Singularity in Potential,Perturbation and Variational Methods

We carefully study the energy eigen-values and splitting of heavy quarkonia as there exist 1/r³ and δ³(r) singular terms in the potential which make a direct numerical solution of the Schrodinger equation impossible. We compare the results obtained in terms of perturbation and variational methods with various treatments.     

Hadamard's Fundamental Solution and Conical Refraction

Conical refraction in anisotropic media shows two different light speeds, hence the charaderistic conoid is composed of two sheets. In a special case that two of the dielectric constants are equal, conic refraction is depicted by a partial differential operator which is factorizable. Thus the singular support of the fundamental solution should also be composed of two sheets. In this paper, the author gives the Hadamard construction of the fundamental solution which is just singular on these two sheets. In case of conic refraction considered, these two sheets are tangent to each other along two bi-characteristic curves, and a special singularity of the boundary-layer type appears there.     

流体与流体混合物的临界与超临界特性

本文第一部分介绍单组分流体的临界与超临界特性.主要内容为临界点的经典与标度状态方程;临界点异常特性的实验证据;热力学与输运性质的计算等等.第二部分介绍二元流体混合物的临界与超临界特性.第三部分介绍超临界气体萃取的工业应用.      

A direct method for deriving fundamental solution of half-plane problem

A fundamental solution for half-plane problems which will play a key role in calculation of the stress concentration around a hole embedded in half-plane is derived by a method combining images with direct integrations. It is wore intuitive than the Fourier transform method used by Gladwell[6]. In addition, the principle and procedure of boundary element method to solve the half-plane problems are also presented by means of Betti’s reciprocal theorem in this paper.It is shown that the. computing procedure for half-plane problems is much more convenient using the fundamental solution presented here than the one adopted by C.A.     

On a Class of Analytic Operator Functions and Their Linearizations

We consider an operator function T in a Krein space which can formally be written as (0.1)but the last term on the right of (0.1) is replaced by a relatively form‐compact perturbation of a similar form. We study relations between the operator function T, a selfadjoint operator M in some Krein space, associated with T, and an operator which can be constructed with the help of the operator function –T^–1. The results are applied to a Sturm‐Liouville problem with a coefficient depending rationally on the eigenvalue parameter.     

Periodic solutions of singular second order equations at resonance

In this paper, we study the existence of positive periodic solutions for singular second order equations x" + n²/4x+h(x) = p(t), where h has a singularity at the origin and n is a positive integer. We give an explicit condition to ensure the existence of positive periodic solutions when h is an unbounded perturbation at infinity by using qualitative analysis and topological degree theory.     

The expression of stress and strain at the tip of three-dimensional notch

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关于奇点类型的讨论

剖析了数学奇点和物理奇点这两个概念及其它们在本质上的差异：数学上的本性奇点只不过是无穷级极点。而物理奇点如Schwarzchild黑洞中的Schwarzchild坐标的原点r=0的奇异性却出现在黎曼曲率张量里,它才真正反映了事物本质上的奇异性。     

Patterns in random binary search trees

In a randomly grown binary search tree (BST) of size n, any fixed pattern occurs with a frequency that is on average proportional to n. Deviations from the average case are highly unlikely and well quantified by a Gaussian law. Trees with forbidden patterns occur with an exponentially small probability that is characterized in terms of Bessel functions. The results obtained extend to BSTs a type of property otherwise known for strings and combinatorial tree models. They apply to paged trees or to quicksort with halting on short subfiles. As a consequence, various pointer saving strategies for maintaining trees obeying the random BST model can be precisely quantified. The methods used are based on analytic models, especially bivariate generating function subjected to singularity perturbation asymptotics. © 1997 John Wiley & Sons, Inc. Random Struct. Alg., 11 : 223–244, 1997     

On the fixed part of pluricanonical systems for surfaces

We show that $|m{K}_X|$ defines a birational map and has no fixed part for some bounded positive integer m for any $\frac{1}{2}$-lc surface X such that $K_X$ is big and nef. For every positive integer $n\ge 3$, we construct a sequence of projective surfaces $X_{n,i}$, such that $K_{X_{n,i}}$ is ample, $\mathrm{mld}(X_{n,i})>\frac{1}{n}$ for every i, ${\lim }_{i\to +\infty }\mathrm{mld}(X_{n,i})=\frac{1}{n}$, and for any positive integer m, there exists i such that $|m{K}_{X_{n,i}}|$ has nonzero fixed part. These results answer the surface case of a question of Xu.