Random Explorations
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Random Explorations
Author: Gregory F. Lawler
language: en
Publisher: American Mathematical Society
Release Date: 2022-12-06
The title “Random Explorations” has two meanings. First, a few topics of advanced probability are deeply explored. Second, there is a recurring theme of analyzing a random object by exploring a random path. This book is an outgrowth of lectures by the author in the University of Chicago Research Experiences for Undergraduate (REU) program in 2020. The idea of the course was to expose advanced undergraduates to ideas in probability research. The book begins with Markov chains with an emphasis on transient or killed chains that have finite Green's function. This function, and its inverse called the Laplacian, is discussed next to relate two objects that arise in statistical physics, the loop-erased random walk (LERW) and the uniform spanning tree (UST). A modern approach is used including loop measures and soups. Understanding these approaches as the system size goes to infinity requires a deep understanding of the simple random walk so that is studied next, followed by a look at the infinite LERW and UST. Another model, the Gaussian free field (GFF), is introduced and related to loop measure. The emphasis in the book is on discrete models, but the final chapter gives an introduction to the continuous objects: Brownian motion, Brownian loop measures and soups, Schramm-Loewner evolution (SLE), and the continuous Gaussian free field. A number of exercises scattered throughout the text will help a serious reader gain better understanding of the material.
Tests and Proofs
Author: Martin Gogolla
language: en
Publisher: Springer Science & Business Media
Release Date: 2011-06-22
This book constitutes the refereed proceedings of the 5th International Conference on Tests and Proofs, TAP 2011, held in Zurich, Switzerland in June/July 2011. The 12 revised full papers presented together with 2 invited papers were carefully reviewed and selected from 27 submissions. Among the topics covered are model checking, testing systems, test generation, symbolic testing, SAT solvers, SMT solvers, property-based testing, automated test generation, learning-based testing, UML, OCL, specification-based testing, and network testing.
Complexity
Author: Melanie Mitchell
language: en
Publisher: Oxford University Press
Release Date: 2009-04-01
What enables individually simple insects like ants to act with such precision and purpose as a group? How do trillions of neurons produce something as extraordinarily complex as consciousness? In this remarkably clear and companionable book, leading complex systems scientist Melanie Mitchell provides an intimate tour of the sciences of complexity, a broad set of efforts that seek to explain how large-scale complex, organized, and adaptive behavior can emerge from simple interactions among myriad individuals. Based on her work at the Santa Fe Institute and drawing on its interdisciplinary strategies, Mitchell brings clarity to the workings of complexity across a broad range of biological, technological, and social phenomena, seeking out the general principles or laws that apply to all of them. Richly illustrated, Complexity: A Guided Tour--winner of the 2010 Phi Beta Kappa Book Award in Science--offers a wide-ranging overview of the ideas underlying complex systems science, the current research at the forefront of this field, and the prospects for its contribution to solving some of the most important scientific questions of our time.