On the Spectral Moment of Quasi-Unicyclic Graphs

A connected graph G=(V(G),E(G)) is called a quasi-unicyclic graph,if there exists u_0∈V(G) such that G-u_0 is a unicyclic graph.Denote Q(n,d_0)={G:G is a quasi-unicyclic graph of order n with G-u_0 being a unicyclic graph and d_G(u_0)=d_0}.Let A(G) be the adjacency matrix of a graph G,and let λ_1(G),λ_2(G),…,λ_n(G) be the eigenvalues in non-increasing order of A(G).The number■(k=0,1,…,n-1) is called the k-th spectral moment of G,denoted by S_k(G).Let S(G)=(S_0(G),S_1(G),…,S_(n-1)(G)) be the sequence of spectral moments of G.For two graphs G_1,G_2,we have■ if for some k(k=1,2,…,n-1), and we have S_i(G_1)=S_i(G_2)(i=0,1,…,k-1) and S_k(G_1)S_k(G_2).In this paper,we determine the second to the fourth largest quasi-unicyclic graphs,in an S-order,in the set Q(n,d_0),respectively.     

SOME PROPERTIES OF SUPER-DECOMPOSABLE OPERATORS

In this paper,the following main resvlts have been proved:1.Let T∈B(X) be a super-decomposable operator,then for any hyperinvariant subspace Y of T,the operator T/Y is super-decomposable.2.The operator T∈B(X) is super-decomposable if and only if T/Y is super-decomposable for any hyperinvariant subspace Y of T.3. If T∈B(X) has on open spectral resolvent,then T is superdecomposable.     

FUZZY REGULAR SUBSEMIGROUPS IN SEMIGROUPS

In this paper,the definitions of fuzzy regular subsemigroup and fuzzy left(right,intra-)regular sub-semigroup in semigroups are introduced.Some characterizations of them are given.Proposition 2.1.A fuzzy set A in a semigroup S is a fuzzy subsemigroup iff for any λ∈[0,1],if A_λ={x∈S|A(x)≥λ}≠ ,then A_λ is a subsemigroup of S.Proposition 2.2.A fuzzy set A in a semigroup S is a fuzzy left(right)ideal iff for any λ∈(0,1],if A={x∈S|A(x)     

《一条摄动定理》一文的补遗

In this paper, a simple proof of the following perturbation theorem is given:Suppose that i) T is a self-adjoint operator in the Hilbert space with discrete spectrum σ(T)={λ_n|n=1,2,…}, λ_1≤λ_2≤… and λ_n=cn~P(1+0(1/n)),c>0, p>0.ii) Let 0≤v<1. Let P be a linear operator such that thedomain of P contains the domain of T~v and PT~(-v) is bounded.iii) p(1-v)>1.Then T+P has Property S and T+P and T have an asymptotically common decomposition. Furthermore, if v≤1/2, T+P has the strong Property S.     

On k-Star Arboricity of Graphs

A star forest is a forest whose components are stars. The star arboricity of a graph G,denoted by sa（ G）,is the minimum number of star forests needed to decompose G. Let k be a positive integer. A k-star forest is a forest whose components are stars of order at most k ＋ 1. The k-star arboricity of a graph G,denoted by sak（ G）,is the minimum number of k-star forests needed to decompose G. In this paper,it is proved that if any two vertices of degree 3 are nonadjacent in a subcubic graph G then sa2（ G） ≤2.For general subcubic graphs G, a polynomial-time algorithm is described to decompose G into three 2-star forests. For a tree T and[Δ k, T）/k]t≤ sak（ T） ≤[Δ（ T）- 1/K]＋1,where Δ（ T） is the maximum degree of T.kMoreover,a linear-time algorithm is designed to determine whether sak（ T） ≤m for any tree T and any positive integers m and k.     

哈密尔顿连通的度和与邻域交条件

Let S^23 denote an independent set with mini dist (u,v)|u, v∈S} = 2 and |S|=3. Our main result is the following theorem: Let G be a 3-connected graph of order n such that d(u) d(v) d(w)≥n 1 |N(u)∩N(v)∩N(w)|for any independent set S^23={u,v,w}, then G is Hamilton-connected.     

满足φ(n)|n-1的合数n(英文)

§1. Introduction In 1932, D. H. Lehmer asked if there are any composite integers n forwhich φ(n) |n-1, (φ)n being Euler's function. The answer to this question isstill not known. If S is any set of positive integers, denote by N (S,x) the number ofmembers of S which do not exceed x. Let a be an arbitrary integer,     

SOME PROPERTIES OF BOUNDED SETS IN INDUCTIVE LIMITS

Let (E,ξ）=indlim (En,ξn） be an inductive limit of a sequence of locally convex spaces,For brevity,denote by (DS) each set Bbounded in (E,ξ） is contained in some En； and (DST) each set B bounded in (E,ξ） is contained and bounded in some (En,ξn）. Theovem 1.(DS) holds provided that (i) for each n∈N,there is a neighborhood Un of o in (En,ξn） and m(n)∈ such that -↑Un^E包含于Em(n),and (ii) for any neighborhood V n of o in (En,ξn）,∞↑Un=1 Vn absorbs every bounded set in (E,ξ）. theorem 2 Let all (En,ξn） be metrizable and (DS) hold,then for each bounded set B IN (E,ξ）and each n ∈N thcrc is a neighborhood U k of o in (Ek,ξk）, 1≤k≤n ,and m(n)∈N such that ——↑(B U1 U2 … Un)^E包含于 Em(n). theorem 3. Let all (En,ξn） be Frechet spaces.Then (DST) holds if and only if (i) for each n ∈N,there is u neighborhood U n of in (En,ξn） and m(n)∈N such that 0↑Un^E包含于Em(n),and (ii) for each each closed ,absosed,absolutely conuex,bounded set B in (E,ξ）,∞↑Un=1(（εnB)∩Un）absorbs B,where U n is any neighborhood of o in (En,ξn） and εn is any positive number for every n ∈N。     

线段连续自映射的迭代极限点

Let I be an interval and f:I→I be continuous. The set p(f) of periodic points, the set R(f) of recurrent points, the set Q(f) of nonwandering points and the orbit O(x, f) of x ∈I are defined as usual. A point y ∈I is called an iterated limit point of the point x ∈I, if there exists a subsequence of the sequence f(x), f~2(x), ..., which converges to y. Denote the set of iterated limit points of x by ω(x, f). We show the following theorems.     

On p-ω-hyponormal operators

In what follows,H means a complex Hilbert space.Call an invertible operator T log-hyponormal if log (T^*T)≥log(TT^*). Let T=U|T|be the polar decomposition of T.Aluthge in [1] defined the operatorT=|T|^1/2U|T|^1/2 which called the Aluthge transformation of T.An operator T is said to be ω-hyponormal if     

Quadratic Number Fields with Class Numbers Divisible by a Prime q

Let q≥5 be a prime number. Let k=Q(√d) be a quadratic number field, where d=(-1)q(q-1)/2. (-(q-1)^q-1UW^q u^2q^q). Then the class number of k is divisible by q for certain integers u,w. Conversely, assume Ω/k is an unramifled cyclic extension of degree q (which implies the class number of k is divisible by q), and Ω2 is the splitting field of some irreducible trinomial f(X)=X^q-aX-b with integer coefficients, k=Q(√D(f)) with D(f) the discriminant of f(X). Then f(X) must be of the form f(X)=X^q-u^q-2 wX-u^q-1 in a certain sense where u,w are certain integers. Therefore, k=Q(√d) with d=(-1)^q(q-1)/2 (-(q-1)^q-1uw^q u^2q^q). Moreover, the above two results are both generalized for certain kinds of general polynomials.     

A general version of the Morse-Sard theorem

Let k, m, n be positive integers, and k≥2, a∈(0,1], 0＜r＜min{m,n} an integer, d=r (m-r)/(k a), and if f∈C^k,a(IR^m,IR^n),A=Cr(f)={x∈IR^m|rank(Df(x))≤r}, then f(A) is d-null. Thus the statement posed by Arthur Sard in 1965 can be completely solved when k≥2.     

Principal component analysis using neural network

The authors present their analysis of the differential equation dX ( t )/dt = AX ( t ) - X^T( t ) BX( t)X( t), where A is an unsymmetrical real matrix, B is a positive definite symmetric real matrix,X E Rn ; showing that the equation characterizes a class of continuous type full-feedback artificial neural network; We give the analytic expression of the solution; discuss its asymptotic behavior; and finally present the result showing that, in almost all cases, one and only one of following eases is true. 1. For any initial value X0∈R^n, the solution approximates asymptotically to zero vector. In thin cane, the real part of each eigenvalue of A is non-positive. 2. For any initial value X0 outside a proper subspace of R^n,the solution approximates asymptoticaUy to a nontrivial constant vector Y( X0 ). In this cane, the eigenvalue of A with maximal real part is the positive number λ=Ⅱ Y (X0)ⅡB^2 and Y (X0) is the corre-sponding eigenvector. 3. For any initial value X0 outsidea proper subspace of R^n, the solution approximates asymptotically to a non-constant periodic function Y( X0 , t ). Then the eigenvalues of A with maximal real part is a pair of conjugate complex numbers which can be computed.     

Exact Hausdorff measure of the lever sets of a multi-parameter stable process

Let X(t)be a(N,d,α)stable process of type A with Nα＞ d.The exact Hausdorff measure of the lever sets of X - 1(x)is evaluated.     

ON STRONGLY M_1-SPACES

In this paper we have obtained the fallowing results: (1)A space X is strongly M1 if and only if X is a paracompact σ-space and every closed subset F of X has a σ-FCP open outer base. (2)If X is strongly M1 and F is closed in X, then the quotient space X/F is strongly M1. (3) The following propositions are equivalent; (Ⅰ) Every closed image of any strongly M1-space is srongly M1. (Ⅱ) In every closed image of any strongly M1-space, each point has a (σ)-FCP open neibourhood (briefly, nbd)base.     

A matrix version of the Wielandt inequality and its application to statistics

Suppose that A is an n × n positive definite Hemitain matrix. Let X and Y be n × p and n × q matrices (p+ q≤n), such that X* Y = 0. The following inequality is proved X*AY( Y* AY)-Y Y*AX≤( (λ1-λn)/(λ1+λn))2 X*AX, where λ1 and λn are respectively the largest and smallest eigenvalues of A, and M- stands for a generalized inverse of M. This inequality is an extension of the well-known Wielandt inequality in which both X and Y are vectors. The inequality is utilized to obtain some interesting inequalities about covari-ance matrix and various correlation coefficients including the canonical correlation, multiple and simple correlation.     

Binding number,minimum degree and bipancyclism in bipartite graphs

Let G =(V_1,V_2,E) be a balanced bipartite graph with2 n vertices.The bipartite binding number of G,denoted by B(G),is defined to be n if G =K_n and min i∈{1,2}|N(S)|n min |N(S)|/|S|otherwise.We call G bipancyclic if it contains a cycle of every even length m for 4 ≤ m ≤ 2n.A theorem showed that if G is a balanced bipartite graph with 2n vertices,B(G) 3 / 2 and n 139,then G is bipancyclic.This paper generalizes the conclusion as follows:Let 0 c 3 / 2 and G be a 2-colmected balanced bipartite graph with 2n(n is large enough) vertices such that B(G) c and δ(G)(2-c)n/(3-c)+2/3.Then G is bipancyclic.     

Super-Edge-Connectivity of G(k,d,s)(s≥k/2)

A graph G is super-edge-connected,for short super-λ,if every minimum edge-cut consists of edges adjacent to a vertex of minimum degree.Alphabet overlap graph G(k,d,s)is undirected,simple graph with vertex set V={v|v=1()kv…v;vi∈{1,2,…,d},i=1,…,k}.Two vertices u=(u1…uk)and v=(v1…vk)are adjacent if and only if us+i=vi or vs+i=ui(i=1,…,k-s).In particular G(k,d,1)is just an undirected de Bruijn graph.In this paper,we show that the diameter of G(k,d,s)is k s,the girth is 3.Finally,we prove that G(k,d,s)(s≥k/2)is super-λ.     

ON THE SIMULTANEOUS TRANSFORMATION OF A HERMITIAN MATRIX AND A SYMMETRIC OR SKEW-SYMMETRIC MATRIX.

Let A1,A2 be two matrices of the same size over the complex field, with A1'= A1 and A1' = sA2 (s= ±1). In the text such a pair of matrices is called a ΦS, pair, and, according as s= +1 or -1, we speak of Φ+-pairs or Φs-pairs. By a transformation @ of a Φs pair (A1, A2) we mean the transformation which changes (A1, A2) into the Φs-pair (PA1P', PA2P'), where P is some non-singular matrix. If (A1, A2) is a Φ-pair, the Hermitian pencil of matricesis called in the text the associated pencil of the pair (A1, A2).     

一个人员指派问题(英文)

Let T be a set of personnel.Suppose that“≤”is a partial ordering relationon T(which expresses,for example,the seniority of the perssonnel)such that(T,≤)is a rooted tree.Let U be a subset of T.If U can be partitioned into groups eachconsisting of one leader,say t,and k ordinary members,say t_1,t_2,…,t_k,with theproperty that the leader must have higher seniority than the k ordinary members,