整数集的唯一加权表示基

Let k1, k2 be nonzero integers with(k1, k2) = 1 and k1k2≠-1. In this paper, we prove that there is a set A■Z such that every integer can be represented uniquely in the form n = k1a1 + k2a2, a1, a2 ∈ A.     

Bipartition of graph under degree constraints

Let G be a graph, let s be a positive integer, and let X be a subset of V(G). Denote δ(X) to be the minimum degree of the subgraph G[X] induced by X. A partition(X, Y) of V(G) is called s-good if min{δ(X), δ(Y)} s. In this paper, we strengthen a result of Maurer and a result of Arkin and Hassin, and prove that for any positive integer k with 2 k |V(G)|- 2, every connected graph G with δ(G) 2 admits a1-good partition(X, Y) such that |X| = k and |Y| = |V(G)|- k, and δ(X) + δ(Y) δ(G)- 1.     

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.     

Movement of Intransitive Permutation Groups Having Maximum Degree

Let G be a permutation group on a set Ω with no fixed points in,and m be a positive integer.Then the movement of G is defined as move(G):=sup Γ {|Γg\Γ| | g ∈ G}.It was shown by Praeger that if move(G) = m,then |Ω| 3m + t-1,where t is the number of G-orbits on.In this paper,all intransitive permutation groups with degree 3m+t-1 which have maximum bound are classified.Indeed,a positive answer to her question that whether the upper bound |Ω| = 3m + t-1 for |Ω| is sharp for every t > 1 is given.     

A REMARK ON LIMINF SETS IN DIOPHANTINE APPROXIMATION

Let Q be an infinite set of positive integers, τ 1 be a real number and let Wτ(Q) = {x ∈ R : |x-p/q| q-τ for infinitely many(p,q) ∈ Z × Q }For any given positive integer m, set Q(m) = {n ∈ N :(n, m) = 1}.If m is divisible by at least two prime factors, Adiceam [1] showed that Wτ(N) \ Wτ(Q(m))contains uncountably many Liouville numbers, and asked if it contains any non-Liouville numbers? In this note, we give an affirmative answer to Adiceam's question.     

Uniform Hypergraphs under Certain Intersection Constraints between Hyperedges

A hypergraph H is an(n,m)-hypergraph if it contains n vertices and m hyperedges,where n≥1 and m≥ 0 are two integers.Let k be a positive integer and let L be a set of nonnegative integers.A hyper graph H is k-uniform if all its hyperedges have the same size k,and H is L-intersecting if the number of common vertices of every two hyperedges belongs to L.In this paper,we propose and investigate the problem of estimating the maximum k among all k-uniform L-intersecting(n,m)-hypergraphs for fixed n,m ...     

A Hybrid of Theorems of Goldbach and Piatetski-Shapiro

It is proved that for almost all sufficiently large even integers n, the prime variable equation n = p1 p2,p1∈Pγis solvable, with 13/15 <γ≤1, where Pγ= {p| p = [mγ/1], for integer to and prime p} is the set of the Piatetski-Shapiro primes.     

某些Ruzsa数$R_m$的新上界

For A ■ Z m and n ∈ Z m ,let σ A (n) be the number of solutions of equation n = x + y,x,y ∈ A.Given a positive integer m,let R m be the least positive integer r such that there exists a set A ■ Z m with A + A = Z m and σ A (n) ≤ r.Recently,Chen Yonggao proved that all R m ≤ 288.In this paper,we obtain new upper bounds of some special type R kp 2 .     

On the Equation and the Number of Solution of Znam's Problem

§1. In 1972,St.Znam posed the problem whether for every s>1 thereexist integers x_i>1,i=1,…,s such that x_i is a proper divisor of the numberx_1…x_(i-1)x_(i 1)…x_s 1 for i=1,…, s.Without loss of generality, we may assume1相似文献   

MULTIPLE SOLUTIONS FOR AN INHOMOGENEOUS SEMILINEAR ELLIPTIC EQUATION IN R~N

In this paper, the authors study the existence and nonexistence of multiple positive solutions for problem(*)μwhere h ∈ H-1(RN), N ≥ 3, |f(x,u)| ≤ C1up-1 + C2u with C1 > 0, C2∈ [0,1) being some constants and 2 < p < ∞. Under some assumptions on f and h, they prove that there exists a positive constant μ* <∞ such that problem (*)μ has at least one positive solution uμ if μ,∈ (0,μ*), there are no solutions for (*)μ if μ, > μ* and uμ is increasing with respect to μ∈ (0,μ*); furthermore, problem (*)μ has at least two positive solution for μ ∈ (0,μ*) and a unique positive solution for μ, =μ* if p ≤2N/N-2.     

On Non-commuting Sets in a Finite p-group with Derived Subgroup of Prime Order

Let G be a finite group. A nonempty subset X of G is said to be noncommuting if xy≠yx for any x, y ∈ X with x≠y. If |X| ≥ |Y| for any other non-commuting set Y in G, then X is said to be a maximal non-commuting set. In this paper, we determine upper and lower bounds on the cardinality of a maximal non-commuting set in a finite p-group with derived subgroup of prime order.     

和图、整和图与模和图的几个结果

Let N denote the set of positive integers. The sum graph G^＋（S） of a finite subset S belong to N is the graph （S, E） with uv ∈ E if and only if u ＋ v ∈ S. A graph G is said to be a sum graph if it is isomorphic to the sum graph of some S belong to N. By using the set Z of all integers instead of N, we obtain the definition of the integral sum graph. A graph G = （V, E） is a mod sum graph if there exists a positive integer z and a labelling, λ, of the vertices of G with distinct elements from {0, 1, 2,..., z - 1} so that uv ∈ E if and only if the sum, modulo z, of the labels assigned to u and v is the label of a vertex of G. In this paper, we prove that flower tree is integral sum graph. We prove that Dutch m-wind-mill （Dm） is integral sum graph and mod sum graph, and give the sum number of Dm.     

Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy–Sobolev–Maz'ya term:-Δu- λu/|y|2=|u|pt-1u/|y|t+ μf(x), x ∈Ω,where Ω is a bounded domain in RN(N ≥ 2), 0 ∈Ω, x =(y, z) ∈ Rk× RN-kand pt =N +2-2t N-2(0 ≤ t ≤2). For f(x) ∈ C1(Ω)\{0}, we show that there exists a constant μ* 0 such that the problem possessesat least two positive solutions if μ∈(0, μ*) and at least one positive solution if μ = μ*. Furthermore,there are no positive solutions if μ∈(μ*, +∞).     

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.     

On the Existence of Nontrivial Solutions of Quasi-asymptotically Linear Problem for the P-Laplacian

In this paper, we study the existence of nontrivial solutions for the following Dirichlet problem for the p-Laplacian (p > 1):where Ω is a bounded domain in Rn (A≥1) and f(x,u) is quasi-asymptotically linear with respect to |u|p-2 u at infinity. Recently it was proved that the above problem has a positive solution under the condition that f(x, s)/sp-1 is nondecrcasing with respect to s for all x ∈Ω and some others. In this paper. by improving the methods in the literature, we prove that the functional corresponding to the above problem still satisfies a weakened version of (P.S.) condition even if f(x, s)/sp-1 isn't a nondecreasing function with respect to s, and then the above problem has a nontrivial weak solution by Mountain Pass Theorem.     

On the Congruence σ(n) ≡ 1(mod n), II

Let k ≥ 2 be an integer, and let σ(n) denote the sum of the positive divisors of an integer n. We call n a quasi-multiperfect number if σ(n) = kn + 1. In this paper, we give some necessary properties of quasi-multiperfect numbers with four different prime divisors.     

Score sets in oriented 3-partite graphs

Let D(U, V, W) be an oriented 3-partite graph with |U|=p, |V|=q and |W|= r. For any vertex x in D(U, V, W), let d x and d-x be the outdegree and indegree of x respectively. Define aui (or simply ai) = q r d ui - d-ui, bvj(or simply bj) = p r d vj - d-vj and Cwk (or simply ck) = p q d wk - d-wk as the scores of ui in U, vj in V and wk in Wrespectively. The set A of distinct scores of the vertices of D(U, V, W) is called its score set. In this paper, we prove that if a1 is a non-negative integer, ai(2≤i≤n - 1) are even positive integers and an is any positive integer, then for n≥3, there exists an oriented 3-partite graph with the score set A = {a1,2∑i=1 ai,…,n∑i=1 ai}, except when A = {0,2,3}. Some more results for score sets in oriented 3-partite graphs are obtained.     

Self-similar fractals: An algorithmic point of view

This paper studies the self-similar fractals with overlaps from an algorithmic point of view.A decidable problem is a question such that there is an algorithm to answer"yes"or"no"to the question for every possible input.For a classical class of self-similar sets{E b.d}b,d where E b.d=Sn i=1(E b,d/d+b i)with b=(b1,...,b n)∈Qn and d∈N∩[n,∞),we prove that the following problems on the class are decidable:To test if the Hausdorff dimension of a given self-similar set is equal to its similarity dimension,and to test if a given self-similar set satisfies the open set condition(or the strong separation condition).In fact,based on graph algorithm,there are polynomial time algorithms for the above decidable problem.     

A Remark on Jesmanowicz' Conjecture for the Non-coprimality Case

Let a, b, c be relatively prime positive integers such that a2+ b2= c2. Je′smanowicz'conjecture on Pythagorean numbers states that for any positive integer N, the Diophantine equation(aN)x+(b N)y=(cN)zhas no positive solution(x, y, z) other than x = y = z = 2. In this paper, we prove this conjecture for the case that a or b is a power of 2.     

On tame operators between non-archimedean power series spaces

Let p ∈ {1, ∞}. We show that any continuous linear operator T from A1 (a) to Ap (b) is tame, i.e., there exists a positive integer c such that sup x||Tx||k/|x|ck ∞ for every k ∈ N. Next we prove that a similar result holds for operators from A∞(a) to Ap(b) if and only if the set Mb,a of all finite limit points of the double sequence (bi /aj ) i,j∈N is bounded. Finally we show that the range of every tame operator from A∞(a) to A∞(b) has a Schauder basis.