On Differentiable Compactifications of the Hyperbolic Plane and Algebraic Actions of SL2(\mathbb R) on Surfaces

Real-analytic actions of SL(2;R) on surfaces have been classified, up to analytic change of coordinates. In particular it is known that there exists countably many analytic equivariant compactification of the isometric action on the hyperbolic plane. In this paper we study the algebraicity of these actions. We get a classification of the algebraic actions of SL(2,R) on surfaces. In particular, we classify the algebraic equivariant compactifications of the hyperbolic plane. An erratum to this article can be found at     

The smallest ideal of βS is not closed

Let S be an infinite discrete semigroup which can be embedded algebraically into a compact topological group and let βS be the Stone–Čech compactification of S. We show that the smallest ideal of βS is not closed.     

On the Cone of Bounded Lower Semicontinuous Functions

We prove that the cone of bounded lower semicontinuous functions defined on a Tychonoff space X is algebraically and structurally isomorphic and isometric to a convex cone contained in the cone of all bounded lower semicontinuous functions defined on the Stone-Cech compactification βX if and only if the space X is normal. We apply this theorem to the study of relationship between a class of multivalued maps and sublinear operators. Using these results, we obtain new proofs of theorems about continuous selections.__________Translated from Matematicheskie Zametki, vol. 77, no. 6, 2005, pp. 886–902.Original Russian Text Copyright ©2005 by Yu. E. Linke.     

Dirichlet Boundary Control of Semilinear Parabolic Equations Part 2: Problems with Pointwise State Constraints

This paper is the continuation of the paper ``Dirichlet boundary control of semilinear parabolic equations. Part 1: Problems with no state constraints.' It is concerned with an optimal control problem with distributed and Dirichlet boundary controls for semilinear parabolic equations, in the presence of pointwise state constraints. We first obtain approximate optimality conditions for problems in which state constraints are penalized on subdomains. Next by using a decomposition theorem for some additive measures (based on the Stone—Cech compactification), we pass to the limit and recover Pontryagin's principles for the original problem. Accepted 21 July 2001. Online publication 21 December 2001.     

Products of Compactifications

A product operation of compactifications is defined and its different properties are studied. Some applications are considered.     

Extension of locally pseudocompact group actions

It is shown that: (1) any action of a Moscow group G on a first countable, Dieudonné complete (in particular, on a metrizable) space X can uniquely be extended to an action of the Dieudonné completion γG on X, (2) any action of a locally pseudocompact topological group G on a b _f -space (in particular, on a first countable space) X can uniquely be extended to an action of the Weil completion on the Dieudonné completion γX of X. As a consequence, we obtain that, for each locally pseudocompact topological group G, every G-space with the b _f -property admits an equivariant embedding into a compact Hausdorff G-space. Furthermore, for each pseudocompact group G, every metrizable G-space has a G-invariant metric compatible with its topology. We also give a direct construction of such an invariant metric. Received: June 22, 2000; in final form: May 22, 2001?Published online: June 11, 2002     

L-拓扑空间的单点超F紧化

给出L-拓扑空间的单点超F紧化的一种具体作法,以及局部超F紧性的定义,并证明了:(1)局部超F紧性是L-好推广;(2)一个L-拓扑空间是局部超F紧T2空间当且仅当其单点超F紧化空间是超F紧T2空间;(3)单点超F紧化在同胚意义下是唯一的。     

Compactifications determined by subsets of C1(X)

It is well known that every compactification of a completely regular space X can be generated, via a Tychonoff-type embedding, by some suitably chosen subset of C¹(X). Different subsets may give rise to equivalent compactifications, and we are concerned with the problem of finding all subsets of C¹(X) which yield a given compactification αX. The problem is easier if generalized: we say that a subset F of C¹(X) “determines” the compactification αX if αX is the smallest compactification to which every element of F extends, and give a simple necessary and sufficient condition for F to determine a given compactification αX. A number of sufficient conditions for two sets to determine the same compactification are given, and the relation between sets which determine αX and those which generate αX (via an embedding) is considered. Generally, a much smaller set of functions is required to determine αX than to generate it; the number needed to determine αX is never more than the weight of αX?X, while the number required to generate it is, if infinite, equal to the weight of αX.     

Parabolic bundles on algebraic surfaces I — The Donaldson-Uhlenbeck compactification

The aim of this paper is to construct the parabolic version of the Donaldson-Uhlenbeck compactification for the moduli space of parabolic stable bundles on an algebraic surface with parabolic structures along a divisor with normal crossing singularities. We prove the non-emptiness of the moduli space of parabolic stable bundles of rank 2.