# Mathematical Modeling

### Why mathematical modeling?

Mathematical modeling is the art of translating problems from an application area into tractable mathematical formulations whose theoretical and numerical analysis provides insight, answers, and guidance useful for the originating application.

Mathematical modeling

is indispensable in many applications

is successful in many further applications

gives precision and direction for problem solution

enables a thorough understanding of the system modeled

prepares the way for better design or control of a system

allows the efficient use of modern computing capabilities

Learning about mathematical modeling is an important step from a theoretical mathematical training to an application-oriented mathematical expertise, and makes the student fit for mastering the challenges of our modern technological culture.

### A list of applications

In the following, I give a list of applications whose modeling I understand, at least in some detail. All areas mentioned have numerous mathematical challenges.

This list is based on my own experience; therefore it is very incomplete as a list of applications of mathematics in general. There are an almost endless number of other areas with interesting mathematical problems.

Indeed, mathematics is simply the language for posing problems precisely and unambiguously (so that even a stupid, pedantic computer can understand it).

Anthropology

• Modeling, classifying and reconstructing skulls

Archeology

• Reconstruction of objects from preserved fragments
• Classifying ancient artifices

Architecture

• Virtual reality

Artificial intelligence

• Computer vision
• Image interpretation
• Robotics
• Speech recognition
• Optical character recognition
• Reasoning under uncertainty

Arts

• Computer animation (Jurassic Park)

Astronomy

• Detection of planetary systems
• Correcting the Hubble telescope
• Origin of the universe
• Evolution of stars

Biology

• Protein folding
• Humane genome project
• Population dynamics
• Morphogenesis
• Evolutionary pedigrees
• Spreading of infectious diseases (AIDS)
• Animal and plant breeding (genetic variability)

Chemical engineering

• Chemical equilibrium
• Planning of production units

Chemistry

• Chemical reaction dynamics
• Molecular modeling
• Electronic structure calculations

Computer science

• Image processing
• Realistic computer graphics (ray tracing)

Criminalistic science

• Finger print recognition
• Face recognition

Economics

• Labor data analysis

Electrical engineering

• Stability of electric curcuits
• Microchip analysis
• Power supply network optimization

Finance

• Risk analysis
• Value estimation of options

Fluid mechanics

• Wind channel
• Turbulence

Geosciences

• Prediction of oil or ore deposits
• Map production
• Earth quake prediction

Internet

• Web search
• Optimal routing

Linguistics

• Automatic translation

Materials Science

• Microchip production
• Microstructures
• Semiconductor modeling

Mechanical engineering

• Stability of structures (high rise buildings, bridges, air planes)
• Structural optimization
• Crash simulation

Medicine

• Computer-aided tomography
• Blood circulation models

Meteorology

• Weather prediction
• Climate prediction (global warming, what caused the ozone hole?)

Music

• Analysis and synthesis of sounds

Neuroscience

• Neural networks
• Signal transmission in nerves

Pharmacology

• Docking of molecules to proteins
• Screening of new compounds

Physics

• Elementary particle tracking
• Quantum field theory predictions (baryon spectrum)
• Laser dynamics

Political Sciences

• Analysis of elections

Psychology

• Formalizing diaries of therapy sessions

Space Sciences

• Trajectory planning
• Flight simulation
• Shuttle reentry

Transport Science

• Air traffic scheduling
• Taxi for handicapped people
• Automatic pilot for cars and airplanes
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