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26076 results in Abstract analysis

Page 998 of 1304

  • Preface

  • Albrecht Pietsch, Friedrich-Schiller-Universität, Jena, Germany, Jörg Wenzel, Friedrich-Schiller-Universität, Jena, Germany
  • Book:
    Orthonormal Systems and Banach Space Geometry
    Published online:
    10 December 2009
    Print publication:
    10 September 1998, pp ix-x

  • Summary

    This book is based on the pioneering work of (in chronological order) R. C. James, S. Kwapień, B. Maurey, G. Pisier, D. L. Burkholder and J. Bourgain.

    We have done our best to unify and simplify the material. All participants of the Jenaer Seminar ‘Operatorenideale’ contributed their ideas and their patience. Above all, we are indebted to A. Hinrichs who made several significant improvements. From S. Geiss we learnt many results and techniques related to the theory of martingales. Particular gratitude goes to H. Jarchow (Zürich) for various helpful remarks.

    For many years, our research on this subject was supported by the Deutsche Forschungsgemeinschaft, contracts Ko 962/3–1 and Pi 322/1–1.

    Finally, we thank CAMBRIDGE UNIVERSITY PRESS for their excellent collaboration in publishing this book. Jena, October 1997 ALBRECHT PIETSCH JÖRG WENZEL

  • The Size of the Giant Component of a Random Graph with a Given Degree Sequence

  • MICHAEL MOLLOY, BRUCE REED
  • Journal:
    Combinatorics, Probability and Computing / Volume 7 / Issue 3 / September 1998
    Published online by Cambridge University Press:
    01 September 1998, pp. 295-305

  • Given a sequence of nonnegative real numbers λ0, λ1, … that sum to 1, we consider a random graph having approximately λin vertices of degree i. In [12] the authors essentially show that if [sum ]i(i−2)λi>0 then the graph a.s. has a giant component, while if [sum ]i(i−2)λi<0 then a.s. all components in the graph are small. In this paper we analyse the size of the giant component in the former case, and the structure of the graph formed by deleting that component. We determine ε, λ′0, λ′1 … such that a.s. the giant component, C, has εn+o(n) vertices, and the structure of the graph remaining after deleting C is basically that of a random graph with n′=n−[mid ]C[mid ] vertices, and with λ′in′ of them of degree i.

  • Generalized Stack Permutations

  • M. D. ATKINSON
  • Journal:
    Combinatorics, Probability and Computing / Volume 7 / Issue 3 / September 1998
    Published online by Cambridge University Press:
    01 September 1998, pp. 239-246

  • Stacks which allow elements to be pushed into any of the top r positions and popped from any of the top s positions are studied. An asymptotic formula for the number un of permutations of length n sortable by such a stack is found in the cases r=1 or s=1. This formula is found from the generating function of un. The sortable permutations are characterized if r=1 or s=1 or r=s=2 by a forbidden subsequence condition.

  • Consistency of Natural Relations on Sets

  • AKIRA MARUOKA, MIKE PATERSON, HIROTAKA KOIZUMI
  • Journal:
    Combinatorics, Probability and Computing / Volume 7 / Issue 3 / September 1998
    Published online by Cambridge University Press:
    01 September 1998, pp. 281-293

  • The natural relations for sets are those definable in terms of the emptiness of the subsets corresponding to Boolean combinations of the sets. For pairs of sets, there are just five natural relations of interest, namely, strict inclusion in each direction, disjointness, intersection with the universe being covered, or not. Let N denote {1, 2, …, n} and (N2) denote {(i, j)[mid ]i, j∈N and i<j}. A function μ on (N2) specifies one of these relations for each pair of indices. Then μ is said to be consistent on M⊆N if and only if there exists a collection of sets corresponding to indices in M such that the relations specified by μ hold between each associated pair of the sets. Firstly, it is proved that if μ is consistent on all subsets of N of size three then μ is consistent on N. Secondly, explicit conditions that make μ consistent on a subset of size three are given as generalized transitivity laws. Finally, it is shown that the result concerning binary natural relations can be generalized to r-ary natural relations for arbitrary r[ges ]2.

Page 998 of 1304