High-dimensional D. H. Lehmer problem over short intervals

Letk be a positive integer and n a nonnegative integer,0 λ1,...,λk+1 ≤ 1 be real numbers and w =(λ1,λ2,...,λk+1).Let q ≥ max{[1/λi ]:1 ≤ i ≤ k + 1} be a positive integer,and a an integer coprime to q.Denote by N(a,k,w,q,n) the 2n-th moment of(b1··· bk c) with b1··· bk c ≡ a(mod q),1 ≤ bi≤λiq(i = 1,...,k),1 ≤ c ≤λk+1 q and 2(b1+ ··· + bk + c).We first use the properties of trigonometric sum and the estimates of n-dimensional Kloosterman sum to give an interesting asymptotic formula for N(a,k,w,q,n),which generalized the result of Zhang.Then we use the properties of character sum and the estimates of Dirichlet L-function to sharpen the result of N(a,k,w,q,n) in the case ofw =(1/2,1/2,...,1/2) and n = 0.In order to show our result is close to the best possible,the mean-square value of N(a,k,q) φk(q)/2k+2and the mean value weighted by the high-dimensional Cochrane sum are studied too.     

A Sufficient Condition for the Genus of an Annulus Sum of Two 3-manifolds to Be Non-degenerate

Let Mi be a compact orientable 3-manifold, and Ai a non-separating incompressible annulus on a component of δMi, say Fi, i = 1, 2. Let h ： A1 → A2 be a homeomorphism, and M→M1 ∪h M2, the annulus sum of Mi and M2 along A1 and A2. Suppose that Mi has a Heegaard splitting Vi ∪Si Wi with distance d（Si） ≥ 2g（Mi） ＋ 2g（F3-i） ＋ 1, i = 1, 2. Then g（M） = g（M1） ＋ g（M2）, and the minimal Heegaard splitting of M is unique, which is the natural Heegaard splitting of M induced from Vi∪S1 Wi and V2 ∪S2 W2.     

A LIL and limit distributions for trimmed sums of random vectors attracted to operator semi-stable laws

Let θ∈ Rdbe a unit vector and let X,X1,X2,...be a sequence of i.i.d.Rd-valued random vectors attracted to operator semi-stable laws.For each integer n ≥ 1,let X1,n ≤···≤ Xn,n denote the order statistics of X1,X2,...,Xn according to priority of index,namely | X1,n,θ | ≥···≥ | Xn,n,θ |,where ·,· is an inner product on Rd.For all integers r ≥ 0,define by(r)Sn = n-ri=1Xi,n the trimmed sum.In this paper we investigate a law of the iterated logarithm and limit distributions for trimmed sums(r)Sn.Our results give information about the maximal growth rate of sample paths for partial sums of X when r extreme terms are excluded.A stochastically compactness of(r)Sn is obtained.     

On the order of the high-dimensional Cochrane sum and its mean value

The main purpose of this paper is to study the high-dimensional Cochrane sum and give a sharp estimate of its order by using properties of hyper-Kloosterman sum and the mean value theorems of Dirichlet L-functions.     

A note on the upper bound estimate of high-dimensional Cochrane sums

An upper bound estimate of high-dimensional Cochrane sums is given in this note using Weinstein's version of Deligne's estimate for the hyper-Kloosterman sum and a mean value theorem of Dirichlet L-functions.     

On a Cochrane Sum and Its Hybrid Mean Value Formula

The main purpose of this paper is to use a mean value theorem of Dirichlet L-functions to study the asymptotic property of a hybrid mean value of a Cochrane sum and to give an interesting mean value formula.     

Characterization of arithmetic functions that preserve the sum-of-squares operation

We characterize all functions f :N → C such thatf(m2+n2)=f(m)2+f(n)2for all m,n∈ N.It turns out that all such functions can be grouped into three families,namelyf≡0,f(n)=±n(subject to some restrictions on when the choice of the sign is possible) and f(n)=±1/2(again subject to some restrictions on when the choice of the sign is possible).     

WEIGHTED NONLINEAR REGRESSION

Minimization of the weighted nonlinear sum of squares of differences may be converted to the minimization of sum of squares. The Gauss-Newton method is recalled and the length of the step of the steepest descent method is determined by substituting the steepest descent direction in the Gauss-Newton formula. The existence of minimum is shown.     

Neighbor sum distinguishing total colorings of graphs with bounded maximum average degree

A proper [h]-total coloring c of a graph G is a proper total coloring c of G using colors of the set [h]={1,2,...,h}.Letw(u) denote the sum of the color on a vertex u and colors on all the edges incident to u.For each edge uv∈E(G),if w(u)≠w(v),then we say the coloring c distinguishes adjacent vertices by sum and call it a neighbor sum distinguishing [h]-total coloring of G.By tndi(G),we denote the smallest value h in such a coloring of G.In this paper,we obtain that G is a graph with at least two vertices,if mad(G)3,then tndi∑(G)≤k+2 where k=max{Δ(G),5}.It partially con?rms the conjecture proposed by Pil′sniak and Wozniak.     

Sums of nine almost equal prime cubes

We prove that each sufficiently large odd integer N can be written as sum of the form N = p1^3 ＋p2^3 ＋... ＋p9^3 with [pj - （N/9）^1/31 ≤ N^（1/3）-θ, where pj, j = 1,2,...,9, are primes and θ = （1/51） -ε.     

On a Cochrane sum and its hybrid mean value formula (II)

The main purpose of this paper is using a mean value theorem of Dirichlet L-functions to study the asymptotic property of a hybrid mean value of a Cochrane sum, and give an interesting mean value formula.     

On a Generalized Cochrane Sum and Its Hybrid Mean Value Formula

The main purpose of this paper is to use a mean value theorem of Dirichlet L-functions to study the asymptotic property of a hybrid mean value of a generalized Cochrane sum, and give an interesting mean value formula.2000 Mathematics Subject Classification: Primary—11F20; Secondary—11F99     

关于D.H.Lehmer问题与超级Kloosterman和

设整数q>2,c与q互素．对于1到q之间与q互素的任意整数a,在1到q之间存在唯一的整数b满足ab≡c mod q．对任意整数k≥2,定义M(q,k,c)为满足1≤ai≤q, (ai,q)=1,i=1,2,…,k,a1a2…ak≡c mod q且2 a1 a2 … ak的正整数组(a1,a2,…,ak)的数目,并设E(q,k,c)=M(q,k,c)-(φk-1(q))/2．本文的主要目的是利用Gauss和与原特征的性质,以及Dirichlet L-函数的均值定理,来研究E(q,k,c)与超级Kloosterman和K(h,k,q)的混合均值,并给出一个均值公式．     

Neighbor sum distinguishing total colorings of K 4-minor free graphs

A total [k]-coloring of a graph G is a mapping φ： V（G） U E（G） →{1, 2, ..., k} such that any two adjacent elements in V（G）UE（G） receive different colors. Let f（v） denote the sum of the colors of a vertex v and the colors of all incident edges of v. A total [k]-neighbor sum distinguishing-coloring of G is a total [k]-coloring of G such that for each edge uv E E（G）, f（u） ≠ f（v）. By tt [G, Xsd（ J, we denote the smallest value k in such a coloring of G. Pilniak and Woniak conjectured X＇sd（G） 〈 A（G） ＋ 3 for any simple graph with maximum degree A（G）. This conjecture has been proved for complete graphs, cycles, bipartite graphs, and subcubic graphs. In this paper, we prove that it also holds for Ka-minor free graphs. Furthermore, we show that if G is a Ka-minor flee graph with A（G） 〉 4, then ＂ Xnsd（G） 〈 A（G） ＋ 2. The bound A（G） ＋ 2 is sharp.     

Mean Values of the Hardy Sum

Let p ≥ 5 be a prime.For any integer h,the Hardy sum is defined by H(h,p) =p-1Σj=1(-1)j+1+[hj/p],which is related to the classical Dedekind sum.The mean values of the Hardy sum in short intervals are studied by using the mean value theorems of Dirichlet L-functions.     

关于广义齐性Cochrane和的加权均值

研究了广义齐性Cochrane和的一些加权均值,并利用不完全区间上特征和的性质,Dirichlet函数的均值估计以及周期Bernoulli多项式的性质,得到一些较强的渐近公式．     

关于Hardy和的混合均值公式

本文的目的是利用Dirichlet L-函数的均值定理研究类似于Dedekind和的和与Ramanujao和的混合均值的分布性质，并给出一个较为精确的渐近公式及一个有趣的恒等式．     

Twisted higher moments of Kloosterman sums

Let be a nontrivial Dirichlet character modulo an odd prime . Write

We shall prove

and, for complex ,

0, \end{displaymath}">

where is a constant depending only on .

     

On Centralizer Subalgebras of Group Algebras

Let G be a finite group, H ≤ G and R be a commutative ring with an identity 1R. Let CRG（H）={α ∈ RG｜αh = hα for all h ∈ H）, which is called the centralizer subalgebra of H in RG. Obviously, if H=G then CRG（H） is just the central subalgebra Z（RG） of RG. In this note, we show that the set of all H- conjugacy class sums of G forms an R-basis of CRG（H）. Furthermore, let N be a normal subgroup of G and γthe natural epimorphism from G to G to G/N. Then γ induces an epimorphism from RG to RG, also denoted by % We also show that if R is a field of characteristic zero, then γ induces an epimorphism from CRG（H） to CRG（H）, that is, 7（CRG（H）） = CRG（H）.     

某些对角方程在有限域上的解数

主要运用Gauss和以及Jacobi和的相关性质给出两类对角方程在有限域上的解数公式,分别是形如s∑(i=1) a_ix_i~(m_i)=c的对角方程,其中a_i,c∈F_q~2~*,(m_i,m_j)=1,m_i|(q+1),m_i为奇数或(q+1)/(m_i)为偶数,i=1,2,…,s,以及形如s∑(i=1) x_i~m=c的对角方程,其中c∈F_q~*,m|(q+1),m为奇数或(q+1)/m为偶数.