Applications of Math
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Mathematics is at the heart of many of today’s advancements in science and technology and are contributing to progress in other fields such as industrial and architectural design, economics, biology, linguistics, and psychology. Studying math can provide you with a competitive advantage in many fields. Below are some examples of how mathematics plays a role in science, nature, technology, and human culture

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Light and Water Shows
The interplay of water,light,and music in some modern fountains is magical to behold, and mathematics is part of that magic.Geometry is used in the overall design, mathematical modeling simulates the fluid-particle interactions, and powerful algorithms drive the software that coordinates thousands of valves and lights through the numerous sequences in a typical show.

The ability to make water act so precisely results from the use of laminar flow streamswhere all particles move in parallel and at the same speed. A complex mathematical analysis of fluid dynamics makes it possible for water to perform feats such as climbing stairs or behaving like individual marbles. The result is both wondrous and efficient: A four-foot column of water wouldn't fill a normal drinking glass.

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Bin Packing
Packing items into bins of given capacities may not sound important (unless you're packing for a trip), but the topic of bin packingincludes situations such as allocating blocks of computer memory and scheduling airline flights as well as traditional problems like loading trucks. Researchers use areas of mathematics (such as number theory, geometry, and probability) to solve packing problems so that time and storage – both physical and electronic – can be used efficiently.

Mathematicians proved that bin packing problems are “complex,” and a practical algorithm that gives an optimal solution to all packing problems appears unlikely. Yet even though there may never be a “fast” general solution,mathematicians still seek to improve packing algorithms,saving industry time and money. One such result demonstrates that one of the simplest packing algorithms, first loading the biggest things that fit, is always within about 20% of the best solution possible.

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Translating Languages
The current pace of document creation (on the Internet, for example) is much greater than the capacity of human translators, which makes machine translationa necessity. Machine translators use probability, statistics, and graph theory in combination with databases of hundreds of millions of words and phrases in many languages to achieve good translations efficiently. Thus,mathematics, often called the universal language, also forms a bridge between languages.

Once a document is translated, the question becomes: How good is the translation? Numerical measures of effectiveness help automate this part of the process as well, saving time and money. Results from the evaluation improve translation algorithms so that the urban legend of a computer translating “The spirit is willing but the flesh is weak” into Russian and back into English as “The vodka is good but the meat is rotten” will remain a legend.

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20 Questions: The Game
How does something the size of a yo-yo successfully play a game of 20 Questions? Although its success tempts players to think that the device is reading their minds, it's not. This sophisticated toy uses mathematics such as probability and fuzzy logic, and mathematical objects such as matrices to determine your animal, vegetable, or mineral more than 75% of the time.

The online version of the game is an example of artificial intelligence,specifically a neural network,which uses feedback loops and weights to "learn" as it gets more information. In this case, answers are given weights,with "unknown" having a weight of zero,and (in the online game) weights are adjusted as necessary after each game. The weights form a matrix,with objects and questions indexing the rows and columns,respectively. The game chooses a question by first determining which objects are still probable and then finding which question has the most desirable set of weights for the remaining candidate objects. What is the most desirable set of weights? Sorry,that's not a Yes-No question.

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Space Travel
The recent discovery of low-energy pathways along which space vechiles can travel using far less fuel has made previously impossible missions feasible. Much of space travel depends on calculus, trigonometry, and vector analysis, but the existence of these routes derives from an area of mathematics called dynamical systems applied to the mutual interaction of the gravities of the sun, nearby planets, and moons.

Calculations of forces between two celestial bodies and their orbits are fairly direct, but to understand orbits and trajectories when more than two bodies are involved, dynamical systems and chaos theory are necessary. Even the simplest extension beyond two bodies, the three-body problem, has been proven to have no explicit general solution. Some special cases, however, have been solved and applied not only to mission design, but also now to atomic physics to study the paths of certain excited electrons. Thus, mathematics is locating new routes for space travel and establishing connections between the atomic and the cosmic.

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Boarding Airplanes
Waiting in line while boarding a plane isn’t just irritating, it’s also costly: The extra time on the ground amounts to millions of dollars each year in lost revenue for the airlines. Research into different boarding procedures uses mathematics such as Lorentzian geometry and random matrix theory to demonstrate that open seating is a quick way to board while back-to-front boarding is extremely slow. In fact, mathematical models show that even people boarding at random get to their assigned seats faster than when boarding back-to-front.

Figuring out your own strategy for boarding a plane is hard enough, but modeling the general problem—which depends on many variables such as distance between rows, amount of carry-on baggage ,and passengers’ waistlines—is substantially more complex. So researchers were pleased when they discovered that their theoretical analysis confirmed simulations conducted by some airlines. An added bonus to the research is that the mathematics used in the boarding problem is similar to that used to improve a disk drive’s data input and output requests. One clear difference: Data doesn’t try to carry on an extra bit.

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Solving Crimes
The CBS-TV program NUMB3RS shows modern mathematics and mathematicians at work—both instrumental each week in solving and preventing crimes. Although the series is fictional, many of the show’s episodes are based on true stories. In fact, statistics, combinatorics, and graph theory are just some of the mathematical fields being used today by real-life investigators to solve actual crimes.

One of the most impressive instances of mathematics solving a crime was a case in which an algorithm pinpointed a serial offender’s location, based on the sites of previous crimes. When DNA samples cleared all the suspects living in the area, however, the natural conclusion was that the mathematics was unsound. Then a tip led investigators to a deputy who had been above suspicion (because of his job) and who had lived in the target area. He was eventually arrested and sentenced, proving that crime doesn’t pay but checking your assumptions does.

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Predicting Storm Surge
Storm surge is often the most devastating part of a hurricane. Mathematical models used to predict surge must incorporate the effects of winds, atmospheric pressure, tides, waves and river flows, as well as the geometry and topography of the coastal ocean and the adjacent floodplain. Equations from fluid dynamics describe the movement of water, but most often such huge systems of equations need to be solved by numerical analysis in order to better forecast where potential flooding will occur.

Much of the detailed geometry and topography on or near a coast require very fine precision to model, while other regions such as large open expanses of deep water can typically be solved with much coarser resolution. So using one scale throughout either has too much data to be feasible or is not very predictive in the area of greatest concern, the coastal floodplain. Researchers solve this problem by using an unstructured grid size that adapts to the relevant regions and allows for coupling of the information from the ocean to the coast and inland. The model was very accurate in tests of historical storms in southern Louisiana and is being used to design better and safer levees in the region and to evaluate the safety of all coastal regions.

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Finding Oil
As high as gas prices are, they would be much higher without modern oil exploration techniques, which make operations more efficient (and cleaner). Drilling a well can cost 20 million dollars, so drillers now rely on mathematical models of reservoirs, rather than hunches, to choose sites. The models approximate a reservoir’s characteristics from data collected using sound waves beamed underground, and from the resulting systems of nonlinear equations. In fact, one company estimates that it solves over 250,000 systems a day.

The reservoir simulations are derived from partial differential equations describing fluid flow and from terabytes of data, but they still contain a good deal of uncertainty. Researchers are using statistics to quantify the uncertainty involved, thus giving planners models that are more descriptive of subsurface properties such as permeability. One thing is certain, however: Finding new sources of energy to meet future energy demand will continue to depend on advances in the mathematical sciences.

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Math Behind Politics
Votes are cast by the full membership in each house of Congress, but much of the important maneuvering occurs in committees. Graph theory and linear algebra are two mathematics subjects that have revealed a level of organization in Congress— groups of committees—above the known levels of subcommittees and committees. The result is based on strong connections between certain committees that can be detected by examining their memberships, but which were virtually unknown until uncovered by mathematical analysis.

Mathematics has also been applied to individual congressional voting records. Each legislator’s record is represented in a matrix whose larger dimension is the number of votes cast (which in a House term is approximately 1000). Using eigenvalues and eigenvectors, researchers have shown that the entire collection of votes for a particular Congress can be approximated very well by a two-dimensional space. Thus, for example, in almost all cases the success or failure of a bill can be predicted from information derived from two coordinates. Consequently it turns out that some of the values important in Washington are, in fact, eigenvalues.

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Artists' Style
Mathematics is not just numbers and brute force calculation—there is considerable art and elegance to the subject. So it is natural that mathematics is now being used to analyze artists’ styles and to help determine the identities of the creators of disputed works. Attempts at measuring style began with literature—based on statistics of word use—and have successfully identified disputed works such as some of The Federalist Papers. But drawings and paintings resisted quantification until very recently. In the case of Jackson Pollock, his paintings have a demonstrated complexity to them (corresponding to a fractal dimension between 1 and 2) that distinguishes them from simple random drips.

A team examining digital photos of drawings used modern mathematical transforms known as wavelets to quantify attributes of a collection of 16th century master’s drawings. The analysis revealed measurable differences between authentic drawings and imitations, clustering the former away from the latter. This is an impressive feat for the non-experts and their model, yet the team agrees that its work, like mathematics itself, is not designed to replace humans, but to assist them.

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Video Games
Video games are a lot of fun, but they’d be much less fun without mathematics. Geometry, calculus, and linear algebra all help make characters, scenes, and action look less two-dimensional and more realistic. And, as one game company executive noted, advancing through mathematics is similar to working through the increasingly more difficult levels of a video game. Who knows, by graduation you may have enough skills to save the world.

Much of a character’s movement involves inverse kinematics: For example, what should the angles of the foot, shin, and upper leg be as a character runs? This is an important area of research that also involves collision and contact detection (obvious in the real world, but requiring explicit calculation in the video world). There can be an infinite number of answers to such problems but fast algorithms must find realistic solutions in less time than you can say “The leg bone’s connected to the hip bone.”

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Music
Mathematics and music have long been closely attached. Now a recent mathematical breakthrough uses topology (a generalization of geometry) to represent musical chords as points in a space called orbifold, which twists and folds back on itself - much like a Mobius strip does. This representation makes sense musically in that sounds that are far apart in one sense yet similar in another, such as two notes that are an octave apart, are identified in the space.

The latest insight provides a way to analyze any type of music. In the case of Western music, pleasing chords lie near the center of the orbifolds and pleasing melodies are paths that link nearby chords. Yet despite the new connection between music and coordinate geometry, music is still more than a connect-the-dots exercise, just as mathematics is more than addition and multiplication.

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