Kloosterman double sums

In this paper estimates of incomplete Kloosterman double sums with weights are obtained. Translated fromMatematicheskie Zametki, Vol. 66, No. 5, pp. 682–687, November, 1999.     

On the generalization of the D. H. Lehmer problem

Let n ≥ 2 be a fixed positive integer, q ≥ 3 and c be two integers with （n, q） = （c, q） = 1. We denote by rn（51, 52, C; q） （δ 〈 δ1,δ2≤ 1） the number of all pairs of integers a, b satisfying ab ≡ c（mod q）, 1 〈 a ≤δ1q, 1 ≤ b≤δ2q, （a,q） = （b,q） = 1 and nt（a＋b）. The main purpose of this paper is to study the asymptotic properties of rn （δ1, δ2, c; q）, and give a sharp asymptotic formula for it.     

On bandwidth sums of graphs

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On Explicit Formulas for the Number of Solutions to the Equation (x1+...+xn)2 =ax1...xn in a Finite Field

We consider the equation of the title in a finite field of q elements. Assuming certain relations between n and q, we obtain explicit formulas for the number of solutions to this equation.     

Upper bounds on the sum of powers of the degrees of a simple planar graph

For a simple planar graph G and a positive integer k, we prove the upper bound 2(n ? 1)^k + 4^k(n ? 4) + 2·3^k ? 2((δ + 1)^k ? δ^k)(3n ? 6 ? m) on the sum of the kth powers of the degrees of G, where n, m, and δ are the order, the size, and the minimum degree of G, respectively. The bound is tight for all m with 0?3n ? 6 ? m≤?n/2? ? 2 and δ = 3. We also present upper bounds in terms of order, minimum degree, and maximum degree of G. © 2010 Wiley Periodicals, Inc. J Graph Theory 67:112‐123, 2011     

On a diophantine inequality involving prime numbers (III)

Let 1<c<11/10. In the present paper it is proved that there exists a numberN(c)>0 such that for each real numberN>N(c) the inequality is solvable in prime numbersp ₁,p ₂,p ₃, wherec ₁ is some absolute positive constant. Project supported by the National Natural Science Foundation of China (grant: 19801021) and by MCSEC     

On κ—ordered Graphs Involved Degree Sum

A graph G is κ-ordered Hamiltonian 2≤κ≤n,if for every ordered sequence S of κ distinct vertices of G,there exists a Hamiltonian cycle that encounters S in the given order,In this article,we prove that if G is a graph on n vertices with degree sum of nonadjacent vertices at least n 3κ-9/2,then G is κ-ordered Hamiltonian for κ=3,4,…,[n/19].We also show that the degree sum bound can be reduced to n 2[κ/2]-2 if κ(G）≥3κ-1/2 or δ（G）≥5κ-4.Several known results are generalized.     

A Game Related to the Number of Hides Game

In this paper, we study a discrete search game on an array of N ordered cells, with two players having opposite goals: player I (searcher) and player II (hider). Player II has to hide q objects at consecutive cells and player I can search p consecutive cells. The payoff to player I is the number of objects found by him. In some situations, the players need to adopt sophisticated strategies if they are to act optimally.     

Coloring of trees with minimum sum of colors

The chromatic sum of a graph is the smallest sum of colors among all proper colorings with natural numbers. The strength is the minimum number of colors needed to achieve the chromatic sum. We construct for each positive integer k a tree with strength k that has maximum degree only 2k − 2. The result is best possible. © 1999 John Wiley & Sons, Inc. J Graph Theory 32: 354–358, 1999     

Exponential sum estimates over subgroups and almost subgroups of \mathbb{Z}_{Q}^{*} , where Q is composite with few prime factors

In this paper we extend the exponential sum results from [BK] and [BGK] for prime moduli to composite moduli q involving a bounded number of prime factors. In particular, we obtain nontrivial bounds on the exponential sums associated to multiplicative subgroups H of size q^δ, for any given δ > 0. The method consists in first establishing a ‘sumproduct theorem’ for general subsets A of . If q is prime, the statement, proven in [BKT], expresses simply that either the sum-set A + A or the product-set A.A is significantly larger than A, unless |A| is near q. For composite q, the presence of nontrivial subrings requires a more complicated dichotomy, which is established here. With this sum-product theorem at hand, the methods from [BGK] may then be adapted to the present context with composite moduli. They rely essentially on harmonic analysis and graph-theoretical results such as Gowers’ quantitative version of the Balog–Szemeredi theorem. As a corollary, we get nontrivial bounds for the ‘Heilbronn-type’ exponential sums when q = p^r (p prime) for all r. Only the case r = 2 has been treated earlier in works of Heath-Brown and Heath-Brown and Konyagin (using Stepanov’s method). We also get exponential sum estimates for (possibly incomplete) sums involving exponential functions, as considered for instance in [KS]. Submitted: October 2004 Revision: June 2005 Accepted: August 2005     

An asymptotic version of a conjecture by Enomoto and Ota

In 2000, Enomoto and Ota [J Graph Theory 34 (2000), 163–169] stated the following conjecture. Let G be a graph of order n, and let n₁, n₂, …, n_k be positive integers with \begin{eqnarray*}\sum\nolimits_{{{i}} = {{1}}}^{{{k}}} {{n}}_{{{i}}} = {{n}}\end{eqnarray*}. If σ₂(G)≥n+ k?1, then for any k distinct vertices x₁, x₂, …, x_k in G, there exist vertex disjoint paths P₁, P₂, …, P_k such that |P_i|=n_i and x_i is an endpoint of P_i for every i, 1≤i≤k. We prove an asymptotic version of this conjecture in the following sense. For every k positive real numbers γ₁, …, γ_k with \begin{eqnarray*}\sum\nolimits_{{{i}} = {{1}}}^{{{k}}} \gamma_{{{i}}} = {{1}}\end{eqnarray*}, and for every ε>0, there exists n₀ such that for every graph G of order n≥n₀ with σ₂(G)≥n+ k?1, and for every choice of k vertices x₁, …, x_k∈V(G), there exist vertex disjoint paths P₁, …, P_k in G such that \begin{eqnarray*}\sum\nolimits_{{{i}} = {{1}}}^{{{k}}} |{{P}}_{{{i}}}| = {{n}}\end{eqnarray*}, the vertex x_i is an endpoint of the path P_i, and (γ_i?ε)n<|P_i|<(γ_i + ε)n for every i, 1≤i≤k. © 2009 Wiley Periodicals, Inc. J Graph Theory 64: 37–51, 2010     

A note on the Lehmer problem over short intervals

Let p be an odd prime, c be an integer with (c, p) = 1, and let N be a positive integer with N ≤ p − 1. Denote by r(N, c; p) the number of integers a satisfying 1 ≤ a ≤ N and 2 ∤ a + ā, where ā is an integer with 1 ≤ ā ≤ p − 1, aā ≡ c (mod p). It is well known that r(N, c; p) = 1/2N + O(p ^1/2log² p). The main purpose of this paper is to give an asymptotic formula for Σ _c=1 ^p−1(r(N, c; p) − 1/2N)².     

On some congruence with application to exponential sums

We will study the solution of a congruence,x ≡g ^1/2)ωg(2 ⁿ ) mod 2 ⁿ , depending on the integersg andn, where ω _g (2 ⁿ ) denotes the order ofg modulo 2 ⁿ . Moreover, we introduce an application of the above result to the study of an estimation of exponential sums.     

Central Units,Class Sums and Characters of the Symmetric Group

In the search for central units of a group algebra, we look at the class sums of the group algebra of the symmetric group S _n in characteristic zero, and we show that they are units in very special instances.     

Kloosterman sum identities and low-weight codewords in a cyclic code with two zeros

We apply relations of n-dimensional Kloosterman sums to exponential sums over finite fields to count the number of low-weight codewords in a cyclic code with two zeros. As a corollary we obtain a new proof for a result of Carlitz which relates one- and two-dimensional Kloosterman sums. In addition, we count some power sums of Kloosterman sums over certain subfields.     

Moments of the maximum of normed partial sums of ρ^--mixing random variables

Let {Xn,n ≥ 1} be a sequence of identically distributed ρ^--mixing random variables and set Sn =∑i^n=1 Xi,n ≥ 1,the suffcient and necessary conditions for the existence of moments of supn≥1 ｜Sn/n^1/r｜^p（0 〈 r 〈 2,p 〉 0） are given,which are the same as that in the independent case.     

Spanning 2‐trails from degree sum conditions

Suppose G is a simple connected n‐vertex graph. Let σ₃(G) denote the minimum degree sum of three independent vertices in G (which is ∞ if G has no set of three independent vertices). A 2‐trail is a trail that uses every vertex at most twice. Spanning 2‐trails generalize hamilton paths and cycles. We prove three main results. First, if σ₃G)≥ n ‐ 1, then G has a spanning 2‐trail, unless G ? K_1,3. Second, if σ₃(G) ≥ n, then G has either a hamilton path or a closed spanning 2‐trail. Third, if G is 2‐edge‐connected and σ₃(G) ≥ n, then G has a closed spanning 2‐trail, unless G ? K_2,3 or K (the 6‐vertex graph obtained from K_2,3 by subdividing one edge). All three results are sharp. These results are related to the study of connected and 2‐edge‐connected factors, spanning k‐walks, even factors, and supereulerian graphs. In particular, a closed spanning 2‐trail may be regarded as a connected (and 2‐edge‐connected) even [2,4]‐factor. © 2004 Wiley Periodicals, Inc. J Graph Theory 45: 298–319, 2004     

Hamiltonian cycles in n‐extendable graphs

A graph G of order at least 2n+2 is said to be n‐extendable if G has a perfect matching and every set of n independent edges extends to a perfect matching in G. We prove that every pair of nonadjacent vertices x and y in a connected n‐extendable graph of order p satisfy deg_G x+deg_G y ≥ p ? n ? 1, then either G is hamiltonian or G is isomorphic to one of two exceptional graphs. © 2002 Wiley Periodicals, Inc. J Graph Theory 40: 75–82, 2002     

On a problem of the Goldbach-waring type

LetE(X)=‖{N≤X;N≠p ₁ ² +p ₂ ³ +p ₃ ⁴ +p ₄ ⁵ for any primesp _i}‖. It is proved in this paper that there exists a positive constant δ>0 such that

An Application of Exponential Sum Estimates

Let p be an odd prime and let δ be a fixed real number with 0<δ<2. For an integer a with 0<a<p, denote by ā the unique integer between 0 and p satisfying a ā≡1 (mod p). Further, let {x} denote the fractional part of x. We derive an asymptotic formula for the number of pairs of integers (a, b with . This work is supported by N. S. F. of P. R. China (10271093)