Articles liés à Applied Mathematics: An Intellectual Orientation
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Synopsis
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 Developments
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