定义在迭代函数系统吸引子上的动力系统的平衡态

在该文中, 令E表示一个迭代函数系统(X,T₁,…, T_m). 的吸引子. 定义连续自映射 f : E→E为f(x)=T^-1_j(x), x∈ T_j(E), j=1, …, m . 给定Given ψ ∈C_R(E), 令 K_ψ(δ, n = sup{∣∑^n-1_k=0[ψ(f ^kx)-ψ(f ^ky)]|:y ∈ B_x (δ, n)}, 这里B_x(δ, n) 表示Bowen球. 取一个扩张常数 ε, 记K_ψ=sup_n K_ψ(ε, n) , 定义ν(E)={ψ : K_ψ < ∞}. 对f : E → E, 作为Ruelle的一个定理^{[3, 定理2.1]}的一个应用, 我们证明每个ψ ∈ν(E)具有惟一的平衡态. 此结果推广了文献[12]中的主要结果.     

带强迫项的二阶非线性椭圆方程的区域振动准则

对二阶非线性椭圆型方程∑ _i,j=1ⁿ D_i[A_ij(x)D_jy]+∑_i=1ⁿ b_i(x)D_iy+q(x)f(y)=e(x)建立了若干新的振动准则, 所得结果仅依赖于方程在外区域Ω С Rⁿ的一个子区域序列的信息而有别于已知的大多数结论.     

独立数的一个下界

设G是一个图,其度序列为(d_v). 若由G的任意邻域导出子图的最大度至多为m, 则G的独立数至少是 ,这里当x>0, 函数f_m+1(x)大于 . 对于加权图G=(V,E,w), 证明了它的加权独立数至少是 ,这里w_v是顶点v的权重.     

对于轮和完全图的 Ramsey 数的渐近上界

该文给出:对于偶数m≥4当n→ ∞时 r(W_m,K_n)≤l(1+o(1))C₁(m) (n/logn ) ^(2m-2)/(m-2)对于奇数m≥5当n→∞时r(W_m,K_n)≤(1+o(1))C₂(m) (n^2m/_m+1/^{_{log n)}(m+1)/(m-1)} .特别地,C₂(5)=12. 以及 c(ⁿ/_logn)^5/2≤r(K₄,K_n)≤ (1+o(1)) ^n3_/(logn)².此外,该文还讨论了轮和完全图的 Ramsey 数的一些推广.     

测度链上p-Laplacian 边值问题的三个正对称解

该文研究了p-Laplacian 动力边值问题 (g(u^△(t)))^▽+a(t)f(t, u(t))=0, t ∈ [0, T] _T, u(0)=u(T)=w, u^△(0)=-u^△(T) 正解的存在性. 其中w是非负实数, g(ν)=|ν| ^p-2ν, p>1 . 根据对称技巧和五泛函不动点定理, 证明了边值问题至少有三个正的对称解, 同时, 给出了一个例子验证了我们的结果.     

关于Hammerstein型非线性积分方程固有值的一点注记

Let Aφ(x)=∫_GK(x,y)f(y,φ(y))dy, where G is a bounded closed domain in Euclidean space, K(x,y) is continuous on G×G, f(x,u) is continuous on G×R, and f(x,0)≡0. Set G_x={x|x∈G,K(x,y)≠0},G_y={y|y∈G,K(x,y)≠0},G₁=G_x∩G_y≠φ.Let K₁(x,y) be the restriction of K(x,y) on G₁×G₁,f₁(x,u)be the restriction of f(x, u) on G₁× R, and A₁φ=∫_G1K₁(x,y)f₁(y,φ₁(y))dy, The main result of this paper is Theorem λ≠0 is an eigenvalue of A, if and only if λ is an eigenvalue of A₁.     

图和有向图的测地数

图G内的任意两点u和v, u-v测地线是指u和v之间的最短路. I(u,v)表示 位于u-v测地线上所有点的集合, 对于子集SÍV(G), I(S)表示所有I(u,v)的并, 这里u,vÎ S. 图 G的测地数g(G)是使得I(S)=V(G)的点集S的最小基数. 对于有向图D, 类似地可定义g(D). 图G 的测地谱是G的所有定向图的测地数的集合, 记为S(G). G的下测地数g^-(G)=minS(G), 上测地数g⁺(G)=maxS(G). 文中主要研究了连通图G的g(G), g^-(G)和g⁺(G)之间的关系. 同时,还给出g(G)和g(G× K₂）相等的充分必要条件, 从而推广了 Chartrand, Harary 和 Zhang 的相关结论.     

脉冲泛函微分方程的渐近性态

该文讨论脉冲泛函微分方程$\left\{\begin{array}{ll}x^,(t)=f(t,x_t), t≥ t₀,△x=I_k(t,x(t^-)), t=t_k,k∈ Z⁺,给出了方程零解渐近稳定性和一致渐近稳定性的充分条件,指出这些条件推广或改进了文献[7--9]的相应结论.     

齐型空间上的一些新的Triebel-Lizorkin空间及其标架

设(X, ρ, μ)_d,θ是齐型空间, ε∈(0,θ), |s|<ε且 max{d/(d+ε), d/(d+s+ε)}<q≤∞. 引进了一类新的Triebel-Lizorkin空间F^s_∞q(X),并通过先建立与空间F^s_∞q(X)的范数相关的Plancherel-Pólya型不等式的方法建立了这些空间的标架特征; 给出了当|s|<ε, max{d/(d+ε), d/(d+s+ε)}<p≤∞且0<q≤∞时, Besov 空间B^s_pq(X),以及当|s|<ε, max{d/(d+ε), d/(d+s+ε)}<p<∞且max{d/(d+ε), d/(d+s+ε)}<q≤∞时, Triebel-Lizorkin空间F^s_∞q(X)的标架特征; 此外, 还引进了与给定仿增函数b相关的新的Triebel-Lizorkin空间bF^s_∞q(X)和HF^s_∞q(X), 并且建立了空间bF^s_∞q(X)和空间HF^s_∞q(X)的相互关系; 进一步证明了如果s=0且q=2, 则HF^s_∞q(X)=F^s_∞q(X). 因为F^s_∞q(X), 所以事实上这也给出了空间BMO(X)一个新的特征刻画.     

图的全符号控制数

本文考虑的图G均为有限简单连通图, 是一个有顶点集合V边集合E的有限简单连通图，用V(G) 和E(G) 分别表示G的顶点集和边集. f 是一个从V(G)∪E(G)→{-1, 1}的函数. f 的权重定义为 w(f)=∑_{x∈V(G)∪E(G)}f(x). 对任一元素x∈V(G)∪E(G), 定义f[x]=∑_y∈NT[x]f(y). 图G的全符号控制函数f : V(G)∪ E(G)→{-1, 1}是一个对所有的x∈ V(G)∪ E(G), 都满足f[x]≥1的函数. G的所有全符号控制函数中最小的权定义为G 的全符号控制数，记作γ_s^*(G). 讨论了图的全符号控制数, 证明了图的全符号控制数的下界, 并对一些特殊的图类C_n 和P_n本文得到了全符号控制数的精确值.     

The w-median of a connected strongly chordal graph

Suppose G = (V, E) is a graph in which every vertex x has a non-negative real number w(x) as its weight. The w-distance sum of a vertex y is D_{G, w}(y) = σ_x?v d(y, x)w(x). The w-median of G is the set of all vertices y with minimum w-distance sum D_G,w(y). This paper shows that the w-median of a connected strongly chordal graph G is a clique when w(x) is positive for all vertices x in G.     

Claw Conditions for Heavy Cycles in Weighted Graphs

A graph is called a weighted graph when each edge e is assigned a nonnegative number w(e), called the weight of e. For a vertex v of a weighted graph, d^w(v) is the sum of the weights of the edges incident with v. For a subgraph H of a weighted graph G, the weight of H is the sum of the weights of the edges belonging to H. In this paper, we give a new sufficient condition for a weighted graph to have a heavy cycle. A 2-connected weighted graph G contains either a Hamilton cycle or a cycle of weight at least c, if G satisfies the following conditions: In every induced claw or induced modified claw F of G, (1) max{d^w(x),d^w(y)} c/2 for each non-adjacent pair of vertices x and y in F, and (2) all edges of F have the same weight.     

A Fan Type Condition For Heavy Cycles in Weighted Graphs

 A weighted graph is a graph in which each edge e is assigned a non-negative number w(e), called the weight of e. The weight of a cycle is the sum of the weights of its edges. The weighted degree d ^w (v) of a vertex v is the sum of the weights of the edges incident with v. In this paper, we prove the following result: Suppose G is a 2-connected weighted graph which satisfies the following conditions: 1. max{d ^w (x),d ^w (y)∣d(x,y)=2}≥c/2; 2. w(x z)=w(y z) for every vertex z∈N(x)∩N(y) with d(x,y)=2; 3. In every triangle T of G, either all edges of T have different weights or all edges of T have the same weight. Then G contains either a Hamilton cycle or a cycle of weight at least c. This generalizes a theorem of Fan on the existence of long cycles in unweighted graphs to weighted graphs. We also show we cannot omit Condition 2 or 3 in the above result. Received: February 7, 2000 Final version received: June 5, 2001     

重图的超f - 边覆盖染色

图G的一个超f - 边覆盖染色就是它的一个f - 边覆盖染色并且使得图G中的重边染上不同的颜色. 令χ^H_fc(G)是图G存在一个超f - 边覆盖染色时所需最大的颜色数k. χ^H_fc(G)称作是图G的超f - 边覆盖染色色数. 本文讨论重图的超f - 边覆盖染色的存在性并且给出了重图的超f - 边覆盖染色的色数下界.     

Cycles in weighted graphs

A weighted graph is one in which each edgee is assigned a nonnegative numberw(e), called the weight ofe. The weightw(G) of a weighted graphG is the sum of the weights of its edges. In this paper, we prove, as conjectured in [2], that every 2-edge-connected weighted graph onn vertices contains a cycle of weight at least 2w(G)/(n–1). Furthermore, we completely characterize the 2-edge-connected weighted graphs onn vertices that contain no cycle of weight more than 2w(G)/(n–1). This generalizes, to weighted graphs, a classical result of Erds and Gallai [4].     

[a,b]-factorization of a graph

Let a and b be integers such that 0 ? a ? b. Then a graph G is called an [a, b]-graph if a ? d_G(x) ? b for every x ? V(G), and an [a, b]-factor of a graph is defined to be its spanning subgraph F such that a ? d_F(x) ? b for every vertex x, where d_G(x) and d_F(x) denote the degrees of x in G and F, respectively. If the edges of a graph can be decomposed into [a.b]-factors then we say that the graph is [2a, 2a]-factorable. We prove the following two theorems: (i) a graph G is [2a, 2b)-factorable if and only if G is a [2am,2bm]-graph for some integer m, and (ii) every [8m + 2k, 10m + 2k]-graph is [1,2]-factorable.     

A wavelike functional equation of Pexider type

Summary. Let (G, +) and (H, +) be abelian groups such that the equation 2u = v 2u = v is solvable in both G and H. It is shown that if f₁, f₂, f₃, f₄, : G ×G ? H f_1, f_2, f_3, f_4, : G \times G \longrightarrow H satisfy the functional equation f₁(x + t, y + s) + f₂(x - t, y - s) = f₃(x + s, y - t) + f₄(x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , then f₁, f₂, f₃, and f₄ are given by f₁ = w + h, f₂ = w - h, f₃ = w + k, f₄ = w - k where w : G ×G ? H w : G \times G \longrightarrow H is an arbitrary solution of f (x + t, y + s) + f (x - t, y - s) = f (x + s, y - t) + f (x - s, y + t) for all x, y, s, t ? G x, y, s, t \in G , and h, k : G ×G ? H h, k : G \times G \longrightarrow H are arbitrary solutions of D_y,t³g(x,y) = 0 \Delta_{y,t}^{3}g(x,y) = 0 and D_x,t³g(x,y) = 0 \Delta_{x,t}^{3}g(x,y) = 0 for all x, y, s, t ? G x, y, s, t \in G .     

Matchings and walks in graphs

The matching polynomial α(G, x) of a graph G is a form of the generating function for the number of sets of k independent edges of G. in this paper we show that if G is a graph with vertex v then there is a tree T with vertex w such that \documentclass{article}\pagestyle{empty}\begin{document}$ \frac{{\alpha (G\backslash v, x)}}{{\alpha (G, x)}} = \frac{{\alpha (T\backslash w, x)}}{{\alpha (T, x)}}. $\end{document} This result has a number of consequences. Here we use it to prove that α(G\v, 1/x)/xα(G, 1/x) is the generating function for a certain class of walks in G. As an application of these results we then establish some new properties of α(G, x).