Applications of Mathematics

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Contents

Applications of mathematics

Mathematics as a calculatory science

Numerical notations

Aggregations (e.g. counting by fives or tens, the dozen etc.). ancient numerical notations. Decimal notation and modern development.

Geometrical aids

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.

Mathematical modes

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

Analogue computation

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 computation

Digital calculators e.g. the abacus, registers, adders. Punched cards. Digital computers

Statistics

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.

Estimation

Point estimation, the method of moments, interval estimation, robust estimation, Bayesian methods.

Hypothesis testing

Structure in data

Use of regression analysis to discover systematic patterns.

Numerical analysis

Finite differences

Applications of numerical analysis

Automata theory

Introduction

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.

Probabilistic questions

Relation between an automaton and its environment. Automata with random components. Computable probability spaces: can an automaton generate a sequence of random numbers.

Classification of automata

Acceptors: Turing, finite state, pushdown, and linear bounded acceptors. Finite transducers. Post machines.


Optimisation Theory

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.

Cybernetics

Control theory

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.

Information theory

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.

Fluid mechanics

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.

Statistical mechanics

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 theory

Electromagnetic wave concept. Maxwell’s field theory. Velocity of electromagnetic waves.

Relativity theory

Riemannian geometry

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.

Quantum mechanics

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.

Dimensional analysis

The pi theorem.

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