Explicit Expressions for Timelike and Spacelike Observables of Quantum Chromodynamics in Analytic Perturbation Theory

We study the possibility of expressing the invariant QCD coupling function (i.e., the effective coupling constant) in an explicit analytic form in two- and three-loop approximations as well as in the case of the Padé-transformed -function. Both the timelike and spacelike domains are investigated. Technical aspects of the Shirkov–Solovtsov analytic perturbation theory are considered. Explicit expressions for the two- and three-loop effective coupling functions in the timelike domain are obtained. In the last case, we apply a new method of expanding functions represented in an arbitrary loop order of perturbation theory in powers of the two-loop function. The comparison with numerical data in the infrared region shows that the obtained explicit expressions for the three-loop functions agree fully with the exact numerical results.     

Analytic Perturbation Theory and Renormalization Analysis of Matter Coupled to Quantized Radiation

For a large class of quantum mechanical models of matter and radiation we develop an analytic perturbation theory for non-degenerate ground states. This theory is applicable, for example, to models of matter with static nuclei and non-relativistic electrons that are coupled to the UV-cutoff quantized radiation field in the dipole approximation. If the lowest point of the energy spectrum is a non-degenerate eigenvalue of the Hamiltonian, we show that this eigenvalue is an analytic function of the nuclear coordinates and of α^3/2, α being the fine structure constant. A suitably chosen ground state vector depends analytically on α^3/2 and it is twice continuously differentiable with respect to the nuclear coordinates. Submitted: November 24, 2008. Accepted: March 4, 2009.     

非线性Klein-Gordon方程的微扰理论

求得了一阶近似下微扰对非线性Klein-Gordon孤子的影响,即求得了孤子参数随时间的缓慢变化及一阶修正的具体表达式。     

Perturbation Theory for Discrete Volterra Equations

Fixed point theory is used to investigate nonlinear discrete Volterra equations that are perturbed versions of linear equations. Sufficient conditions are established (i) to ensure that stability (in a sense that is defined) of the solutions of the linear equation implies a corresponding stability of the zero solution of the nonlinear equation and (ii) to ensure the existence of asymptotically periodic solutions.     

Perturbation Theory for Critically Damped Nonlinear Systems

A simple analytical method is developed to obtain the time response of the second order critically damped nonlinear systems based on the method of variation of parameters. Solutions obtained for different initial conditions for a second order system whose linear equation has equal roots (negative) shows good agreement with those obtained by numerical method.AMS: 34E05     

关于n次积分C半群的扰动

研究了n次积分C半群的可交换扰动问题,得到其扰动定理;并根据n次积分C半群与C半群的关系得到了n次积分C半群的两个乘积扰动定理.     

Perturbation Theory for Simultaneous Bases of Singular Subspaces

New perturbation theorems are proved for simultaneous bases of singular subspaces of real matrices. These results improve the absolute bounds previously obtained in [6] for general (complex) matrices. Unlike previous results, which are valid only for the Frobenius norm, the new bounds, as well as those in [6] for complex matrices, are extended to any unitarily invariant matrix norm. The bounds are complemented with numerical experiments which show their relevance for the algorithms computing the singular value decomposition. Additionally, the differential calculus approach employed allows to easily prove new sin perturbation theorems for singular subspaces which deal independently with left and right singular subspaces.     

Perturbation Theory for Linearly Perturbed Algebraic Riccati Equations

The perturbation results for the solutions of two linearly perturbed algebraic Riccati equations are derived. We generalize the results of Sun [SIAM J. Matrix Anal. Appl., 19 (1998):39–65] for continuous (CARE) and discrete (DARE) algebraic Riccati equations, respectively. The results are illustrated by numerical examples.     

定态微扰论的一种处理新方法

利用渐近方法,直接由Ｌａｇｒａｎｇｅ常数变易法求解含微扰项的Ｓｔｕｒｍ－Ｌｉｏｕｖｉｌｅ本征值问题．最后举一实例证明,与传统的展开法（结果为无穷多项）相比较,两种方法完全一致,等价,适用范围也相同,而且本文所述方法简明（一般只含两项）更具有一定的物理意义     

A Perturbation Theory for Core Operators of Hilbert-Schmidt Submodules

We discuss a perturbation theory for core operators of Hilbert-Schmidt submodules in the Hardy space over the bidisk. It is shown that eigenvalues of core operators perturbed by automorphisms of the bidisk have a certain continuity, and also we show that the family of eigenspaces of simple eigenvalues has a local cross section.     

Homological Perturbation Theory and Mirror Symmetry

We explain how deformation theories of geometric objects such as complex structures,Poisson structures and holomorphic bundle structures lead to differential Gerstenhaber or Poisson al-gebras.We use homological perturbation theory to construct A∞ algebra structures on the cohomology,and their canonically defined deformations.Such constructions are used to formulate a version of A∞ algebraic mirror symmetry.     

解析数论在中国（迎接ICM2002特约文章）

本文综述了解析数论近70年来在中国的发展及中国数学家在这一领域的重要贡献,特别是华罗庚及陈景润的重大成就。     

多频常微分方程的几何奇异扰动理论

在该文中,作者把常微分方程的几何奇异扰动理论推广到具多个频率的系统,同时给出了一个例子来说明主要理论．     

Integral-equation Perturbation Theory for Electrostatic Torus Problems with Applied Fields

A perturbation method has been given for solving the Fredholmintegral equations of electrostatics as applied to tori of revolution,but under the restriction that there be in no case an appliedfield other than perhaps one that is uniform and parallel tothe axis of the torus. In the present paper this restrictionis removed, both for the conductor situation (Robin's equation)and the dielectric one (Durand's equation), and the generalizedmethod is illustrated by three applications. Two of these areto new problems which prove to be of considerable interest inthemselves, and one of the two leads to an example of significantfailure of the perturbation method.     

Blockwise Perturbation Theory for 2x2 Block Markov Chains

Let P be a transition matrix of a Markov chain and be of the form $$P=\Bigg( \begin{matrix} P_{11} &P_{12} \\P_{21} &P_{22} \end{matrix} \Bigg).$$ The stationary distribution $π^T$ is partitioned conformally in the form $(π^T_1, π^T_2)$. This paper establish the relative error bound in $π^T_i (i=1,2)$ when each block $P_{ij}$ get a small relative perturbation.     

On the Renormalization Group Approach to Perturbation Theory for PDEs

We investigate the rigorous application of the renormalization group method to (singular) perturbation theory for nonlinear partial differential equations. As a paradigm, we consider the concrete example of the nonlinear Schrödinger equation with quadratic nonlinearity in three spatial dimensions. We obtain an approximate solution using the RG method together with an estimate of the difference between the true and approximate solutions. Our analysis applies to cases where (space–time) resonances are present.     

New Perturbation Theory for Quantum Field Theory: Convergent Series Instead of Asymptotic Expansions

We present a new approach to perturbation theory for quantum field theory based on convergent series instead of asymptotic expansions. This approach could be considered as the next step after traditional perturbation theory calculations, which allows more comprehensive use of previously obtained information in finding numerical values with greater accuracy.