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First-passage percolation (FPP) is a fundamental model in probability theory that has a wide range of applications to other scientific areas (growth and infection in biology, optimization in computer science, disordered media in physics), as well as other areas of mathematics, including analysis and geometry. FPP was introduced in the 1960s as a random metric space. Although it is simple to define, and despite years of work by leading researchers, many of its central problems remain unsolved. In this book, the authors describe the main results of FPP, with two purposes in mind. First, they give self-contained proofs of seminal results obtained until the 1990s on limit shapes and geodesics. Second, they discuss recent perspectives and directions including (1) tools from metric geometry, (2) applications of concentration of measure, and (3) related growth and competition models. The authors also provide a collection of old and new open questions. This book is intended as a textbook for a graduate course or as a learning tool for researchers.
This book constitutes the refereed proceedings of the 20th International Conference on Analytical and Stochastic Modelling and Applications, ASMTA 2013, held in Ghent, Belgium, in July 2013. The 32 papers presented were carefully reviewed and selected from numerous submissions. The focus of the papers is on the following application topics: complex systems; computer and information systems; communication systems and networks; wireless and mobile systems and networks; peer-to-peer application and services; embedded systems and sensor networks; workload modelling and characterization; road traffic and transportation; social networks; measurements and hybrid techniques; modeling of virtualization; energy-aware optimization; stochastic modeling for systems biology; biologically inspired network design.
These Proceedings have been assembled from papers presented at the Conference on Models of Biological Growth and Spread, held at the German Cancer Research Centre Heidelberg and at the Institute of Applied Mathematics of the University of Heidelberg, July 16-21, 1979. The main theme of the conference was the mathematical representation of biolog ical populations with an underlying spatial structure. An important feature of such populations is that they and/or their individual com ponents may interact with each other. Such interactions may be due to external disturbances, internal regulatory factors or a combination of both. Many biological phenomena and processes including embryogenesis, cell growth, chemotaxis, cell adhesion, carcinogenesis, and the spread of an epidemic or of an advantageous gene can be studied in this con text. Thus, problems of particular importance in medicine (human and veterinary), agriculture, ecology, etc. may be taken into consideration and a deeper insight gained by utilizing (more) realistic mathematical models. Since the intrinsic biological mechanisms may differ considerably from each other, a great variety of mathematical approaches, theories and techniques is required. The aims of the conference were (i) To provide an overview of the most important biological aspects. (ii) To survey and analyse possible stochastic and deterministic approaches. (iii) To encourage new research by bringing together mathematicians interested in problems of a biological nature and scientists actively engaged in developing mathematical models in biology.
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Covers the proceedings of the 1984 AMS Summer Research Conference. This work provides a summary of results from some of the areas in probability theory; interacting particle systems, percolation, random media (bulk properties and hydrodynamics), the Ising model and large deviations.