ON THE SYMPLECTIC INVARIANTS OF A SUBSPACE OF A VECTOR SPACE~*

Let F be any commutative field. Let v be an integer≥1 and be a fixed 2v × 2v nonsingular alternate matrix over F. Define Sp_(zv)(F)={T: 2v×2v matrix over F|TKT~T=K}. It is well-known that Sp_(2v)(F) is a group with respect to the matrix multiplication and is called the symplectic group of degree 2v over F     

中心对称本原矩阵的k点指数集

Let G = (V, E) be a primitive digraph. The vertex exponent of G at a vertex v ∈ V, denoted by expG(v), is the least integer p such that there is a v → u walk of length p for each u ∈ V. We choose to order the vertices of G in the k-point exponent of G and is denoted by expG(k), 1 ≤ k ≤ n. We define the k-point exponent set E(n, k) := {expG(k)| G = G(A) with A ∈ CSP(n)}, where CSP(n) is the set of all n × n central symmetric primitive matrices and G(A) is the associated graph of the matrix A. In this paper, we describe E(n,k) for all n, k with 1 ≤ k ≤ n except n ≡ 1(mod 2) and 1 ≤ k ≤ n - 4. We also characterize the extremal graphs when k = 1.     

The k-point Exponent Set of Primitive Digraphs with Girth 2

Let D =(V,E)be a primitive digraph.The vertex exponent of D at a vertex v∈V,denoted by exPD(V),is the least integer p such that there is a v→u walk of length p for each u∈V.Following Brualdi and Liu,we order the vertices of D so that exPD(v_1)≤exPD(v_2)≤…≤exPD(v_n).Then exPD(v_k)is called the k- point exponent of D and is denoted by exP_D(k),1≤k≤n.In this paper we define e(n,k):=max{exp_D(k)|D∈PD(n,2)} and E(n,k):= {expD(k)|D∈PD(n,2)},where PD(n,2)is the set of all primitive digraphs of order n with girth 2.We completely determine e(n,k)and E(n,k)for all n,k with n≥3 and 1≤k≤n.     

On a Proof of Roth''''s Theorem

We denote by M_(n,m)(F) the set of all n×m matrices over the field F and by M_n(F) the set of all n×n matrices over the field F. W. E. Roth has shown the following theorem in 1952, [1]. Theorem Let A∈M_n(F),B∈M_m(F) and C∈M_(n,m)(F), then the matrix equation AX-YB=C (1) has a solution X, Y∈M_(n,m)(F) if and only if the matrices     

INTERSECTION OF TRANSLATIONS OF CANTOR TRIADIC SET

Let C be the Cantor triadic set and let Cα=C+α={β+α: β∈C} for -1≤α≤ 1. Let Hp={α∈C: dim H(Cα∩C)=dim B(C α∩C)=(1-p)log 2/log 3}, 0＜ p＜ 1. The authors give the dimensions of C α∩C and Hp. In additionthe characteristic of Hp is described by means of some measure μ supported on C.     

TWO WEIGHT CHARACTERIZATIONS FOR THE MULTILINEAR LOCAL MAXIMAL OPERATORS

Let 0<β<1 andΩbe a proper open and non-empty subset of Rⁿ.In this paper,the object of our investigation is the multilinear local maximal operator Mβ,defined by M_β((f))(x)=sup_Q(∈xQ∈Fβ)Π_i=1^m1/|Q|∫_Q|f_i(y_i)|dy_i,where F_β={Q(x,l):x∈Ω,l<βd(x,Ω^c)},Q=Q(x,l)is denoted as a cube with sides parallel to the axes,and x and l denote its center and half its side length.Two-weight characterizations for the multilinear local maximal operator M_βare obtained.A formulation of the Carleson embedding theorem in the multilinear setting is proved.     

THE KERNELS OF PRODUCT TYPE ON LOCAL FIELDS

Let K be a local field,that is.K is a locally compactnon-discrete complete and totally disconnected field.A non-Archimedean norm is endowed on K:x→|x|is a mapping from K intoR~+,such that(i)|X|=0 iff X=0;(ii)|xy|=|x||y|;(iii)|x+y|≤max{|x|,|y|}.Then|x|is called the absolute value of x.Theset={x∈K:|x|≤1}is the ring of integers in K,and={x∈K:     

Multiplicities of Fixed Points of Iterates of Holomorphic Maps From C" Into C"

In this announcement,C" denotes the complexspace of dimension v, 0 the origin of C",△v the unit ball in C" with center 0, F:△v→ a holomorphic map %with F(0)=0, JF(0) the Jacobian matrix of F at 0 and Fk,κ∈N,denotesthe κ-th iterate defined asF0=id, F1=F,…,Fk=FοFk-1successively, where id is the identity and s the set of allpositive integers.     

On the adjacent vertex-distinguishing equitable edge coloring of graphs

Let G(V, E) be a graph. A k-adjacent vertex-distinguishing equatable edge coloring of G, k-AVEEC for short, is a proper edge coloring f if (1) C(u)≠C(v) for uv ∈ E(G), where C(u) = {f(uv)|uv ∈ E}, and (2) for any i, j = 1, 2,… k, we have ||Ei| |Ej|| ≤ 1, where Ei = {e|e ∈ E(G) and f(e) = i}. χáve (G) = min{k| there exists a k-AVEEC of G} is called the adjacent vertex-distinguishing equitable edge chromatic number of G. In this paper, we obtain the χáve (G) of some special graphs and present a conjecture.     

Constructing quasi-random subsets of ZN by using elliptic curves

Let ε:y2 =x3 + Ax + B be an elliptic curve defined over the finite field Zp(p ＞ 3)and G be a rational point of prime order N on ε.Define a subset of ZN,the residue class ring modulo N,as S ∶={n ∶n ∈ZN,...     

图映射的非游荡集

§ 1　IntroductionLet N be the set of all natural numbers.Write Z+=N∪ { 0 } ,Nn={ 1 ,2 ,...,n} andZn={ 0 }∪Nnfor any n∈N.Let X be a topological space and f:X→X be a continuous map.Forx∈X,O(x,f) ={ fk(x) :k∈ Z+} is called the orbit of x.The set of periodic points,the set of recurrentpoints,the set ofω-limit points for some x∈X and the set of non-wandering points of fare denoted by P(f) ,R(f) ,ω(x,f) andΩ(f) ,respectively(for the definitions see[1 ] ) .Let A X,we use int(A) ,A…     

Γ-环的诸根的关系及Kyuno的一个结果

Let M={a, b, c,…} and Γ={α,β,γ,…} be additive abefian groups. If for all a, b, C∈M and all a, β∈Γ, the following conditions are satisfied: (0) aab∈M, (1) (a b)ac=aac bar, a(α β)b=aab aβb, aa(b e)=aab aac, (2) (aab)[βc=aa(bβc),then M is called a Γ-ring. If for all a, b, ,∈M ahd all tx, β∈Γ, the following conditions are satisfied:     

圈和完全图的点可区别VE-全染色(英文)

Let G be a simple graph of order at least 2.A VE-total-coloring using k colors of a graph G is a mapping f from V (G) E(G) into {1,2,···,k} such that no edge receives the same color as one of its endpoints.Let C(u)={f(u)} {f(uv) | uv ∈ E(G)} be the color-set of u.If C(u)=C(v) for any two vertices u and v of V (G),then f is called a k-vertex-distinguishing VE-total coloring of G or a k-VDVET coloring of G for short.The minimum number of colors required for a VDVET coloring of G is denoted by χ ve vt (G) and it is called the VDVET chromatic number of G.In this paper we get cycle C n,path P n and complete graph K n of their VDVET chromatic numbers and propose a related conjecture.     

乘积图的Fractional控制

Let G =(V,E) be a simple graph.For any real function g :V-→ R and a subset S V,we write g(S) =∑v∈Sg(v).A function f :V-→ [0,1] is said to be a fractional dominating function(F DF) of G if f(N [v]) ≥ 1 holds for every vertex v ∈ V(G).The fractional domination number γf(G) of G is defined as γf(G) = min{f(V)|f is an F DF of G }.The fractional total dominating function f is defined just as the fractional dominating function,the difference being that f(N(v)) ≥ 1 instead of f(N [v]) ≥ 1.The fractional total domination number γ0f(G) of G is analogous.In this note we give the exact values ofγf(Cm × Pn) and γ0f(Cm × Pn) for all integers m ≥ 3 and n ≥ 2.     

TO SOLVE MATRIX EQUATION sum (A~iXBi=c) BY THE SMITH NORMAL FORM

§ 1　IntroductionLet F be a field,F[λ] be the polynomial ring over F,Fm× n( or Fm× n[λ] ) be the setofall m×n matrices over F( or F[λ] ) .Let M(i) be the ith column of M∈Fm× m[λ] ,i=1 ,...,n.A g-inverse of M∈Fm× n will be denoted by M- and understood as a matrix for whichMM- M=M.In this paper,we discuss the linear matrix equation ki=0Ai XBi =C, ( 1 )where A∈Fm× m,Bi∈Fn× q,i=0 ,1 ,...,k,and C∈Fm× q.Equation( 1 ) is called universally solvable if ithas a solution f…     

空间中渐近非扩张型映射的渐近行为

Let X be a uniformly convex Banach space X such that its dual X^＊ has the KK property. Let C be a nonempty bounded closed convex subset of X and G be a directed system. Let ={Tt ： t ∈ G} be a family of asymptotically nonexpansive type mappings on C. In this paper, we investigate the asymptotic behavior of {Ttx0 ： t∈ G} and give its weak convergence theorem.     

阶极小渐近基

Let N denote the set of all nonnegative integers and A be a subset of N.Let W be a nonempty subset of N.Denote by F~*(W) the set of all finite,nonempty subsets of W.Fix integer g≥2,let A_g(W) be the set of all numbers of the form sum f∈Fa_fg~f where F∈F~*(W)and 1≤a_f≤g-1.For i=0,1,2,3,let W_i = {n∈N|n≡ i(mod 4)}.In this paper,we show that the set A = U_i~3=0 A_g(W_i) is a minimal asymptotic basis of order four.     

单圈图的极小能量

For a simple graph G, the energy E(G) is defined as the sum of the absolute values of all eigenvalues of its adjacency matrix. Let Undenote the set of all connected unicyclic graphs with order n, and Ur n= {G ∈ Un| d(x) = r for any vertex x ∈ V(Cl)}, where r ≥ 2 and Cl is the unique cycle in G. Every unicyclic graph in Ur nis said to be a cycle-r-regular graph.In this paper, we completely characterize that C39(2, 2, 2) ο Sn-8is the unique graph having minimal energy in U4 n. Moreover, the graph with minimal energy is uniquely determined in Ur nfor r = 3, 4.     

Matrix Algebra Motivated by Essentially Stochastic Matrices

A matrix of order n whose row sums are all equal to 1 is called an essentially stochastic matrix (see Johnsen [4]). We extend this notion as the following. Let F be a field of characteristic 0 or a prime greater than n. M_n(F) denotes the set of all n×n matrices over F. Let t be an elernent of F. A matrix A=(a_(ij)) in M_n(F) is called essentially t-stochastic' provided its row sums are each equal to t. We denote by R_n(t) the set of all essentially t-stochastic matrices over F. We shall mainly study R_n(0) and. Our main references are Johnson [2,4] and Kim [5].     

A Class of Constrained Inverse Eigenproblem and Associated Approximation Problem for Symmetric Reflexive Matrices

Let S∈Rn×n be a symmetric and nontrival involution matrix. We say that A∈E R n×n is a symmetric reflexive matrix if AT = A and SAS = A. Let S R r n×n(S)={A|A= AT,A = SAS, A∈Rn×n}. This paper discusses the following two problems. The first one is as follows. Given Z∈Rn×m (m < n),∧= diag(λ1,...,λm)∈Rm×m, andα,β∈R withα<β. Find a subset (?)(Z,∧,α,β) of SRrn×n(S) such that AZ = Z∧holds for any A∈(?)(Z,∧,α,β) and the remaining eigenvaluesλm 1 ,...,λn of A are located in the interval [α,β], Moreover, for a given B∈Rn×n, the second problem is to find AB∈(?)(Z,∧,α,β) such that where ||.|| is the Frobenius norm. Using the properties of symmetric reflexive matrices, the two problems are essentially decomposed into the same kind of subproblems for two real symmetric matrices with smaller dimensions, and then the expressions of the general solution for the two problems are derived.