The Mathematical Sciences: A Report (1968) / Chapter Skim
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5 Applied Mathematical Sciences
5 Applied Mathematical Sciences
Pages 84-100
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From page 84... ...
Like computer science, statistics is both a rr~athematical science and something else. At the other extreme are herds mathematical economics, mathematical psychology, mathematical linguistics, among others that deal with the mathematical aspects of rather specific areas.
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From page 85... ...
FLUID DYNAMICS: AN EXAMPLE OF PHYSICAL MATHEMATICS To illustrate the nature of physical mathematics (classical applied mathematics) , we shall describe in some detail one of its typical and central subdivisions the mathematical theory of fluid motion.
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From page 86... ...
Fluid-flow theories applicable to physical reality appeared only in the nineteenth and early twentieth centuries, largely as a response to the needs of aeronautics. A major achievement was Prandtl's theory of the boundary layer, which showed in which cases one can use classical fluid dynamics to obtain physically valid results, and how in such cases the viscosity effects are limited to a small layer of fluid next to a moving body (whence the name "boundary layered.
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From page 87... ...
Yet some problems in fluid dynamics, which are almost as old as the science itself, still resist all efforts at mathematical treatment. An outstanding example is turbulence, a phenomenon easily observed by anybody watching the disintegration of the smoke jet from a factory chimney or from a cigarette.
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From page 88... ...
Once the theory is put into mathematical form, even in a form not yet deserving serious interest of core mathematics, it could be of basic importance to other branches of applied mathematics. Equilibrium, stability and instability, wave motion, linear versus nonlinear processes, reversible versus irreversible processes, entropy and one can name many more are certainly concepts whose applications are general.
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From page 89... ...
and digest data in a wide Karl Pearson's willingness to attack the dimensions of crabs, the inheritance of intelligence, and the mechanisms of evolution with mathematical techniques, however, signaled the opening of a new era, one in which new problems and mathematical theories have shared responsibility for intellectual stimulation, and where in one field after another, the statistician has done much to introduce workers in that field to new ideas and new ways of thinking as well as to new techniques. The basic ideas of quantitative scientific method, of uncertainty, and of elementary mathematical models have been brought into many fields by statisticians.
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From page 90... ...
The selection of test material and the study of individual arid interrelated behavior of tests has had to be carried out in a statistical way, demanding and receiving the development of new statistical techniques. Qualitative studies of a national economy and of the economic environment of single firms share the difficulties of (1)
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From page 91... ...
Yet most of the large-scale challenges involved in harnessing modern computing systems to the effective analysis of data have hardly been tackled. lust as it is still a methodological science, a computational science, and a behavioral science, statistics continues to be a mathematical science.
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From page 92... ...
Communicating results to people can be much more effective once we understand how to use computer-produced graphs and pictures. Adequate treatment for the nonexperimental observations of behavioral science and medicine requires rethinking of the ideas whose formalizations underlie statistical techniques.
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From page 93... ...
During any rocket launch, for instance, dozens of nonlinear ordinary differential equations are being solved in a completely automatic manner. Numerical analysis, the branch of mathematics concerned with the invention and evaluation of methods for mathematical calculation, has been revitalized by the advent of the computer but has so far been unable to keep pace with technological developments.
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From page 94... ...
If many details must be treated, examples of behavior can be obtained by direct simulation when general results cannot be obtained from similarly complicated lists of mathematical formulas. Examples of problems that have been treated by direct simulation include highway traffic control, the design of telephone networks, and the design of concrete shields for nuclear reactors.
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From page 95... ...
Both government and universities tend to view it as such, as is evidenced by dozens of university computer science departments and a separate Office for Computing Activities within the National Science Foundation. As a mathematical science, computer science emphasizes the constructive, problem-solving, algorithmic aspect of mathematics, in contrast with the structure aspects, often emphasized in core mathematics.
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From page 96... ...
By now the number of algebraic languages has grown out of hand, and research is actively going on in automating the writing of compilers by other ntn~rr~mc called compiler-compilers. ~ ~t ~ ~ ~ ~ s r ' ~ ~111~9 my us speed of processing information and making deri.sionc the electronic digital computer can be an extension of the human mind with capacities in this direction incomparably greater than those of any mental aid heretofore available.
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From page 97... ...
, and those of engineering operation for patterns of control (whether in trajectory choice or the feedback control of production machinery) all lead to a single interrelated complex of problems, concepts, and results.
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From page 98... ...
~v ~7 ~ Certain types of these problems have fortunately been found to be reducible to problems of linear and nonlinear programming and have thus become much more easily soluble. Others have yielded to special algorithms, and the remainder pose serious problems for the techniques of combinatorial analysis and the capacities of modern computing systems.
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From page 99... ...
In many ways, these problems of control theory are continuous-time analogs of the problems of dynamic programming. Both lines of development have contributed ideas to one another, with control theory more frequently taking the lead, since continuous timewith a decision every instant requires much more sophisticated mathematical techniques and much deeper mathematical results.
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From page 100... ...
100 The State of the Mathematical Sciences provide instruction in core mathematics, including its own latest and most useful concepts, approaches, and results, which alone will strain their capacities. In addition, they must provide instruction that identifies and illustrates the multifaceted role of mathematics in our society.
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