On Convergence to Infinity

 By a metric mode of convergence to infinity in a regular Hausdorff space X, we mean a sequence of closed subsets of X with and , and a sequence (or net) in X is convergent to infinity with respect to provided for each contains eventually. Modulo a natural equivalence relation, these correspond to one-point extensions of the space with a countable base at the ideal point, and in the metrizable setting, they correspond to metric boundedness structures for the space. In this article, we study the interplay between these objects and certain continuous functions that may determine the metric mode of convergence to infinity, called forcing functions. Falling out of our results is a simple proof that each noncompact metrizable space admits uncountably many distinct metric uniformities. (Received 2 March 1999)     

Relative equidimensionality and stability of actions¶of a reductive algebraic group

In order to inquire into invariants of non-semisimple groups, we introduce and study relative versions of equidimensionality and stabilty, which are called relative quasi-equidimensionality and relative stability, of actions of affine algebraic groups, especially of reductive groups, on affine varieties. As an application of our results, for complex reductive groups of semisimple rank one, we characterize, respectively, relatively stable representations and relatively equidimensional representations and, consequently, show that every equidimensional representation is cofree. Received: 23 October 1998     

The algebraic torus theorem

A structure theorem is proved for a finitely generated group with a finitely generated virtually polycyclic codimension one subgroup. Oblatum 12-II-1999 & 18-XI-1999?Published online: 21 February 2000     

Classification of Homomorphisms from C (X) to Simple C*-Algebras of Real Rank Zero

Abstract Let A be a unital simple C*-algebra of real zero, stable rank one, with weakly unperforated K ₀₍ A) and unique normalized quasi-trace τ, and let X be a compact metric space. We show that two monomorphisms φ, ψ : C(X)→A are approximately unitarily equivalent if and only if φ and ψ induce the same element in KL(C(X), A) and the two lineal functionals τ∘φ and τ∘ψ are equal. We also show that, with an injectivity condition, an almost multiplicative morphism from C(X) into A with vanishing KK-obstacle is close to a homomorphism. Research partially supported by NSF Grants DMS 93-01082 (H.L) and DMS-9401515(G.G). This work was reported by the first named author at West Coast Operator Algebras Seminar (Sept. 1995, Eugene, Oregon)     

Properly immersed minimal disks bounded by straight lines

Let and be two distinct parallel planes in . Let and denote two points such that the segment meets and orthogonally. Let be a straight line containing , and denote as the set of straight lines in containing . Then there exists an analytic family of proper pairwise non congruent minimal immersions satisfying: 1. is homeomorphic to , where 2. , where . 3. is contained in the slab determined by and . 4. If and are the two connected components of , then is injective, . 5. The parameter is an analytic determination of the angle that the orthogonal projection of on makes with and is invariant under the reflection around a straight line not contained in the surface. 6. If is a proper minimal immersion satisfying 1, 2, 3 and4, then, up to a rigid motion, . Received December 28, 1997 / Revised November 20, 1999 / Published online October 11, 2000     

Commutant lifting and interpolation: The time-varying case

We show how the commutant lifting theorem for nest algebras due to Paulsen and Power can be used to give a unified framework for the treatment of a variety of interpolation problems for nest algebras which have been considered recently in the literature. Applications include the treatment of robust control for time-varying systems.Partially supported by NSF grant DMS-9500912     

Convex nonagons with five intervertex distances

The vertices of a convex planar nonagon determine exactly five distances if and only if they are nine vertices of a regular 10-gon or a regular 11-gon. This result has important ties to related concerns, including the maximum number of points in the plane that determine exactly five distances and, for each n7, the samllest t for which there exists a convex n-gon whose vertices determine t distances and are not all on one circle.     

Crowns,cutsets, and valuable partial orders

We consider the problem of recognizing those partial orders which admit a valuation: this is a linear-algebraic condition which arises naturally in an algebraic/geometric context. We show that a partial order has at most one valuation (which is integer-valued) and present various structural conditions which are either necessary or sufficient for a partial order to be valuable. The first main result is a reduction theorem which allows us to restrict attention to those partial orders which do not have a bounded cutset. We use this and a theorem of Kelly and Rival to prove the second main result: that every contraction of a bounded partial order is fibre-valuable if and only if the partial order is a dismantlable lattice. This result has a geometric interpretation.This research was supported by the Natural Sciences and Engineering Research Council of Canada under operating grant OGP0105392.     

Flocks and ovals

An infinite family of q-clans, called the Subiaco q-clans, is constructed for q=2^e. Associated with these q-clans are flocks of quadratic cones, elation generalized quadrangles of order (q ², q), ovals of PG(2, q) and translation planes of order q ² with kernel GF(q). It is also shown that a q-clan, for q=2^e, is equivalent to a certain configuration of q+1 ovals of PG(2, q), called a herd.W. Cherowitzo gratefully acknowledges the support of the Australian Research Council and has the deepest gratitude and warmest regards for the Combinatorial Computing Research Group at the University of Western Australia for their congenial hospitality and moral support. I. Pinneri gratefully acknowledges the support of a University of Western Australia Research Scholarship.