无穷次可重整二次多项式的共形测度

利用无穷次可重整化二次多项式三维拼图的技巧, 证明了有复界无分支的无穷次可重整化二次多项式共形测度的遍历性.     

带状区域中渐近周期曲率流方程的整体解

该文研究了具有渐近周期系数的曲率流方程的Neumann边值问题.首先,考虑一列初值问题及其相应的全局解,通过一致的先验估计取一个收敛子列,得到其极限就是一个整体解的结论.其次,向负无穷时间方向进行重整化,使用强极值原理证明了整体解的唯一性.最后,为了研究整体解的ω-和α-极限,再次使用重整化方法,通过构造拉回函数、进行...     

带变异的分枝交互粒子系统和相关的极限定理

本文用Poisson随机测度驱动的随机积分方程构造一类带变异的分枝交互粒子系统.首先证明在某些条件下其重整化极限是同时具有局部和非局部分枝机制的超过程,其底运动是平凡的;其次证明在另外的一些条件下,其重整化极限是具有局部分枝机制和非平凡底运动的超过程.     

三阶重整化群方程的解

为揭示产生Feigenbaum现象的数学基础,有关重整化群方程(Renormalization Group Equatjon)的解的存在与性态的考察成为引人注目的研究对象。本文运用类似于文献[2]的方法,考虑另一类三阶重整化群方程。由于得到这类方程的任一单谷连续解必产生Li-Yorke意义下的浑沌(Chaos)现象的结论,研究这类方程的解更显得必要。本文得到了这类重整化群方程的单谷连续解的一些性质并用构造性方法给出了所有单谷连续解,进一步也指出了C′解的存在性。     

从离散速度模型到矩方法

为了求解动理学方程,我们通过研究一维情形下的离散速度模型,发现通过对离散速度点使用自适应技术可以直接得到Grad矩方程组.作为一个统一的认识,矩方程组可以看作是对离散速度点自适应的离散速度模型,而离散速度模型可以看作是取特别形式的"矩"的矩方程组.这使得我们可以在一致的框架下来理解离散速度模型和矩方法,而不是将它们对立起来.为了建立这样的一致框架,最近在[2]中发展的正则化理论是根本性的.     

自旋s=1的半无限无规场伊辛模型的临界行为

该文应用平均场重整化群方法,对s=1的半无限简立方伊辛铁磁体在表面无规场(三峰分布)存在时的表面临界行为进行了研究,得到了系统的表面相图.     

多指标独立随机变量的重对数律及其应用

对于多指标独立同分布的随机变量序列,在某些更广泛的正则化序列下,本文给出了重对数律成立的充分必要条件.作为应用,本文讨论了正则和极大值函数的矩存在的充分必要条件.     

利用Dirac理论进行Poisson流形及Jacobi流形的约化(英文)

通过将可约的Dirac以及Jacobi-Dirac结构分别分为两种类型,给出对应于Poisson流形和Jacobi流形的约化定理.这些约化定理的证明只需要进行一些直接的计算,而不需要借助于矩映射或者相容函数等复杂概念的引入.另外,给出了一些相应的例子和应用.     

具有高消失矩的3-进正交对称小波的构造(英文)

本文给出一类具有高消失矩的对称3-进正交小波的构造.首先通过解非线性方程组构造出高消失矩的尺度函数.其次,基于矩阵扩充的方法,我们给出新的定理和算法.相应的关于原点对称/反对称的小波由我们提出的方法构造,并且,该方法便于计算机程序化实现.最后,我们通过3个实例来验证相关的结论,该类小波为应用提供了好的选择.     

基于广义Hermite展开的全局双曲矩方程组

在最近的研究中,Cai等人(2014)使用本征热通量代替通常使用的沿坐标轴方向的热通量作为变量获得了一个改进的13矩方程组.该方程组比使用坐标轴方向的热通量获得的Grad 13矩方程组具有更好的性质,例如,局部平衡态是双曲区域的内点.该改进的13矩方程组是通过对分布函数使用广义的各向异性Hermite展开获得,其中该各向异性展开是指,使用完全的温度张量代替局部平衡态的温度.本文将该方法推广到高阶广义Hermite展开从而获得任意阶的新的矩方程组,并类似Cai等人(2014)的方法提出了一个正则化方案,使得所得方程组全局双曲,从而保证所得方程组的局部适定性.此外,本文还深入研究该方程组的系数矩阵的特征结构,并剖析了方程组的所有特征波.该矩系统提供了一套系统的可看成是Euler方程组的推广的动力学模型,并可通过提高展开阶来逼近Boltzmann方程本身.     

重正化核的Poisson随机积分表示

本文考虑具有有限矩的1维无穷可分分布的正交多项式的母函数,通过“一步提升”原则得到的重正化核的显式表示,建立重正化核运算与Poisson随机积分之间的关系.     

On a multivariate renewal-reward process involving time delays and discounting: applications to IBNR processes and infinite server queues

This paper considers a particular renewal-reward process with multivariate discounted rewards (inputs) where the arrival epochs are adjusted by adding some random delays. Then, this accumulated reward can be regarded as multivariate discounted Incurred But Not Reported claims in actuarial science and some important quantities studied in queueing theory such as the number of customers in \(G/G/\infty \) queues with correlated batch arrivals. We study the long-term behaviour of this process as well as its moments. Asymptotic expressions and bounds for quantities of interest, and also convergence for the distribution of this process after renormalization, are studied, when interarrival times and time delays are light tailed. Next, assuming exponentially distributed delays, we derive some explicit and numerically feasible expressions for the limiting joint moments. In such a case, for an infinite server queue with a renewal arrival process, we obtain limiting results on the expectation of the workload, and the covariance of queue size and workload. Finally, some queueing theoretic applications are provided.     

Nonlinear stability of spatially-periodic traveling-wave solutions of systems of reaction-diffusion equations

Using spatial domain techniques developed by the authors and Myunghyun Oh in the context of parabolic conservation laws, we establish under a natural set of spectral stability conditions nonlinear asymptotic stability with decay at Gaussian rate of spatially periodic traveling waves of systems of reaction-diffusion equations. In the case that wave-speed is identically zero for all periodic solutions, we recover and slightly sharpen a well-known result of Schneider obtained by renormalization/Bloch transform techniques; by the same arguments, we are able to treat the open case of nonzero wave-speeds to which Schneider?s renormalization techniques do not appear to apply.     

Asymptotic results for time-changed Lévy processes sampled at hitting times

We provide asymptotic results for time-changed Lévy processes sampled at random instants. The sampling times are given by the first hitting times of symmetric barriers, whose distance with respect to the starting point is equal to ε. For a wide class of Lévy processes, we introduce a renormalization depending on ε, under which the Lévy process converges in law to an α-stable process as ε goes to 0. The convergence is extended to moments of hitting times and overshoots. These results can be used to build high frequency statistical procedures. As examples, we construct consistent estimators of the time change and, in the case of the CGMY process, of the Blumenthal-Getoor index. Convergence rates and a central limit theorem for suitable functionals of the increments of the observed process are established under additional assumptions.     

Renormalization group approach to function approximation and to improving subsequent approximations

We establish a relation between bijective functions and renormalization group transformations and find their renormalization group invariants. For these functions, taking into account that they are globally one-to-one, we propose several improved approximations (compared with the power series expansion) based on this relation. We propose using the obtained approximations to improve the subsequent approximations of physical quantities obtained, in particular, by one of the main calculation techniques in theoretical physics, i.e., by perturbation theory. We illustrate the effectiveness of the renormalization group approximation with several examples: renormalization group approximations of several analytic functions and calculation of the nonharmonic oscillator ground-state energy. We also generalize our approach to the case of set maps, both continuous and discrete.     

Dimensional renormalization inp-adic models of field theory

We define fractional-dimensional p-adic Feynman amplitudes and construct a dimensional renormalization with minimum subtractions. In the fermionic model case, another dimensional renormalization procedure is defined as the inversion of the normalizing transformation at the trivial stable point for the hierarchical renormalization group transformation. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 123, No. 3, pp. 462–475, June, 2000.     

Differential Algebraic Birkhoff Decomposition and the renormalization of multiple zeta values

In the Hopf algebra approach of Connes and Kreimer on renormalization of quantum field theory, the renormalization process is viewed as a special case of the Algebraic Birkhoff Decomposition. We give a differential algebra variation of this decomposition and apply this to the study of multiple zeta values.     

Renormalization in classical mechanics and many body quantum field theory

We attempt to give apedagogical introduction to perturbative renormalization. Our approach is to first describe, following Linstedt and Poincaré, the renormalization of formal perturbation expansions for quasi-periodic orbits in Hamiltonian mechanics. We then discuss, following [FT1, FT2], the renormalization of the formal ground state energy density of a many Fermion system. The construction of formal quasi-periodic orbits is carried out in detail to provide a relatively simple model for the considerably more involved, and perhaps less familiar, perturbative analysis of a field theory. As we shall see, quasi-periodic orbits and many Fermion systems have a number of important features in common. In particular, as Poincaré observed in the classical case and [FT1, FT2] pointed out in the latter, the formal expansions considered here both contain divergent subseries. Dedicated to Professor Shmuel Agmon Research supported in part by the Natural Sciences and Engineering Research Council of Canada.     

Dynamics of renormalization group in the lower half-plane of the coupling constants of the fermionic hierarchical model

We consider four-component fermionic (Grassmann-valued) field on the hierarchical lattice. The Gaussian part of the Hamiltonian in the model is invariant under the block-spin renormalization group transformation with given degree of normalization factor (renormalization group parameter). The non-Gaussian part of the Hamiltonian is given by the sum of the selfinteraction forms of the 2-nd and 4-th order. The action of the renormalization group transformation in this model is reduced to the rational map in the plane of coupling constants. We investigate the global dynamics of this map in the case when the coupling constant of the 4-th order form is less than zero (lower half-plane) and the renormalization group parameter belongs to the interval [1, 3/2).