On a problem of D.H. Lehmer over short intervals

Let p be an odd prime and a be an integer coprime to p. Denote by N(a,p) the number of pairs of integers b,c with bc≡a (mod p), and with b,c having different parity. The main purpose of this paper is to study the sum , and obtain a sharp asymptotic formula.     

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 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.     

A problem of D.H. Lehmer and its mean square value formula

Let q be an odd positive integer and let a be an integer coprime to q. For each integer b coprime to q with 1?b<q, there is a unique integer c coprime to q with 1?c<q such that . Let N(a,q) denote the number of solutions of the congruence equation with 1?b,c<q such that b,c are of opposite parity. The main purpose of this paper is to use the properties of Dedekind sums, the properties of Cochrane sums and the mean value theorem of Dirichlet L-functions to study the asymptotic property of the mean square value , and give a sharp asymptotic formula.     

On a Problem of D. H. Lehmer and General Kloosterman Sums

Abstract Let q ≥ 3 be an odd number, a be any fixed positive integer with (a, q) = 1. For each integer b with 1 ≤ b < q and (b, q) = 1, it is clear that there exists one and only one c with 0 < c < q such that bc ≡ a (mod q). Let N(a, q) denote the number of all solutions of the congruent equation bc ≡ a (mod q) for 1 ≤ b, c < q in which b and c are of opposite parity, and let . The main purpose of this paper is to study the distribution properties of E(a, q), and give a sharper hybrid mean-value formula involving E(a, q) and general Kloosterman sums. This work is supported by the NSF and the PSF of P. R. China     

On Cochrane sums over short intervals

The main purpose of this paper is to study the mean square value problem of Cochrane sums over short intervals by using the properties of Gauss sums and Kloosterman sums, and finally give a sharp asymptotic formula.     

On a Problem of D. H. Lehmer

The main purpose of this paper is to use the Fourier expansion for character sums and the mean value theorem of Dirichlet L-functions to study the asymptotic property of the difference between a D. H. Lehmer number and its inverse modulo p （an odd prime）, A interesting mean square value formula is also given.     

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．     

一个无k次幂因子D.H.Lehmer数的渐近性质

设素数p>2,对任何满足条件1 a相似文献   

On the mean value of a sum analogous to character sums over short intervals

The main purpose of this paper is to study the mean value properties of a sum analogous to character sums over short intervals by using the mean value theorems for the Dirichlet L-functions, and to give some interesting asymptotic formulae. This work is supported by the N.S.F. (60472068) of P.R. China.     

关于模q的原根中D.H.Lehmer数的分布

设奇数ｑ≥３存在原根，本文研究模ｑ的原根中Ｄ．Ｈ．Ｌｅｈｍｅｒ数的分布性质，并给出一个较强的渐近公式。     

The design centering problem as a D.C. programming problem

The following problem is studied: Given a compact setS inR ⁿ and a Minkowski functionalp(x), find the largest positive numberr for which there existsx S such that the set of ally R ⁿ satisfyingp(y–x) r is contained inS. It is shown that whenS is the intersection of a closed convex set and several complementary convex sets (sets whose complements are open convex) this design centering problem can be reformulated as the minimization of some d.c. function (difference of two convex functions) overR ⁿ . In the case where, moreover,p(x) = (x ^T Ax)^1/2, withA being a symmetric positive definite matrix, a solution method is developed which is based on the reduction of the problem to the global minimization of a concave function over a compact convex set.     

A method for solving d.c. programming problems. Application to fuel mixture nonconvex optimization problem

We present a numerical method for solving the d.c. programming problem

Robust H{infty} dynamic output feedback control for 2D linear parameter-varying systems

James Lam Department of Mechanical Engineering, The University of Hong Kong, Pokfulam Road, Hong Kong Changhong Wang Space Control and Inertial Technology Research Center, Harbin Institute of Technology, Harbin 150001, People's Republic of China Corresponding author. Email: ligangwu{at}hit.edu.cn Email: james.lam{at}hku.hk Email: cwang{at}hit.edu.cn Received on June 28, 2006; Accepted on October 1, 2007 This paper is concerned with the problem of robust dynamicoutput feedback control for 2D discrete-time linear parameter-varyingsystems. Given a Fornasini–Marchesini local state-spacesystem with linear varying parameters, our attention is focussedon the design of full-order dynamic output feedback controller,which guarantees the closed-loop system to be asymptoticallystable and has a prescribed disturbance attenuation performance.A sufficient condition for the existence of a desired robustoutput feedback controller is established in terms of parameterizedlinear matrix inequalities, and the corresponding controllersynthesis is cast into a convex optimization problem which canbe efficiently handled by using standard numerical software.A numerical example is provided to illustrate the effectivenessof the proposed design method.