The Neighborhood Union of Essential Sets and Hamiltonicity of Graphs

Let G be a graph. An independent set Y in G is called an essential independent set (or essential set for simplicity) if there is {y1,y2} (?) Y such that dist (y1,y2) = 2. In this paper, we use the technique of the vertex insertion on l-connected (l = k or k + 1, k≥ 2) graphs to provide a unified proof for G to be hamiltonian, or hamiltonian-connected. The sufficient conditions are expressed     

关于H-cordial图和semi-H-cordial图的若干结论（英文）

I. Cahit calls a graph H-cordial if it is possible to label the edges with the numbers from the set{1,-1} in such a way that, for some k, at each vertex v the sum of the labels on the edges incident with v is either k or-k and the inequalities |v(k)-v(-k)| ≤ 1 and|e(1)-e(-1)| ≤ 1 are also satisfied. A graph G is called to be semi-H-cordial, if there exists a labeling f, such that for each vertex v, |f(v)| ≤ 1, and the inequalities |e_f(1)-e_f(-1)| ≤ 1 and |vf(1)-vf(-1)| ≤ 1 are also satisfied. An odd-degree(even-degree) graph is a graph that all of the vertex is odd(even) vertex. Three conclusions were proved:(1) An H-cordial graph G is either odd-degree graph or even-degree graph;(2) If G is an odd-degree graph, then G is H-cordial if and only if |E(G)| is even;(3) A graph G is semi-H-cordial if and only if |E(G)| is even and G has no Euler component with odd edges.     

On All Fractional （a,b, k）-Critical Graphs

Let a,b,k,r be nonnegative integers with 1≤a≤b and r≥2.LetG be a graph of order n with n(a+b)(r(a+b)-2)+ak/a.In this paper,we first show a characterization for all fractional(a,b,k)-critical graphs.Then using the result,we prove that G is all fractional(a,b,k)-critical if δ(G)≥(r-1)b2/a+k and |NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b for any independent subset {x1,x2,...,xr} in G.Furthermore,it is shown that the lower bound on the condition|NG(x1)∪NG(x2)∪···∪NG(xr)|≥bn+ak/a+b is best possible in some sense,and it is an extension of Lu's previous result.     

NSD随机变量阵列的L_r收敛性（英文）

Let {X_(nk), k ≥ 1, n ≥ 1} be an array of rowwise negatively superadditive dependent random variables and {a_n, n ≥ 1} be a sequence of positive real numbers such that a_n↑∞. Under some suitable conditions,L_r convergence of 1/an max 1≤j≤n |j∑k=1 X_(nk)| is studied. The results obtained in this paper generalize and improve some corresponding ones for negatively associated random variables and independent random variables.     

L(d1, d2,..., dt)-Number λ(Cn; d1, d2,...,dt) of Cycles

An L(d1,d2,...,dt)-labeling of a graph G is a function f from its vertex set V(G) to the set {0, 1,..., k} for some positive integer k such that {f(x) - f(y)| ≥ di, if the distance between vertices x and y in G is equal to i for i = 1,2,...,t. The L(d1,d2,...,dt)-number λ(G;d1,d2,... ,dt) of G is the smallest integer number k such that G has an L(d1,d2,... ,dt)labeling with max{f(x)|x ∈ V(G)} = k. In this paper, we obtain the exact values for λ(Cn; 2, 2,1) and λ(Cn; 3, 2, 1), and present lower and upper bounds for λ(Cn; 2,..., 2,1,..., 1)     

A Note on Matchings of Bipartite Graphs

In this paper it is proved that let G(X,Y) be a bipartite graph with A(?)X and B(?)Y, and let k be an integer such that max{|A|, |B|}≤k≤min {|X|,|Y|}, then G contains a matching of k edges saturating both A and B iff for all S(?)X, (i) min{|N(S)| |X|-k, |N_B(S)| |X|-|B|}≥|S| and (ii) min{|N(S∩A)|,|N_B(S∩A)| k-|B|}≥|B∩A|, where N_B(S∩A)=N(S∩A∩B).It extends the theorems of Ore, Chen, Hoffman and Kuhn.     

环状硅酸盐图的友好指数集

Let G be a graph with vertex set V(G) and edge set E(G). A labeling f : V(G) →Z2 induces an edge labeling f*: E(G) → Z2 defined by f*(xy) = f(x) + f(y), for each edge xy ∈ E(G). For i ∈ Z2, let vf(i) = |{v ∈ V(G) : f(v) = i}| and ef(i) = |{e ∈ E(G) : f*(e) =i}|. A labeling f of a graph G is said to be friendly if |vf(0)- vf(1)| ≤ 1. The friendly index set of the graph G, denoted FI(G), is defined as {|ef(0)- ef(1)|: the vertex labeling f is friendly}. This is a generalization of graph cordiality. We investigate the friendly index sets of cyclic silicates CS(n, m).     

Sufficient and Necessary Condition for the Measurability of Fallable Fuzzy Sets

Let(X,)be a measurable space,and,=σ-field generated by {x|x∈X},where x={A∈ |x∈A}.(Y,)another measurable space,let ρ(X, Y,)={∈ |§ be measurable}.∈ρ(X,Y,),we define ()(y)=~(-1)(y),y∈y. Defination 1.T is an index set,f:{0,1}~T→{0,1},then,O~T:( (Y)~x)~T→ (Y)~x is called the operation derived from f if for any { }_(t∈T)∈((Y)~x)~Tand any(x,y)∈X×Y,it holds     

Resolvability in circulant graphs

A set W of the vertices of a connected graph G is called a resolving set for G if for every two distinct vertices u, v ∈ V (G) there is a vertex w ∈ W such that d(u, w) ≠ d(v, w). A resolving set of minimum cardinality is called a metric basis for G and the number of vertices in a metric basis is called the metric dimension of G, denoted by dim(G). For a vertex u of G and a subset S of V (G), the distance between u and S is the number min s∈S d(u, s). A k-partition Π = {S 1 , S 2 , . . . , S k } of V (G) is called a resolving partition if for every two distinct vertices u, v ∈ V (G) there is a set S i in Π such that d(u, Si )≠ d(v, Si ). The minimum k for which there is a resolving k-partition of V (G) is called the partition dimension of G, denoted by pd(G). The circulant graph is a graph with vertex set Zn , an additive group of integers modulo n, and two vertices labeled i and j adjacent if and only if i-j (mod n) ∈ C , where CZn has the property that C =-C and 0 ■ C. The circulant graph is denoted by Xn, Δ where Δ = |C|. In this paper, we study the metric dimension of a family of circulant graphs Xn, 3 with connection set C = {1, n/2 , n-1} and prove that dim(Xn, 3 ) is independent of choice of n by showing that dim(Xn, 3 ) ={3 for all n ≡ 0 (mod 4), 4 for all n ≡ 2 (mod 4). We also study the partition dimension of a family of circulant graphs Xn,4 with connection set C = {±1, ±2} and prove that pd(Xn, 4 ) is independent of choice of n and show that pd(X5,4 ) = 5 and pd(Xn,4 ) ={3 for all odd n ≥ 9, 4 for all even n ≥ 6 and n = 7.     

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:     

Estimates of stability of characterization of additive types of distributions

An estimate of stability of characterization of distribution types is obtained for the case of additive types. Under some conditions, the estimate has the order ε^1/3L(ε), where L(ε) is a slowly varying function. Proceedings of the Seminar on Stability Problems for Stochastic Models, Moscow, Russia, 1996, Part I.     

Study of the blow-up of solutions of systems of equations of hydrodynamic type

Mathematical Notes - We study the initial boundary-value problem for three-dimensional systems of equations of pseudoparabolic type. The system is similar to the Oskolkov system, but differs from...     

Characterization of types of asymptotic discernibility of families of hypotheses

We give a characterization of the types of asymptotic discernibility of families of hypotheses in the case of hypothetical measures that are not, in general, mutually absolutely continuous. The case when the logarithm of the likelihood ratio admits an asymptotic expansion of the type of an expansion with local asymptotic normality is examined in detail. Examples are studied.Translated fromTeoriya Sluchainykh Protsessov, Vol. 15, pp. 64–71, 1987.     

一类完全正则半群簇子簇格的性质

本文研究了完全正则半群簇的子簇格［Ｖ＋∩ＰＶ，Ｖ＋∩ＰＶ］的某些格运算性质，我们证明了簇Ｖ＋∩ＰＶ可分解为Ｖ与Ｖ＋∩ＰＶ的并；对任意完全正则半群簇Ｗ，有Ｗ∩（Ｖ∨Ｖ＋∩ＰＶ）＝（Ｗ∩Ｖ）∨（Ｗ∩Ｖ＋∩ＰＶ）．特别地，我们得到了等式Ｖ＋∩ＰＶ＝Ｖ成立的若干条件．     

Distribution of types of symmetry of tensors of high degree

The asymptotic distribution of tensors of degree N in symmetry types is studied in this paper.Translated from Zapiski Nauchnykh Seminarov Leningradskogo Otdeleniya Matematicheskogo Instituta im. V. A. Steklova AN SSSR, Vol. 155, pp. 181–186, 1986.     

Birational aspects of the geometry of Varieties of Sums of Powers

Varieties of Sums of Powers describe the additive decompositions of a homogeneous polynomial into powers of linear forms. The study of these varieties dates back to Sylvester and Hilbert, but only few of them, for special degrees and number of variables, are concretely identified. In this paper we aim to understand a general birational behavior of VSP. To do this we birationally embed these varieties into Grassmannians and prove the rational connectedness of many VSP in arbitrary degrees and number of variables.     

Estimate of growth of Lebesgue functions of linear methods of summation

Sufficient conditions are obtained in terms of matrix elements for the boundedness of Lebesgue functions of linear methods of summation of expansions in orthogonal systems of polynomial type.Translated from Matematicheskie Zametki, Vol. 6, No. 3, pp. 277–286, September, 1969.