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Improved Bonferroni inequalities via abstract tubes. Inequalities and identities of inclusion-exclusion type. (English) Zbl 1026.05009
Lecture Notes in Mathematics. 1826. Berlin: Springer. viii, 113 p. EUR 29.95/net; sFr. 51.50; £ 23.00; $ 44.95 (2003).
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.

05A19Combinatorial identities, bijective combinatorics
05A20Combinatorial inequalities
60C05Combinatorial probability
60E15Inequalities in probability theory; stochastic orderings
68R10Graph theory in connection with computer science (including graph drawing)
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