LIMITING BEHAVIOR OF UNIFORM RECURSIVE TREES

The authors consider the limiting behavior of various branches in a uniform recursive tree with size growing to infinity.The limiting distribution ofζ_(n,m),the number of branches with size m in a uniform recursive tree of order n,converges weakly to a Poisson distribution with parameter 1/m with convergence of all moments.The size of any large branch tends to infinity almost surely.     

Bifurcation and Isochronicity at Infinity in a Class of Cubic Polynomial Vector Fields

In this paper, we study the appearance of limit cycles from the equator and isochronicity of infinity in polynomial vector fields with no singular points at infinity. We give a recursive formula to compute the singular point quantities of a class of cubic polynomial systems, which is used to calculate the first seven singular point quantities. Further, we prove that such a cubic vector field can have maximal seven limit cycles in the neighborhood of infinity. We actually and construct a system that has seven limit cycles. The positions of these limit cycles can be given exactly without constructing the Poincare cycle fields. The technique employed in this work is essentially different from the previously widely used ones. Finally, the isochronous center conditions at infinity are given.     

On the Chromatic Numbers of Random Hypergraphs

Doklady Mathematics - The asymptotic behavior of the chromatic number of the binomial random hypergraph $$H(n,k,p)$$ is studied in the case when $$k \geqslant 4$$ is fixed, n tends to infinity, and...     

On the Positivity of Scattering Operators for Poincaré-Einstein Manifolds

In this paper, we mainly study the scattering operators for a Poincaré-Einstein manifold $(X^{n+1}, g_+)$, which define the fractional GJMS operators $P_{2\gamma}$ of order $2\gamma$ for $0<\gamma<\frac{n}{2}$ for the conformal infinity $(M, [g])$. We generalise Guillarmou-Qing's positivity results in [8] to the higher order case. Namely, if $(X^{n+1}, g_+)$ $(n\geq 5)$ is a hyperbolic Poincaré-Einstein manifold and there exists a smooth representative $g$ for the conformal infinity such that the scalar curvature $R_g$ is a positive constant and $Q_4$ is semi-positive on $(M, g)$, then $P_{2\gamma}$ is positive for $\gamma\in [1,2]$ and the first real scattering pole is less than $\frac{n}{2}-2$.     

一个柔性边界条件约束下的随机变量的分布

讨论n个受柔性边界条件约束的随机变量的概率分布．理论解显示其概率密度函数随变量值增大而减小,当n趋於无穷大时收敛于Delta函数．在有序统计的理论框架下,同时得到最小值分布的解析解．     

Resampling student'st-type statistics

The present paper establishes conditional and unconditional central limit theorems for various resampling procedures for thet-statistic. The results work under fairly general conditions and the underlying random variables need not to be independent. Specific examples are then them(n) (double) bootstrap out ofk(n) observations, the Bayesian bootstrap and two-samplet-type permutation statistics. In case whenm(n)/k(n) is bounded away from zero and infinity necessary and sufficient conditions for the conditional central limit law of the bootstrapt-statistics are established. For high resampling intensity whenm(n)/k(n) tends to infinity the following general result is obtained. Without further other assumptions the bootstrap makes the resampledt-statistic automatically normal. The results are based on a general conditional limit theorem for weighted resampling statistics which is of own interest.     

Spectrum and states of the bcs hamiltonian in a finite domain. I. Spectrum

The BCS Hamiltonian in a finite cube with periodic boundary condition is considered. The special subspace of pairs of particles with opposite momenta and spin is introduced. It is proved that, in this subspace, the spectrum of the BCS Hamiltonian is defined exactly for one pair, and for n pairs the spectrum is defined through the eigenvalues of one pair and a term that tends to zero as the volume of the cube tends to infinity. On the subspace of pairs, the BCS Hamiltonian can be represented as a sum of two operators. One of them describes the spectra of noninteracting pairs and the other one describes the interaction between pairs that tends to zero as the volume of the cube tends to infinity. It is proved that, on the subspace of pairs, as the volume of the cube tends to infinity, the BCS Hamiltonian tends to the approximating Hamiltonian, which is a quadratic form with respect to the operators of creation and annihilation.     

On the rate of decay of concentration functions of <Emphasis Type="Italic">n</Emphasis>-fold convolutions of probability distributions

Concentration functions of n-fold convolutions of probability distributions is shown to exhibit the following behavior. Let φ(n) be an arbitrary sequence tending to infinity as n tends to infinity, and ψ(x) be an arbitrary function tending to infinity as x tends to infinity. Then there exists a probability distribution F of a random variable X such that the mathematical expectation E _ψ(|X|) is infinite and, moreover, the upper limit of the sequence \(\sqrt n \phi \left( n \right)Q_n\) is equal to infinity, where Q _n is the maximal atom of the n-fold convolution of distribution F. Thus, no infinity conditions imposed on the moments can force the concentration functions of n-fold convolutions decay essentially faster than o(n ^?1/2).     

关于随机环境中分支过程物种总体的一个注记

本文研究随机环境中分支过程直流到n代的物种总体的渐近行为问题.利用概率生成函数的方法,得到了其期望和方差的具体表达式,并且在n趋于无穷大时,给出了灭绝概率和全部物种个数的生成函数之间的关系式.     

相伴随机变量序列的泛函型几乎处处中心极限定理

对于均值为零的平稳相伴随机变量序列,首先证明了在L(n)=EX_1~2 2 sum from n to j=2 Cov(X_1,X_j)是一个缓变函数的条件下的泛函型几乎处处中心极限定理.另外还给出了正则化部分和函数的对数平均几乎处处收敛性.     

关于Smarandache对偶函数

定义Smarandache对偶函数S*(n)为最大的正整数m使得m!|n.定义另一种双阶乘函数S**(n)为最大的正整数2m-1使得(2m-1)!!|n,其中2 n;且当2|n时,为最大的正整数2m使得(2m)!!|n.本文的主要目的是利用初等方法研究一个包含S**(n)的无穷级数的收敛性,并给出一个有趣的恒等式.     

一类非牛顿流体模型解的渐近性态

本文主要讨论一类带p(1 2n/(n 2)■p＜3)幂增长耗散位势的非牛顿流体模型解的渐近性态,利用改进的Fourier分解方法,证明了其解在L~2范数下衰减率为(1 t)~(-n/4)．     

Uniform decay rates for nonlinear viscoelastic Marguerre-von Kármán equations

In this paper, we consider a model of nonlinear viscoelastic shallow shell that is referred to as the full Marguerre-von Kármán under the presence of long-time memory. We show that the energy functional associated with the system decays exponentially to zero as time goes to infinity.     

两端固定的奇异梁方程的多重正解

设 n 是一个任意的自然数. 证明了一个两端固定的奇异梁方程的 n 个正解的存在性, 其中非线性项是一个Carathéodory 函数. 主要工具是涉及非线性项的高度函数与锥压缩锥拉伸型的 Krasnoselskii不动点定理. 进一步的研究表明,如果非线性项在零点和无穷远处的增长极限均为无界函数, 该方程仍可能具有正解.     

Weak Convergence of a Planar Random Evolution to the Wiener Process

The weak convergence of the distributions of a symmetrical random evolution in a plane controlled by a continuous-time homogeneous Markov chain with n, n3, states to the distribution of a two-dimensional Brownian motion, as the intensity of transitions tends to infinity, is proved.     

Sparre Andersen风险模型的破产问题(英)

本文主要研究了一类Sparre Andersen模型,其索赔时间间隔的分布为指数分布与Erlang(n) 分布的混合.得到了当初始资金u趋于无穷大时,破产概率ψ(u)的确切表达式和渐近表达式.     

Selection of the number of regression variables; A minimax choice of generalized FPE

Summary A generalized Final Prediction Error (FPE_α)_ criterion is considered. Based onn observations, the numberk of regression variables is selected from a given range 0≦k≦K, so as to minimize . It is shown that if α tends to infinity withn, the selection is consistent but the maximum of the mean squared error of estimates of parameters diverges to infinity with the same order of divergence as that of α. A meaningful minimax choice of α exists for a regret type mean squared error, while for simple mean squared error it is trivially 0. The minimax regret choice of α converges to a constant, approximately 3.5 forK≧8 ifn−K increases simultaneously withn, otherwise it diverges to infinity withn.     

一类非牛顿流体模型解的渐近性态(英)

本文主要讨论一类带 $p \,\,( 1+\frac{2n}{n+2} \leq p<3 )\,$ 幂增长耗散位势的非牛顿流体模型解的渐近性态, 利用改进的 Fourier分解方法, 证明了其解在$L^2$ 范数下衰减率为 $(1+t)^{-\frac{n}{4}}$.     

一类共形不变摄动积分方程正解的存在性

本文讨论了一类共形不变摄动积分方程正解的存在性. 我们证明了:当参数对(p, q) 属于集合(-n, 0) × (0,∞) 且pq + p + 2n = 0 时, 对应摄动积分方程存在正解; 而当参数对(p, q) 属于集合(0,∞)×(-∞, 0) 也满足pq +p+2n = 0 时, 摄动积分方程不存在非负解. 这与原共形不变积分方程有着本质的不同, 此结果隐含着这类积分方程正解的存在性取决于解在无穷远处的性态.