Sturm-liouville operators with singular potentials

This paper deals with Sturm-Liouville operators generated on a finite interval and on the whole axis by the differential expressionl(y)=−y ^" +q(x)y, whereq(x) is a distribution of first order, such that . The minimal and maximal operators corresponding to potentials of this type on a finite interval are constructed. All self-adjoint extensions of the minimal operator are described and the asymptotics of the eigenvalues of these extensions is found. It is proved that the constructed operator coincides with the norm resolvent limit of the Sturm-Liouville operators generated by smooth potentialsq _n , provided that the condition holds. The convergence of the spectra of these operators to the spectrum of the limit operator is also proved. Similar results are obtained in the case of the whole axis. Translated fromMatematicheskie Zametki, Vol. 66, No. 6, pp. 897–912, December, 1999.     

综合函数与扩展原理

本文首先对合情扩展映射的公理化条件的完备性与独立性进行讨论,之后给出了扩展原理的一般形式及合情扩展映射的等价定义。最后,利用综合函数,特别是变维综合函数列,给出了扩展原理的一些具体形式     

Extensions of perfect GO-spaces and -discrete dense sets

In this paper, we prove that if a perfect GO-space has a -discrete dense set, then has a perfect linearly ordered extension. This answers a problem raised by H. R. Bennett, D. J. Lutzer and S. Purisch. And the result is also a partial answer to an old problem posed by H. R. Bennett and D. J. Lutzer.

     

包含轻子结构的夸克模型

对标准模型作手征性扩充后,模型中平行地存在手征性相反的夸克态.相应于构成强子的f代普通夸克,新加的同样代数、手征性相反的夸克可能构成轻子.对扩充后的模型超对称化之后,使得渐近自由性质的破坏所要求的代的数目从大于33/8减少到大于9/4,从而有可能解释手征相反的夸克构成的轻子为什么比普通夸克构成的强子大小要小得多等特点.     

On Asymptotic Dimension of Groups Acting on Trees

We prove the following.THEOREM. Let be the fundamental group of a finite graph of groups with finitely generated vertex groups G _v having asdim G _v n for all vertices v. Then asdim n+1.This gives the best possible estimate for the asymptotic dimension of an HNN extension and the amalgamated product.     

Singular Sturm-Liouville problems whose coefficients depend rationally on the eigenvalue parameter

Let −Dω(·,z)D+q be a differential operator in L²(0,∞) whose leading coefficient contains the eigenvalue parameter z. For the case that ω(·,z) has the particular form

     

The first cohomology group of trivial extensions of special biserial algebras

Given a finite dimensional special biserial algebra A with normed basis we obtain the dimension formulae of the first Hochschild homology groups of A and the vector space Alt(DA). As a consequence, an explicit dimension formula of the first Hochschild cohomology group of trivial extension TA = A×DA in terms of the combinatorics of the quiver and relations is determined.     

On extension of isometries between unit spheres of -spaces

In this paper, we study the extension of isometries between unit spheres of atomic -spaces . We find a condition under which an isometry between unit spheres can be linearly isometrically extended. Moreover, we prove that every onto isometry between unit spheres of atomic -spaces can be linearly isometrically extended to the whole space.

     

The generalized Roper-Suffridge extension operator

Let f(z) be a normalized convex (starlike) function on the unit disc D. Let , where z=(z₁,z₂,…,z_n), z₁∈D, , p_i?1, i=2,…,n, are real numbers. In this note, we prove that Φ(f)(z)=(f(z₁),f′(z₁)^1/p₂z₂,…,f′(z₁)^1/p_nz_n) is a normalized convex (starlike) mapping on Ω, where we choose the power function such that (f′(z₁))^1/p_i|_z1=0=1, i=2,…,n. Some other related results are proved.     

Lie Groupoids, Gerbes, and Non-Abelian Cohomology

We observe that any regular Lie groupoid G over a manifold M fits into an extension K G E of a foliation groupoid E by a bundle of connected Lie groups K. If F is the foliation on M given by the orbits of E and T is a complete transversal to F , this extension restricts to T, as an extension K _T G _T E _T of an étale groupoid E _T by a bundle of connected groups K _T . We break up the classification problem for regular Lie groupoids into two parts. On the one hand, we classify the latter extensions of étale groupoids by (non-Abelian) cohomology classes in a new ech cohomology of étale groupoids. On the other hand, given K and E and an extension K _T G _T E _T over T, we present a cohomological obstruction to the problem of whether this is the restriction of an extension K G E over M; if this obstruction vanishes, all extensions K G E over M which restrict to a given extension over the transversal together form a principal bundle over a group of bitorsors under K.