Several Results on Systems of Residue Classes

Let (m,n) and a(n) denote the g.c.d, of m, n and the residue class {x∈Z∶x≡α (mod n)} respectively. Any period of the characteristic function ofkU a_i(n_i) is called a covering period of {a_i(n_i)}_(i-1)~k.i-ITheorem Let A = {a_i(n_i)}_(i-1)~k. be a disjoint system (i. e. a_I(n_I,...,a_k(n_k) are pairwise disjoint). Let [n_I,...,n_k] (the I.c.m. of n_1,...,n_k) have the prime faetorization [n_1,...,n_k] = Πp_i~ai and T = Πp_iβi(β_i≥0 be the smallest positive covering period of A. Then     

The Prediction Theory of Stationary Random Field (Ⅲ) Four-fold Wold Decompositions

Let{X(m,n)}_(m,m=0,±1,±2…)be a stationary random field.The closed linearspace spanned by all X(m,n):m,n=0,±1,±2,…is denoted by L(X).Throughoutthe following pages L_1(x:s)will denote the subspace generated by all X(m,n):m≤s,-∞相似文献   

关于布尔矩阵行空间基数的若干存在区间

Let B_n be the set of all n×n Boolean Matrices;R(A) denote the row space of A∈B_n,|R(A)| denote the cardinality of R(A),m,n,k,l,t,i,γ_i be positive integers,S_i,λ_i be non negative integers.In this paper,we prove the following two results: (1)Let n≥13,n-3≥k > S_l,S_(i+1)> S_i,i = 1,2,…,l-1.if k+l≤n,then for any m=2~k+2~(S_(l)) + 2~(S_(l-1))+…+ 2~(S_(1)),there exists A∈B_n,such that |R(A)|= m. (2)Let n≥13,n-3≥k>S_(n-k-1)> S_(n-k-2)>…>S_1>λ_t>λ_(t-1)>…>λ_1,2≤t≤n-k.If existγ_i(k+1≤γ_i≤n-1,i=1,2,…,t-1)γ_i<γ_...     

高阶Hermite-Fejer型插值的渐近估计

1. Introduction Let f∈C[-1,1] and X_k=X_(kn)=COSθ_k=COS(2k-1)π/(2n)(k=1,…,n) be the zeros of the Chebyshev polynomial T_n(x)=cosnθ(x=cosθ). Let ω(t) be a given modulus of continuity and H_ω={f;ω(f,t)≤ω(t),for all.t≥0}. In this paper, c will always denote different constant independent of x, n and f and the sign"A～B" means that there exist two positive constants c_1相似文献   

HOLOMORPHIC VECTOR FIELDS AND CHARACTERISTIC FORMS ON A HERMITIAN MANIFOLD

Let M be a compact Hermtian manifold, dim_cM=m, Ω be the curvature form of the Hermitian connection. F is a U(m)-invariant polynomial of degree k相似文献   

The SL(V)-ample Cone of Product of Flag Varieties

Let k be an algebraically closed field, and V be a vector space of dimension n over k. For a set ω =(d(1),...,d(m)) of sequences of positive integers, denote by Lω the ample line bundle corresponding to the polarization on the product X =∏m i=1Flag(V, n(i)) of flag varieties of type n(i)determined by ω. We study the SL(V)-linearization of the diagonal action of SL(V) on X with respect to Lω. We give a sufficient and necessary condition on ω such that Xss(Lω) ≠φ(resp., Xs(Lω) ≠φ).As a consequence, we characterize the SL(V)-ample cone(for the diagonal action of SL(V) on X),which turns out to be a polyhedral convex cone.     

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 .     

关于同余式$\sigma(n)\equiv 1\pmod n$

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 them.     

ON THE ESTIMATIONS OF BOUNDS FOR DETERMINANT OF HADAMARD PRODUCT OF H-MATICES

1. IntroductionLet Rmxn denote the set of m x n real matrices, s: dellote the at of n x n positive delhatereal symmetric matrices. For A = (a.) and B = (hi,) E Rm", the Hadamard product of Aand B is defined as an m x n matriX denoted by A o B: (A o B). = a.bt,.We write A 2 B if at; 2 hi, for all i,j. A real n x n matriX A is called a nonsingularM-~ac if A = sl ~ B satisfied: 8 > 0, B 2 0 and 8 > P(B), the spectral rebus of B, let Madenote the s6t of all n x n nonsingular M-mstrices…     

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.     

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.     

ON THE APPROXIMATION OF (0,2) INTERPOLATION

Let $-1=x_{n,n}相似文献   

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.     

二项式展开的抵消系数以及$q$-摸拟

Let {An}∞n=0 be an arbitary sequence of natural numbers. We say A(n,k;A) are the Convolution Annihilation Coefficients for {An}n∞=0 if and only if n k=0 A(n,k;A)(x - Ak)n-k = xn. (0.1) Similary, we define B(n,k;A) to be the Dot Product Annihilation Coefficients for {An}n∞=0 if and only if n k=0 B(n,k;A)(x - Ak)k = xn. (0.2) The main result of this paper is an explicit formula for B(n,k;A), which depends on both k and {An}∞n=0. This paper also discusses binomial and q-analogs of Equations (0.1) and (0.2).     

本刊英文版2015年58卷第7期摘要（英文）

<正>On universal sums of polygonal numbers SUN Zhi-Wei Abstract For m=3,4,...,the polygonal numbers of order m are given by p_m(n)=(m-2)(_2~n)+n(n=0,1,2,...).For positive integers a,b,c and i,j,k≥3 with max{i,j,k}≥5,we call the triple(ap_i,bp_j,cp_k)universal if for any n=0,1,2,...,there are nonnegative integers x,y,z such that n=ap_i(x)+bp_j(y)+cp_k(z).     

A Counter Example on Faudree-Schelp Conjecture

Let simple graph G=(V, E),V=n,E=m. If there exists a path containing i vertices connecting u and v in V, then property P_i(u,v) will be said to told.For 2≤i≤n, let S_i be the set of all unordered pairs of distinct u and v for which property P_i(u.v) holds, and Let S_1 be the set of all unordered pairs of vertices which are not connected by any path. A graph G satisfies property P_i if |S_i|=n(n-1)/2.     

A LIMIT LAW FOR FUNCTIONALS OF MULTIPLE INDEPENDENT FRACTIONAL BROWNIAN MOTIONS

Let B = {B~H(t)}t≥0 be a d-dimensional fractional Brownian motion with Hurst parameter H ∈(0, 1). Consider the functionals of k independent d-dimensional fractional Brownian motions■where the Hurst index H = k/d. Using the method of moments, we prove the limit law and extending a result by Xu [19] of the case k = 1. It can also be regarded as a fractional generalization of Biane [3] in the case of Brownian motion.     

On universal sums of polygonal numbers

For m = 3, 4,..., the polygonal numbers of order m are given by pm(n) =(m- 2) n2 + n(n= 0, 1, 2,...). For positive integers a, b, c and i, j, k 3 with max{i, j, k} 5, we call the triple(api, bpj, cpk)universal if for any n = 0, 1, 2,..., there are nonnegative integers x, y, z such that n = api(x) + bpj(y)+ cpk(z). We show that there are only 95 candidates for universal triples(two of which are(p4, p5, p6) and(p3, p4, p27)), and conjecture that they are indeed universal triples. For many triples(api, bpj, cpk)(including(p3, 4p4, p5),(p4, p5, p6) and(p4, p4, p5)), we prove that any nonnegative integer can be written in the form api(x) + bpj(y) + cpk(z) with x, y, z ∈ Z. We also show some related new results on ternary quadratic forms,one of which states that any nonnegative integer n ≡ 1(mod 6) can be written in the form x2+ 3y2+ 24z2 with x, y, z ∈ Z. In addition, we pose several related conjectures one of which states that for any m = 3, 4,...each natural number can be expressed as pm+1(x1) + pm+2(x2) + pm+3(x3) + r with x1, x2, x3 ∈ {0, 1, 2,...}and r ∈ {0,..., m- 3}.     

套链分解

Let X1,X2,...,Xk be k disjoint subsets of S with the same cardinality m.Define H(m,k) = {X (C) S: X (C) Xi for 1 ≤I ≤k} and P(m,k) = {X (C) S : X ∩ Xi ≠φ for at least two Xi's}.Suppose S = Uki=1 Xi,and let Q(m,k,2) be the collection of all subsets K of S satisfying|K ∩ Xi|≥ 2 for some 1 ≤ I ≤ k.For any two disjoint subsets Y1 and Y2 of S,we define F1,j = {X (C) S : either |X ∩ Y1|≥ 1 or |X ∩ Y2|≥ j}.It is obvious that the four posers are graded posets ordered by inclusion.In this paper we will prove that the four posets are nested chain orders.     

On the Congruence σ(n)≡1(mod n)

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 them.