Gaussian fluctuations of eigenvalues in log-gas ensemble: Bulk case I

We study the central limit theorem of the k-th eigenvalue of a random matrix in the log-gas ensemble with an external potential V = q2mx2 m. More precisely, let Pn(d H) = Cne-nTrV(H)dH be the distribution of n × n Hermitian random matrices, ρV(x)dx the equilibrium measure, where Cnis a normalization constant, V(x) = q2mx2m with q2m=Γ(m)Γ(12)/Γ(2m+1/2), and m ≥ 1. Let x1 ≤···≤ xnbe the eigenvalues of H. Let k := k(n) be such that k(n)/n∈ [a, 1- a] for n large enough, where a ∈(0,12).Define G(s) :=∫s-1ρV(x)dx,- 1 ≤ s ≤ 1,and set t := G-1(k/n). We prove that, as n →∞,xk- t log n1/2 2π21/2nρV(t)→ N(0, 1)in distribution. Multi-dimensional central limit theorem is also proved. Our results can be viewed as natural extensions of the bulk central limit theorems for GUE ensemble established by J. Gustavsson in 2005.     

A Bishop–Phelps–Bollobás Theorem for Asplund Operators

By characterizing Asplund operators through Fréchet differentiability property of convex functions, we show the following Bishop–Phelps–Bollobás theorem: Suppose that X is a Banach space,T : X → C(K) is an Asplund operator with ║T║= 1, and that x_0 ∈ S_X, 0 ε satisfy ║T(x_0)║ 1-ε~2/2.Then there exist x_ε∈ S_X and an Asplund operator S : X → C(K) of norm one so that ║S(x_ε)║ = 1, x_0-x_ε ε and ║T-S║ ε.Making use of this theorem, we further show a dual version of Bishop–Phelps–Bollobás property for a strong Radon–Nikodym operator T : ?_1 → Y of norm one: Suppose that y_0~*∈ S_(Y~*), ε≥ 0 satisfy T~*(y_0~*) 1-ε~2/2. Then there exist y_ε~*∈ S_(Y~*), x_ε∈(±e_n), y_ε∈ S_Y, and a strong Radon–Nikodym operator S : ?_1 → Y of norm one so that (ⅰ)║S(x_ε)║= 1;(ⅱ) S(x_ε) = y_ε;(ⅲ)║T-S║ ε;(ⅳ)║S~*(y_ε~*)║=y_ε~*, y_ε= 1;(ⅴ)║y_0~*-y_ε~*║ ε and (ⅵ)║T~*-S~*║ ε,where(e_n) denotes the standard unit vector basis of ?_1.     

非齐型空间上分数型Marcinkiewicz积分算子的加权估计

令（X，d，μ）为满足所谓上倍双倍条件和几何双倍条件的度量测度空间.设M_β，ρ，q为（X，d，μ）上的分数型Marcinkiewicz积分算子.在本文中，作者证明了若β ∈[0，∞），ρ ∈（0，∞），q ∈（1，∞）且M_β，ρ，q在L²（μ）上有界，则M_β，ρ，q是从加权Lebesgue空间L^p（w）到加权弱Lebesgue空间L^p，∞（w）上有界和从加权Morrey空间L^p，κ，η（ω）到加权弱Morrey空间WL^p，κ，η（ω）上有界.     

Multiple Positive Solutions for a Nonlinear Elliptic Equation Involving Hardy–Sobolev–Maz'ya Term

In this paper, we study the existence and nonexistence of multiple positive solutions for the following problem involving Hardy–Sobolev–Maz'ya term:-Δu- λu/|y|2=|u|pt-1u/|y|t+ μf(x), x ∈Ω,where Ω is a bounded domain in RN(N ≥ 2), 0 ∈Ω, x =(y, z) ∈ Rk× RN-kand pt =N +2-2t N-2(0 ≤ t ≤2). For f(x) ∈ C1(Ω)\{0}, we show that there exists a constant μ* 0 such that the problem possessesat least two positive solutions if μ∈(0, μ*) and at least one positive solution if μ = μ*. Furthermore,there are no positive solutions if μ∈(μ*, +∞).     

Existence of semiclassical states for a quasilinear Schrödinger equation on ℝ<Superscript><Emphasis Type="Italic">N</Emphasis></Superscript> with exponential critical growth

We study a quasilinear Schrödinger equation -ε^NΔ_Nu+V(x)|u|^N-2u=Q(x)f(u) in R^N, 01,N(R^N),u(x)0, where V, Q are two continuous real functions on R^N and ε>0 is a real parameter. Assume that the nonlinearity f is of exponential critical growth in the sense of Trudinger-Moser inequality, we are able to establish the existence and concentration of the semiclassical solutions by variational methods.     

Hom-derivations in C*-ternary Algebras

In this paper, we introduce and solve the following additive (ρ₁, ρ₂)-functional inequalities ||f(x+y+z)-f(x)-f(y)-f(z)|| ≤ ||ρ₁(f(x+z)-f(x)-f(z))||+||ρ₂(f(y+z)-f(y)-f(z))||, where ρ₁ and ρ₂ are fixed nonzero complex numbers with|ρ₁|+|ρ₂|< 2. Using the fixed point method and the direct method, we prove the Hyers-Ulam stability of the above additive (ρ₁, ρ₂)-functional inequality in complex Banach spaces. Furthermore, we prove the Hyers-Ulam stability of hom-derivations in C^*-ternary algebras.     

小麦茎秆性状与单穗重的相关分析与通径分析

利用2010-2011连续两年河南科技学院小麦试验田所得的试验资料,对6个不同小麦品种的茎秆性状与单穗重进行相关分析与通径分析.相关分析结果表明,所考查的茎秆性状与单穗重的相关程度依次为:茎重>茎粗>株高>茎长>壁厚;茎重与单穗重(r_(5y)=0.8213~(**))和茎粗与茎重(r_(25)=0.7218~(**))、茎长与株高(r_(13)=0.9582~(**))的相关分别呈极显著,茎粗与单穗重(r_(2y)=0.5990~*)、茎重与株高(r_(13)=0.6550~*)均达显著水平.通径分析结果显示,考查的茎部性状对单穗重的直接效应(绝对值)依次为:茎重>株高>茎长>壁厚>茎粗.茎重的直接正效应(p_(x_5→x_5→y)=1.3986~(**))最大,茎长(p_(x_1→x_1→y)=0.6297~*)次之,壁厚(p_(x_3→x_3→y)=0.2323)居三,株高的直接负效应(p_(x_3→x_3→y)=-1.1617~(**))极显著;5个茎部性状的决策系数依次为:茎重(0.3414)>壁厚(-0.0030)>茎粗(-0.1959)>茎长(一0.2063)>株高(一1.9800).据此明确了茎重、壁厚为提高小麦单穗重的主选因素,株高是主控因素.在注重对基部节间短、粗、茎壁厚性状选择的同时,适当控制株高,是选育高产抗倒小麦品种的有效途径.     

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   

具有蝴蝶型相图的三次近Hamilton系统Abelian积分的零点个数

考虑了如下近Hamilton系统{x=2y(ax~2+2cy~2)+εf(x,y),y=2x(1-2bx~2)+εg(x,y),其中a0,c0,4bca~2,0|ε|■1,且f(x,y)和g(x,y)是关于x和y的3次多项式.得到了其相应Abelian积分孤立零点个数的上界.     

奇异摄动问题在Bakhvalov-Shishkin 网格上的流线扩散有限元逼近

本文采用线性插值的流线扩散有限元在Bakhvalov-Shishkin网格上求解一维对流扩散型的奇异摄动问题. 在ε ≤ N^-1的前提下,可以得到,关于扰动参数ε 是一致收敛的. 在离散的SD范数下,其u-u_I的误差阶提高到N^-2,u-u_h的误差阶达到N^-2（lnN）^0.5. 最后,通过数值算例,验证了理论分析.     

一类带约束的二维弱奇异积分方程的解*

本文找出二维弱奇异第一类积分方程作用着约束方程的解p.p=p(r,θ)={2/[π2k(φ₀]}√F(r,θ)-c_*(0≤r≤r_*)其中是(s,φ)原点在M(r,θ)的局部极坐标,(r,θ)是原点在O(0,0)的总体极坐标:k和F是给出的连续函数;φ₀是一常数;F(_*,θ)=c_*(常数)是研究域Q的边界围线。所用方法可推广到三维情形。     

核范数和谱范数下广义Sylvester方程最小二乘问题的有效算法

本文从数值角度讨论Schatten q-范数下的广义Sylvester方程约束最小二乘问题min x∈s‖N∑i=1A_iXB_i—C‖_q,其中S为闭凸约束集合,Schatten q-范数定义为‖M‖_q~q=∑_(i=1)~nσ_i~q(M),其中σ_i(M)为M∈R~(n×n)的奇异值.该问题的几类特殊情形在图像处理、控制论等领域有广泛的应用.q=2即Frobenius范数下该问题已被充分研究,故本文着重讨论q=1,+∞,即核范数和谱范数下该问题的数值求解.采用的数值方法是非精确标准容易执行的部分非精确交替方向法,并结合奇异值阈值算法,Moreau-Yosida正则化算法,谱投影算法和LSQR算法等求解相应子问题.给出算法的收敛性证明,并用数值算例验证其高效可行性.     

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.     

von Neumann代数上的局部可乘Lie n-导子

A₁，…，A_n的（n-1）-换位子记为p_n（A₁，…，A_n）.令M是von Neumann代数，n ≥ 2是任意正整数，L：M → M是一个映射.本文证明了，若M不含I₁型中心直和项，且L满足L（p_n（A₁，…，A_n））=∑_k=1ⁿ p_n（A₁，…，A_k-1，L（A_k），A_k+1，…，A_n）对所有满足条件A₁A₂=0的A₁，A₂，…，A_n ∈ M成立，则L（A）=φ（A）+f（A）对所有A ∈ M成立，其中φ：M → M和f：M → Z（M）（M的中心）是两个映射，且满足φ在P_iMP_j上是可加导子，f（p_n（A₁，A₂，…，A_n））=0对所有满足A₁A₂=0的A₁，A₂，…，A_n ∈ P_iMP_j成立（1 ≤ i，j ≤ 2），P₁ ∈ M是core-free投影，P₂=I-P₁；若M还是因子且n ≥ 3，则L满足条件L（p_n（A₁，A₂，…，A_n））=∑_k=1ⁿ p_n（A₁，…，A_k-1，L（A_k），A_k+1，…，A_n）对所有满足A₁A₂A₁=0的A₁，A₂，…，A_n ∈ M成立当且仅当L（A）=φ（A）+h（A）I对所有A ∈ M成立，其中φ是M上的可加导子，h是M上的泛函且满足h（p_n（A₁，A₂，…，A_n））=0对所有满足条件A₁A₂A₁=0的A₁，A₂，…，A_n ∈ M成立.     

Jacobian-determinant equation in the critical Besov spaces

Let Ω be a bounded domain in R~n with smooth boundary. Here we consider the following Jacobian-determinant equation det u(x)=f(x),x∈Ω;u(x)=x,x∈?Ω where f is a function on Ω with min_Ω f = δ 0 and Ωf(x)dx = |Ω|. We prove that if f ∈B_(p1)~(np)(Ω) for some p∈(n,∞), then there exists a solution u ∈ B_(p1)~(np+1)(Ω)C~1(Ω) to this equation. On the other hand, we give a simple example such that u ∈ C_0~1(R~2, R~2) while detu does not lie in B_(p1)~(2p)(R~2) for any p∞.     

Nodal Sets and Doubling Conditions in Elliptic Homogenization

This paper is concerned with uniform measure estimates for nodal sets of solutions in elliptic homogenization. We consider a family of second-order elliptic operators {L_ε} in divergence form with rapidly oscillating and periodic coefficients. We show that the(d-1)-dimensional Hausdorff measures of the nodal sets of solutions to L_ε(u_ε) = 0 in a ball in Rdare bounded uniformly in ε 0. The proof relies on a uniform doubling condition and approximation of u_ε by solutions of the homogenized equation.     

The Degree of Proper Holomorphic MappingsBetween Special Domains in C^n

Let Gi be closed Lie groups of U (n), Ω i be bounded Gi-invariant domains in C^n which contains 0, and O(C^n)^Gi = C, for i = 1, 2. It is known that if f : Ω 1 → Ω 2 is a proper holomorphic mapping, and f^-1{0} = {0}, then f is a polynomial mapping. In this paper, we provide an upper bound for the degree of such a polynomial mapping using the multiplicity of f .     

Comparison and Extremal Results on Three Eccentricity-based Invariants of Graphs

The first and second Zagreb eccentricity indices of graph G are defined as:E1(G)=∑(vi)∈V(G)εG(vi)~2,E2(G)=∑(vivj)∈E(G)εG(vi)εG(vj)whereεG(vi)denotes the eccentricity of vertex vi in G.The eccentric complexity C(ec)(G)of G is the number of different eccentricities of vertices in G.In this paper we present some results on the comparison between E1(G)/n and E2(G)/m for any connected graphs G of order n with m edges,including general graphs and the graphs with given C(ec).Moreover,a Nordhaus-Gaddum type result C(ec)(G)+C(ec)(■)is determined with extremal graphs at which the upper and lower bounds are attained respectively.     

Parameter Estimation for the Discretely Observed Vasicek Model with Small Fractional Lévy Noise

The statistical inference of the Vasicek model driven by small Levy process has a long history.In this paper,we consider the problem of parameter estimation for Vasicek model dX_t=(μ-θX_t)dt+εdL_t^d,t∈[0,1],X_0=x_0,driven by small fractional Lévy noise with the known parameter d less than one half,based on discrete high-frequency observations at regularly spaced time points{t_i=i/n,i=1,2,...,n}.For the general case and the null recurrent case,the consistency as well as the asymptotic behavior of least squares estimation of unknown parametersμandθhave been established as small dispersion coefficientε→0 and large sample size n→∞simultaneously.     

Trivial and simple spectrum for SL(d, ℝ) cocycles with free base and fiber dynamics

Let AC_D(M,SL(d,R)) denote the pairs (f,A) so that f ∈ A ⊂ Diff¹(M) is a C¹-Anosov transitive diffeomorphisms and A is an SL(d,R) cocycle dominated with respect to f. We prove that open and densely in AC_D(M,SL(d,R)), in appropriate topologies, the pair (f,A) has simple spectrum with respect to the unique maximal entropy measure μ_f. Then, we prove prevalence of trivial spectrum near the dynamical cocycle of an area-preserving map and also for generic cocycles in AutLeb(M) × L^p(M,SL(d,R)).