1 Introduction
The strength of a nation and the wellbeing of its citizens are determined to a substantial degree by its technological development and its level of economic organization. These determinants of national position are of serious concern precisely because they are now under challenge. Highlevel reviews of U.S. advanced technology are in agreement with this point of view: consider the 12 emerging technologies identified by the Department of Commerce, the grand challenges associated with the Federal High Performance Computing Program, the 20 critical technologies identified by the departments of Defense and Energy, and the report by the Technology Policy Task Force of the Committee on Science, Space and Technology of the U.S. House of Representatives (Appendix A). Running through these studies and the accompanying public discussion is a concern for the future viability of the longheld technological leadership of the United States.
A range of opinions exists concerning the importance of factors shaping a country's ability to be economically competitive. Such discussions are outside the scope of this report, which takes as its starting point the broad consensus that advanced technology is a vital component of economic competitiveness. The report documents the importance of quantitative reasoning, supported by computational and mathematical models, to all aspects of the complete product cycle and to economic competitiveness. In addition, it identifies some prominent opportunities for enhancing U.S. competitiveness in the future.
Dividing the U.S. economy into sectors provides useful insight. Service makes up approximately 70 percent, manufacturing 20 percent, and construction, mining, and agriculture the remaining 10 percent (Economic Report of the President, 1989 [1]). From the perspective of
world trade, the division of the U.S. economy is very different. The share of manufacturing is 57 percent, mining and agriculture are each about 10 percent, transportation and travel are each about 6 percent, and the balance is 11 percent (Passel, 1990 [2]). Manufacturing, the area in which international economic competition is most intense, plays a dominant role in export and import, and thus is given major emphasis in this report. Also discussed is the role of technology and quantitative thinking in the service sector, because the benefits of increases in productivity will be greatly multiplied if they are realized in this sector as well. Most sectors of our economy (service, agriculture, government, national security and defense, and manufacturing) have already benefited greatly from the introduction of U.S. advanced technology.
Major U.S. industries with a trade deficit (automobiles, oil, consumer electronics), those with a trade surplus (aircraft manufacture, chemicals), and those that are emerging or have a strategic importance for the future (biotechnology, computers) depend on advanced technology and quantitative reasoning, not only for research and development, but also through all stages of the product cycle, including especially the manufacturing process itself.
Both advanced technology and economic analysis are implemented in quantitative terms. From the strategic planning for large organizations, to the design of novel or superior products, to the assessment of risk and safety factors and the engineering of reliable products, quantitative thinking has been critical to success. Quantitative thinking typically implies the formulation and modeling of a problem in mathematical terms and the simulation and solution of the model equations, often using computational methods. It necessarily requires validation and parameter adjustment through laboratory and field data. It is often interdisciplinary in nature.
Examples of clear quantitative and mathematical successes that have an impact on advanced technology and economic competitiveness are widespread. This fact can be verified by examining the complete product cycle, from strategic planning to research, engineering design, manufacturing efficiency, process control, quality improvement, marketing, inventory, transportation, distribution, and product maintenance. There are similar examples within the technology base, outside of specific product cycles, such as the advanced computational meth
ods that serve weather forecasting needs. The simulation of physical phenomena, optimization, scheduling methods, nonlinear partial differential equations, and statistical and mathematical models are ideas that arise both in the support of the technology base and in specific economic and technological endeavors. The report will show that

The quantitative, mathematical, and computational approach is enabling, making possible accomplishments that would otherwise not occur. The design of fuelefficient transonic aircraft is an example.

This approach allows successive and dramatic improvement. The rapid evolution of computer technology is an example.

This approach is essential to achieving, maintaining, and improving product quality. An example is the maintenance of quality through process control.

This approach addresses novel problems and objectives. The redesign of the automotive engine to emphasize emission control and fuel efficiency is an example.

This approach is an ongoing and longterm process. To compete successfully in the global marketplace requires a continuous supply of highvalueadded products. Our nation must, therefore, emphasize high technology, which is intensively quantitative, from planning to design to production. There is no serious challenge to the validity or the success of this approach, nor to the requirement for a strong technology base, which this strategy implies.
The many changes in worldwide political structures, such as the coming economic unification of western Europe, the growing economic power of the nations of the Pacific rim, and the changing trade patterns and decrease in military tensions in central Europe, have led to a vision of a new world order. We can expect increasing economic competition among nations, and that this competition will be increasingly important. These changes only serve to emphasize the invariance of the central fact that leadership in advanced technology and the maintenance of a strong technology base are essential to the continued strength and competitive position of our nation. One competitive advantage of the
United States is its world leadership in the theory and application of computing. The importance of this fact to economic competitiveness is illustrated throughout this report. Further advantages of the United States are its position in basic research and its strong university system. The mathematical sciences are an especially strong component of U.S. science. These strengths must be preserved, and they must be brought to bear on problems of competitiveness, if this country is to maintain its position into the next century.
The central conclusion of this report is that the mathematical sciences are vital to economic competitiveness and that they are a critical, generic, and enabling technology. This conclusion is examined from three points of view. In chapter 2, key industries are examined from the point of view of their competitive positions, their dependence on advanced technology, and the role that the mathematical sciences have played in support of advanced technology. The industries selected are not exceptional, and the same points could have been made through the examination of a number of different industries. Chapter 3 considers key economic functions and activities; 11 steps in the product cycle are examined in some detail for their mathematical components. These steps cover the entire product cycle, from economic planning for new products to maintenance and repair of the manufactured goods. Case studies illustrate the mathematical contributions found in the entire product cycle. In chapter 4 the mathematical sciences technology base for economic competitiveness is considered, and the role of mathematics in technology transfer and education is discussed.
The uses of mathematics described in this report generally involved the creation of new mathematics, a proof of principle, or an initial reduction of a mathematical result to practice. All three of these activities are legitimate mathematical research issues. In almost every case, these applications raise new mathematical issues, which, the board believes, is the reason they were carried out by mathematicians, rather than by scientists or engineers from other fields.
Industrial mathematics is defined in this report as applicable mathematics that is used in an industrial context. Its value to a final product or service varies from one context to another but in general can be categorized as helpful, enabling, or critical. The board does not attempt to determine which of the examples cited in this report fall into each
of these three categories.
This report discusses representative selections of the many mathematical areas that have been used to develop advanced technology and and thus have contributed to improving U.S. economic competitiveness. This discussion cannot be complete for a simple reason: most of mathematics would then be included, either directly or indirectly.
The major findings and recommendations of this report are summarized in chapter 5. The report concludes with a list of references and two appendices, one with a list of pertinent governmental, industrial and academic policy studies, the other with studies conducted by the mathematical sciences community over the past 8 years.