单位上三角矩阵群的注记

记Tr_1(n,Z)是整数环Z上对角线元素全是1的全体上三角矩阵组成的群,k_(ij)(1≤i相似文献   

非负全连续算子的谱模

设 A=(a_(ij))是 l_2中一个全连续算子,其中a_(i_1j)≥0.当 A~*A 为不可约时,本文证明了|||A|||+2=min{r(B)c_1(C)∶A=BoC},其中 A=BoC 表示对一切 i,j,a_(ij)=b_(ji)c_(ji),r(B)=sup(sum from j=1 to ∞ |b_(ij)|~2)~(1/2),c_1(C)=(sum from i=1 to ∞ (c_(ji)~2)~(1/2),并给出极小解的具体形式.文中所有结果均适用于 A_(mn)为一 m×n 矩阵的情形     

基于混合l_2/l_1范数极小化方法的块稀疏信号重构条件

研究压缩感知中的块稀疏信号重构问题,主要对混合l_2/l_1极小化方法建立了一类改进的可重构条件.具体地说,本文证明若测量矩阵满足条件δ_k+θ_(k,k)1,则混合l_2/l_1极小化方法可精确重构(无噪声情形)或鲁棒重构(有噪声情形)原始块k-稀疏信号.进而表明本文给出的新条件弱于现有文献所给出的条件.     

Ramsey数的几个新下界公式

本文通过构造循环图,得到并证明了公式:r(3,q)≥5(q-3)+2,r(3,q)≥7(q-5)+2,(q为奇数),又由所给引理:若r(l_1,k_1)>t_1,r(l_2,k_2)>t_2,则r(l_1-1·l_2-1+1,k_1-1·k_2-1+1)>t_1t_2,归纳出又一公式:r(3~n+1,3~n+1)≥17~n+1     

平行直线系中λ的几何意义

我们知道,由两条平行直线。l_1 Ax+By+C_1=0及 l_2 Ax+By+C_2=0所成的平行直线系方程是(Ax+By+C_1)+λ(Ax+By+C_2)=0(1)(其中λ为任意实数,且λ≠-1)l_1和 l_2的距离是 d=丨C_2-C_1丨(A~2+B~2)~(1/2)。1.若我们把(C_2-C_1)(A~2+B~2)~(1/2)叫做两平行直线 l_1到 l_2间的问隔量。显然这是一个有方向的量,且 l_2到 l_1的间隔量是 l_1到 l_2的间隔量的相     

一类高斯和显式公式的直接求法

本文讨论了指数2情形下的一类阶数为偶数的高斯和.异于现有的方法,不借助其他指数2情形高斯和的结果,直接利用Stickelberger理想分解定理得到了相应高斯和的显式计算公式.     

问题征解

一、本期问题征解 1.求证对于所有正整数m,不等式 1/(m 1) 1/(m 2) … 1/ (2m 1)m>1成立。2. 求证不等式,(2-(2 (2 (2 … 2~(1/2))~(1/2))~(1/2))~(1/2))/(2-(2 (2 … 2~(1/2))~(1/2))~(1/2))>1/4(其中分子有n个根号,而分母包含有n-1个根号) 以上二题由汉川刘之英译供 3.已知f(x)=x~2-2mx b, (1)求抛物线y=f(x)的顶点轨迹方程; (2)说出l_1:f(x)-f(y)=3, l_2:y=f(x 1)-f(x)的形状; (3)当l_1与l_2相切时,求m的值。 4.某校先后举行数、理、化三科竞赛,学     

带有非线性边界条件的非线性抛物型方程组的爆破速率估计

In this paper, the estimate on blow-up rate of the following nonlinear parabolic system is considered: We will prove that there exist two positive constants such that: where l_1= l_(21)α／α_2 l_(22),r=α_1／α_2＞1,α_1≤α_2＜0.     

ON INTERPOLATION BY S_3~1(△_mn~(1))(I)

§0 Introduction Let △_(mn)~(1) be a subdivision of D: [a,b]x[c,d] (Fig.1). l_1,_2be lengths of [a,b] and [c,d] respectively, and h_1=l_1/m, h_2=l_2/n,l=max(l_1,l_2). Suppose that S∈S1/3(△_(mn)~(1)). In §1-§5 we consider thefollowing interpolation problem:     

在非自然指数下Beltrami方程的L~p可积性

对于n维的 Beltrami方程,存在指数 r_1和r_2,满足1相似文献   

Sign or root of unity ambiguities of certain Gauss sums

Gauss sums play an important role in number theory and arithmetic geometry. The main objects of study in this paper are Gauss sums over the finite field with q elements. Recently, the problem of explicit evaluation of Gauss sums in the small index case has been studied in several papers. In the process of the evaluation, it is realized that a sign (or a root of unity) ambiguity unavoidably occurs. These papers determined the ambiguities by the congruences modulo L, where L is certain divisor of the order of Gauss sum. However, such method is unavailable in some situations. This paper presents a new method to determine the sign (root of unity) ambiguities of Gauss sums in the index 2 case and index 4 case, which is not only suitable for all the situations with q being odd, but also comparatively more efficient and uniform than the previous method.     

Complete solving of explicit evaluation of Gauss sums in the index 2 case

Let p be a prime number,N be a positive integer such that gcd(N,p) = 1,q = pf where f is the multiplicative order of p modulo N.Let χ be a primitive multiplicative character of order N over finite field Fq.This paper studies the problem of explicit evaluation of Gauss sums G(χ) in the "index 2 case"(i.e.[(Z/NZ):p] = 2).Firstly,the classification of the Gauss sums in the index 2 case is presented.Then,the explicit evaluation of Gauss sums G(χλ)(1 λ N-1) in the index 2 case with order N being general even integer(i.e.N = 2r·N0,where r,N0 are positive integers and N0 3 is odd) is obtained.Thus,combining with the researches before,the problem of explicit evaluation of Gauss sums in the index 2 case is completely solved.     

Strongly regular Cayley graphs and rationality of relative Gauss sums

In this note, we give a construction of strongly regular Cayley graphs. The presented construction is based on choosing cyclotomic classes in finite fields, and our results generalize ten of the eleven sporadic examples of cyclotomic strongly regular graphs given by Schmidt and White [B. Schmidt, C. White, All two-weight irreducible cyclic codes, Finite Fields Appl. 8 (2002), 321–367] into infinite families. These infinite families of strongly regular graphs have new parameters. The main tools that we employed are relative Gauss sums instead of explicit evaluations of Gauss sums.     

Gauss sums over some matrix groups

In this note, we give explicit expressions of Gauss sums for general (resp. special) linear groups over finite fields, which involve classical Gauss sums (resp. Kloosterman sums). The key ingredient is averaging such sums over Borel subgroups, i.e., the groups of upper triangular matrices. As applications, we count the number of invertible matrices of zero-trace over finite fields and we also improve two bounds of Ferguson, Hoffman, Luca, Ostafe and Shparlinski in [R. Ferguson, C. Hoffman, F. Luca, A. Ostafe, I.E. Shparlinski, Some additive combinatorics problems in matrix rings, Rev. Mat. Complut. 23 (2010) 501–513].     

Cyclotomic problem, Gauss sums and Legendre curve

In this paper, explicit determination of the cyclotomic numbers of order l and 2l, for odd prime l ≡ 3 (mod 4), over finite field Fq in the index 2 case are obtained, utilizing the explicit formulas on the corresponding Gauss sums. The main results in this paper are related with the number of rational points of certain elliptic curve, called "Legendre curve", and the properties and value distribution of such number are also presented.     

Some results on strongly regular graphs from unions of cyclotomic classes

In this paper, we construct some families of strongly regular graphs on finite fields by using unions of cyclotomic classes and index 2 Gauss sums. New infinite families of strongly regular graphs are found.     

Estimates on polynomial exponential sums

In this paper we establish new bounds on exponential sums of high degree for general composite moduli. The sums considered are either Gauss sums or ‘sparse’ and we rely on earlier work in the case of prime modulus.     

三次高斯和与Kloosterman和的线性递推公式

应用三角和方法以及高斯和的若干性质,研究三次高斯和与Kloosterman和的一类高次混合均值的计算问题,本文给出该混合均值的一个有趣的线性递推公式.同时,还应用该递推公式,得到三次高斯和与Kloosterman和的高次混合均值的一系列较强的渐近公式.     

Strongly regular Cayley graphs,skew Hadamard difference sets,and rationality of relative Gauss sums

In this paper, we give constructions of strongly regular Cayley graphs and skew Hadamard difference sets. Both constructions are based on choosing cyclotomic classes in finite fields. Our results generalize ten of the eleven sporadic examples of cyclotomic strongly regular graphs given by Schmidt and White (2002) [23] and several subfield examples into infinite families. These infinite families of strongly regular graphs have new parameters. The main tools that we employed are relative Gauss sums instead of explicit evaluations of Gauss sums.