The primary objective of the course presented here is orientation for those interested in applying mathematics, but the course should also be of value or in using math to those interested in mathematical research and teaching ematics in some other professional context. The course should be suitable for college seniors and graduate students, as well as for college juniors who have had mathematics beyond the basic calculus sequence. Maturity is more significant than any formal prerequisite. The presentation involves a number of topics that are significant for applied mathematics but that normally do not appear in the curriculum or are depicted from an entirely different point of view. These topics include engineering simulations, the experience patterns of the exact sciences, the conceptual nature of pure mathematics and its relation to applied mathe matics, the historical development of mathematics, the associated conceptual aspects of the exact sciences, and the metaphysical implications of mathe matical scientific theories. We will associate topics in mathematics with areas of application. This presentation corresponds to a certain logical structure. But there is an enormous wealth of intellectual development available, and this permits considerable flexibility for the instructor in curricula and emphasis. The prime objective is to encourage the student to contact and utilize this rich heritage. Thus, the student's activity is critical, and it is also critical that this activity be precisely formulated and communicated.
1. Introduction.- 1.1. Vocational Aspects.- Applied mathematics is the vocational use of mathematics other than in teaching or mathematical research..- 1.2. Intellectual Attitudes.- In a technical effort, understanding cannot be disjointed into pieces corresponding to the academic disciplines. Technical understanding has a basically algorithmic character..- 1.3. Opportunities in Applied Mathematics.- Many of the possibilities for applied mathematics occur as part of research and development programs of the Federal Government. Technology advances may also open opportunities for applied mathematics in industry..- 1.4. Course Objectives.- The exercises and the student projects are an essential part of the course..- Exercises.- 2. Simulations.- 2.1. Organized Efforts.- Applied mathematics is usually part of a large effort under contract with the Federal Government and based on scientific and technical understanding. It is a team effort and documentation is essential..- 2.2. Staging.- The efficient use of resources requires that such efforts proceed in stages, each of which provides a decision basis for the next..- 2.3. Simulations.- Technical simulations permit decisions to be based on the scientific and technical understanding of the original situation..- 2.4. Influence Block Diagram and Math Model.- The basic understanding is expressed in the influence block diagram and the math model..- 2.5. Temporal Patterns.- The block diagram and the math model are supplemented by the flow chart, which describes the relations in time of the original situation. Specific scenarios are also used..- 2.6. Operational Flight Trainer.- The notions of influence block diagram and math model are illustrated in this example..- 2.7. Block Diagrams.- Block diagrams originally referred to equipment. In analog computers these became associated with the math model..- 2.8. Equipment.- The equipment includes the computer and the input and output devices required for the simulation. The objectives of the simulation determine the requirements..- 2.9. The Time Pattern of the Simulation.- The basic time pattern of the simulation is based either on an advance by fixed time intervals or by critical events. Provision must be made for input and output..- 2.10. Programming.- The structure of the program should be modular and subject to an executive program. The numerical procedures must be determined with the required accuracy, stability, and range..- 2.11. Management Considerations.- The total effort in the simulations must be scheduled to permit the efficient use of resources such as manpower and facilities..- 2.12. Validity.- The mathematical formulation of understanding can best be understood in terms of its historical development..- Exercises.- References.- 3. Understanding and Mathematics.- 3.1. Experience and Understanding.- Understanding permits us to cope with an environment by using past experience patterns in a mental exploration of possibilities..- 3.2. Unit Experience.- A flow diagram for a unit experience indicates the adjustment between understanding and the interaction with the environment. The validity of knowledge is associated with this adjustment..- 3.3. The Exact Sciences.- For situations in their milieus, the exact sciences produce a block diagram analysis whose blocks correspond to concepts based on patterns of experience and whose math model yields prediction and control..- 3.4. Scientific Understanding.- Scientific understanding is effective because it represents a long-range adjustment of concepts and math model to match experience. But this adjustment involves complications that must be understood..- 3.5. Logic and Arithmetic.- Many aspects of experience can be usefully formulated in terms of the concepts associated with finite sets and the natural numbers..- 3.6. Algebra.- Algebra represents an abstraction of the properties of numbers that greatly supplements the logical possibilities for elementary arithmetic..- 3.7. Axiomatic Developm
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