杭州大学一九八三年攻读硕士学位研究生入学考试试题

1.给定函数f∈C~2[a,b]和分划a=x_0相似文献   

Multivariate discriminant and iterated resultant

In this paper,we study the relationship between iterated resultant and multivariate discriminant.We show that,for generic form f(x_n) with even degree d,if the polynomial is squarefreed after each iteration,the multivariate discriminant △(f) is a factor of the squarefreed iterated resultant.In fact,we find a factor Hp(f,[x_1,...,x_n]) of the squarefreed iterated resultant,and prove that the multivariate discriminant △(f) is a factor of Hp(f,[x_1,...,x_n]).Moreover,we conjecture that Hp(f,[x_1,...,x_n]) = △(f) holds for generic form/,and show that it is true for generic trivariate form f(x,y,z).     

Nonexistence and symmetry of solutions to some fractional laplacian equations in the upper half space

In this article, we consider the fractional Laplacian equation(-△)~(α/2)u = K(x)f(u), x ∈ R_+~n,u ≡ 0, x/∈R_+~n,where 0 α 2, R_+~n:= {x =(x_1, x_2, ···, x_n)|x n 0}. When K is strictly decreasing with respect to |x′|, the symmetry of positive solutions is proved, where x′=(x_1, x_2, ···, x_(n-1)) ∈R~(n-1). When K is strictly increasing with respect to x n or only depend on x n, the nonexistence of positive solutions is obtained.     

Multiple solutions for weighted nonlinear elliptic system involving critical exponents

In this paper,by using the idea of category,we investigate how the shape of the graph of h(x)affects the number of positive solutions to the following weighted nonlinear elliptic system:-div(|x|-2au)-μu|x|2(a+1)=αα+βh(x)|u|α-2|v|βu|x|b2*(a,b)+λK1(x)|u|q-2u,in,-div(|x|-2av)-μv|x|2(a+1)=βα+βh(x)|u|α|v|β-2v|x|b2*(a,b)+σK2(x)|v|q-2v,in,u=v=0,on,where 0∈is a smooth bounded domain in RN(N 3),λ,σ0 are parameters,0μμa(N-2-2a2)2;h(x),K1(x)and K2(x)are positive continuous functions in,1 q2,α,β1 andα+β=2*(a,b)(2*(a,b)2N N-2(1+a-b),is critical Sobolev-Hardy exponent).We prove that the system has at least k nontrivial nonnegative solutions when the pair of the parameters(λ,σ)belongs to a certain subset of R2.     

Existence of Nonnegative Solutions for a Class of Systems Involving Fractional(p,q)-Laplacian Operators

The authors study the following Dirichlet problem of a system involving fractional(p, q)-Laplacian operators:{(-△)_p~su=λa(x)|u|+~(p-2)u+λb(x)|u|~(α-2)|u|~βu+μ(x)/αδ|u|~(γ-2)|v|~δu in Ω,(-△)_p~su=λc(x)|v|+~(q-2)v+λb(x)|u|~α|u|~(β-2)v+μ(x)/βγ|u|~γ|v|~(δ-2)v in Ω,u=v=0 on R~N\Ω where λ 0 is a real parameter, ? is a bounded domain in RN, with boundary ?? Lipschitz continuous, s ∈(0, 1), 1 p ≤ q ∞, sq N, while(-?)s pu is the fractional p-Laplacian operator of u and, similarly,(-?)s qv is the fractional q-Laplacian operator of v. Since possibly p = q, the classical definitions of the Nehari manifold for systems and of the Fibering mapping are not suitable. In this paper, the authors modify these definitions to solve the Dirichlet problem above. Then, by virtue of the properties of the first eigenvalueλ_1 for a related system, they prove that there exists a positive solution for the problem when λ λ_1 by the modified definitions. Moreover, the authors obtain the bifurcation property when λ→λ_1~-. Finally, thanks to the Picone identity, a nonexistence result is also obtained when λ≥λ_1.     

等距节点三次插值样条导数误差的最佳估计

设是[0,1]上的均匀分划。s(x)是插值于F(x)的Ⅰ型三次插值样条,即满足(ⅰ)s(x)∈C~2[0,1];(ⅱ)s(x)在每一子区间上是三次多项式;(ⅲ) s(x_i)=f(x_i);(ⅳ) s′(x_0)=f′(x_0),s′(x_n)=f′(x_n)。 C.A.Hall与W.W.Meyer研究了最佳误差界他们得到了c_1=1/24。在本文中,我们求得了。     

EXISTENCE OF SOLUTIONS TO THE PARABOLIC EQUATION WITH A SINGULAR POTENTIAL OF THE SOBOLEV-HARDY TYPE

We study the existence of solutions to the following parabolic equation{ut-△pu=λ/|x|s|u|q-2u,(x,t)∈Ω×(0,∞),u(x,0)=f(x),x∈Ω,u(x,t)=0,(x,t)∈Ω×(0,∞),(P)}where-△pu ≡-div(|▽u|p-2▽u),1相似文献   

实轴上Hermite插值算子的收敛性

The present paper investigates the convergence of Hermite interpolation operators on the real line.The main result is: Given 0 <δ0 < 1/2,0 < ε0 < 1.Let f ∈ C(-∞,∞) satisfy |yk| = O(e(1/2-δ0)x2k) and |f(x)| = O(e(1-ε0)x2).Then for any given point x ∈R,we have limn→∞ Hn(f,x) = f(x).     

计算方程重根及其重数的算法

给出一种计算方程重根及重数的迭代算法,分别具有平方收敛和线性收敛.(i)迭代:x_(n+1)=x_n-f x_n (f'(x_n))/((f'(x_n))~2-(f(x_n)f~n(x_n)),m_n=((f'(x_n)))~2/((f'(x_n))~2-f(xn_)f″(x_n)),n=0,1,2,…,重数m≈mn;(ii)加速迭代:x_(n+1)=x_n-(f~((m-1))(x_n))/(f(~m)(x_n)).     

A Remark on the Generalized Roper-Suffridge Extension Operator for Spirallike Mappings of Type β and Order α

Suppose f is a spirallike function of type β and order α on the unit disk D.let Ωn,p1,p2,…,pn={z=(z1,z2,…,zn)'∈Cn:nΣj=1|zj|pj<1},where 1≤p1≤2,pj≥1,j=2,…,n,are real numbers. In this paper,we will prove that Фn,β2,у2…βn,уn(f)(z)=(f(z1),(f(z1)z1)β2(f'(z1))у2z2,…,(f(z1)z1)βn(f'(z1))уnzn)' Preserves spirallikeness of type β and order α on Ωn,p1,p2,…pn.     

形如2kp+1的数的素性检验

本文初步探讨了如何快速检验一个大数n是素数(这里n-1含有大的素因子)的算法问题以及如何生成一个大素数p使得p-1有大的素因子q的算法问题.我们给出了形如n=2kp+1的数的素性检验的多项式时间算法,这里p是一个给定的大素数,k是正整数满足22k＜2kp.该算法的计算量为O(log32n).然后我们给出了生成一个大素数p使得p-1有大的素因子q的算法,其中q满足q＞(p-1)/log2(p-1).特别地,我们给出了判定并生成一个安全素数p的算法.     

平方函数算子的弱1—1有界性

利用非紧一秩对称空间上热核的表达式和Ｓｔｒｏｍｂｅｒｇ定理，我们在本文中得到了这种空间上Ｌｉｔｔｌｅｗｏｏｄ－Ｐａｌｅｙ平方函数算子的弱１－１有界性。     

Grünwald插值算子的L_1收敛速度

先给出了以第二类Ｔｃｈｅｂｙｃｈｅｆ多项式的零点为插值结点组的Ｇｒｕｎｗａｌｄ插值多项式于Ｌ１下的收敛速度，然后给出了一种修改的Ｇｒｕｎｗａｌｄ插值多项式及其于Ｌ１下的收敛速度．     

一类线性正算子对多元函数的点态逼近度

本文将所谓的ＳＢＫ算子推广为更为一般的多项式算子，研究了它对所谓Ｂ－有界变差的多元函数的点态逼近，改进并推广了文［６］和文［７］的结果。     

不含 l_1-Copy 的空间的鞅特征

自从 Jamesc 将 Asplund 空间与不含 l_1-Copy 的空间明确地区分之后,划分两种空间界限的问题引起了广泛的兴趣。特别地,Riddle,Uhl 利用向量值鞅的某些性质很好地给出了两者之间的界限。本文在[1]的基础上,应用鞅和鞅型序列的某些性质进一步刻画了不 l_1-Copy 的空间的特征,从某些方面划分了 Asplund 空间与不含 l_1-Copy空间的界限(本文略去了 Asplund 空间情况的相应定理)。     

A NEW CHARACTERIZATION OF NONDETERMINIST1CALLY RECOGNIZABLE FAMILIES OF LANGUAGES

In this paper, the author establishes the concepts of relative self-compatibility and relative finite derivability of languages and obtains the relation between self-compatibility and relative self-compatibility. From this it is proved that a family of languages is nondeterministically recognizable if and only if it is relatively self-compatible and relatively finitely derivable to some set of families of langugaes.     

实二次域上理想类的zeta-函数在-1处的值

用连分数给出了实二次域理想类的zeta-函数-1处值的一个具体的计算公式.     

实二次域上理想类的zeta-函数在-1处的值

用连分数给出了实二次域理想类的zeta-函数-1处值的一个具体的计算公式.     

关于半完全环的K1群

本文研究了半完全环的K1群. 利用半局部环的K1群的已知结果和半完全环的构造, 证明了半完全环R的K1群是R的一些子环的K1群的直和与R的单位群的一个子群的商群.     

正规实型对称空间上平方函数算子的弱1——1有界性

利用正规实型对称空间上热核的一个上界估计及非紧致Riemann对称空间上热核 的一个梯度估计,证明了正规实型对称空间上的Littlewood-Paley平方函数算子的弱1-1 有界性.