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l ~ Applied Mathematical Sciences Fields of mathematical science outside the core are of various kinds. Both computer science and statistics have dual sources of identity and intellectual force, only one of which is mathematical; hence they are more accurately described as partly mathematical sciences. These two already apply to almost as wide a variety of activities as does mathematics itself. Computer science is both a mathematical science and something else. In its present form it could not exist without mathematics. But it would be utterly unproductive without a piece of machinery- the computer, both as we have access to it at the moment and as we can envisage it in the future. Modern statistics could not operate without mathematics, espe- cially without the theory of probability. Equally, it could not exist without the challenge of inference in the face of uncertainty and the stimulus of the quantitative aspects of the scientific method. Like computer science, statistics is both a rr~athematical science and something else. At the other extreme are herds mathematical economics, mathe- matical psychology, mathematical linguistics, among others that deal with the mathematical aspects of rather specific areas. At an intermediate level of breadth and identity stand fields that have derived their nature from some area of application, usually a quite broad one. These fields have initially chosen their tools to meet the application area's problems but have grown and developed far enough to have a mathematical character of their own, thereby gaining a certain independence of existence and 84

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Applied Mathematical Sciences 85 breadth of application they could not otherwise have had. The outstanding example is physical mathematics, or classical applied mathematics, which has grown out of the mathematics of classical physics to a point where it deals with a wide variety of applications of mathematical analysis. A similarly intermediate field is today growing rapidly in an area stimulated by applications to pure and applied economics and to management and the conduct of oper- ations of all kinds business, governmental, and military. As yet there is little agreement about this held's name, but its concentra- tion on problems of optimizing any or all of allocation, control and decision is so evident that we can label it accordingly. 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. The description will be followed by a few general comments on related areas. Both liquids and gases (neutral or ionized) are fluids. The study of fluid motion is, strictly speaking, a part of physics. It has been left, however, for many decades now, largely in the care of applied mathematicians, engineers, astrophysicists, and geophysicists (in- cluding meteorologists and oceanographers). Applied mathematics has found in the study of fluid motion some of its most spectacular triumphs and also has suffered some of its, hopefully temporary, notable defeats. Attempts to describe the motion of fluids mathematically are almost as old as mathematical analysis. The early history of fluid dynamics is associated with names such as Bernoulli, Euler, and D'Alembert. Classical fluid dynamics led to some of the most im- portant developments in mathematics and to techniques and ideas that have found many other applications. Riemann contributed to the basic theory of propagation of strong compression waves. His ideas led to the development of the notion of "characteristics," which greatly influenced the theory of partial differential equations. Much of Hadamard's work was concerned with wave propagation. All this work laid the basis for many of the recent developments in the theory of supersonic flow arid shock waves, which occur on the

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86 The State of the Mathematical Sciences earth (sonic boom associated with supersonic aircraft) as well as in the stars (novas and supernovas). Much of the classical fluid dynamics of incompressible flow seemed originally to be completely useless as a model for physical reality. For instance, D'Alembert proved that a body moving through fluid experiences no drag. Of course, D'Alembert's deriva- tion neglected viscosity (the tendency of one part of a fluid to pull adjacent parts along with it); but his result was contrary to experi- ence even in cases where viscosity effects seemed to be very insig . Decant. 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. Boundary-layer theory led, by the way, to important developments in the general theory of differential equations. Compressibility of air was neglected in aerodynamics that proved sufficient for the engineering needs of the period up to World War II. Interest in high-speed flight required that compressibility be taken into account and that the essentially nonlinear theory of gas flow be faced. While certain mathematical points in that theory are still unexplained, today sufficient theoretical understanding is avail- able for practical needs. The "sound barrier" turned out to be largely imaginary, but the high altitudes achieved in rocket flight necessitated the creation of a new type of fluid dynamics, the dy- namics of rarefied gases. This theory contributed to the successful solution of the re-entry problem for earth satellites. A young and very active discipline of modern fluid dynamics is magnetohydrodynamics, which describes the interaction between electromagnetic fields and conducting liquids or ionized gases (plasmas). We can hope that this subject will contribute as much to the future course of mathematics as did the individual subjects of electromagnetic theory, fluid dynamics, and statistical mechanics. Magnetohydrodynamics has applications to astrophysics (stellar atmospheres and interstellar matter), to the search for a controlled thermonuclear reaction, and to a promising use of ionized-gas "armatures" in generating electric power. The striking spiral struc- ture of manY galaxies (including our own) can now be understood

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Applied Mathematical Sciences 87 in terms of magnetohydrodynamics and statistical dynamics of stellar systems. Here the very old subject of self-gravitating fluids, studied by MacLaurin, Jacobi, Riemann, and Dedekind in another connection, plays a predominant role. The behavior of stellar systems and rarefied plasmas has been known to be governed by a type of "collisionless Boltzmann equation" and to exhibit "collective behavior." On a still larger scale, the distribution of matter in the universe is often treated in terms of fluids governed by Einstein's general theory of relativity. The question of the existence of gravitational radiation offers chal- lenging nonlinear partial differential equations begging for solu- tion. In recent years, there has been a great upsurge of the study of fluid mechanics for the purpose of describing phenomena in geo- physics (meteorology and oceanography), where the rotation of the earth often plays an essential role. Much of the work was done by applied mathematicians; Von Neumann was among the pioneers in the use of numerical methods for weather prediction (see An Example from the Environmental Sciences: Numerical Weather Prediction, page 108~. Problems of hydrodynamic stability, on which work is still flourishing, occur in a variety of ways. 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 ob- served by anybody watching the disintegration of the smoke jet from a factory chimney or from a cigarette. Although some of the best minds in physics and mathematics have applied themselves to this problem and have achieved some partial successes, there is at the present little hope for a satisfactory solution in the near future. It is now clear that the existing studies and the still unresolved problems both relate to the broader area of nonlinear random processes. At the same time, the theory of the fundamental partial differ ential equations governing the motion of incompressible viscous flow has not been fully developed. It remains a challenging and interesting problem to pure mathematicians and a source of much current stimulation for work. As mentioned above, there is a similarity between the behavior of stellar systems and that of rarefied plasmas in exhibiting collective behavior. The similarity is not only in the basic ideas but also, sur- prisingly, in the details. Concepts used in fluid mechanics are also

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88 The State of the Mathematical Sciences useful when continuum mechanics deals with other material- elastic solids, plastic solids, and bodies exhibiting both flow and elastic properties. The study of the basic laws of continuum mechanics depends heavily on the concepts of "covariance" (i.e., the assertion that physical laws would remain in the same form no matter what observer is making the description). Indeed, the concepts and methods that originate in one branch of applied mathematics are often applicable to other branches. 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 proc- esses, reversible versus irreversible processes, entropy and one can name many more are certainly concepts whose applications are general. The mathematical theory of economics makes use of several of them. The new science of information theory, with its fascinating coding theorems and important applications to practical problems in communication engineering, also uses the concept of entropy. These observations demonstrate that much of classical applied mathematics has a tendency to merge into a coherent whole across traditional scientific disciplines, adding to the strength and useful- ness of the entire area of activity. STATISTICS Statisticians easily recognize other statisticians as such, though they are often surprised by the fields in which they work and the tools that they use. It is now almost three quarters of a century since Karl Pearson turned from graphical mechanics to the application of algebra and differential equations to problems arising in the analysis of experimental and observational data. The mathemati- cal theory of probability, now considered part of core mathematics, has applications to the "adjustment of data," first in surveying and astronomy and then in many other fields, going back to the work of Gauss and Laplace a century earlier. A variety of traditions have merged in contemporary statistics, not only the application of the theory of probability and the more quantitative aspects of scientific method but the numerical skills of the economist and the meteorologist and the data-handling skills

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Applied Mathematical Sciences 89 that have been served by one form or another of automation since the invention by Hollerith of the punched-card machine in the last century. Today, statisticians are concerned with how it would seem we might well make inferences, how we do make inferences, and how we might and do collect, process variety of fields. 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. Agriculture, biology, and medicine; insurance and actuarial techniques; indus- trial production and market research; psychology, sociology, and anthropology; all have benefited. Once statistics was thought of as confined to the study and manipulation of large masses of data. Such problems are still among the most challenging, but the problems of doing a good job of con- verting three or five or ten numbers into suitably qualified conclu- sions are today at least equally important. Individuals, corporate bodies, governments, scientists, engineers, economists all use a wide variety of statistical techniques to assess "what the data say" and thus to guide them in their deliberations and actions. Wherever the evidence is less than complete or bears only indirectly on the point at issue in truth, almost everywhere-there is a place for statistics in untangling the data and assessing the conclusion. Karl Pearson, "Student" (W. S. Gossett), and R. A. Fisher took leading parts in the early development of a mathematical statistics of wide application. During the years between the two world wars, new statistical methods and new methods for arranging experiments made major contributions to the development of agricultural tech- nology. (One of the three leaders of the first American seminar on R. A. Fisher's analysis of variance was later president of the Pioneer Hybrid Corn Breeding Company.) Statistical techniques made our agricultural progress possible, while the needs of agricultural ex- perimentation called whole areas of statistics into existence. Beginning in the early 1920's, farsighted men at Bell Telephone

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JO The State of the Mathematical Sciences Laboratories recognized the importance of statistics' potential in industrial production. Walter Shewhart founded and developed the theory of statistical quality control, and Harold Dodge led in the development of sampling schemes for inspecting mass-produced items. Both of these techniques are sometimes given major credit for significant contributions to the success of the Allied production effort in World War II. In particular, statistical quality control was a major factor in the early success of the plutonium-production process at Hanford. After the war, the application of statistical techniques to the experimental improvement of industrial produc- tion processes made outstanding progress, particularly in connection with chemical processes. Stimulated by Student and Shewhart, i. Neyman and E. S. Pear- son developed the first of the modern mathematical theories of formally optimized inference in the latter 1930's. During the same years and in the decade that followed, the mathematical theory of sampling developed to meet the needs of studying large groups of people, farms, business firms, and many other things by observing relatively small samples. The sample survey thus became effective, efficient, and irreplaceable. Wartime needs for even more efficient inspection led Abraham Wald, once a pure mathematician, in one sleepless night, to the basic theory of sequential analysis and seeded his later work on statistical decision theory (to which a brief introduction is given in the essay by Kiefer in reference 7~. The needs of a variety of prob- lems in geophysics (waves on the ocean, earthquakes, gusts that disturb airplanes) and in engineering (e.g., tracking radars) de- manded new tools to isolate phenomena tagged by their frequencies of oscillation. Statistical spectrum analysis developed rapidly, spreading into a wide range of applications, including recent con- gressional inquiries into the proper adjustment of economic series, such as unemployment, for the season of the year. The implications of psychological and educational testing have often been discussed and will be discussed many times again, demonstrating the ever-increasing importance of these techniques in our lives. The selection of test material and the study of indi- vidual 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) complexities

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Applied Mathematical Sciences 91 requiring explicit treatment of many variables and (~) limitation of available data almost to the point where there are not enough to obtain meaningful answers. As a result, econometricians have developed, and have stimulated others to develop, another broad area of statistical technique econometrics. For instance, important issues in the recent Federal Communications Commission hearing about telephone rates came down to questions of statistical tech- nique and statistical appropriateness. Most areas of biological and medical research have come to de- pend on statistical techniques. Experimenters routinely use methods that grew up in agricultural research, supplemented by special statistical techniques appropriate to the effectiveness of drugs and poisons. The study of epidemic and occupational disease calls on other techniques and poses very difficult problems. The study of medical and surgical performance is an even more difficult field. The recent national study of one of the anesthetics most used for surgery fortunately showed certain specific suspicions to be unfounded. Yet it turned up suggestive differences between postoperative death rates that may prove to be associated with the anesthetics used. These apparent differences could not have been seen without the use of statistical techniques not in existence when the study began; the untangling of their true nature and causes will require both better statistical techniques and better data, each of which stimulates the other. The arrival of modern electronic computation opened many new opportunities in almost every field of statistics. Routine sea- sonal adjustment of economic time series no longer calls for an expert; a dollar's worth of computer time does it. The periods of the natural vibrations of the earth would never have been measured without computer processing, and we would have known much less about the earth's interior. 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 sci- ence, and a behavioral science, statistics continues to be a mathe- matical science. Its health depends on effective exchanges of results and problems with workers in a wide variety of fields. The increas- ing variety of applications has increased the needs for the inven- tion of new techniques appropriate to new problems. Throughout the areas of statistical technique that have been developed over the last four or five decades there are great needs for modernization.

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92 The State of the Mathematical Sciences There is a need to develop probabilistic models for new phenomena, to study the properties of these models, and to investigate statistical procedures appropriate for them. Most formal optimization of statistical methods has been carried out for classes of distributions that are too narrow (normal or Gaussian) or too broad (all con- tinuous distributions) to correspond satisfactorily to Practice. The :~ . r .1 . . ~ ~c o~ ~e computer nor one on what is noccibl~ half ~1~, an what is reasonable in statistical techniques has been only barely thought about and only in very special cases. 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. These are but four of many direc- tions in which progress is being demanded. '-In ^^ ~ 7 ~^ ~ ~ ^~ ~ ~ ~ ~1 ~ ~ 1_ ~ I at # ~ J I I COMPUTER SCIENCE Automatic high-speed computing is scarcely 20 years old. Neverthe- less, it already constitutes one of the nation's most valuable scientific and economic resources. Until automatic electronic digital computers first became com- mercially available around 1950, most persons took at least 10 sec- onds to multiply two moderately large numbers. With today's elec- tronic computers, the same multiplication can be done nearly 10,000,000 times faster. In comparison, the speedup in travel be- tween walking and going by jet is by a factor of approximately 100, while that in communication between sound and radio waves is by a factor of approximately 1,000,000. These changes in rates of transportation and communication have completely remade the world. The even greater speedups in information processing are remaking our world again. It will be many years before our capacity to exploit the new computers fully will catch up with even their present capabilities. Meanwhile, rapid advances in speed and in the capacity of computers to store information continue. The development of computing is accelerating the penetration of mathematics into old and new fields of human endeavor. In fact, the original purpose for developing computers was the numerical solution of mathematical problems. The success achieved in this direction has been phenomenal. Calculations that required years

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Applied Mathematical Sciences 93 of effort by teams of people can now be done in a matter of minutes. Mathematical problems that were thought inaccessible to effective numerical treatment are now solved as a matter of routine. 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 calcula- tion, has been revitalized by the advent of the computer but has so far been unable to keep pace with technological developments. (See Davis' essay in reference 7.) Computers vastly increased the im- portance of parts of numerical analysis and made others completely obsolete. Carrying out a stupendous number of arithmetic oper- ations in solving a single problem introduced a host of difficulties associated with the limited precision and range of computer num- bers. (See the essay by Forsythe in reference 7.) Meeting these diffi- culties has led to an array of new and powerful algorithms for solv- ing mathematical problems. It is important to realize the variety of fields in which computing has become an important tool. One of these is mathematics, but this is a relatively minor fraction of the total volume of computing done today. Others include experimental and theoretical physics, business- data processing, economic planning, library work, engineering de- sign (from transportation systems to computers themselves), edu- cation, inventory management, police operations, space science, musical performance, and content analyses of documents. One can speak without reservations of a computerization of our culture that is already broader, though less deep, than its mathematization. The modern computer can do many things. The logical possibil- ity of building a universal computing machine that in principle can do anything any computing machine can do was first recognized, as a theorem in mathematical logic, by A. M. Turing. This discovery influenced the work of the mathematician Von Neumann, who contributed the crucially important suggestion that computers should store their instructions together with their data. The practical importance of computing to our society can be judged from the direct annual cost to the federal government of acquiring and operating electronic computers in 1967. It was in the range of $2 billion. (Source: The New York Times, 25 January 1966, p. 21, quoting the President's budget message.) The near-universality of a modern computer means that it can

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94 The State of the Mathematical Sciences perform almost any manipulation of symbols. The breathtaking future perspectives to which this may someday lead are described in an essay by J. T. Schwartz on nro.snertc nit r~mn,~t~r criP=rp 7 TO _~.:_ 1~ --- ~~~-r~ ~. ... ,~ Purr, computers can be used for direct simulation of complicated technological, natural, and social phenomena, where computer instructions play the role of formulas in setting up a formal model. Such models are as truly mathematical as those ex- pressed in formulas. Lists of computer instructions can be long and interrelated in complicated ways and still be executable. 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 high- way traffic control, the design of telephone networks, and the design of concrete shields for nuclear reactors. lust as zoology is the study of animals and animal behavior, so computer science is the study of computers and computation. Three aspects are currently of outstanding importance: 1. The design and analysis of computer hardware the com- ponents and total electronic and mechanical systems that comprise computers; 2. The design and analysis of computer software- the basic languages and resident programs essential to convert bare hardware into productive computing systems, including control programs, compilers, and time-sharing executives; 3. The methodology of solving problems with computers- those techniques that are common to solving broad classes of problems, as opposed to the preparation of individual programs to solve single problems. One of these techniques is the appropriate representation of complex information. Today, computer science is, among other things, a mathematical science. As Professor Allen Newell describes it in his memorandum to COSRIMS: Computer science shares with mathematics a concern with formalism and a concern with the manipulation of symbols. It also shares with mathematics the role of handmaiden to all of science and technology. It shares with elec- trical engineering the concern with the design and construction of informa- tion processing systems that accomplish ends. It shares with all of engineering a concern with the process of design,

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Applied Mathematical Sciences 95 considered as an intellectual endeavor. It shares with linguistics a concern with language and communication. It shares with psychology a special con- cern with forms of information processing that result in intelligent behavior, broadly viewed. It shares with the library sciences a concern with how to store and retrieve large amounts of information, either as documents or as facts. It has both theoretical aspects, as in the study of automata, and ex- perimental ones, as in the discovery of new types of systems through pro- gramming them and exploring their behavior. All of these shared problems with other parts of science and technology imply that the future status of computer science is still indeterminate. It may permanently become one of the mathematical sciences. It may become an autonomous science, such as geology. The result is genuinely ire doubt, because "science" is a social con- struct, shaped as much by social forces as by anything intrinsic to its subject matter. Lying as it does across the research fields of mathematics, elec- tronic engineering, linguistics, psychology, among others, computer science, though still somewhat formless and unfocused, is also an independent entity. 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, com- puter science emphasizes the constructive, problem-solving, algo- rithmic aspect of mathematics, in contrast with the structure aspects, often emphasized in core mathematics. As a young field that must respond to soaring demands for knowledge at all levels of speciali- zation, computer science can use the cooperation of mathematicians in creating a solid body of knowledge. Computer science is at once abstract and pragmatic. The focus on actual computing systems introduces the pragmatic component; the central questions are economic ones like the relations among speed, accuracy, and cost of a proposed computation, and the hardware and software organization required. The often better understood theoretical questions of existence and computability provide an important conceptual basis for the study of the more pressing ques- tions. (And these in turn have led to new mathematical research in such questions as degrees of finite computability.) On the other hand, computer science deals with information in an abstract way. The meanings of symbols and numbers may change from appli- cation to application, just as they do in every application of mathe- matics. Thus computer science shares the main goal of mathe- matics to create a basic structure in terms of inherently defined concepts that is not bound to any particular application. Computer

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96 The State of the Mathematical Sciences scientists have barely begun the creation of such a basic structure and are still mainly concerned with exploring what computers can and cannot do economically. So far the most mathematical components of computer science are numerical analysis and the theory of programming languages. The first computers were programmed in machine language, a language that might be likened to microsyllables of human speech. Typical of machine language is an instruction saying in effect: "Add the number in cell 4565 to the number in arithmetic register A, and leave the answer in register A." The solution of substantial mathe- matical problems in terms of such instructions expended many man- months of tedious, repetitive human labor. After some years, it was realized that computers, with their infinite capacity for carrying out details, could themselves be instructed to translate an algebraic 7'r ~7 are / ~ rr ~ ; ~ ~ ~ ~ ~ _~: _ ~ 1 ~ ~ ~ ~ _ _ ~, ~ ~ . 'ww<~ ^~ `~ACL~11111~ `"ll~"a~C. I 111~.~; Ilil`~l;ln n,~ln OCR for page 84
Applied Mathematical Sciences 97 require the combined skills of mathematicians, operations analysts, computer scientists, and engineers, since extremely complex prob- lems are involved. This is a field of very active research in computer science today. In addition to the reasonably successful methods developed for numerical computation and programming languages, algebraic (nonnumerical) methods for some classes of problems are now emerging. Methods for handling pictorial data have been formalized for large classes of pictures. There still remains an almost unlimited array of topics that are not being treated systematically. This is the frontier of computer science. It includes a large number of non- numerical problems, as well as such problems as executive control (scheduling, allocation, handling, interruptions) and simulation of intelligent behavior. OPTIMIZED ALLOCATION, CONTROL, AND DECISIONS The needs of economic theory for results about allocation, those of management practice for approaches to decisions (which can usually be cast in terms of allocation), 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. In almost every case, these problems can be described in terms of maximizing or mini . . . . . . . . . . . mung be., optimizing- some criterion subject to constraints on the variables. Separately developed traditions are now being merged into a coherent body of concepts and results. Workers in the field have come to recognize its unity as more important than its diversity. The needs of economists, managers, and engineers for explicit guid- ance in choosing answers, and for as many explicit answers as pos- sible, have anchored this area close to the frontier. There is interest in a procedure that in principle describes a solution, but ordinarily only as a step toward a feasible procedure for finding actual solu- tions. Since feasibility is a matter of computation, exactly what computing tools are available is vitally important. This whole area is therefore intimately bound up with computer systems and with machine arithmetic. An interesting mathematical development in this area was Von Neumann's and Morgenstern's "theory of games and economic be

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98 The State of the Mathematical Sciences havior," which tackled the strategic and mathematical problems of .CouS optimization of rewards by two or more name Dlaver.c or economic competitors. A ~;~ ^1 ~ 4_ _ .. 1 ~ . , ~ 1 J ~ IIlalOr unread reran with linear nro~nmmin~ th'` Nil ^' V - ~ _ 1 ~-Act ~ ~-~^ ~ ~-51A VA optimizing a linear function of many variables when subject to very many simple linear inequalities. Linear programming problems were first systematically attacked in the 1930's, in Russia by Kantorovich, whose results escaped notice for many years, and in the United States at about the same time by Koopmans and Hitch- cock. The held's growth in the United States was greatly stimulated ~ n A ~ 1 ~ . ~ v 1n 1Y9/ by L)antzlgrs discovery of a method that made noccibl~ the ~1~ ~:~ ~r ~ ~ ~ 1_ _ ~ OWl"Ll~ll Ql Icy complex problems with then available com puter facilities. A burst of incll]strin1 ~nnlir~timnc crier, c~'h^~ A ~ both the practical and the theoretical boundaries of the subject. Nonlinear programming was introduced, in which the function to be maximized was generalized from a linear function of the given variables, and the constraining inequalities were also generalized. The fact that many practical problems involve units that cannot be subdivided (how many dresses can be cut from a bolt of cloth?) now demanded, and received, an effective theory of integer program mlng. ~ r~ r~~~~ ~O~V14 ~ LO ~ L~11~t A third line of development involved problems of simultaneous allocations: How can n persons be best assigned to n jobs? How can many factories best ship their products to many customers? Some al- location problems have proved to be intrinsically very difficult: What pattern of warehouse locations will minimize shipping costs? What Is one shortest route passing through every one of a list of cities? _ _ . 1_ _ _ 1~ Treated directly, such problems have a characteristic irreducible complexity, since the entire solution must be examined simultane- ously. ~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. A fourth line of development involved dynamic programming or the theory of multistage decision processes, which treats problems in which a number of decisions are made over a period of time. The maintenance of inventories in the face of uncertain demand, where

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Applied Mathematical Sciences 99 there must be repeated decisions of whether and how much to order, poses a typical problem, as do the scheduling of production, equip- ment replacement, the conversion of stored water to power, and a wide variety of queuing processes. As these lines of development continued, perspectives broadened, and the field matured, linear and nonlinear programming and dynamic programming fused together into mathematical program- m~ng. A fifth line of development, carried on vigorously both in the Soviet Union and in the United States, involved the optimum con- trol of systems reasonably thought of as tracing a path. When choosing a rocket trajectory that will require least use of fuel, the path is physical. When deciding how the temperatures in a chemical reaction vessel should change to maximize production of the desired chemical, the path is symbolic. The general mathematical problem is the same, and the same mathematical results are applicable. 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 time- with a decision every instant requires much more sophisticated mathematical techniques and much deeper mathematical results. Allocation is the immediate end of the theory of games and a consequence of most applications of mathematical programming. Control is the aim of control theory and many applications of dynamic programming. Decisions have to be made in every case. Optimization of some prescribed function is the common thread that links all these lines of development together. Opinions differ on whether the area thus described has already become a clearly identifiable field of mathematical science. If not yet, it soon will become one. OTHER AREAS The fields of mathematical science outside the core will continue to make increasing contributions. They will require deepening mathematical education and broadening mathematical literacy among their own workers. This places a great responsibility on uni- versity and college faculty in the mathematical sciences. They must

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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.