Hostname: page-component-78c5997874-xbtfd Total loading time: 0 Render date: 2024-11-03T01:28:17.542Z Has data issue: false hasContentIssue false

Rearrangement of vector series. I

Published online by Cambridge University Press:  01 March 2001

C. St. J. A. NASH-WILLIAMS
Affiliation:
Department of Mathematics, University of Reading, Whiteknights, P.O. Box 220, Reading RG6 6AX. e-mail: [email protected]; e-mail: [email protected]
D. J. WHITE
Affiliation:
Department of Mathematics, University of Reading, Whiteknights, P.O. Box 220, Reading RG6 6AX. e-mail: [email protected]; e-mail: [email protected]

Abstract

Let ℝd* = ℝd ∪ {[midast ]} be the one-point compactification of Euclidean space ℝd and d [ges ] 2. Given a permutation f of the set ℕ of positive integers, let [Cscr ]f(ℝd*) denote the set of all sets C ⊆ ℝd* for which there is a series [sum ]an in ℝd with zero sum such that C is the cluster set in ℝd* of the sequence of partial sums of [sum ]af(n). Every C ∈ [Cscr ]f(ℝd*) is non-empty, connected and closed in ℝd*. We give a combinatorial characterization of the permutations f for which all non-empty closed connected subsets of ℝd* belong to [Cscr ]f(ℝd*). For every permutation f of ℕ, we determine all C ∈ [Cscr ]f(ℝd*) which contain [midast ].

Type
Research Article
Copyright
2001 Cambridge Philosophical Society

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)