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Rearrangement of vector series. II

Published online by Cambridge University Press:  06 March 2001

C. St. J. A. NASH-WILLIAMS  and
D. J. WHITE
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]
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Abstract

Let [sum ]∞an denote the set of cluster points of the sequence of partial sums of a series [sum ]an with terms in ℝd. For any permutation f of the set ℕ of positive integers, [Cscr ]f (ℝd) denotes the set of all sets [sum ]∞af(n) arising from series [sum ]an with terms in ℝd and sum 0. For each f, we use the Max-Flow Min-Cut Theorem to determine all convex sets in [Cscr ]f(ℝd) which are symmetric about a point. These sets depend only on a parameter w(f) ∈ ℕ ∪ {0, ∞}, called the width of f. We show that w(f), when it is a positive integer, falls far short of completely determining [Cscr ]f(ℝd) but, for each q ∈ ℕ, we find the largest of the sets [Cscr ]f(ℝd) arising from permutations f of width q. We also describe the smallest of these sets when q = 1.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 130 , Issue 1 , January 2001 , pp. 111 - 134
Copyright
2001 Cambridge Philosophical Society

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