Strengthening the Linkages Between the Sciences and the Mathematical Sciences (2000) / Chapter Skim
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Appendix A: Ten Case Studies of Math/Science Interactions
Appendix A: Ten Case Studies of Math/Science Interactions
Pages 43-74
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From page 45... ...
Some years ago, ~ became interested in developing the theory for weakly nonlinear, unstable baroclinic waves as models of weather systems in the atmosphere and eddies in the ocean. The weakly nonlinear theories existing at the time followed the work of Stuart and Watson (for the Orr-Sommerfeld problem)
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From page 46... ...
It was the first time that ~ know of that the Lorenz equations were derived as a systematic asymptotic approximation to weakly nonlinear theory instead of an arbitrary, low-order, truncation of a Fourier expansion of a strongly nonlinear problem (where it usually fails as a solution to the original problem when the nonlinearity is important enough to be interesting this is the part of the Lorenz theory of convection nobody mentions)
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From page 47... ...
Applied mathematician Chris Jones of Brown University and T have been working together over the past 5 years to develop the new approach and use it to gain new insights about ocean mixing. We first met in ~ 992, when Chris was visiting the Woods Hole Oceanographic Institution (WHOl)
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From page 48... ...
Applied mathematicians do in fact receive credit for publishing in, say, the Journal of Physical Oceanography, since the work is, as a result, perceived by their colleagues as relevant. The general problem of reforming the criteria for professional recognition to accommodate interdisciplinary collaborations remains unsolved.
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From page 49... ...
Secondly, the mathematician should glean from the work some deeper mathematical problems that promise to further develop mathematical theory and to suggest future directions, enriching the profession. The collaboration that developed between me, a physical oceanographer, and Chris, an applied mathematician, owes a lot to the presence of a third party outside of either discipline the Of lice of Naval Research.
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From page 50... ...
Out of it emerged a versatile new mathematical too! waveless, which are being used by everyone from theoretical physicists and neuroscientists to electrical engineers and image-processing experts.
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From page 51... ...
And that has cycled back into the mathematical development of wavelet theory. For example, waveless led to a new mathematical understanding of the subband filtering algorithms first developed by electrical engineers.
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From page 52... ...
Consider Martin Vetterli at Berkeley. Like Meyer, he overcame his first reaction, shared by many electrical engineers, that waveless seemed but a rediscovery of their own subband filtering algorithms; when he looked at the wavelet papers more closely, he saw there was more.
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From page 53... ...
Similarly, the algorithms that underlie the fast wavelet transform were known almost only to mostly electrical engineers. Until the wavelet synthesis took place almost by accident it was not clear these ideas could be useful outside their own communities.
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From page 54... ...
Pacala, a plant and theoretical ecologist, and Levin, a theoretical ecologist and mathematical biologist, share a common interest in forest ecosystems and in the scaling problems inherent in forest dynamics. In forests, the individual is the fundamental unit of ecological interaction and the natural scale at which to make measurements on demography, dispersal, etc.
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From page 55... ...
Durrett and Levin, 1994; Neuhauser and Pacala, 1998~. The mathematical work includes the study of connections between the behavior of spatial stochastic models and the behavior of differential equations.
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From page 56... ...
1997. Mathematical and computational challenges in population biology and ecosystems science.
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From page 57... ...
Examples of effective interactions between mathematicians and epidemiologists can be found in the study of sexually transmitted diseases such as gonorrhea and AIDS. In the late ~ 970s and early ~ 980s, mathematicians Jim Yorke (University of Maryland)
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From page 58... ...
A more recent example at the interface of mathematics and epidemiology is the collaborative work of epidemiologist Jim Koopman, mathematician Car! Simon, and physiologist/mathematical modeler John Jacquez on the transmission of the human immunodeficiency virus (HIV)
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From page 59... ...
1997. Mathematical and computational challenges in population biology and ecosystems science.
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From page 60... ...
For example, some years ago, he started a collaboration with an engineer, John Thwaites, from the University of Cambridge, whose expertise in fiber science enabled them to undertake detailed studies of the basic mechanical properties of the bacterial filaments (Thwaites and Mendelson, ~ 991~.2 Mendelson's collaboration with mathematicians started about 6 years ago with the arrival at Arizona of Michael Tabor, who had been recruited from Columbia University to become head of the University's renowned Applied Mathematics Program. Tabor was keen to develop interactions between biologists and mathematicians and had a personal interest in elasticity theory.
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From page 61... ...
1997. Nonlinear dynamics of filaments Il: Nonlinear analysis.
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From page 62... ...
1997. Nonlinear dynamics of filaments IlI: Instabilities of helical rods.
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From page 63... ...
We found that coating DNA knots and links with a particular protein made the path of the DNA traceable by electron microscopy. For the first time one could determine the topology of DNA knots and links.
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From page 64... ...
lames White answered my letter and said that topologists could indeed solve my problems. He introduced me to the world of topological invariants and to topologists including Ken Millet, Lou Kauffman, Vaughan Jones, and De Witt Sumners who worked with them.
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From page 65... ...
We went to DeLill and said we were ready to submit our application to NSF, but she wisely counseled us to broaden out beyond DNA topology and geometry. In the next phase we brought Eric Lander, Mike Waterman, Gene Lawler, and Mike Levitt into the group.
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From page 66... ...
, Mike Waterman (University of Southern California) , and Jim White (University of California, Los Angeles)
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From page 67... ...
My own doctoral thesis in 1990 focused on the challenge of making simulated annealing calculations faster using parallel computers. After completing my doctorate, ~ worked for several years on a variety of scheduling problems in parallel computation in an industrial research laboratory before making the transition to another area-computational biology where mathematics and science come together in a rich intersection.
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From page 68... ...
One barrier centers on how research in computational biology should be evaluated and rewarded. Can a computer scientist in a computer science department receive tenure if she publishes a significant part of her research in biology journals?
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From page 69... ...
Stephen Berry Department of Chemistry University of Chicago Spectra of Nonrigid Molecules The spectra of molecules-the pattern of wavelengths the molecules absorb or emit- reveal to us the structures of molecules and how they rotate, vibrate, and change their shapes. Interpreting molecular spectra is not always straightforward, and sophisticated mathematical tools frequently come into play when chemists extract the hidden messages of such spectra.
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From page 70... ...
The research turned out to be remarkably successful. The scientific knowledge of the professor and postdoctoral associates combined well with the graduate student's ability to learn and master new mathematics quickly.
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From page 71... ...
Doob worked out much of the interrelationship between martingale theory and potential theory, and the role that Brownian motion plays in the solution of certain partial differential equations. He summarized this research and provided extensive historical commentary in his 800-page book Classical Potential Theory and Its Probabilistic Counterpart, published in 1983.
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From page 72... ...
Martingale theory also plays an important role in stochastic control theory. Martingales have continued to serve scientists in a variety of ways that have circled back into the mathematical sciences.
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From page 73... ...
The interaction between mathematical science and the other sciences catalyzed by martingale theory clearly has borne much fruit for all parties involved.
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