MATHEMATICS IN INDUSTRY - A SIAM PROJECT REPORT 1: SOME VIEWS OF MATHEMATICS IN INDUSTRY Society for Industrial and Applied Mathematics 3600 University City Science Center Philadelphia, PA 19104-2688 Prepared by Paul Davis Mathematical Sciences Department Worcester Polytechnic Institute Worcester, MA 01609ABSTRACT
The culture and the values of mathematics in business, industry, and government differ radically from those of academia, as conversations with more than forty nonacademic mathematicians vividly demonstrate.
A harsh reality is the lack of a market niche for mathematics outside of academia. Few organizations in business, industry, or government have departments that must be staffed with mathematicians. Mathematics is seldom the dominant technical discipline. Demonstrating relevance is a key to survival outside of academia.
Teamwork, communication skills, and breadth of scientific interest are valued highly in industry, in contrast to the single academic investigator working at the frontier of a subdiscipline. Critical judgment and problem solving ability are essential. Generalists can often function more productively in industry than narrower specialists, particularly when a small group of mathematicians serves a diverse clientele.
Formal training in mathematics is much less a prerequisite for employment in business and industry than in academia. Many of those trained in mathematics pursue activities that would not be classified as mathematics by many university faculty.
Industrial problems are commonly selected by relevance to corporate needs rather than by the natural evolution of classical scholarship. Good problems need not be elegant, new, or well posed, just necessary to corporate welfare.
The working environment ranges from large groups similar in size to university mathematics departments to isolated individuals. In any case, much of the work is joint, sometimes in rather large interdisciplinary teams.
These and other differences suggested by this informal study offer opportunities and challenges to the educational community. A careful- ly drawn sample of workers and employers in business, industry and government would provide an even more reliable understanding of nonacademic careers upon which to base thoughtful educational innovation.
From that improved understanding of the nonacademic environment, one could extract design principles for educational experiences that met nonacademic career needs. Specific curricula could be developed and implemented to nurture such key attributes as problem solving, communication, and teamwork. More ambitiously, genuine interdisciplinary educational experiences might yield graduates who would become agents for technology transfer. An improved understanding of mathematics outside of academia could also contribute to its broader and more productive use.
INTRODUCTION
"Where's Wallace ?" [1] is a colorfully illustrated children's book that tells the exploits of an active, inventive orangutan named Wallace. Repeatedly slipping away from the traditional bounds of his cage at the zoo, he disguises himself and appears in a variety of surprising settings, shopping in a large department store, swimming at the beach, sharing a picnic in the park, and so on. After hearing the tale of each adventure, the reader is challenged to find Wallace among the flurry of activities depicted in a colorful two-page panorama of Wallace's latest exploits.
The observer of mathematics and its industrial practitioners is much like the reader searching for Wallace. Outside of academia, mathematics appears in unusual places and disguises. It can be difficult to pinpoint in the myriad of industrial disciplines and activities. Facing different values and different demands than in academia, its practitioners develop skills that vary from those of classical scholarship. In industry, mathematics is out of the cage and into the world.
This mathematical "Where's Wallace?" report provides some glimpses of mathematics and its practitioners in industrial environments. Its vignettes illustrate the working environment of these mathematicians, their professional backgrounds, the sources of their problems, the kinds of solutions they find and the disposition of those solutions, their sources of support, the skills and attributes that underlie successful careers in industry, and the balance between demands for breadth and depth.
In addition, the observations of these practitioners illuminate the remarkable differences between the culture and values of academia and those of business, industry and government. For example, there is no assured niche for mathematicians in business, industry or government as there is in a university, which never lacks a mathematics department. When there is a place for mathematics outside of academia, it may hold only one or two isolated mathematicians pursuing work that university faculty might not even label as mathematics.
To the extent that this report suggests radical differences in values between academia on one hand and business, industry and government on the other, it mirrors similar conclusions in economics [2, 3] and in operations research [4].
The experiences upon which these observations are based are not perfectly balanced. They are the results of interviews with twenty six industrial mathematicians, visits or extended conversation with ten other groups or individuals, and briefer telephone conversations with three recent master's graduates. Five of the twenty six inter- viewed held master's degrees, the rest doctorates. Six of this group held terminal degrees in other than mathematics, applied mathematics, or statistics. Among the ten who were visited, only four held terminal degrees in mathematics. (Note: Peter E. Castro, Supervisor, Applied Mathematics and Statistics, Eastman Kodak Co. and I. E. Block, Managing Director, SIAM, joined the author for most of the interviews and for some of the visits.) By happenstance rather than design, individuals working in non-manufacturing businesses and in government agencies were largely omitted from these informal samples.
In the future, SIAM's Mathematics in Industry project will build on this informal study to reach two goals: helping mathematics faculty shape educational experiences that better prepare graduates for successful careers outside of academia and helping industry make better use of the productive potential of mathematics.
JOBS AND SKILLS
Formal Training
Formal training in mathematics is much less a prerequisite for employment in business, industry or government than in academia. Furthermore, a good deal of what those trained as mathematicians do in industry might not qualify as mathematics by the standards of many universities. Literacy in some other field of science or engineering is often essential, however.
A consultant and head of a scientific computing group at a major pharmaceutical concern has a doctorate in chemistry. A member of the research group for one of the big three auto manufactures has a Ph.D. in differential topology; others have more conventional backgrounds in engineering, scientific computing, or applied mathematics.
With a touch of hyperbole, one mathematician with both academic and industrial experience addresses industry's demand for breadth: "You need literacy in some field of science or engineering to get credibility or you need computer expertise. Unless it already had an institutionalized mathematics effort, (my company) wouldn't hire Leonard Euler without a chemistry course!"
Mathematicians who enter industry at the master's level may find themselves learning a specialty determined by their employer's needs, not by the details of their graduate training. That new specialty often has a large computational component. It may be the result of formal in-house courses or simply a natural response to the pressures of job requirements. An applied mathematician in the aircraft industry suggests that the "most successful master's graduates are those who might as well have been Ph.D. candidates... The best master's hires grow to be almost Ph.D.'s."
Hiring Mathematicians
A harsh reality is the absence of an assured role for mathematics outside of academia. Few organizations in business, industry, or government have large departments that must be staffed with mathematicians.
Demonstrating relevance is a key to survival outside of academia. For example, the paperweight on the desk of the physicist who is Director of IBM Research says, "Be Vital to IBM" [5]. A staff member in an automotive research laboratory says simply, "There is not a market niche for mathematicians."
A semiconductor manufacturer "doesn't usually hire mathematicians specifically. It hires to fill technical weaknesses -- for example, device physics and modeling. The new hire may be a mathematician, or a physicist, or an electrical engineer. We know which universities produce the right people (at the doctoral level)."
Teamwork, communication skills, and breadth of scientific interest are valued highly in industry, in contrast to the single academic investigator working at the frontier of a subdiscipline. Indeed, generalists can often function more productively in industry than narrower specialists, particularly when a small group is called upon to serve a diverse clientele. Critical judgment and problem solving ability are always essential.
When there is a conscious decision to hire a mathematician, the desired characteristics might include "a broad background and interest in a variety of mathematical areas, computation, and science in general." The other criteria this mathematician in a pharmaceutical concern had when hiring included the ability "to take a problem out of the blue" and the promise of "day-by-day professional development motivated by intellectual curiosity."
In the same vein, the technical center of a diversified metals manufacturer will look for "general problem solving skills" and "the ability to communicate across disciplines". Whether among members of multidisciplinary teams or in the consulting environment in which many industrial mathematicians work, that ability to communicate across disciplines has a multitude of professional and interpersonal dimensions: listening skillfully, translating from another discipline into mathematics, maintaining visibility, integrating personal and professional relationships, and even simple salesmanship.
Attributes of Successful Industrial Mathematicians
Much to the surprise of academic mathematicians, their industrial colleagues seldom list knowledge of specific subject matter when asked about the attributes of successful industrial applied mathematicians. Technical competence and (usually) knowledge of computation are taken for granted. The defining criteria are more cultural than purely intellectual.
In the aircraft industry, it may take six months to two years before new applied mathematicians are really contributing. It takes time to "teach them how to work in a group, to learn the corporate culture, to learn the nature of the company, and to master the nature of the problems and finding good solutions."
A prospective industrial applied mathematician must "show curiosity and the ability to penetrate." "The key is an open mind and flexibility." Such individuals need to "know about the politics of work in industry (as opposed to the academic environment in which they were trained), and they need to have more realistic expectations." In addition, they need "taste for good methods and for good problems. We are all problem solvers in industry, whether we are mathematicians or marketers."
Communication and Interpersonal Skills
One group of applied mathematicians quickly agreed among themselves, "Communication skills are key." Another mathematician observed that beyond technical skills, "You need visibility for success. You must show others how and why your ideas work."
"A mathematician needs communication skills to interact with chemists, physicists and engineers of various stripes. Some cross training helps you to get involved in problems at a much earlier stage. The cross training that's important is not in a particular discipline. It is in the ability to approach a problem with an open mind, learning to translate from other disciplines into yours."
As suggested earlier, successful industrial work demands the ability to integrate personal and professional relationships. A consulting mathematician in a chemical company first learned at a cocktail party of a colleague's problem involving an important product for the analysis of blood chemistry. The friendship with that biochemist led to sharing the problem and then to the professional interactions of model building, analysis, and interpretation.
Another mathematician specializing in computation points out the importance of early involvement with software projects, a position that depends on well developed relationships with colleagues: "A lot of intelligence about the applications must go into software. Being in on the ground floor is essential to success. The more you see of a model, the more insight you have and the more political advantage you have."
The financial facts of life can make salesmanship essential. To find new funding in the face of budget cuts, one group at a government contractor now "must make friends and let them know our capabilities."
Listening is important, not just when consulting. One applied mathematician describes a group of colleagues in a pure research organization "who were insulated from the real requirements of (the petroleum industry). They looked at real problems with disdain. They preferred model problems, and they didn't know what the (real) business needs were. (People in the field) would ask `What do I get for it? I can't use toy codes.'" This group's research charter lasted only as long as management was willing to protect it.
PROBLEMS AND SOLUTIONS
Sources of Problems
The sources of industrial problems and the definition and disposition of their solutions distinguish industrial mathematicians from their academic colleagues.
Many problems are posed to industrial mathematicians by colleagues in other disciplines, who may not yet understand the real problems they face. Good problems need not be elegant, new, or well posed, just necessary to corporate welfare.
Industrial problems are seldom selected by the natural evolution of classical scholarship. For example, two applied mathematicians (one originally trained as a chemist) at a pharmaceutical manufacturer include in their suite of problems developing a model of tumor heterogeneity. That problem was posed by the head of their laboratory.
Many mathematicians work either explicitly or implicitly in a consulting environment that can provide a natural flow of problems. Since their clients "may not yet know the real problem, small questions can grow into big problems."
An applied mathematician working with a major computer graphics manufacturer observes that the problems "don't even have to be interesting --- just necessary. If a group has hit the wall and their code release is next week, it's a good feeling when they make their deadline because you helped. You can go back to your other work with a sense of satisfaction." This same mathematician observes, "Every week I ask myself `Is my job secure from what I'm doing? Am I relevant? Am I known by others?'"
Finding the real problem
The key to a good solution is identifying the real problem that needs to be solved.
In a few cases simply pushing back the frontier, either for internal dissemination or for external publication, is enough.
In most cases, an acceptable solution is a new piece that fits nicely into a larger puzzle that a multidisciplinary team is working to solve. Relevance and the quality of the fit can determine the value of the solution. Good solutions answer the question that really should have been asked, and they often are the consequence of deep involvement with problem formulation.
Communicating the solution to the user is important. External publication, with a few exceptions, is much less highly valued than in academia and may even be restricted by corporate policy.
Occasionally, a simple one-way transfer of information may be adequate. But for one applied mathematician in the aircraft industry, who wants mathematics seen as an essential component of the company's success, that is not enough: "Simply saying, `Here's the solution' sets the customer and the mathematician apart. It doesn't build a team. It doesn't contribute to having mathematics viewed as a key discipline like structural mechanics."
A basic requirement of a good solution is understanding the problem itself. At a major chemical manufacturer, that may mean "getting on the wave length of a physical polymer chemist." For an applied mathematician with a large government contractor, understanding the problem requires "getting involved in the problem at a much earlier stage in order to capture its salient features."
An experienced consultant within a photographic products manufacturer observes that an applied mathematician "must hear the question that's really being asked. You must lead clients to see the real problems, not just dump a quick answer to the first question they ask." Others warn, "Be prepared to ask questions. (Ask the client), `What do you really want?'" "You must speak to engineers in a variety of disciplines and understand what their problems really are."
Using Solutions
An essential outcome of solving an industrial problem is communicating the solution to the customer, whether internal or external. In only a few cases does that communication include publication in a refereed journal.
In crafting a solution, mathematicians can not be insulated from the competitive requirements of their businesses. Mathematicians "can't look at real world problems with disdain or prefer model problems or not know what their company's business needs are." "Someone has got to pay the bills."
THE WORKING ENVIRONMENT
Working with Colleagues
The working environment ranges from large groups similar in size to university mathematics departments to isolated individuals. In any case, much of the work is joint, sometimes in rather large teams. The challenge of teamwork continues throughout a career, in contrast with the personally directed research path a tenured faculty member can choose to follow.
The dictates of teamwork "may mean you have to do what you don't want to do for a while." "You must have tolerance for a range of abilities and the wisdom to navigate the demands of teamwork and a diversity of personalities."
As one experienced woman makes clear, gender can be a factor in collegial relations. "Its tough for women if they are not aggressive. They must make sure people know what they did. They can't be afraid to say, `That's my idea.'" (But those recommendations appear to apply equally well to men.)
One mathematician's prescription for success describes the realities of a common working environment: "Learn how to work together in teams, have an openness of mind and people skills. Bring in customers and understand what they want. But understand that neither you nor they can know everything."
Sources of Suppor
Broadly speaking, industrial practitioners of mathematics are supported in three ways. (The rare exception is the laboratory with a pure research charter only loosely related to corporate productivity.) They may be part of a staff whose mission is directly linked to the company's product, production cycle, or service. Examples would be a mathematician developing signal processing algorithms for a defense contractor or a statistician responsible for quality control in a manufacturing plant.
The other two modes of support hinge on consulting. Those who function as consultants may be funded either directly from the corporate operating budget, often called funding from overhead, or they may be supported by billing their time to sponsors inside or outside the organization. Regardless of the source of funding and of problems, most industrial mathematicians agree that "being in the middle of the action pays."
Much like their academic colleagues who worry about the way the dean is treating their department, industrial mathematicians find their support is often closely tied to the apparent value upper management places on the contributions of mathematics. In any case, mathematics is seldom the dominant technical discipline. At the corporate laboratories of a major, diversified chemical manufacturer, "Mathematics is always in the background. It is never in front with the physical problem. It is never in the limelight."
Consequently, at one extreme, if management practices intellectual apartheid, favoring one or two disciplines above all others, then mathematicians are best hidden under the cloak of some other discipline. In happier situations, continuing efforts to point out the concrete contributions of mathematics can coincide with competitive forces to strengthen the support for mathematics. One major chemical company, for example, is expanding its group of mathematical consultants because a competitive analysis has shown that it can no longer afford unguided, Edisonian build and bust experiments. More intelligent modeling must inform its experimentation.
DIFFERING VALUES
Exploiting the Trivial and Training Others
What the academic may scornfully dismiss as trivial, the industrial mathematician may need to exploit fully. Trivial problems can be important because they allow demonstrations of success and because their solution can build bridges to more important problems. An applied mathematician in the computer industry says, "You need a Mickey Mouse project where you can quantify progress."
Assisting in the solution of easy problems also provides opportunities to train new users, and hence additional advocates, of mathematics. For one internal consultant in a diversified chemical manufacturer, a chemist's request for help with the numerical solution of a system of seven ordinary differential equations was the beginning of a productive relationship that led to more challenging mathematics and significant contributions to profitable products. The easy response for the mathematician would have been "That's trivial. Use one of the packages in the computer center." But that answer would have pushed the chemist back across the disciplinary divide. An applied mathematician in a group developing software tools for engineering colleagues strikes a middle ground. "We usually provide the tools rather than teach the mathematical details. But for users to know when they have a mathematical problem, they need either good intuition of their own or salesmanship from our group." A mathematician in pharmaceuticals makes a case for developing "math awareness" but acknowledges that "we will hand-hold when the value is there," that value being the development of another advocate for mathematics.
The relationships with professionals from other disciplines cover a spectrum of communication, teaching, indoctrination, and selling. A key to building the case for mathematics and for training nonmathematicians is openness and a participatory style. An applied mathematician working in computer graphics argues, "You have to use their language. It helps to talk out loud. They can understand the process you are going through. They can see that abstraction and turning back to the original context of the problem can be useful. People like to listen to how you think. You can't be just a black box. They must understand your solution."
An applied mathematician in the pharmaceutical industry argues that the statistical consulting model is not appropriate for most other forms of industrial mathematics. A statistical group that does "over the wall problem solving" can find itself with "high-priced people computing means because they don't train their users." In other settings it may be appropriate to "train the users, both as advocates for the solution itself and as advocates for mathematics in general."
The matter of training colleagues can go far beyond simple interchange of technical information. In one government contract laboratory, for example, a computational implementation of a complex model of ground water flow and contaminant behavior is the basis for policy decisions affecting the safety of community water supplies many centuries into the future.
The final results of this modeling will guide decisions by government executives with no conception of the power or limits of mathematical modeling. In this setting, applied mathematicians become advocates for their profession as much as technical specialists. Nothing can be dismissed as trivial.
Another dimension of training is technology transfer. The importance of abetting effective integration of new ideas can not be dismissed quickly. Many users of mathematics are alert to new ideas, but "the technology transfer can't be one-way. You want the user to become a champion of the mathematics that's produced. You need a joint commitment to making the solution work."
Breadth versus Depth
Although industrial employers do rely on narrow expertise, they often want breadth as well. In the pharmaceutical industry (and certainly elsewhere), "You do need years of experience to develop your craft," but practitioners also need breadth. "Industry wants breadth but relies heavily on narrow expertise as well."
The interdisciplinary work that is so common in industry also demands balance between breadth of knowledge and depth of knowledge. Of course, the latter alone is often the measure of mastery in an academic setting.
At a major corporate research laboratory, "The range of disciplines is so broad it doesn't matter what you know. Can you talk to others?" A mathematician who is an internal consultant to a petroleum company says, "I serve as a consultant. I can't specialize."
The size of the group with which the individual associates may determine the relative needs for depth and breadth. Larger groups of mathematicians can usually support a greater number of narrow specialists than smaller groups.
Cultural Barriers
Corporate culture may favor certain disciplines (typically, an engineering discipline) over mathematics. That bias can make the introduction of mathematical approaches quite difficult. A kind of glass ceiling in the management structure may pass into leadership roles those trained in one or two anointed disciplines but hold back mathematicians. In response to questions about favoring other disciplines over mathematics, one might hear explanations like "Nothing replaces the physical background."
Beyond those labeled as mathematicians, there is a larger community of users of mathematics and developers of computational tools. They are potential employers of mathematicians, and their work could be advanced by collaboration with well educated interdisciplinary applied mathematicians.
Among such users, there may also be significant cultural barriers to introducing individuals trained primarily in mathematics. For example, engineers at a prominent defense contractor can tell stories of lost competitive bids and design disasters that cry out for simple analyses and simulation. However, the corporate culture is not ready for mathematics. Facing the strains of the end of the cold war, management has little interest in gambling on an unproven (and perhaps secretly threatening) discipline.
From a different perspective, the vice-president for research and development at a medium-sized manufacturer of precision optical instruments comments, "We barely use arithmetic in our own quality studies. The central issue is culture --- corporate and on the manufacturing floor --- not mathematics."
A cultural gap also separates academic and nonacademic mathematicians. Careers in industrial mathematics are often viewed as less acceptable than academic pursuits. An experienced independent consultant argues, "We may need to nurture attitude changes among ourselves that produce a comprehensive acceptance of a wide range of professional needs, not just those of the academic research mathematician." Another spoke of that lack of acceptance: "My advisor and faculty treated me like I was lost when I decided to go into industry."
EDUCATIONAL IDEAS FROM NONACADEMIC MATHEMATICIANS
Many of the mathematicians who spoke of their experiences in business, industry and government offered a variety of educational ideas.
A numerical analyst in the aircraft industry suggests to the mathematics community, "We must make the idea of stopping with a master's and going into industry more attractive." A "solid background in fundamentals" would be important for such a graduate.
Problem solving experiences are important, but co-operative employment might also be a "dangerous waste of time" if it fails to capture the experiences, expectations, challenges, and disappointments of industrial work. Employment experiences like summer internships can help a student develop "more realistic expectations" and learn "more about the politics of work in industry". One industrial mathematician suggests, "Academicians would benefit from this process as much as their students."
Connecting mathematics with physical experience is important. One engineer observed, "There is nothing quite like watching a 5,000 pound hammer strike an electronics module whose durability you have simulated numerically."
Taking courses outside of mathematics seems essential to developing the breadth of scientific interest demanded by most industrial environments. Computational ability, if not sophistication, is nearly as important. Mathematicians educating students for industry must accept the idea that "there are few people in the world who can do pure mathematics in industry."
Communication skills are simply too important to ignore. Graduates headed for industry must be able to write and speak effectively. They must also listen well if they are to be productive consultants.
Problem formulation is seldom addressed explicitly in university curricula, but it is an integral part of the work of most industrial applied mathematicians.
CONCLUSIONS
The academic mathematician might be seen as a single organism that has adapted to different environments. Those environments range from teaching multitudes in a community college to nurturing a few graduate students in a research university.
In contrast, the nonacademic mathematician exists in a variety of distinct forms. These include serving as a specialist among a large group of mathematicians, serving as a wide-ranging consultant, either singly or in a team, or even working in an environment where mathematics is not explicitly recognized in titles or job descriptions. The niche of the nonacademic mathematician is never as assured as a in a university mathematics department. Because these nonacademic cultures are so different from academia, their value systems differ as well.
Except perhaps for a common thread of problem solving, the skills valued in industry differ dramatically from those of classical academic scholarship. Teamwork, communication skills (speaking, writing, and listening), and learning new disciplines are valued in industry, but they are seldom critical to most academic mathematics. In industry, breadth can be more important than depth, and a timely, incomplete answer to a complex but crucial question may be worth more than a lengthy, complete solution of a model problem. Computational skills and scientific interests outside of mathematics are commonly valued. In many industrial environments, mathematicians can seek out their own problems, but only within the limits defined by customers, clients or business needs. They can not follow just the path of their own investigations. Intellectual flexibility is essential. Priorities for problems are determined by corporate welfare. Taking care to solve the right problem and communicating the solution to the user are as important as mastery of a particular mathematical technique.
Concerns for funding are as common as in academia but without tenure to offer job security. Industrial mathematicians often need to find funded problems, and they are vexed by the difficulties of communicating the importance of mathematics to their managers. The most successful industrial mathematicians are always conscious of their relevance to the corporate mission, reflecting the hard fact that mathematics is seldom valued for its own sake outside of academia.
Many of the industrial mathematicians whose views shaped this report offered specific educational recommendations. These suggestions, which centered on problem solving and the central importance of communication skills, are summarized in the previous section. Beyond the specifics of curriculum, the experiences of those mathematicians also suggest some directions for self- examination by the academic community.
Academic mathematicians might reassess their own views of the industrial work place. They could then better prepare their students who will pursue careers other than academic research, and they could adopt other experiences and approaches of value. Certainly, students who pursue nonacademic careers should not be consigned to a lower caste than those who enter the cloister of academic mathematics.
Interdisciplinary problem solving is an activity upon which a great many mathematically sophisticated individuals build satisfying careers in industry. Given the value of that activity, might not problem solving be a stimulating addition to many educational programs, particularly if it could be integrated into daily classroom experiences as it is into the daily professional life of the industrial applied mathematician?
In a similar vein, industrial mathematicians who are serving as advocates for their profession among their nonmathematical colleagues might offer an alternate model for communicating mathematics. These individuals are enlisting apostles from among the doubtful by solving their problems, not by delivering sermons to captives in a classroom.
A more ambitious goal of such alternate educational experiences might be developing students who could become active agents for technology transfer. Their interdisciplinary training, their skill in mathematics, and their intellectual predisposition could enable them to communicate across disciplinary boundaries. Their skills could create an enlarged role for mathematicians as a primary vehicle for technology transfer in industry.
It remains to validate the anecdotes given here using carefully drawn samples of nonacademic mathematicians. The more reliable picture that would emerge from such testing could form the basis for a thoughtful, carefully organized educational response. That response could encompass the enunciation of principles for the design of curricula which address nonacademic needs as well as the implementation of innovative curricula based on those principles. The improved understanding of mathematics in industry could also contribute to its broader and more productive use. Future phases of the SIAM Mathematics in Industry Project will address these issues.
REFERENCES
1. H. Knight, "Where's Wallace?, Harper and Row, New York, 1991.
2. W. L. Hansen, Report of the Commission on Graduate Education in Economics, J. Economic Literature, XXIX (September 1991), 1035--1053.
3. W. L. Hansen, The Education and Training of Economics Doctorates, ibid., 1054--1087
4. P. Horner, Where Do We Go From Here?, OR/MS Today, April 1992, 20-30.
5. R. Wrubel, Down From the Mountain --- How IBM Research Woke Up and Reorganized Itself for the Nineties, Financial World, vol. 161(22), Nov. 10, 1992, 36--44; reprinted with permission, SIAM News, May 1993, pp. 10, 15.
Source: http://www.siam.org/about/mii.htm
March 1995