Applications of Mathematics
Applications of mathematics
Mathematics as a calculatory science
Aggregations (e.g. counting by fives or tens, the dozen etc.). ancient numerical notations. Decimal notation and modern development.
Early application of geometry in response to practical problems in geography and astronomy. Instruments for observation and navigation e.g. quadrants, sextants. Mapping: geometry to terrestrial measurement. Celestial measurement: spherical trigonometry, stereographic mappping, astrolabe. Optical instruments e.g. verniers, theodolites, transit telescopes. Drawing instruments e.g. straight edges, rules, compasses, T-squares.
Physical constructions to aid the visualisation of mathematical ideas, e.g. polyhedra, models to illustrate identiteis or topological concepts.
Calculatory aspects of algebra
Algebraic notation with use of letters and symbols to denote algebraic variables. Logarithms.
Calculation using tables and graphs
Mathematical tables e.g. integral tables, tables of function. Graphs and graphical procedures e.g. Cartesian and polar graphs
Types of problems solveable by analogue computation. Types of mechanical analogue devices e.g. resolvers, multipliers, electromechanical and direct current analogue computers, hybrid computer systems.
Digital calculators e.g. the abacus, registers, adders. Punched cards. Digital computers
Basic principles of statistical inference
Probability theory application to the analysis of data. Distribution of functions e.g. the median, mean, variance, and standar deviation of a distribution, the gaussian or normal distribution.
Point estimation, the method of moments, interval estimation, robust estimation, Bayesian methods.
Structure in data
Use of regression analysis to discover systematic patterns.
Applications of numerical analysis
Analogy automata and the nervous system of living organisms.
Neural nets and automata
The finite automata of McCulloch and Pitts, neurophysiological model and its description in mathematical terms. Logical organs, automata that corrrespond to the binary operations of disjunction and conjunction and the unary operatoin of negation or complementation. The generalised automaton and Turing’s machine. Input.
Relation between an automaton and its environment. Automata with random components. Computable probability spaces: can an automaton generate a sequence of random numbers.
- New forms of randomness with the internet, slowing down of the net, concurrence of resources, various machines sending packages over same lines.
Classification of automata
Acceptors: Turing, finite state, pushdown, and linear bounded acceptors. Finite transducers. Post machines.
Collection of mathematical principles and methods used for solving quantitative problems in many disciplines, including physics, biology, engineering, economics, and business. The subject grew from a realization that quantitative problems in manifestly different disciplines have important mathematical elements in common. Cf. Management Theory and Operations Research in Management. Includes calculus of variations, control theory, convex optimisation theory, decision theory, game theory, linear and nonlinear programming, Markov chains, network analysis, queuing systems, Cybernetics etc.
The theory of games: analysis of strategic features of conflict situations
Classification of games. Concept of pure strategy. Games with finite and infinite numbers of strategies. Concept of utility. Normalised dual games, matrix games, symmetric games, and games on a square, minimax theorem. Extensive games, pure strategies and behaviour strategies. Plural games. Game playing programs.
Linear and nonlinear programming
Simplex method of solution. The dual of a linear program. Nonlinear programming, solutions based on Kuhn-Tucker conditions, methods of steepest descent.
Definition, examples of modern control systems. Control of linear systems with constant coefficients: the determination of the explicit form of the controllability condition. Optimal control, optimal filtering, and state estimation. Nonlinear control systems.
See Information Sciences.
Mathematical aspects of physical theories
Mechanics of particles and systems
Newton’s laws and their mathematical formulation. The principles of conservation of linear and angular momentum. Formal definition of work and energy and their relation. Dynamics of a system of particles.
Mathematical description of three-dimensional flow. Velocity and acceleration of a fluid. Equation of continuity. Dynamic equation for an inviscid fluid. Steady flow of an inviscid fluid: Bernouill’s equation
Mechanics of solids
The theory of elasticity. Concept of stress, equilibrium conditions. Tensile and compressive stress. Triaxial and plane stress. Concept of strain. Relation between stress and strain.
The state of thermodynamic system. The role of probability. Pressure and energy density of a perfect gas. Maxwel/Boltzmann distribution law. Gibbs ensembles and the concept of ensemble average. The hamiltonian. Liouville’s theorem. Quantum statistics.
Electromagnetic wave concept. Maxwell’s field theory. Velocity of electromagnetic waves.
Gilles Deleuze 1995. Negotiations : Riemannian space as involving setting up 'little neighbouring portions that can be joined up in an infinite number of ways' - has made possible theory of relativity ; Deleuze draws linkage to cinema of Luc Bresson, in that it creates neighbourhoods joined up in infinite number of ways ; also : Baker's transformation ; a square pulled out to rectangle, cut in two, the resulting square again pulled out and repeating the process infinitely, : any two points, however close initially, are bound to end up in two different halves. resulting in probabilistic phsyics theory. (Prigogine and Stenger analysed this) ; Deleuze parallel to cinema of Resnais, Je t'aime, je t'aime with layers of context on each other and changing. ; science and arts (and philosophy) resonates.. see Pro-Knowledge.
Planck’s quantum hypothesis. Wave mechanics: de Broglie waves, Schrödinger wave equation. Matrix form of quantum mechanicsÖ the thoery of Born and Heisenberg and its relation to wave theory. The tranformation theory of Jordan and Dirac. Probability distribution in momentum space: the uncertainty principle. Theory of relativistic quantum fields: quantum electrodynamics.
The pi theorem.