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Zbl 1026.05009
Dohmen, Klaus
Improved Bonferroni inequalities via abstract tubes. Inequalities and identities of inclusion-exclusion type. (English)
[B] Lecture Notes in Mathematics. 1826. Berlin: Springer. viii, 113 p. EUR 29.95/net; sFr. 51.50; \sterling 23.00; \$ 44.95 (2003). ISBN 3-540-20025-8/pbk

This work is based on the author's habilitation prepared at Berlin Humboldt University in the years 1994-2000. The recent theory of abstract tubes is the framework for establishing improved inclusion-exclusion identities and Bonferroni inequalities. The role of closure and kernel operators is emphasized, and applications are given to system and network reliability, reliability covering problems and chromatic graph theory. Topics also covered are Zeilberger's abstract lace expansion, matroid polynomials and Möbius functions. The list of the chapters includes: Preliminaries, Bonferroni inequalities via abstract tubes, Abstract tubes via closure and kernel operators, Recursive schemes, Reliability applications, Combinatorial applications and related topics (Inclusion-exclusion on partition lattices, Chromatic polynomials and broken circuits, Sums over partially ordered sets, Matroid polynomials and the $\beta $ invariant, Euler characteristics and Möbius functions). The book is well written and contains many original results published by the author.
[Ioan Tomescu (Bucureşti)]
MSC 2000:
*05A19 Combinatorial identities
05A20 Combinatorial inequalities
60C05 Combinatorial probability
60E15 Inequalities in probability theory
68R10 Graph theory in connection with computer science

Keywords: Bonferroni inequalities; abstract tubes; closure operator; kernel operator; semilattice; chromatic polynomial; matroid polynomial; Euler characteristics; Möbius functions

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