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Mathematics is a universal language for describing reality:
- the physical world
- chemical reactions
- movement of the planets and other space objects
- population growth
- weather
- man-made world
- finance
- engineering
- music
It can also be used to describe some more abstract ideas:
- complex numbers
- topology
- string theory
Inspired from http://www.ams.org/careers/mathapps.html, below is a index of applications of Mathematics. Each example is followed by the key mathematics areas involved
Index of applications
A B C C
D
E
F
I
M
N O P Q
R
S
T
U
V
W
A
- Aircraft landing field length : ordinary differential equations
- Aircraft survivability and effectiveness : probability and statistics, optimization
- Airflow patterns in the respiratory tract (Toxicology) : computational geometry, three-dimensional grid generation, numerical solution of Navier-Stokes equations, visualization of numerical results
C
- Color balance in a photographic system : geometry and linear algebra, multivariate statistical distributions
- Computer aided design of mechanical parts and assemblies : computational geometry, differential geometry, approximation theory
- Computer network design : graph theory, queuing theory, statistics, probability
- Cost analysis : regression theory, economic analysis, forecasting
- Cryptology : linear algebra, (esp. finite fields), Galois Theory, representation theory of finite groups, computational number theory, effective methods in algebraic geometry (elimination theory, Grobner Bases), elliptic curves, statistical inference
D
- Design and analysis of control systems for aircraft : control theory, modeling and simulation, numerical solution of ODEs
E
- Ecology: dynamical system, differential equations, numerical methods.
- Electromagnetics : vector calculus, partial differential equations, linear algebra, numerical analysis
- Electromagnetics analysis for detection by radar : numerical solution of PDEs
- Engagement analysis(Defence) : calculus, DE, probability, calculus of variations, optimization, physics
- Estimation of ocean currents : data visualization, geostatistics, numerical analysis
- Evaluating the effectiveness of different emergency notification methods : probability distribution functions, Laplace transform, convolution
F
- Fiber optics : linear algebra, optimization, partial differential equations, approximation theory, numerical analysis, applied mathematics
I
- Image compression : wavelets
- Image deblurring inverse problems : inverse problems
- Immuno-assay chemistry for developing new blood tests : dynamical systems, systems of coupled, nonlinear, autonomous ODEs
- Ionic transport in biology: perturbation methods, differential equations, boundary value problems.
L
- Large scale shock wave physics : hyperbolic (and other) differential equations, difference schemes and assorted numerical techniques, and material modeling and the mathematics of implementing material models in computer programs
- Lung models : how we breathe and how it goes wrong. Finite element methods, differential equations, modelling, statistics, data fitting.
M
- Materials science : partial differential equations, probability, optimization
- Mission and campaign analysis (Defence) : probability, numerical analysis, game theory, mathematical programming
- Modeling combustion flow in a coal power plant : fluid flow, numerical methods
- Modeling gravitational forces : orthogonal polynomials (Lagrange polynomials)
- Modeling of airflow over airplane bodies : numerical solution of PDEs, including complex grid generation
- Modeling semi conductors : linear algebra, differential equations, approximation theory, parallel programming, numerical analysis, applied mathematics
- Models and algorithms that allow sensors and other systems to process and assess information automatically and autonomously : stochastic processes, stochastic filtering, harmonic analysis, statistics, numerical methods
- Models of acoustic and electro-magnetic sensors; predict power requirements and performance : differential equations, numerical methods, physics
- Molecular and cellular mechanisms of toxicity : pattern recognition, numerical solution of systems of ODEs and PDEs
- Motion of a space vehicle : systems of ordinary differential equations, initial value problems, boundary value problems; parameter optimization problems
N
- Naval antenna modeling : H-infinity theory
O
- Optics for design of a reflector : geometry, bivariate curve fitting, optimization
P
- Paint stripping using lasers : optimal control theory, statistics, data visualization
- Photographic development : reaction-diffusion equations, parabolic PDEs with nonlinear forcing terms
- Physiology : differential equations, numerical methods, dynamical systems, modelling. Statistics and data fitting. Stochastic methods.
- Population biology: differential equations, modelling, stochastic methods. Numerical solution of differential equations.
- Predict the evolution of crystals growing in an industrial crystallizer : hyperbolic differential-delay equations
- Predicting performance of proposed sonar, radar, or communication systems : stochastic processes, harmonic analysis, differential equations, statistics, numerical methods
Q
- Queuing networks : linear algebra, numerical analysis
R
- Radar, image or stocastic processing : Harmonic analysis: Fourier and non-Abelian
- Radio interferometry : linear algebra, Fourier analysis, (measure theoretic) probability, statistical inference, information theory, optimization and mathematical modeling
- Reentry simulations for the Space Shuttle : numerical integration tools to solve stiff systems of ordinary differential equations
- Regional uptake of inhaled materials by respiratory tract (Toxicology) : numerical solution of transport equations, visualization of results
- Rocket launch trajectory analysis : optimal control, boundary value problems of ODEs
S
- Scheduling tasks on processors in a heterogeneous multiprocessor computing network : graph theory, combinatorial optimization and stochastic processes
- Scientific programming : mathematical modeling and numerical analysis
- Search for submarines probability, game theory, optimization, mathematical programming, data visualization : probability, game theory, optimization, mathematical programming, data visualization
- Signal processing : linear algebra, Fourier analyis, probability, statistics, time series analysis
- Signal processing and classification : Linear algebra: subspace angles, regularization, tensors
- Signals processing : linear and multilinear algebra, statistical inference, modeling, Fourier Analysis
- Speech recognition : automata theory, time series analysis, Fourier analysis, statistics, dynamic programming
T
- Terrain modeling : Computational geometry
- Tomography : numerical optimization, statistics, data structures
- Tracking submarines : statistics, estimation, filtering, signal processing, optimization
- Trajectory prescribed path control and optimal control problems : numerical integration tools to solve differential-algebraic equations
- Transport and disposition of chemicals through the body : numerical solution of systems of ODEs and PDEs, reaction-diffusion equations, parameter estimation techniques, Monte Carlo simulations
U
- Underwater acoustic signal processing : linear algebra, Fourier analysis, (measure theoretic) probability, statistical inference, partial differential equations and mathematical modeling
V
- Vulnerability analysis (Defence) : Markov chains, probability, statistics, Monte Carlo simulation
W
- Waves in composite media : nonlinear mixed-type PDEs
- Waves in biology: nonlinear waves, reaction-diffusion equations, parabolic systems
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