneously, contains a cycle of every odd length from n-k ＋ 2 to 2^n-3 where n （≥ 3） and k have the different parity. Furthermore, when ｜Fv｜ = fv ≤ n - 2, we prove that there exists the longest fault-free cycle, which is of even length 2^n - 2fv whether n （n ≥ 3） and k have the same parity or not; and there exists the longest fault-free cycle, which is of odd length 2^n - 2fv ＋ 1 in Qn,k - Fv where n （≥ 3） and k have the different parity.     

PATHS AND CYCLES EMBEDDING ON FAULTY ENHANCED HYPERCUBE NETWORKS

Let Qn,k(n≥3,1≤k≤n-1) be an n-dimensional enhanced hypercube which is an attractive variant of the hypercube and can be obtained by adding some complementary edges,fv and fe be the numbers of faulty vertices and faulty edges,respectively.In this paper,we give three main results.First,a fault-free path P [u,v] of length at least 2n-2fv-1(respectively,2n-2fv-2) can be embedded on Qn,k with fv+fe≤n-1 when d Qn,k(u,v) is odd(respectively,d Qn,k(u,v) is even).Secondly,an Qn,k is(n-2) edgefault-free hyper Hamiltonian-laceable when n(≥3) and k have the same parity.Lastly,a fault-free cycle of length at least 2n-2fv can be embedded on Qn,k with fe≤n-1 and fv+fe≤2n-4.     

Volumes Spanned by Random Points in the Hypercube

Consider the hypercube [0, 1]ⁿ in Rⁿ. This has 2ⁿ vertices and volume 1. Pick N = N(n) vertices independently at random, form their convex hull, and let V_n be its expected volume. How large should N(n) be to pick up significant volume? Let k=2/√≈1.213, and let ? > 0. We shall show that, as n→∞, V_n→0 if N(n)?(k??)ⁿ →1 if N(n) ? (k + ?)ⁿ. A similar result holds for sampling uniformly from within the hypercube, with constant .     

Vertex-fault-tolerant cycles embedding on enhanced hypercube networks

In this paper, we focus on the vertex-fault-tolerant cycles embedding on enhanced hypercube, which is an attractive variant of hypercube and is obtained by adding some complementary edges from hypercube. Let F _v be the set of faulty vertices in the n-dimensional enhanced hypercube Q _n,k (1 ≤ k ≤ n?1). When |F _v | = 2, we showed that Q _n,k ? F _v contains a fault-free cycle of every even length from 4 to 2 ⁿ ?4 where n (n ≥ 3) and k have the same parity; and contains a fault-free cycle of every even length from 4 to 2 ⁿ ? 4, simultaneously, contains a cycle of every odd length from n ? k + 2 to 2 ⁿ ? 3 where n(≥ 3) and k have the different parity. Furthermore, when |F ^v | = f ^v ≤ n ? 2, we proof that there exists the longest fault-free cycle, which is of even length 2 ⁿ ? 2f ^v whether n(n ≥ 3) and k have the same parity or not; and there exists the longest fault-free cycle, which is of odd length 2 ⁿ ? 2f ^v ? 1 in Q ^n,k ? F ^v where n(≥ 3) and k have the different parity.     

1-vertex-fault-tolerant cycles embedding on folded hypercubes

In this paper, we focus on a hypercube-like structure, the folded hypercube, which is basically a standard hypercube with some extra links between its nodes. Let f be a faulty vertex in an n-dimensional folded hypercube FQ_n. We show that FQ_n−{f} contains a fault-free cycle of every even length from 4 to 2ⁿ−2 if n≥3 and, furthermore, every odd length from n+1 to 2ⁿ−1 if n≥2 and n is even.     

Survey on path and cycle embedding in some networks

To find a cycle (resp. path) of a given length in a graph is the cycle (resp. path) embedding problem. To find cycles of all lengths from its girth to its order in a graph is the pancyclic problem. A stronger concept than the pancylicity is the panconnectivity. A graph of order n is said to be panconnected if for any pair of different vertices x and y with distance d there exist xy-paths of every length from d to n. The pancyclicity or the panconnectivity is an important property to determine if the topology of a network is suitable for some applications where mapping cycles or paths of any length into the topology of the network is required. The pancyclicity and the panconnectivity of interconnection networks have attracted much research interest in recent years. A large amount of related work appeared in the literature, with some repetitions. The purpose of this paper is to give a survey of the results related to these topics for the hypercube and some hypercube-like networks.      

Rainbows in the Hypercube

Let Q_n be a hypercube of dimension n, that is, a graph whose vertices are binary n-tuples and two vertices are adjacent iff the corresponding n-tuples differ in exactly one position. An edge coloring of a graph H is called rainbow if no two edges of H have the same color. Let f(G,H) be the largest number of colors such that there exists an edge coloring of G with f(G,H) colors such that no subgraph isomorphic to H is rainbow. In this paper we start the investigation of this anti-Ramsey problem by providing bounds on f(Q_n,Q_k) which are asymptotically tight for k = 2 and by giving some exact results.     

Restricted Fault Diameter of Hypercube Networks

This paper studies restricted fault diameter of the n-dimensional hypercube networks Qn (n ≥ 2).It is shown that for arbitrary two vertices x and y with the distance d in Qn and any set F with at most 2n-3 vertices in Qn - {x, y}, if F contains neither of neighbor-sets of x and y in Qn, then the distance between x andy in Qn - F is given by D(Qn-F;x,y){=1 , for=1;≤d 4 , for 2≤d≤n-2,n≥4;≤n 1, for d=n-1,n≥3; =n, for d=n. Furthermore, the upper bounds are tight. As an immediately consequence, Qn can tolerate up to 2n-3 vertices failures and remain diameter 4 if n = 3 and n 2 if n ≥ 4 provided that for each vertex x in Qn, all the neighbors of x do not fail at the same time. This improves Esfahanian‘s result.     

Covering the vertices of a graph by cycles of prescribed length

The main theorem of that paper is the following: let G be a graph of order n, of size at least (n² - 3n + 6)/2. For any integers k, n₁, n₂,…,n_k such that n = n₁ + n₂ +. + n_k and n_i ? 3, there exists a covering of the vertices of G by disjoint cycles (C_i) =l…k with |C_i| = ni, except when n = 6, n₁ = 3, n₂ = 3, and G is isomorphic to G₁, the complement of G₁ consisting of a C₃ and a stable set of three vertices, or when n = 9, n₁ = n₂ = n₃ = 3, and G is isomorphic to G₂, the complement of G₂ consisting of a complete graph on four vertices and a stable set of five vertices. We prove an analogous theorem for bipartite graphs: let G be a bipartite balanced graph of order 2n, of size at least n² - n + 2. For any integers s, n₁, n₂,…,n_s with n_i ? 2 and n = n₁ + n₂ + ? + n_s, there exists a covering of the vertices of G by s disjoint cycles C_i, with |C_i| = 2n_i.