Towards the generalized purely wild inertia conjecture for product of alternating and symmetric groups

We obtain new evidence for the Purely Wild Inertia Conjecture posed by Abhyankar and for its generalization. We show that this generalized conjecture is true for any product of simple Alternating groups in odd characteristics, and for any product of certain Symmetric or Alternating groups in characteristic two. We also obtain important results towards the realization of the inertia groups which can be applied to more general set up. We further show that the Purely Wild Inertia Conjecture is true for any product of perfect quasi p-groups (groups generated by their Sylow p-subgroups) if the conjecture is established for individual groups.     

On the effect of perturbation of conditional probabilities in total variation

A celebrated result by A. Ionescu Tulcea provides a construction of a probability measure on a product space given a sequence of regular conditional probabilities. We study how the perturbations of the latter in the total variation metric affect the resulting product probability measure.     

The Kullback-Leibler risk of the Stein estimator and the conditional MLE

The decomposition of the Kullback-Leibler risk of the maximum likelihood estimator (MLE) is discussed in relation to the Stein estimator and the conditional MLE. A notable correspondence between the decomposition in terms of the Stein estimator and that in terms of the conditional MLE is observed. This decomposition reflects that of the expected log-likelihood ratio. Accordingly, it is concluded that these modified estimators reduce the risk by reducing the expected log-likelihood ratio. The empirical Bayes method is discussed from this point of view.     

Arnold’s web and diffusion in the Stark-Quadratic-Zeeman problem

The Arnold web and the Arnold diffusion arise when an integrable Hamiltonian system is slightly perturbed: the first concerns the peculiar topology characterizing the set of the resonance lines in phase space, the latter the extremaly slow motion (if any) along these lines. While Arnold has proved the possibility of diffusion, it is still unknown if the phenomenon is generic in realistic physical systems. The system we consider is the Hydrogen atom (or Kepler problem) subject to the combined action of a constant electric and magnetic field, which is known as Stark-Zeeman problem. We describe the results of numerical experiments: the Arnold web is clearly highlighted and, looking at the behaviour of the KAM frequencies on orbits of 10⁸ revolutions, evidence for the diffusion existence is reached.      

Stability of an expanding bubble in the Rayleigh model

A bubble expands adiabatically in an incompressible, inviscid liquid. The variation of its radiusR with time is given by the Rayleigh’s equation. We find that the bubble is stable at the equilibrium point in this model.     

中国八个城市居民食品摄入量与胃癌死亡率的对应分析

为探讨食品摄入量及胃癌死亡率与中国部分城市之间的关系,用对应分析方法对食品摄入量及胃癌死亡率与中国部分城市之间的关系进行分析.对应分析将8个城市16项食品摄入量及胃癌死亡率分成四个点群:点群族I城市包括上海、南京、苏州,食品包括大米、豆制品、绿色蔬菜、咸菜、猪肉、鱼虾类、绽粉类;点群族II城市包括北京、天津、西安,食品包括面粉、浅色蔬菜、水果、乳类、植物油;点群族III城市包括长春,食品包括干豆类、蛋类;点群族IV城市包括沈阳,食品包括薯类.说明中国部分城市与胃癌死亡率存在地区性差异,饮食中应减少精制米类及食盐的摄入,而增加面类和植物油的摄入.     

Prime witnesses in the Shor algorithm and the Miller-Rabin algorithm

We prove that prime witnesses in the Miller-Rabin algorithm coincide with those in the Shor algorithm which satisfy the condition of Fermat’s little theorem. We describe the set of natural numbers, whose prime witnesses in the Miller-Rabin algorithm coincide with those in the Shor algorithm. We find all such numbers less than 100,000,000 and experimentally study the rate of increase of the ratio of the quantity of such numbers to the quantity of Carmichael numbers.     

Some classes of projected dynamical systems in Banach spaces and variational inequalities

We introduce some Projected Dynamical Systems based on metric and generalized Projection Operator in a strictly convex and smooth Banach Space. Then we prove that critical points of these systems coincide with the solution of a Variational Inequality.     

The equality cases for the inequalities of Oppenheim and Schur for positive semi-definite matrices

In this paper, necessary and sufficient conditions for equality in the inequalities of Oppenheim and Schur for positive semidefinite matrices are investigated. Supported by National Natural Science Foundation of China (No. 10531070), National Basic Research Program of China 973 Program (No. 2006CB805900), National Research Program of China 863 Program (No. 2006AA11Z209) and the Natural Science Foundation of Shanghai (Grant No. 06ZR14049).     

Estimating the Scale Parameter of a Lévy-stable Distribution via the Extreme Value Approach

The characteristic exponent α of a Lévy-stable law S _α (σ, β, μ) was thoroughly studied as the extreme value index of a heavy tailed distribution. For 1 < α < 2, Peng (Statist. Probab. Lett. 52: 255–264, 2001) has proposed, via the extreme value approach, an asymptotically normal estimator for the location parameter μ. In this paper, we derive by the same approach, an estimator for the scale parameter σ and we discuss its limiting behavior.      

On the Equivalence and Generalized of Weyl Theorem Weyl Theorem

We know that an operator T acting on a Banach space satisfying generalized Weyl＇s theorem also satisfies Weyl＇s theorem. Conversely we show that if all isolated eigenvalues of T are poles of its resolvent and if T satisfies Weyl＇s theorem, then it also satisfies generalized Weyl＇s theorem. We give also a sinlilar result for the equivalence of a-Weyl＇s theorem and generalized a-Weyl＇s theorem. Using these results, we study the case of polaroid operators, and in particular paranormal operators.     

Estimation of the Cholesky decomposition in a conditional independent normal model with missing data

We investigate the problem of estimating the Cholesky decomposition in a conditional independent normal model with missing data. Explicit expressions for the maximum likelihood estimators and unbiased estimators are derived. By introducing a special group, we obtain the best equivariant estimators.     

Beurling algebra analogues of the classical theorems of Wiener and Lévy on absolutely convergent fourier series

Letf be a continuous function on the unit circle Γ, whose Fourier series is ω-absolutely convergent for some weight ω on the set of integersZ. If f is nowhere vanishing on Γ, then there exists a weightv onZ such that 1/f hadv-absolutely convergent Fourier series. This includes Wiener’s classical theorem. As a corollary, it follows that if φ is holomorphic on a neighbourhood of the range off, then there exists a weight Χ on Z such that φ ◯f has Χ-absolutely convergent Fourier series. This is a weighted analogue of Lévy’s generalization of Wiener’s theorem. In the theorems,v and Χ are non-constant if and only if ω is non-constant. In general, the results fail ifv or Χ is required to be the same weight ω.     

Asymptotic periodicity of recurrent flows in infinite networks

We consider a transport process on an infinite network and, using the corresponding flow semigroup as in Dorn (Semigroup Forum 76:341–356, 2008), investigate its long term behavior. Combining methods from functional analysis, graph theory and stochastics, we are able to characterize the networks for which the flow semigroup converges strongly to a periodic group.     

Representation for the Green’s function of the Dirichlet problem for polyharmonic equations in a ball

We explicitly construct the Green’s function for the Dirichlet problem for polyharmonic equations in a ball in a space of arbitrary dimension. The formulas for the Green’s function are of interest in their own right. In particular, the explicit representations for a solution to the Dirichlet problem for the biharmonic equation are important in elasticity.     

The statistics of particle trajectories in the homogeneous Sinai problem for a two-dimensional lattice

In this paper, we generalize and refine some results by F. P. Boca, R. N. Gologan, and A. Zaharescu on the asymptotic behavior as h → 0 of the statistics of the free path length until the first hit of the h-neighborhood (a disk of radius h) of a nonzero integer for a particle issuing from the origin. The established facts imply that the limit distribution function for the free path length and for the sighting parameter (the distance from the trajectory to the integer point in question) does not depend on the particle escape direction (the property of isotropy).     

Boundary Layer Theory and the Zero-Viscosity Limit of the Navier-Stokes Equation

Abstract A central problem in the mathematical analysis of fluid dynamics is the asymptotic limit of the fluid flow as viscosity goes to zero. This is particularly important when boundaries are present since vorticity is typically generated at the boundary as a result of boundary layer separation. The boundary layer theory, developed by Prandtl about a hundred years ago, has become a standard tool in addressing these questions. Yet at the mathematical level, there is still a lack of fundamental understanding of these questions and the validity of the boundary layer theory. In this article, we review recent progresses on the analysis of Prandtl's equation and the related issue of the zero-viscosity limit for the solutions of the Navier-Stokes equation. We also discuss some directions where progress is expected in the near future. Also at Courant Institute, New York University     

Extension of Isometries Between the Unit Spheres of Normed Space E and C（Ω）

In this paper, we study the extension of isometries between the unit spheres of normed space E and C（Ω）. We obtain that any surjective isometry between the unit spheres of normed space E and C（Ω） can be extended to be a linear isometry on the whole space E and give an affirmative answer to the corresponding Tingley＇s problem （where Ω be a compact metric space）.     

Fluctuations in the mean of Euler’s phi function

We consider the error term in the mean value estimate of Euler’s phi function ψ(n), and show that it is Ω_+- (x(log log x)^1/2). This improves on the earlier results of Pillai and Chowla, and of Erdös and Shapiro.