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Materials from Mathematics

By Richard D. James
Like much of mathematics, the mathematical study of materials begins with Euler [7], or perhaps with Hooke’s models of crystals as periodic arrays of balls ([8], Schem. 7). Some readers might know of a more recent historical touchstone, the N – 6 rule of von Neumann and Mullins [11, 16]. Scientists trained on both sides of the increasingly blurred line between mathematics and materials science have been attracted by the striking beauty of microstructure, the extreme nonlinearity, nonconvexity, and even nonexistence exhibited by theories of materials, and the surprising links between the atomic structure of materials and a host of mathematical subjects, including  geometry, calculus of variations, partial differential equations, group theory, graph theory, topology, and harmonic analysis. Mathematics is now guiding the discovery of materials using principles that in some cases run counter to accepted beliefs in materials science, and materials are inspiring new mathematics. 
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Periodic Table of the Finite Elements

By Douglas Arnold, Anders Logg
The finite element method is one of the most powerful and widely applicable techniques for the numerical solution of partial differential equations and, therefore, for the simulation of the physical world. First proposed by engineers in the 1950s as a practical numerical method for predicting the deflection and stress of structural components of aircraft, the method has since been continuously extended and refined. It is now used in almost all application areas modeled by PDEs: solid and fluid dynamics, electromagnetics, biophysics, and even finance, to name just a few.
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Emerging Disease Dynamics

By Sherry Towers, Oscar Patterson-Lomba, Carlos Castillo-Chavez
Sir Ronald Ross introduced the first mathematical model for the transmission of malaria in 1911; this was the de facto creation of the field of mathematical epidemiology as we know it today. Kermack and McKendrick formulated the classic Susceptible, Infected, Recovered (SIR) compartmental model of the spread of disease in 1927. In the ensuing decades these models have been expanded in the broader context of host–parasite dynamics and disease evolution into the robust field of mathematical epidemiology [2, 7].
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"Shareability Networks"

By Paolo Santi
The burgeoning “sharing economy” phenomenon––the collaborative consumption of shared resources made possible by the pervasiveness of information technologies and Internet connectivity [4, 6]––is rapidly taking hold in the context of urban transportation. Business is booming for vehicle-sharing companies, such as Zipcar and Car2Go, and for companies that offer ride-sharing services, such as Bandwagon and Uber, which recently launched the new UberPool application. New sharing services/companies are popping up in cities worldwide.
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Opportunities at the Mathematics/Future Cities Interface

By Peter Grindrod, Desmond J. Higham, Robert S. MacKay
More than half of the world’s population lives in cities, a proportion that is estimated by the World Health Organization to reach 60% by 2030 and 70% by 2050. Thanks to the proliferation of smart devices and interconnected services, cities are gushing data, much of it related to human behavior. City life generates data streams around online social media, telecommunication, geolocation, crime, health, transport, air quality, energy, utilities, weather, CCTV, wi-fi usage, retail footfall, and satellite imaging. The powerful new concept of urban centres as “Living Labs” is inspiring novel research that could lead to improved well-being and economic growth.
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By David H. Bailey, Jonathan M. Borwein
“Experimental mathematics” has emerged in the past 25 years or so as a competing paradigm for research in the mathematical sciences. Challenges in 21st Century Experimental Mathematical Computation, an exciting workshop held at ICERM (the Institute for Computational and Experimental Research in Mathematics), July 21–25, explored emerging challenges of experimental mathematics in the rapidly changing era of modern computer technology. We summarize the workshop findings in this article; information about the research presentations can be found at http://icerm.brown.edu/tw14-5-cemc/.

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The late William Davidon, longtime professor of mathematics and physics at Haverford College, may be known to readers of SIAM News for his part in the DFP (Davidon–Fletcher–Powell) family of quasi-Newton methods. Those familiar with the SIAM Journal on Optimization might remember him as the author of the first paper––actually a reprint of his never-before-published 1959 Argonne report––in the first (February 1991) issue of the journal.
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September 08, 2014 To December 05, 2014

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