实二次域Q(P(1/2))(p≡3(mod 4))类数的上界

设p是适合p≡3（Pod4）的奇素数,h,ε分别是实二次域Q的类数和基本单位．本文运用初等方法证明了：εh＜（p a 2)a 2/4(a 2)!,其中     

实二次域Q(√p)(p≡3(mod4))类数的上界

设p是适合p≡3(mod4)的奇素数,h,分别是实二次域Q(√p)的类数和基本单位.本文运用初等方法证明了:εh<(p+a+2)a+2/4(a+2)!,其中a=[(√p+1)/2].     

混合幂的素变数丢番图逼近

证明了,如果λ1,λ2,λ3,λ4是正实数,λ1/λ2是无理数和代数效,V是well-spaced序列,δ>0,那么对于ν,∈V,ν≤X,ε>0,使得|λ1p21+λ2p22+λ3p33+λ4p34-ν|<ν-δ没有素数解P1,p2,p3,p4的ν的个数不超过O(X20/21+2δ+ε).     

关于一个素数和一个素数的k次方和问题

设k≥2,Hk表示一个正整数n的集合,使对任意的正整数q,同余方程a+b2三n(modq)在模q的既约剩余系中有解a,b.Dk(N)表示n≤N,n∈Hk,但不能表成p1+p22=n的数的个数,其中p1,p2表示素数.则在GRH下,Dk(N)<<N1-1/k(h(k)+1)+ε,这里k=2,3;h(2)=2,h(3)=8.     

关于一个素数和一个素数的k次方和问题

设k≥2,且Hk表示一个正整数n的集合,使得该集合中的元素满足a+bk≡n(modq)对任意的q,在模q的既约剩余系中有解,令Dk(N)表示所有的n≤N,且n∈Hk且不能表成p1+p2k=n形式的整数.那么在GRH下, Dk(N)相似文献   

借用向量分解定理巧求f(x0)的范围

已知函数 f ( xi) ( i =1,2 ,3 ,… )的范围 ,求 f( x0 )的范围 .笔者在同行们研究的基础上 ,借用向量分解定理 ,使这类问题的解决更加简单、明了 ,可操作性强 ,便于实施 .例 1　已知一次函数 f( x) ,1≤ f ( 1)≤2 ,3≤ f ( 2 )≤ 4,试确定 f( 5 )的范围 .解　设一次函数为 f( x) =ax + b,则　 f( 1) =a+ b,f( 2 ) =2 a+ b,f( 5 ) =5 a+ b.记　 p1→ =a+ b,p2→ =2 a+ b,p=5 a+ b显然 p1→ ,p2→ 不共线 ,根据向量分解定理p=λ1 p1→ +λ2 p2→ 　 (λ1 ,λ2 为实数 ) ,即　 5 a+ b=λ1 ( a+ b) +λ2 ( 2 a+ b…     

The exceptional set for sums of unlike powers of primes

It is established that all even positive integers up to N but at most O(N15/16+ε) exceptions can be expressed in the form p21+ p32+ p43+ p54,where p1,p2,p3 and p4 are prime numbers.     

非线性临界Kirchhoff型问题的正基态解

该文研究如下Kirchhoff型方程-(a+b∫R3 |▽u|2dx)Δu+V(x)u=|u|p-2U+ε|u|4u,x∈ R3,u∈ H1(R3),其中a＞0,b＞0,4＜p＜6,V(x)∈L3/2c(R3)是一个给定的非负函数且满足 lim V(x):=V∞.对V(x)给定适当的假设条件,当ε充分小时,证明了基态解...     

混合幂的素变数丢番图逼近

证明了:如果λ_1,λ_2,λ_3,λ_4是正实数,λ_1/λ_2是无理数和代数数,V是well-spaced序列,δ0,那么对于v∈V,v≤X,ε0,使得|λ_(1p_1~2)+λ_(2p_2~2)+λ_(3p_3~3)+λ_(4p_4~3)-v|v~(-δ)没有素数解p1,p2,p3,p4的v的个数不超过O(X~(20/21+21δ+ε)).     

8-ranks of Class Groups of Some Imaginary Quadratic Number Fields

Let F = Q（√-p1p2） be an imaginary quadratic field with distinct primes p1 = p2 = 1 mod 8 and the Legendre symbol （p1/p2） = 1. Then the 8-rank of the class group of F is equal to 2 if and only Pl if the following conditions hold： （1） The quartic residue symbols （p1/p2）4 = （p2/p1）4 = 1; （2） Either both p1 and p2 are represented by the form a^2 ＋ 32b^2 over Z and p^h2＋（2p1）/4=x^2-2p1y^2,x,y∈Z,or both p1 and p2 are not represented by the form a^2 ＋ 32b^2 over Z and p^h2＋（2p1）/4=ε（2x^2-p1y^2）,x,y∈Z,ε∈{±1},where h＋（2p1） is the narrow class number of Q（√2p1）,Moreover, we also generalize these results.     

Bifurcations of limit cycles from a quintic Hamiltonian system with a heteroclinic cycle

In this paper,we consider Li′enard systems of the form dx/dt=y,dy/dt=x+bx3-x5+ε(α+βx2+γx4)y,where b∈R,0|ε|1,(α,β,γ)∈D∈R3 and D is bounded.We prove that for |b|1(b0) the least upper bound of the number of isolated zeros of the related Abelian integrals I(h)=∮Γh(α+βx2+γx4)ydx is 2(counting the multiplicity) and this upper bound is a sharp one.     

Diophantine方程(a-1)x~2+f(a)=4a~n的一点注记

设a是大于1的正整数,f(a)是a的非负整系数多项式,f(1)=2rp+4,其中r是大于1的正整数,p=2~l-1是Mersenne素数.本文讨论了方程(a-1)x~2+f(a)=4a~n的正整数解(x,n)的有限性,并且证明了:当f(a)=91a+9时,该方程仅当a=5,7和25时分别有解(x,n)=(3,3),(11,3)和(3,4).     

REGULARITY OF H~1 ∩L~(n((γ-1)/(2-γ))) WEAK SOLUTIONS FOR NONLINEAR ELLIPTIC SYSTEMS

Let us consider the following elliptic systems of second order-D_α(A_i~α(x, u, Du))=B_4(x, u, Du), i=1, …, N, x∈Q(?)R~n, n≥3 (1) and supposeⅰ) |A_i~α(x, u, Du)|≤L(1+|Du|);ⅱ) (1+|p|)~(-1)A_i~α(x, u, p)are H(?)lder-continuous functions with some exponent δ on (?)×R~N uniformly with respect to p, i.e.ⅲ) A_i~α(x, u, p) are differentiable function in p with bounded and continuous derivativesⅳ)ⅴ) for all u∈H_(loc)~1(Ω, R~N)∩L~(n(γ-1)/(2-γ))(Ω, R~N), B(x, u, Du)is ineasurable and |B(x, u, p)|≤a(|p|~γ+|u|~τ)+b(x), where 1+2/n<γ<2, τ≤max((n+2)/(n-2), (γ-1)/(2-γ)-ε), (?)ε>0, b(x)∈L2n/(n+2), n~2/(n+2)+e(Ω), (?)ε>0.Remarks. Only bounded open set Q will be considered in this paper; for all p≥1, λ≥0, which is clled a Morrey Space.Let assumptions ⅰ)-ⅳ) hold, Giaquinta and Modica have proved the regularity of both the H~1 weak solutions of (1) under controllable growth condition |B|≤α(|p|~γ+|u|~((n+2)/(n-2))+b, 0<γ≤1+2/n and the H~1∩L~∞ weak solutions of (1) under natural     

B-值随机变量序列大数定律的收敛速度

设{X_n:n≥1}是在某可分Banach空间B上取值的独立随机变量序列,S_n=X_1+…+X_n,n≥1,对某00和>1,定义P_n(ε)=P(‖S_n‖/n~(1/p)≥ε)。本文的目的是研究当n→+∞时,P(ε)→0的速度,在Banach空间上推广了Heyde和Rohatgi的结果;同时,本文还讨论了P_n(ε)→0的速度与S_n/n~(1/p)→0 a.s.的关系问题。     

广义Ramanujan-Nagell方程x~2+D~m=p~n的解数

设a是正整数,D=3a2+1,P=4a2+1,其中p是素数.本文证明了:如果a不是4的倍数,则除了当(D,p)=(4,5)时方程x2+Dm=pn恰有3组正整数解(x,m,n)=(1,1,1),(3,2,2),(11,1,3)以外,该方程恰有2组正整数解(x,m,n)=(a,1,1)和(8a3+3a,1,3).     

Quasi-maximum likelihood estimators in generalized linear models with autoregressive processes

The paper studies a generalized linear model(GLM)y_t = h(x_t~T β) + ε_t,t = l,2,...,n,where ε_1 = η_1,ε_1 =ρε_t +η_t,t = 2,3,...;n,h is a continuous differentiable function,η_t's are independent and identically distributed random errors with zero mean and finite variance σ~2.Firstly,the quasi-maximum likelihood(QML) estimators of β,p and σ~2 are given.Secondly,under mild conditions,the asymptotic properties(including the existence,weak consistency and asymptotic distribution) of the QML estimators are investigated.Lastly,the validity of method is illuminated by a simulation example.     

关于半线性椭圆方程正解的唯一性

本文讨论方程正解的唯一性 作者证明了若f(u)=-u+u~p,则存在ε~*>0,使当10,使当|ε|<ε~*时,方程(Ⅰ)的正解是唯一的。     

Geometric and topological rigidity for compact submanifolds of odd dimension

We investigate rigidity problems for odd-dimensional compact submanifolds.We show that if Mn(n 5)is an odd-dimensional compact submanifold with parallel mean curvature in Sn+p,and if RicM(n-2-1n)(1+H2)and Hδn,whereδn is an explicit positive constant depending only on n,then M is a totally umbilical sphere.Here H is the mean curvature of M.Moreover,we prove that if Mn(n 5)is an odd-dimensional compact submanifold in the space form Fn+p(c)with c 0,and if RicM(n-2-εn)(c+H2),whereεn is an explicit positive constant depending only on n,then M is homeomorphic to a sphere.     

Homogenization of Elliptic Problems with Neumann Boundary Conditions in Non-smooth Domains

We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating and periodic coefficients in Lipschitz and convex domains in R~n. We are able to show that the uniform W~(1,p) estimate of second order elliptic systems holds for 2n/(n+1)-δ p 2n/(n-1)+ δ where δ 0 is independent of ε and the ranges are sharp for n = 2, 3. And for elliptic equations in Lipschitz domains, the W~(1,p) estimate is true for 3/2-δ p 3 + δ if n ≥ 4, similar estimate was extended to convex domains for 1 p ∞.     

广义Ramanujan-Nagell方程x~2+D~m=p~n

设D是大于1的正整数,p是不能整除D的素数.本文证明了:当D=3a~2+1,p=4a~2+1,其中a是正整数时,除了(D,p)=(4,5)这一情况以外,方程x~2+D~m=p~n仅有2组正整数解(x,m,n)=(a,1,1)和(8a~3+3a,1,3).根据上述结果得到了该方程解数的最佳上界.