用有限局部环Z/2kZ上m阶斜对称矩阵构作的卡氏验证码

设R=Z/ 2kZ(k＞1),Wm(R)(m=2v 2≥4)是R上所有m阶斜对称矩阵构成的集合,WmA(R,Hr1r2)={A∈Wm(R)|PAP'=Hr1r2}是Wm(R)中一切与Hr1r2合同的斜对称矩阵构成的集合,令F=∪0≤r1＜r2≤k-1WmA(R,Hr1r2),其中Hr1r2=Dr1(◎)Δr2,Dr1=02-r1I(v)-2-r1I(v)0,Δr2=2-k-12-r2-2-r20-,v≥1,0≤r1＜r2≤k-1.利用F中矩阵构作一个Cartesian验证码,计算其全部参数.     

关于Legendre猜想的一个判定准则

Legendre猜想是数论中一个与素数间隙有关的猜想,它断言对于每个正整数n,在2n和2(n+1)之间都有一个素数2.为此,文章首先研究了夹在2n和(n+1)之间的奇数,刻画了小于一个给定平方数的奇素数的特点.然后证明了如下的判定准则:若nΣr=3,r∈odd(「(n+1)2-r2/2r■ -「n2-r2/2r■)n,则Legendre猜想成立;反之,若nΣr=3,r∈odd(「(n+1)2-r2/2r■ -「n2-r2/2r■)≥n,则Legendre猜想不成立.     

某类Nehari函数族的边界性质

为更全面地了解不同参数Nehari函数族的边界性质，利用Nehari函数及其导数的偏差性质，采用分析构造法研究参数α∈［1,2］的函数族的边界性质。研究发现:对于该类函数族中满足规范条件的任一函数的一阶和二阶导数，除了可列个点外，均满足lim infr→1(1-r2)Reζf″f′(rζ)<2α,ζ=1；同时，对于ζ=1，r∈(0,1)，有|f(ζ)-f(rζ)|≤(1-r2)|f′(rζ)|αr-1-r22Reζf″f′(rζ)。当α=1时，以上两个结论与Chuaqui的结论一致。     

连续时间下非参数回归模型的回归函数核估计量的一致均方收敛速度

研究了连续时间下非参数回归的回归函数核估计量的收敛速度，给出了一定条件下回归函数估计量rt(x)的一致均方收敛速度，详细证明了两组条件下rT(X)分别满足：E(supx∈S|rT(X)-r(x)|)^2=0(T^-1/2)和E(supx∈S|rT(x)-r(x)|)^2=0(lnT)^2/S T^-1/2),其中r(x)表示未知的回归函数。     

城市人口空间分布密度衰减模型的一个理论证明

基于理想地表形态和圆形城市假设,借助最大熵方法导出了关于城市人口空间分布密度衰减规律的Clark模型ρ(r)=ρoexp(-r/r0),从而解决了该模型的理论来源问题.式中P(r)为距离市中心为r处的人口密度,参数ρ0=P0/2πγ02在理论上为市中心(r=0)的人口密度,O0为城市总人口,γ0被定义为城市人口空间分布的特征半径.     

一类非线性方程的死角问题

应用初等方法 ,证明了边值问题u″ =λh(u′) ,u≥ 0 ,x∈ (- 1,1) ,u(± 1) =1存在唯一非平凡解 (在C2 [- 1,1]中 )的充分必要条件是∫101h(s) ds<∞ .而且非平凡解有死角的充分必要条件是λ<∫λ0 G- 1(t)dt.这里 ,λ>0 ,h∈C(R) ,h(0 ) =0 ,h(s) >0 , s≠ 0 ,G- 1表示G(t) =∫t01h(s) ds的反函数 ,死角是 [-r ,r],r满足λ=∫λ( 1-r)0 G- 1(t)dt .特别 ,若h(s) =｜s｜p,则存在唯一非平凡解的充分必要条件是 0

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