Part Ten X › Division II II › Section 10/23 10/23

Applications of Mathematics

Subject Outline

AMathematics as a calculatory science
- 1Numerical notations
  - aAggregations, or units used to assist counting or grouping of objects
  - bAncient numerical notations
  - cDecimal notation and modern notational developments
- 2Geometrical aids
  - aEarly applications of geometry
  - bInstruments for observation and navigation
  - cMapping
  - dApplications of geometry to celestial measurement
  - eOptical instruments
  - fDrawing instruments
- 3Mathematical models: physical constructions used to aid the visualization of mathematical ideas or relationships
- 4Calculatory aspects of algebra
  - aAlgebraic notation
  - bLogarithms
  - cSlide rules
- 5Calculation using tables and graphs
  - aMathematical tables
  - bGraphs and graphical procedures
- 6Analogue computation [see also 735.D.]
  - aTypes of problems solvable by analogue computation
  - bAnalogue computers
- 7Digital computation [see also 735.D.]
  - aDigital calculators
  - bPunched cards
  - cProgrammed machines (digital computers)
BStatistics
- 1The basic principles of statistical inference: application of the concepts and techniques of probability theory to the analysis of data
  - aThe concept of a statistical experiment: mathematical description of experiments in terms of random variables
  - bDistribution functions and their properties: the median, mean, variance, and standard deviation of a distribution; the Gaussian or normal distribution
- 2Estimation: techniques for approximating the parameters of families of distributions of random variables
- 3Hypothesis testing: techniques for determining the correctness of alternative hypotheses concerning given data and an assumed probability model
- 4Structure in data: use of regression analysis to discover systematic patterns
CNumerical analysis
- 1Introduction: definition, origins, and basic concepts of numerical analysis
- 2Errors: round-off and truncation
- 3Approximation of functions: linear and polynomial interpolation, least squares approximation
- 4Solution of equations: linear, nonlinear, and differential equations
- 5Applications and implementation of numerical analysis: optimization, large-scale scientific computation, mathematical software
DAutomata theory [see also 712.A.6.]
- 1Introduction: definition and examples of automata, development of the basic concepts of automata theory, the analogy between automata and the nervous systems of living organisms
- 2Neural nets and automata
- 3Probabilistic questions: random effects in the operation of automata
- 4Classification of automata
EMathematical theory of optimization
- 1The theory of games: analysis of the strategic features of conflict situations
- 2Linear and nonlinear programming (mathematical programming)
- 3Cybernetics
- 4Control theory
FInformation theory [see also 735.A.]
- 1Origins and definitions of information theory
- 2Central problems of information theory
- 3Principles of information theory
- 4Applications of information theory to cryptography, linguistics, and other fields
GComputer science
- 1Synthesis and analysis of algorithms
- 2Theory of computation
- 3Computer architecture
- 4Programming languages and methodology
- 5Numeric computation
- 6Artificial intelligence
HMathematical aspects of physical theories
- 1Mechanics of particles and systems [see also 126.A.]
- 2Fluid mechanics [see also 126.F.]
- 3Mechanics of solids [see also 126.D.]
- 4Statistical mechanics [see also 124.A.9.]
- 5Electromagnetic theory [see also 127.E.]
- 6Relativity theory: space and time as a four-dimensional continuum [see also 131.D ]
- 7Riemannian geometry
- 8Quantum mechanics [see also 111.A.4.C.]