The spectrum of the field is perhaps best illustrated by observing the
role of mathematics as it applies to different products. For further
information, explore the
Mathematical Moments Program
which offers more examples of the role mathematics plays in science,
nature, technology, and human culture.
Chlorofluorocarbons (CFCs), like the freon used in aerosol cans and air
conditioning systems, could destroy stratospheric ozone, which protects
the earth from biologically damaging ultraviolet radiation. Mathematical
models, simulations and the numerical solution of a special set of
differential equations, called "stiff" differential equations, are used to
identify safer replacements from the members of hydrohalocarbon (HHC)
Accurate models of oil
reservoirs, including the simulations of oil and water moving through
porous rock, sometimes covering hundreds of acres, are used by the
petroleum industry to make decisions on where to drill. These problems are
solved by reducing complex multidimensional differential equations to a
sequence of simpler one-dimensional problems that are solved numerically
Operations research is
used throughout the airline industry to make sure seats are sold and the
airlines make money. Yield management, including mathematical models,
optimization techniques, and probability calculations, is used for setting
up automated reservation systems and complex systems of connecting routes.
Models based on
computing solutions to partial differential equations are used to solve
problems in signal processing and filtering of noise.
The design of a circuit
uses the concept of a graph, like a schematic map, with lines, called
edges and intersections, called nodes. Systematic searches of the nodes
are used to determine the most efficient connection from one node to
The design of an
aircraft requires computational fluid dynamics, partial differential
equations, and grid generation on complex geometries.
CD players, digital
audio tape and digital television read digital information that consists
of "bits" -- 0's and 1's. Occasionally these devices confuse the two and
error-correction codes, like Reed-Soloman codes, are needed.
Mathematically, Reed-Solomon codes are based on the arithmetic of finite
The law enforcement
community is interested in developing quick ways to match fingerprints
with the database of fingerprints held by the Federal Bureau of
Investigation. The problem is the FBI holds approximately 200 million
fingerprint cards. They have now adopted a standard for digital
fingerprint image compression that will allow the fingerprints to be
The equations of motion
of a space vehicle are systems of ordinary differential equations. One may
wish to solve an initial value problem, say where the initial position and
velocity of the spacecraft is given and you want to determine the
trajectory for some period of time. Boundary value problems also arise,
for example when you want to design an orbit transfer maneuver between two
different orbits. In that case you have beginning and end point
constraints on the maneuver.
The search for enemy
submarines requires the application of a number of fields of mathematics,
including probability, game theory and optimization.
The problem of
separation of the space shuttle fuel tank involves ordinary differential
equations and numerical solution methods.
Color is the result of
the combination of a light source, an object that it illuminates, and a
visual system to perceive the color which is usually the eye and the brain
of a human being working together. Color is commonly described by the
attributes of lightness, chroma, and hue. Standardized color descriptions
use values assigned to these three attributes to identify the color.
Crystal growth can be
modeled using partial differential equations.
Some resources in this section are provided by the
American Mathematical Society,
Mathematical Association of America,
Society for Industrial and Applied
and the US Department
of Labor, Bureau of Labor Statistics.