Fueling Innovation and Discovery The Mathematical Sciences in the 21st Century (2012) / Chapter Skim
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Mathematical Simulations / When the Lab Isn't Big Enough
Mathematical Simulations / When the Lab Isn't Big Enough
Pages 11-19
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From page 11... ...
That is where mathematical sciences enter the story, via computer simulation. In scores of applications, from physics to biology to chemistry to engineering, scientists use computer models -- whose construction requires the formulation of mathematical and statistical models, the development of algorithms, and the creation of software -- to study phenomena that are too big, too small, too fast, too slow, too rare, or too dangerous to study in a laboratory.
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From page 12... ...
/ FUELING innovation and discovery 12
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From page 13... ...
More generally, advances in mathematics and statistics and improved algorithms provide leapfrog advances in computational capabilities. Scholarly studies have estimated that at least half of the improvement in high-performance computing capabilities over the past 50 years can be traced to advances in mathematical sciences algorithms and numerical methods rather than to hardware developments alone.
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From page 14... ...
In cystic fibrosis the channels that are supposed to transport chlorine ions don't work correctly, possibly resulting in a buildup of fluid in the lungs; in certain kinds of heart arrhythmias, the potassium channels do not properly regulate the movement of potassium ions, which can interfere with the normal muscle contractions that create each heartbeat. At present, nobody knows how to take the chemical formula for a protein and predict the shape it will fold into.
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From page 15... ...
First, the precise but computationally intractable force field must be replaced by a good approximation based on empirical data and on simpler systems, and mathematical analysis is necessary to characterize the adequacy of the approximation. Second, computational algorithms have been developed to speed the computation of the interactions between atoms.
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From page 16... ...
For instance, the core collapse of a supernova takes milliseconds, while the crucial convection step takes place over a span of seconds and the aftermath of the explosion lasts for centuries. Spatially, the thermonuclear flame of the supernova varies from millimeters to hundreds of meters during the explosion.
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From page 17... ...
Apparent advances in raw computation speed do not translate directly, and perhaps not even indirectly, to simulations that are faster or more accurate. Today's expectation is that the high-end computers of the future will have huge numbers of very fast "cores" -- processing units operating individually at extremely high speed -- but that communication between cores will be relatively slow.
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From page 18... ...
More fundamentally, the mathematical sciences help to map the topography of the ocean floor and infer large-scale wave behavior from independent ocean tide gauges that are irregularly spaced and can be hundreds of miles apart. This knowledge is behind emergency warnings and evacuations, which help to avoid potentially devastating consequences.
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From page 19... ...
These estimates help to identify evacuation zones and routes. The impacts of tsunamis vary widely, due to local topography, long-term sea level rise, annual climate variability, monthly tidal cycles, and short-term meteorological events.
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