This introduction to the recent theory of abstract tubes describes the framework for establishing improved inclusion-exclusion identities and Bonferroni inequalities, which are provably at least as sharp as their classical counterparts while involving fewer terms. All necessary definitions from graph theory, lattice theory and topology are provided. The role of closure and kernel operators is emphasized, and examples are provided throughout to demonstrate the applicability of this new theory. Applications are given to system and network reliability, reliability covering problems and chromatic graph theory. Topics also covered include Zeilberger's abstract lace expansion, matroid polynomials and Möbius functions.

In the tradition of EuroComb'01 (Barcelona), Eurocomb'03 (Prague), EuroComb'05 (Berlin), Eurocomb'07 (Seville), Eurocomb'09 (Bordeaux), and Eurocomb'11 (Budapest), this volume covers recent advances in combinatorics and graph theory including applications in other areas of mathematics, computer science and engineering. Topics include, but are not limited to: Algebraic combinatorics, combinatorial geometry, combinatorial number theory, combinatorial optimization, designs and configurations, enumerative combinatorics, extremal combinatorics, ordered sets, random methods, topological combinatorics.

Multivariate Bonferroni-Type Inequalities: Theory and Applications presents a systematic account of research discoveries on multivariate Bonferroni-type inequalities published in the past decade. The emergence of new bounding approaches pushes the conventional definitions of optimal inequalities and demands new insights into linear and Frechet optimality. The book explores these advances in bounding techniques with corresponding innovative applications. It presents the method of linear programming for multivariate bounds, multivariate hybrid bounds, sub-Markovian bounds, and bounds using Hamilton circuits. The first half of the book describes basic concepts and methods in probability inequalities. The author introduces the classification of univariate and multivariate bounds with optimality, discusses multivariate bounds using indicator functions, and explores linear programming for bivariate upper and lower bounds. The second half addresses bounding results and applications of multivariate Bonferroni-type inequalities. The book shows how to construct new multiple testing procedures with probability upper bounds and goes beyond bivariate upper bounds by considering vectorized upper and hybrid bounds. It presents an optimization algorithm for bivariate and multivariate lower bounds and covers vectorized high-dimensional lower bounds with refinements, such as Hamilton-type circuits and sub-Markovian events. The book concludes with applications of probability inequalities in molecular cancer therapy, big data analysis, and more.

Focuses on fields such as consensus and voting theory, clustering, location theory, mathematical biology, and optimization that have seen an upsurge of exciting works over the years using discrete models in modern applications. This book discusses advances in the fields, highlighting the approach of cross-fertilization of ideas across disciplines.

This book contains selected chapters on recent research in topology. It bridges the gap between recent trends of topological theories and their applications in areas like social sciences, natural sciences, soft computing, economics, theoretical chemistry, cryptography, pattern recognitions and granular computing. There are 14 chapters, including two chapters on mathematical economics from the perspective of topology. The book discusses topics on function spaces, relator space, preorder, quasi-uniformities, bitopological dynamical systems, b-metric spaces and related fixed point theory. This book is useful to researchers, experts and scientists in studying the cutting-edge research in topology and related areas and helps them applying topology in solving real-life problems the society and science are facing these days.

This book constitutes the proceedings of the 5th International Conference on Mathematical Software, ICMS 2015, held in Berlin, Germany, in July 2016. The 68 papers included in this volume were carefully reviewed and selected from numerous submissions. The papers are organized in topical sections named: univalent foundations and proof assistants; software for mathematical reasoning and applications; algebraic and toric geometry; algebraic geometry in applications; software of polynomial systems; software for numerically solving polynomial systems; high-precision arithmetic, effective analysis, and special functions; mathematical optimization; interactive operation to scientific artwork and mathematical reasoning; information services for mathematics: software, services, models, and data; semDML: towards a semantic layer of a world digital mathematical library; miscellanea.

Reliability and Maintenance: Networks and Systems gives an up-to-date presentation of system and network reliability analysis as well as maintenance planning with a focus on applicable models. Balancing theory and practice, it presents state-of-the-art research in key areas of reliability and maintenance theory and includes numerous examples and exercises. Every chapter starts with theoretical foundations and basic models and leads to more sophisticated models and ongoing research. The first part of the book introduces structural reliability theory for binary coherent systems. Within the framework of these systems, the second part covers network reliability analysis. The third part presents simply structured maintenance policies that may help with the cost-optimal scheduling of preventive maintenance. Each part can be read independently of one another. Suitable for researchers, practitioners, and graduate students in engineering, operations research, computer science, and applied mathematics, this book offers a thorough guide to the mathematical modeling of reliability and maintenance. It supplies the necessary theoretical and practical details for readers to perform reliability analyses and apply maintenance policies in their organizations.

Fractional calculus was first developed by pure mathematicians in the middle of the 19th century. Some 100 years later, engineers and physicists have found applications for these concepts in their areas. However there has traditionally been little interaction between these two communities. In particular, typical mathematical works provide extensive findings on aspects with comparatively little significance in applications, and the engineering literature often lacks mathematical detail and precision. This book bridges the gap between the two communities. It concentrates on the class of fractional derivatives most important in applications, the Caputo operators, and provides a self-contained, thorough and mathematically rigorous study of their properties and of the corresponding differential equations. The text is a useful tool for mathematicians and researchers from the applied sciences alike. It can also be used as a basis for teaching graduate courses on fractional differential equations.

What is the shape of data? How do we describe flows? Can we count by integrating? How do we plan with uncertainty? What is the most compact representation? These questions, while unrelated, become similar when recast into a computational setting. Our input is a set of finite, discrete, noisy samples that describes an abstract space. Our goal is to compute qualitative features of the unknown space. It turns out that topology is sufficiently tolerant to provide us with robust tools. This volume is based on lectures delivered at the 2011 AMS Short Course on Computational Topology, held January 4-5, 2011 in New Orleans, Louisiana. The aim of the volume is to provide a broad introduction to recent techniques from applied and computational topology. Afra Zomorodian focuses on topological data analysis via efficient construction of combinatorial structures and recent theories of persistence. Marian Mrozek analyzes asymptotic behavior of dynamical systems via efficient computation of cubical homology. Justin Curry, Robert Ghrist, and Michael Robinson present Euler Calculus, an integral calculus based on the Euler characteristic, and apply it to sensor and network data aggregation. Michael Erdmann explores the relationship of topology, planning, and probability with the strategy complex. Jeff Erickson surveys algorithms and hardness results for topological optimization problems.