INVERSE EIGENVALUE PROBLEM OF HERMITIAN GENERALIZED ANTI-HAMILTONIAN MATRICES

§1　IntroductionWe considerthe following inverse eigenvalue problem offinding an n-by-n matrix A∈S such thatAxi =λixi,i =1,2 ,...,m,where S is a given set of n-by-n matrices,x1 ,...,xm(m≤n) are given n-vectors andλ1 ,...,λmare given constants.Let X=(x1 ,...,xm) ,Λ=(λ1 ,λ2 ,...,λm) ,then the above inverse eigenvalue problemcan be written as followsProblem Given X∈Cn×m,Λ=(λ1 ,...,λm) ,find A∈S such thatAX =XΛ,where S is a given matrix set.We also discuss the so-called opti…     

High-dimensional D. H. Lehmer problem over short intervals

Letk be a positive integer and n a nonnegative integer,0 λ1,...,λk+1 ≤ 1 be real numbers and w =(λ1,λ2,...,λk+1).Let q ≥ max{[1/λi ]:1 ≤ i ≤ k + 1} be a positive integer,and a an integer coprime to q.Denote by N(a,k,w,q,n) the 2n-th moment of(b1··· bk c) with b1··· bk c ≡ a(mod q),1 ≤ bi≤λiq(i = 1,...,k),1 ≤ c ≤λk+1 q and 2(b1+ ··· + bk + c).We first use the properties of trigonometric sum and the estimates of n-dimensional Kloosterman sum to give an interesting asymptotic formula for N(a,k,w,q,n),which generalized the result of Zhang.Then we use the properties of character sum and the estimates of Dirichlet L-function to sharpen the result of N(a,k,w,q,n) in the case ofw =(1/2,1/2,...,1/2) and n = 0.In order to show our result is close to the best possible,the mean-square value of N(a,k,q) φk(q)/2k+2and the mean value weighted by the high-dimensional Cochrane sum are studied too.     

带有限位势的非线性Schrdinger方程组的无穷多变号解

本文考虑非线性Schrdinger方程组-?u j+λj(x)u j=k i=1β_(ij) u_i~2 u_j,x∈R~N,u_j(x)→0,当|x|→∞时,j=1,...,k,其中N=2,3,β_(ij)是常数,满足β_(jj)0(j=1,...,k),β_(ij)=β_(ji)0(1≤ij≤k),λ_j(j=1,...,k)是位势函数.首先考虑带强制位势的方程组,利用流不变集方法证明带强制位势的方程组有无穷多变号解;然后在位势λ_j具有一定渐近性质(见正文(V_1)–(V_4))时,通过集中紧性分析,证明带强制位势扰动方程组的解趋于原来有限位势的方程组的解,从而证明原方程组有无穷多变号解.     

二阶非线性偏微分方程组的“拟椭圆”条件和部分正则性

如果对某λ>0、所有有界开域 T■R~n、所有矩阵 A∈M~(n×N)所有φ∈C_0~1(T,R~N),成立,∫λ|Dφ|~2dx≤∫_TA_i~α(A+Dφ)D_αφ~idx,我们就称二阶非线性偏微分方程组(*)-D_aA_i~a(Du)=0,i=1,2,…,N 满足“拟椭圆”条件。这个条件远远弱于通常的椭圆条件。■本文只假定方程组(*)满足“拟椭圆”条件(不需要椭圆条件),我们用 blowup 技巧证明了方程组的部分正则性。     

BOUNDEDNESS OF COMPOSITION OPERATORS ON Dτ SPACES FOR τ>1

In this note, some conditions of composition operators on Dτspaces to be bounded are given by means of Carleson measures and pointwisemultipliers, for some ranges of τ. The authors prove that (i) Let 1<τ n+2 and 2k<τ 2k+1 (or 2k-1<τ 2k) for somepositive integer k. Suppose φ=(φ1,...,φn) be aunivalent mapping from B into itself, denote dμj(l)(z)=R(l)φj(z)2(k-l+2)(1-|z|2)2k-τ+1 dν(z) for l=1,2,...,k+1. If μj(l)φ-1 are (τ-2k+2l-4)-Carleson measures for all l, then the composition operator Cφon Dτis bounded; (ii) Let 1<τ n+2, φ=(φ1,...,φn)be univalent and the Fr echet derivativeof φ-1 be bounded onφ(B). If Rφj∈M(Dτ-2) forall j, then the composition operator Cφ on Dτis bounded; (iii) Let τ>n+2 and φ as in (ii).If φj∈Dτ for all j, then the compositionoperator Cφ on Dτ is bounded.     

Short-time Asymptotics of the Heat Kernel on Bounded Domain with Piecewise Smooth Boundary Conditions and Its Applications to an Ideal Gas

The asymptotic expansion of the heat kernel Θ(t)=sum from ∞to j=1 exp(-tλ_j) where {λ_j}_(j=1)~∞are the eigen-values of the negative Laplacian -Δ_n=-sum from n to k=1((?))~2 in R~n(n=2 or 3) is studied for short-time t for a generalbounded domain Ωwith a smooth boundary (?)Ω.In this paper,we consider the case of a finite number of theDirichlet conditions φ=0 on Γ_i (i=1,...,J) and the Neumann conditions (?)=0 on Γ_i (i=J 1,...,k) andthe Robin conditions ((?) γ_i)φ=0 on Γ_i (i=k 1,...,m) where γ_i are piecewise smooth positive impedancefunctions,such that (?)Ωconsists of a finite number of piecewise smooth components Γ_i(i=1,...,m) where(?)Ω=(?)Γ_i.We construct the required asymptotics in the form of a power series over t.The senior coefficients inthis series are specified as functionals of the geometric shape of the domain Ω.This result is applied to calculatethe one-particle partition function of a“special ideal gas”,i.e.,the set of non-interacting particles set up in abox with Dirichlet,Neumann and Robin boundary conditions for the appropriate wave function.Calculationof the thermodynamic quantities for the ideal gas such as the internal energy,pressure and specific heat revealsthat these quantities alone are incapable of distinguishing between two different shapes of the domain.Thisconclusion seems to be intuitively clear because it is based on a limited information given by a one-particlepartition function;nevertheless,its formal theoretical motivation is of some interest.     

Bounds of the Estrada index of graphs

Let G be a graph of order n and let λ1,λ2,...,λn be its eigenvalues. The Estrada index[2] of G is defined as EE = EE(G) =∑n i=1 eλi. In this paper, new bounds for EE are established, as well as some relations between EE and graph energy E.     

TO SOLVE MATRIX EQUATION sum (A~iXBi=c) BY THE SMITH NORMAL FORM

§ 1　IntroductionLet F be a field,F[λ] be the polynomial ring over F,Fm× n( or Fm× n[λ] ) be the setofall m×n matrices over F( or F[λ] ) .Let M(i) be the ith column of M∈Fm× m[λ] ,i=1 ,...,n.A g-inverse of M∈Fm× n will be denoted by M- and understood as a matrix for whichMM- M=M.In this paper,we discuss the linear matrix equation ki=0Ai XBi =C, ( 1 )where A∈Fm× m,Bi∈Fn× q,i=0 ,1 ,...,k,and C∈Fm× q.Equation( 1 ) is called universally solvable if ithas a solution f…     

混合幂为2,3,a和b的丢番图不等式

运用Davenport-Heilbronn方法证明了如果η是实数,λ1,μ1,μ2,μ3,μ4,θ1,θ2是非零实数,并且不同一符号,且至少一个λ1/μi(i=1,2,3,4)是无理教,假设(i)a=3,3≤b≤11,或者(ii)a=4,4≤b≤5,那么对某些σ=σ(a,b)>0,混合幂为2,3,a和b的丢番图不等式有无穷多正整数解x1,...,x7.     

任意阶Laplace算子的特征值估计

设D为n维Euclid空间Rn的一个有界区域,且0＜λ1≤λ2≤…≤λk≤…是l阶Laplace算子的Dirichlet问题{(-△)lu=λu, 在D中,u=(e)u/(e)n=…=(e)l-1u/(e)nl-1=0,在(e)D上的特征值.得到了该问题用其前k个特征值来估计第(k+1)个特征值λk+1的不等式k∑i=1(λk+1-λi)≤1/n(4l(n+2l-2)]1/2{k∑i=1(λk+1-λi)1/2λil-1/lk∑i=1(λk+1-λi)1/2λi1/l}1/2,此不等式不依赖于区域D.对l≥3,上述不等式比所有已知的结果都要好.陈庆民与杨洪苍考虑了l=2的情形.我们的结果是他们结果的自然推广.当l=1时,我们的不等式蕴含杨洪苍不等式的弱形式.文中还给出了陈和杨的一个断言的直接证明.     

基于几何连续的AT-β-Spline曲线曲面的构造

为了更好地修改给定的样条曲线曲面,构造了满足几何连续的带两类形状参数的代数三角多项式样条曲线曲面,简称为AT-β-Spline.这种代数三角曲线曲面不仅具有普通三角多项式的性质,而且具有全局的和局部的形状可调性.同时还具备较为灵活的连续性.当两类形状参数在给定的范围内任意取值时,这种带两类形状参数的AT-β-Spline曲线满足一阶几何连续性;如果给定两段相邻曲线段中的两类形状参数满足-1≤α≤1,μ_i=λ_(i+1)或μ_i=λ_i=μ_(i+1)=λ_(i+1)时,则带两类形状参数的AT-β-Spline曲线满足C~1∩G~2连续.另外利用奇异混合的思想,构造了满足C~1∩G~2插值AT-β-Spline曲线,解决曲线反求的几何连续性等问题.同时还给出了旋转面的构造,描述了两类形状参数对旋转面的几何外形的影响;当形状参数取特殊值时,这种AT-β-Spline曲线曲面可以精确地表示圆锥曲线曲面.从实验的结果来看,本文构造的AT-β-Spline曲线曲面是实用的有效的.     

离散型p次Dirichlet型第一特征值

设μ=(μ_i)_i≥0为Z_+上的测度且p 1,考虑下述离散型p次Dirichlet型D_p(f)=Σ_(i=0)~∞μ_ib_i(f_i-f_(i+1))(f_i~(p-1)-f_(i+1)~(p-1)),f≥0,其中(b_i)_(i≥0)为Z_+上的正序列.本文旨在给出空间L~p(μ)上p次Dirichlet型D_p(f)所对应的第一特征值λ_(0,p)=inf{D_p(f):‖f‖_p=1,f非负且具有紧支撑}的上下界精细估计.     

幂次为2,3,4,5的素变量非线性型的整数部分

考虑了一个混合幂次为2,3,4,5的素变量非线性型的整数部分表示无穷多素数的问题.运用Davenport-Heilbronn方法证明了:如果λ_1,λ_2,λ_3,λ_4是正实数,至少有一个λ_i/λ_j(1≤ij≤4)是无理数,那么存在无穷多素数p_1,p_2,p_3,p_4,p,使得[λ_1p_1~2+λ_2p_2~3+λ_3p_3~4+λ_4p_4~5]=p.     

关于有限秩的幂零群的自同构

设G是有限秩的幂零群,1=ζ_0Gζ_1G …ζ_cG=G是G的上中心列,End(ζ_iG/ζ_(i-1)G)是Abel群ζ_iG/ζ_(i-1)G的自同态环(1≤i≤c),End(ζ_iG/ζ_(i-1)G)可以自然地作成一个Lie环.α_1,α_2,…,α_n是G的n个自同构,把它们在ζ_iG/ζ_(i-1)G上的诱导自同构分别记为α_(1i),α_(2i),…,α_(ni)(1≤i≤c).如果由α_(1i),α_(2i),…,α_(ni)生成的Lie环End(ζ_iG/ζ_(i-1)G)的Lie子环都是完全可解的,那么α_1,α_2,…,α_n生成的AutG的子群具有良好的幂零性质.考虑G的下中心列,可以得到对偶的结果.     

On Graph-Lagrangians and clique numbers of 3-uniform hypergraphs

The paper explores the connection of Graph-Lagrangians and its maximum cliques for 3-uniform hypergraphs.Motzkin and Straus showed that the Graph-Lagrangian of a graph is the Graph-Lagrangian of its maximum cliques.This connection provided a new proof of Turán classical result on the Turán density of complete graphs.Since then,Graph-Lagrangian has become a useful tool in extremal problems for hypergraphs.Peng and Zhao attempted to explore the relationship between the Graph-Lagrangian of a hypergraph and the order of its maximum cliques for hypergraphs when the number of edges is in certain range.They showed that if G is a 3-uniform graph with m edges containing a clique of order t-1,then λ(G)=λ([t-1]~((3))) provided (t-13)≤m≤(t-13)+_(t-22).They also conjectured:If G is an r-uniform graph with m edges not containing a clique of order t-1,then λ(G)λ([t-1]~((r))) provided (t-1r)≤ m ≤(t-1r)+(t-2r-1).It has been shown that to verify this conjecture for 3-uniform graphs,it is sufficient to verify the conjecture for left-compressed 3-uniform graphs with m=t-13+t-22.Regarding this conjecture,we show: If G is a left-compressed 3-uniform graph on the vertex set [t] with m edges and |[t-1]~((3))\E(G)|=p,then λ(G)λ([t-1]~((3))) provided m=(t-13)+(t-22) and t≥17p/2+11.     

GPSD迭代法和Jacobi迭代法的敛散关系

证明了当Jacobi迭代矩阵B非负时,解线性方程组Ax=b(A为不可约矩阵)的GPSD迭代法(0＜ωi＜Ti≤1,i=1,2,…,n)和Jacobi迭代法同时敛散,给出了其谱半径p(ST,Ω)和ρ(B)之间的关系.     

相依误差广义线性模型的M估计的Bahadur表示

考虑如下广义线性模型y_i=h(x~T_i,β)+e_i=1,2,…,n,其中e_i=G(…,ε_(i-1),ε_i),h是一个连续可导函数,ε_i是独立同分布的随机变量,并且它的期望为0,方差σ~2有限.本文给出了参数β的M估计,并且得到了该估计的Bahadur表示,该结论推广了线性模型的相关结论.应用M估计的Bahadur表示,得到了相依误差的线性回归模型,poisson模型,logistic模型和独立误差的广义线性模型等模型的渐近性质.     

素变量混合幂丢番图逼近

设λ_1,λ_2,λ_3,λ_4为不全为负的非零实数,λ_1/λ_2是无理数和代数数.■是具有良好间隔的序列,δ>0.本文证明了:对于任意ε>0及v∈■,v≤X,使得不等式|λ_1p_1~2+λ_2p_2~2+λ_3p_3~3+λ_4p_4~3-v|相似文献   

三角代数上Jordan高阶导子的刻画

设u=Tri(A,M,B)是含单位元I的三角代数,()={()_n}_(n∈N)是u上一簇线性映射.本文证明了:如果对任意U,V∈u且UV=VU=I,有()_n(UV+VU)=∑_(i+j=n)(()_i(U)_(()_j)(V)+()_i(V)()_j(U)),则()={()_n}_(n∈N)是u上高阶导子.作为应用,得到了套代数上Jordan高阶导子的一个刻画.     

素环上强保持2-Jordan乘积的映射

对于给定的正整数k≥1,环R上的元x,y的k-Jordan乘积定义为{x,y}_k={{x,y}_(k-1),y}_1,其中{x,y}_0=x,{x,y}_1=xy+yx.假设R是包含有单位元与一非平凡幂等元的素环.本文证明了R上的满射f满足{f(x),f(y)}2={x,y}_2对所有x,y∈R成立当且仅当存在λ∈l(R的可扩展中心)且λ~3=1,使得下列之一成立:(1)若R的特征不为2,则f(x)=λx对所有x∈R成立;(2)若R的特征为2,则f(x)=λx+μ(x)对所有x∈R成立,其中μ:R→l是一个映射.作为应用,得到了因子von Neumann代数上保持上述性质映射的结构.