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## Finding Friends: Part 2

Facebook has over 700 million users with almost 70 billion connections. The hard part isnâ€™t people making friends; rather itâ€™s Facebookâ€™s computers storing and accessing relevant data, including information about friends of friends. The latter is important for recommendations to users (People You May Know). Much of this work involves computer science, but mathematics also plays a significant role. Subjects such as linear programming and graph theory help cut in half the time needed to determine a personâ€™s friends of friends and reduce network traffic on Facebookâ€™s machines by about two-thirds. Whatâ€™s not to like? The probability of people being friends tends to decrease as the distance between them increases. This makes sense in the physical world, but itâ€™s true in the digital world as well. Yet, despite this, the enormous network of Facebook users is an example of a small-world network. The average distance between Facebook users the number of friend-links to connect people is less than five. And even though the collection of users and their connections may look chaotic, the network actually has a good deal of structure. For example, itâ€™s searchable. That is, two people who are, say, five friend-links away, could likely navigate from one person to the other by knowing only the friends at each point (but not knowing anyoneâ€™s friends of friends). For More Information: Networks, Crowds, and Markets: Reasoning about a Highly Connected World, David Easley and Jon Kleinberg, 2010.…

## Catching and Releasing: Part 1

Thereâ€™s more mathematics involved in juggling than just trying to make sure that the number of balls (or chainsaws) that hits the ground stays at zero. Subjects such as combinatorics and abstract algebra help jugglers answer important questions, such as whether a particular juggling pattern can actually be juggled. For example, can balls be juggled so that the time period that each ball stays aloft alternates between five counts and one? The answer is Yes. Math also tells you that the number of balls needed for such a juggling pattern is the average of the counts, in this case three.

Once a pattern is shown to be juggleable and the number of balls needed is known, equations of motion determine the speed with which each ball must be thrown and the maximum height it will attain. Obviously the harder a juggler throws, the faster and higher an object will go. Unfortunately hang time increases proportionally to the square root of the height, so the difficulty of keeping many objects in the air increases very quickly. Both math and juggling have been around for millennia yet questions still remain in both subjects. As two juggling mathematicians wrote, â€œA juggler, like a mathematician, is never finished: there is always another great unsolved problem.

…

## Catching and Releasing: Part 2

Thereâ€™s more mathematics involved in juggling than just trying to make sure that the number of balls (or chainsaws) that hits the ground stays at zero. Subjects such as combinatorics and abstract algebra help jugglers answer important questions, such as whether a particular juggling pattern can actually be juggled. For example, can balls be juggled so that the time period that each ball stays aloft alternates between five counts and one? The answer is Yes. Math also tells you that the number of balls needed for such a juggling pattern is the average of the counts, in this case three.

Once a pattern is shown to be juggleable and the number of balls needed is known, equations of motion determine the speed with which each ball must be thrown and the maximum height it will attain. Obviously the harder a juggler throws, the faster and higher an object will go. Unfortunately hang time increases proportionally to the square root of the height, so the difficulty of keeping many objects in the air increases very quickly. Both math and juggling have been around for millennia yet questions still remain in both subjects. As two juggling mathematicians wrote, â€œA juggler, like a mathematician, is never finished: there is always another great unsolved problem.

…

## Describing the Oceans

Imagine trying to describe the circulation and temperatures across the vast expanse of our oceans. Good models of our oceans not only benefit fishermen on our coasts but farmers inland as well. Until recently, there were neither adequate tools nor enough data to construct models. Now with new data and new mathematics, short-range climate forecasting for example, of an upcoming El NiÃ±o is possible.There is still much work to be done in long-term climate forecasting, however, and we only barely understand the oceans. Existing equations describe ocean dynamics, but solutions to the equations are currently out of reach. No computer can accommodate the data required to approximate a good solution to these equations. Researchers therefore make simplifying assumptions in order to solve the equations. New data are used to test the accuracy of models derived from these assumptions. This research is essential because we cannot understand our climate until we understand the oceans. For More Information: Whatâ€™s Happening in the Mathematical Sciences, Vol 1, Barry Cipra.…

## Finding Friends: Part 1

Facebook has over 700 million users with almost 70 billion connections. The hard part isnâ€™t people making friends; rather itâ€™s Facebookâ€™s computers storing and accessing relevant data, including information about friends of friends. The latter is important for recommendations to users (People You May Know). Much of this work involves computer science, but mathematics also plays a significant role. Subjects such as linear programming and graph theory help cut in half the time needed to determine a personâ€™s friends of friends and reduce network traffic on Facebookâ€™s machines by about two-thirds. Whatâ€™s not to like? The probability of people being friends tends to decrease as the distance between them increases. This makes sense in the physical world, but itâ€™s true in the digital world as well. Yet, despite this, the enormous network of Facebook users is an example of a small-world network. The average distance between Facebook users the number of friend-links to connect people is less than five. And even though the collection of users and their connections may look chaotic, the network actually has a good deal of structure. For example, itâ€™s searchable. That is, two people who are, say, five friend-links away, could likely navigate from one person to the other by knowing only the friends at each point (but not knowing anyoneâ€™s friends of friends). For More Information: Networks, Crowds, and Markets: Reasoning about a Highly Connected World, David Easley and Jon Kleinberg, 2010.…

## Putting the auto in automobile

It may be hard to accept but itâ€™s likely that weâ€™d all be much safer in autonomous vehicles driven by computers, not humans. Annually more than 30,000 Americans die in car crashes, almost all due to human error. Autonomous vehicles will communicate position and speed to each other and avoid potential collisions-without the possibility of dozing off or road rage. There are still many legal (and insurance) issues to resolve, but researchers who are revving up the development of autonomous vehicles are relying on geometry for recognizing and tracking objects, probability to assess risk, and logic to prove that systems will perform as required. The advent of autonomous vehicles will bring in new systems to manage traffic as well, for example, at automated intersections. Cars will communicate to intersection-managing computers and secure reservations to pass through. In a matter of milliseconds, the computers will use trigonometry and differential equations to simulate vehiclesâ€™ paths through the intersection and grant entry as long as there is no conflict with other vehiclesâ€™ paths. Waiting wonâ€™t be completely eliminated but will be substantially reduced, as will the fuel--and patience--currently wasted. Although the intersection at the left might look wild, experiments indicate that because vehicles would follow precise paths, such intersections will be much safer and more efficient than the ones we drive through now.…

## Forecasting Crime Part 1

No one can predict who will commit a crime but in some cities math is helping detect areas where crimes have the greatest chance of occurring. Police then increase patrols in these "hot spots" in order to prevent crime. This innovative practice, called predictive policing, is based on large amounts of data collected from previous crimes, but it involves more than just maps and push pins. Predictive policing identifies hot spots by using algorithms similar to those used to predict aftershocks after major earthquakes. Just as aftershocks are more likely near a recent earthquakeâ€™s epicenter, so too are crimes, as criminals do indeed return to, or very close to, the scene of a crime. Cities employing this approach have seen crime rates drop and studies are underway to measure predictive policingâ€™s part in that drop. One fact that has been determined concerns the nature of hot spots. Researchers using partial differential equations and bifurcation theory have discovered two types of hot spots, which respond quite differently to increased patrols. One type will shift to another area of the city while the other will disappear entirely. Unfortunately the two appear the same on the surface, so mathematicians and others are working to help police find ways to differentiate between the two so as to best allocate their resources.…

## Forecasting Crime Part 2

No one can predict who will commit a crime but in some cities math is helping detect areas where crimes have the greatest chance of occurring. Police then increase patrols in these "hot spots" in order to prevent crime. This innovative practice, called predictive policing, is based on large amounts of data collected from previous crimes, but it involves more than just maps and push pins. Predictive policing identifies hot spots by using algorithms similar to those used to predict aftershocks after major earthquakes. Just as aftershocks are more likely near a recent earthquakeâ€™s epicenter, so too are crimes, as criminals do indeed return to, or very close to, the scene of a crime. Cities employing this approach have seen crime rates drop and studies are underway to measure predictive policingâ€™s part in that drop. One fact that has been determined concerns the nature of hot spots. Researchers using partial differential equations and bifurcation theory have discovered two types of hot spots, which respond quite differently to increased patrols. One type will shift to another area of the city while the other will disappear entirely. Unfortunately the two appear the same on the surface, so mathematicians and others are working to help police find ways to differentiate between the two so as to best allocate their resources.…

## Being on the Cutting Edge

Cutters of diamonds and other gemstones have a high-pressure job with conflicting demands: Flaws must be removed from rough stones to maximize brilliance but done so in a way that yields the greatest weight possible. Because diamonds are often cut to a standard shape, cutting them is far less complex than cutting other gemstones, such as rubies or sapphires, which can have hundreds of different shapes. By coupling geometry and multivariable calculus with optimization techniques, mathematicians have been able to devise algorithms that automatically generate precise cutting plans that maximize brilliance and yield. The goal is to find the final shape within a rough stone. There are an endless number of candidates, positions, and orientations, so finding the shape amounts to a maximization problem with a large number of variables subject to an infinite number of constraints, a technique called semi-infinite optimization. Experienced human cutters create finished gems that average about 1/3 of the weight of the original rough stone. Cutting with this automated algorithm improved the yield to well above 40%, which, given the value of the stones, is a tremendous improvement. Without a doubt, semi-infinite optimization is a girlâ€™s (or boyâ€™s) best friend.…

## Getting a Handle on Obesity

Once a problem only in the developed world, obesity is now a worldwide epidemic. The overwhelming cause of the epidemic is a dramatic increase in the food supply and in food consumption not a surprise. Yet there are still many mysteries about weight change that canâ€™t be answered either inside the lab, because of the impracticality of keeping people isolated for long periods of time, or outside, because of the unreliability of dietary diaries. Mathematical models based on differential equations can help overcome this roadblock and allow detailed analysis of the relationship between food intake, metabolism, and weight change. The modelsâ€™ predictions fit existing data and explain such things as why it is hard to keep weight off and why obese people are more susceptible to further weight gain.

Researchers are also investigating why dieters often plateau after a few months and slowly regain weight. A possible explanation is that metabolism slows to match the drop in food consumed, but models representing food intake and energy expenditure as a dynamical system show that such a weight plateau doesnâ€™t take effect until much later. The likely culprit is a combination of slower metabolism and a lack of adherence to the diet. Most people are in approximate steady state, so that long-term changes are necessary to gain or lose weight. The good news is that each (enduring) drop of 10 calories a day translates into one pound of weight loss over three years, with about half the loss occurring in the first year.

For More Information: Quantification of the effect of energy imbalance on bodyweight, Hall et al. Lancet, Vol. 378 (2011), pp. 826-837.…

## Keeping Things in Focus - Part 2

Some of the simplest and most well-known curves parabolas and ellipses, which can be traced back to ancient Greece are also among the most useful. Parabolas have a reflective property that is employed in many of todayâ€™s solar power technologies. Mirrors with a parabolic shape reflect all entering light to a single point called the focus, where the solar power is converted into usable energy. Ellipses, which have two foci, have a similar reflecting property that is exploited in a medical procedure called lithotripsy. Patients with kidney stones and gallstones are positioned in a tank shaped like half an ellipse so that the stones are at one focus. Acoustic waves sent from the other focus concentrate all their energy on the stones, pulverizing them without surgery. Math can sometimes throw you a curve, but thatâ€™s not necessarily a bad thing. Parabolas and ellipses are curves called conic sections. Another curve in this category is the hyperbola, which may have the most profound application of all the nature of the universe. In plane geometry, points that are a given distance from a fixed point form a circle. In space, points that are a given spacetime distance from a fixed point form one branch of a hyperbola. This is not an arbitrary mandate but instead a natural conclusion from the equations that result when the principle of relativity is reconciled with our notions of distance and causality. And although a great deal of time has elapsed since the discovery of conic sections, they continue to reap benefits today. For More Information: Practical Conic Sections: The Geometric Properties of Ellipses, Parabolas and Hyperbolas, J. W. Downs, 2010.…

## Keeping Things in Focus - Part 1

Some of the simplest and most well-known curves parabolas and ellipses, which can be traced back to ancient Greece are also among the most useful. Parabolas have a reflective property that is employed in many of todayâ€™s solar power technologies. Mirrors with a parabolic shape reflect all entering light to a single point called the focus, where the solar power is converted into usable energy. Ellipses, which have two foci, have a similar reflecting property that is exploited in a medical procedure called lithotripsy. Patients with kidney stones and gallstones are positioned in a tank shaped like half an ellipse so that the stones are at one focus. Acoustic waves sent from the other focus concentrate all their energy on the stones, pulverizing them without surgery. Math can sometimes throw you a curve, but thatâ€™s not necessarily a bad thing.

Parabolas and ellipses are curves called conic sections. Another curve in this category is the hyperbola, which may have the most profound application of all the nature of the universe. In plane geometry, points that are a given distance from a fixed point form a circle. In space, points that are a given spacetime distance from a fixed point form one branch of a hyperbola. This is not an arbitrary mandate but instead a natural conclusion from the equations that result when the principle of relativity is reconciled with our notions of distance and causality. And although a great deal of time has elapsed since the discovery of conic sections, they continue to reap benefits today.

For More Information: Practical Conic Sections: The Geometric Properties of Ellipses, Parabolas and Hyperbolas, J. W. Downs, 2010.…

## Harnessing Wind Power - Part 2

Mathematics contributes in many ways to the process of converting wind power into usable energy. Large-scale weather models are used to find suitable locations for wind farms, while more narrowly focused models incorporating interactions arising from factors such as wake effects and turbulence specify how to situate individual turbines within a farm. In addition, computational fluid dynamics describes air flow and drag around turbines. This helps determine the optimal shapes for the blades, both structurally and aerodynamically, to extract as much energy as possible, and keep noise levels and costs down.

Mathematics also helps answer two fundamental questions about wind turbines. First, why three blades? Turbines with fewer blades extract less energy and are noisier (because the blades must turn faster). Those with more than three blades would capture more energy but only about three percent more, which doesnâ€™t justify the increased cost. Second, what percentage of wind energy can a turbine extract? Calculus and laws of conservation provide the justification for Betz Law, which states that no wind turbine can capture more than 60% of the energy in the wind. Modern turbines generally gather 40-50%. So the answer to someone who touts a turbine that can capture 65% of wind energy is "All Betz" are off.

For More Information: Wind Energy Explained: Theory, Design and Application, Manwell, McGowan, and Rogers, 2010.…

## Harnessing Wind Power - Part 1

Mathematics contributes in many ways to the process of converting wind power into usable energy. Large-scale weather models are used to find suitable locations for wind farms, while more narrowly focused models incorporating interactions arising from factors such as wake effects and turbulence specify how to situate individual turbines within a farm. In addition, computational fluid dynamics describes air flow and drag around turbines. This helps determine the optimal shapes for the blades, both structurally and aerodynamically, to extract as much energy as possible, and keep noise levels and costs down.

Mathematics also helps answer two fundamental questions about wind turbines. First, why three blades? Turbines with fewer blades extract less energy and are noisier (because the blades must turn faster). Those with more than three blades would capture more energy but only about three percent more, which doesnâ€™t justify the increased cost. Second, what percentage of wind energy can a turbine extract? Calculus and laws of conservation provide the justification for Betz Law, which states that no wind turbine can capture more than 60% of the energy in the wind. Modern turbines generally gather 40-50%. So the answer to someone who touts a turbine that can capture 65% of wind energy is "All Betz" are off.

For More Information: Wind Energy Explained: Theory, Design and Application, Manwell, McGowan, and Rogers, 2010.…

## Keeping the beat - Part 2

The heartâ€™s function of pumping blood may seem fairly simple but the underlying mechanisms and electrical impulses that maintain a healthy rhythm are extremely complex. Many areas of mathematics, including differential equations, dynamical systems, and topology help model the electrical behavior of cardiac cells, the connections between those cells and the heartâ€™s overall geometry. Researchers aim to gain a better understanding of the normal operation of the heart, as well as learn how to diagnose the onset of abnormalities and correct them.

Of the many things that can go wrong with a heartâ€™s rhythm, some measure of unpredictability is (surprisingly) not one of them. A healthy heartbeat is actually quite chaotic not regular at all. Furthermore, beat patterns become less chaotic as people age and heart function diminishes. In fact, one researcher recommends that patients presented with a new medication should ask their doctors, "What is this drug going to do to my fractal dimensionality?"

For More Information: Taking Mathematics to Heart: Mathematical Challenges in Cardiac Electrophysiology, John W. Cain, Notices of the AMS, April 2011, pp. 542-549.…

## Keeping the beat - Part 1

The heartâ€™s function of pumping blood may seem fairly simple but the underlying mechanisms and electrical impulses that maintain a healthy rhythm are extremely complex. Many areas of mathematics, including differential equations, dynamical systems, and topology help model the electrical behavior of cardiac cells, the connections between those cells and the heartâ€™s overall geometry. Researchers aim to gain a better understanding of the normal operation of the heart, as well as learn how to diagnose the onset of abnormalities and correct them.

Of the many things that can go wrong with a heartâ€™s rhythm, some measure of unpredictability is (surprisingly) not one of them. A healthy heartbeat is actually quite chaotic not regular at all. Furthermore, beat patterns become less chaotic as people age and heart function diminishes. In fact, one researcher recommends that patients presented with a new medication should ask their doctors, "What is this drug going to do to my fractal dimensionality?"

For More Information: Taking Mathematics to Heart: Mathematical Challenges in Cardiac Electrophysiology, John W. Cain, Notices of the AMS, April 2011, pp. 542-549.…

## Sustaining the Supply Chain - Part 2

Itâ€™s often a challenge to get from Point A to Point B in normal circumstances, but after a disaster it can be almost impossible to transport food, water, and clothing from the usual supply points to the people in desperate need. A new mathematical model employs probability and nonlinear programming to design supply chains that have the best chance of functioning after a disaster. For each region or country, the model generates a robust chain of supply and delivery points that can respond to the combination of disruptions in the network and increased needs of the population.

Math also helps medical agencies operate more efficiently during emergencies, such as an infectious outbreak. Fluid dynamics and combinatorial optimization are applied to facility layout and epidemiological models to allocate resources and improve operations while minimizing total infection within dispensing facilities. This helps ensure fast, effective administering of vaccines and other medicines. Furthermore, solution times are fast enough that officials can input up-to-the-minute data specific to their situation and make any necessary redistribution of supplies or staff in real time.

For More Information: Supply Chain Network Economics: Dynamics of Prices, Flows, and Profits, Anna Nagurney, 2006.…

## Sustaining the Supply Chain - Part 1

Itâ€™s often a challenge to get from Point A to Point B in normal circumstances, but after a disaster it can be almost impossible to transport food, water, and clothing from the usual supply points to the people in desperate need. A new mathematical model employs probability and nonlinear programming to design supply chains that have the best chance of functioning after a disaster. For each region or country, the model generates a robust chain of supply and delivery points that can respond to the combination of disruptions in the network and increased needs of the population.

Math also helps medical agencies operate more efficiently during emergencies, such as an infectious outbreak. Fluid dynamics and combinatorial optimization are applied to facility layout and epidemiological models to allocate resources and improve operations while minimizing total infection within dispensing facilities. This helps ensure fast, effective administering of vaccines and other medicines. Furthermore, solution times are fast enough that officials can input up-to-the-minute data specific to their situation and make any necessary redistribution of supplies or staff in real time.

For More Information: Supply Chain Network Economics: Dynamics of Prices, Flows, and Profits, Anna Nagurney, 2006.…

## Answering the Question, and Vice Versa

Experts are adept at answering questions in their fields, but even the most knowledgeable authority canâ€™t be expected to keep up with all the data generated today. Computers can handle data, but until now, they were inept at understanding questions posed in conversational language. Watson, the IBM computer that won the Jeopardy! Challenge, is an example of a computer that can answer questions using informal, nuanced, even pun-filled, phrases. Graph theory, formal logic, and statistics help create the algorithms used for answering questions in a timely mannerâ€”not at all elementary. Watsonâ€™s creators are working to create technology that can do much more than win a TV game show. Programmers are aiming for systems that will soon respond quickly with expert answers to real-world problemsâ€”from the fairly straightforward, such as providing technical support, to the more complex, such as responding to queries from doctors in search of the correct medical diagnosis. Most of the research involves computer science, but mathematics will help to expand applications to other industries and to scale down the size and cost of the hardware that makes up these modern question-answering systems. For More Information: Final Jeopardy: Man vs. Machine and the Quest to Know Everything, Stephen Baker, 2011.…

## Sounding the Alarm - Part 2

Nothing can prevent a tsunami from happening they are enormously powerful events of nature. But in many cases networks of seismic detectors, sea-level monitors and deep ocean buoys can allow authorities to provide adequate warning to those at risk. Mathematical models constructed from partial differential equations use the generated data to determine estimates of the speed and magnitude of a tsunami and its arrival time on coastlines. These models may predict whether a trough or a crest will be the first to arrive on shore. In only about half the cases (not all) does the trough arrive first, making the water level recede dramatically before the onslaught of the crest.

Mathematics also helps in the placement of detectors and monitors. Researchers use geometry and population data to find the best locations for the sensors that will alert the maximum number of people. Once equipment is in place, warning centers collect and process data from many seismic stations to determine if an earthquake is the type that will generate a dangerous tsunami. All that work must wait until an event occurs because it is currently very hard to predict earthquakes. People on coasts far from an earthquake-generated tsunami may have hours to take action, but for those closer itâ€™s a matter of minutes. The crest of a tsunami wave can travel at 450 miles per hour in open water, so fast algorithms for solving partial differential equations are essential.

For More Information: Surface Water Waves and Tsunamis,

Walter Craig, Journal of Dynamics and Differential Equations, Vol. 18, no. 3 (2006), pp. 525-549.…

## Sounding the Alarm - Part 1

Nothing can prevent a tsunami from happening they are enormously powerful events of nature. But in many cases networks of seismic detectors, sea-level monitors and deep ocean buoys can allow authorities to provide adequate warning to those at risk. Mathematical models constructed from partial differential equations use the generated data to determine estimates of the speed and magnitude of a tsunami and its arrival time on coastlines. These models may predict whether a trough or a crest will be the first to arrive on shore. In only about half the cases (not all) does the trough arrive first, making the water level recede dramatically before the onslaught of the crest.

Mathematics also helps in the placement of detectors and monitors. Researchers use geometry and population data to find the best locations for the sensors that will alert the maximum number of people. Once equipment is in place, warning centers collect and process data from many seismic stations to determine if an earthquake is the type that will generate a dangerous tsunami. All that work must wait until an event occurs because it is currently very hard to predict earthquakes. People on coasts far from an earthquake-generated tsunami may have hours to take action, but for those closer itâ€™s a matter of minutes. The crest of a tsunami wave can travel at 450 miles per hour in open water, so fast algorithms for solving partial differential equations are essential.

For More Information: Surface Water Waves and Tsunamis,

Walter Craig, Journal of Dynamics and Differential Equations, Vol. 18, no. 3 (2006), pp. 525-549.…

## Putting Another Cork in It - Part 2

A triple cork is a spinning jump in which the snowboarder is parallel to the ground three times while in the air. Such a jump had never been performed in a competition before 2011, which prompted ESPNâ€™s Sport Science program to ask math professor Tim Chartier if it could be done under certain conditions. Originally doubtful, he and a recent math major graduate used differential equations, vector analysis, and calculus to discover that yes, a triple cork was indeed possible. A few days later, boarder Torstein Horgmo landed a successful triple cork at the X-Games (which presumably are named for everyoneâ€™s favorite variable).

Snowboarding is not the only sport in which modern athletes and coaches seek answers from mathematics. Swimming and bobsledding research involves computational fluid dynamics to analyze fluid flow so as to decrease drag. Soccer and basketball analysts employ graph and network theory to chart passes and quantify team performance. And coaches in the NFL apply statistics and game theory to focus on the expected value of a play instead of sticking with the traditional Square root of 9 yards and a cloud of dust.

…

## Putting Another Cork in It - Part 1

A triple cork is a spinning jump in which the snowboarder is parallel to the ground three times while in the air. Such a jump had never been performed in a competition before 2011, which prompted ESPNâ€™s Sport Science program to ask math professor Tim Chartier if it could be done under certain conditions. Originally doubtful, he and a recent math major graduate used differential equations, vector analysis, and calculus to discover that yes, a triple cork was indeed possible. A few days later, boarder Torstein Horgmo landed a successful triple cork at the X-Games (which presumably are named for everyoneâ€™s favorite variable).

Snowboarding is not the only sport in which modern athletes and coaches seek answers from mathematics. Swimming and bobsledding research involves computational fluid dynamics to analyze fluid flow so as to decrease drag. Soccer and basketball analysts employ graph and network theory to chart passes and quantify team performance. And coaches in the NFL apply statistics and game theory to focus on the expected value of a play instead of sticking with the traditional Square root of 9 yards and a cloud of dust.

…

## Assigning Seats - Part 2

As difficult as it is to do the census, the ensuing process of determining the number of congressional seats for each state can be even harder. The basic premise, that the proportion of each state's delegation in the House should match its proportion of the U.S. population, is simple enough. The difficulty arises when deciding what to do with the fractions that inevitably arise (e.g., New York can't have 28.7 seats). Over the past 200 years, several methods of apportioning seats have been used. Many sound good but can lead to paradoxes, such as an increase in the total number of House seats actually resulting in a reduction of a state's delegation. The method used since the 1940s, whose leading proponent was a mathematician, is one that avoids such paradoxes.

A natural question is Why 435 seats? Nothing in the Constitution mandates this
number, although there is a prohibition against having more than one seat per
30,000 people. One model, based on the need for legislators to communicate with their constituents and with each other, uses algebra and calculus to show that the ideal assembly size is the cube root of the population it represents. Remarkably, the size of the House mirrored this rule until the early 1900s. To obey the rule now would require an increase to 670, which would presumably both better represent the population and increase the chances that the audience in the seats for those late speeches would outnumber the speaker.

For More Information: "E pluribus confusion", Barry Cipra, American Scientist, July-August 2010.…

## Assigning Seats - Part 1

As difficult as it is to do the census, the ensuing process of determining the number of congressional seats for each state can be even harder. The basic premise, that the proportion of each state's delegation in the House should match its proportion of the U.S. population, is simple enough. The difficulty arises when deciding what to do with the fractions that inevitably arise (e.g., New York can't have 28.7 seats). Over the past 200 years, several methods of apportioning seats have been used. Many sound good but can lead to paradoxes, such as an increase in the total number of House seats actually resulting in a reduction of a state's delegation. The method used since the 1940s, whose leading proponent was a mathematician, is one that avoids such paradoxes.

A natural question is Why 435 seats? Nothing in the Constitution mandates this
number, although there is a prohibition against having more than one seat per
30,000 people. One model, based on the need for legislators to communicate with their constituents and with each other, uses algebra and calculus to show that the ideal assembly size is the cube root of the population it represents. Remarkably, the size of the House mirrored this rule until the early 1900s. To obey the rule now would require an increase to 670, which would presumably both better represent the population and increase the chances that the audience in the seats for those late speeches would outnumber the speaker.

For More Information: "E pluribus confusion", Barry Cipra, American Scientist, July-August 2010.…

## Knowing Rogues - Part 2

It doesn't take a perfect storm to generate a rogue wave-an open-ocean wave much steeper and more massive than its neighbors that appears with little or no warning. Sometimes winds and currents collide causing waves to combine nonlinearly and produce these towering walls of water. Mathematicians and other researchers are collecting data from rogue waves and modeling them with partial differential equations to understand how and why they form. A deeper understanding of both their origins and their frequency will result in safer shipping and offshore platform operations.

Since rogue waves are rare and short lived (fortunately), studying them is not easy. So some researchers are experimenting with light to create rogue waves in a different medium. Results of these experiments are consistent with sailors' claims that rogues, like other unusual events, are more frequent than what is predicted by standard models. The standard models had assumed a bell-shaped distribution for wave heights, and anticipated a rogue wave about once every 10,000 years. This purported extreme unlikelihood led designers and builders to not account for their potential catastrophic effects.

Today's recognition of rogues as rare, but realistic, possibilities could save the shipping industry billions of dollars and hundreds of lives.

For More Information: "Dashing Rogues", Sid Perkins, Science News, November
18, 2006.…

## Knowing Rogues - Part 1

It doesn't take a perfect storm to generate a rogue wave-an open-ocean wave much steeper and more massive than its neighbors that appears with little or no warning. Sometimes winds and currents collide causing waves to combine nonlinearly and produce these towering walls of water. Mathematicians and other researchers are collecting data from rogue waves and modeling them with partial differential equations to understand how and why they form. A deeper understanding of both their origins and their frequency will result in safer shipping and offshore platform operations.

Since rogue waves are rare and short lived (fortunately), studying them is not easy. So some researchers are experimenting with light to create rogue waves in a different medium. Results of these experiments are consistent with sailors' claims that rogues, like other unusual events, are more frequent than what is predicted by standard models. The standard models had assumed a bell-shaped distribution for wave heights, and anticipated a rogue wave about once every 10,000 years. This purported extreme unlikelihood led designers and builders to not account for their potential catastrophic effects.

Today's recognition of rogues as rare, but realistic, possibilities could save the shipping industry billions of dollars and hundreds of lives.

For More Information: "Dashing Rogues", Sid Perkins, Science News, November
18, 2006.…

## Creating Something out of (Next to) Nothing

Normally when creating a digital file, such as a picture, much more information is recorded than necessary-even before storing or sending. The image on the right
was created with compressed (or compressive) sensing, a breakthrough technique based on probability and linear algebra. Rather than recording excess
information and discarding what is not needed, sensors collect the most significant information at the time of creation, which saves power, time, and memory. The potential increase in efficiency has led researchers to investigate employing compressed sensing in applications ranging from missions in space, where minimizing power consumption is important, to MRIs, for which faster image creation would allow for better scans and happier patients.

Just as a word has different representations in different languages, signals (such as images or audio) can be represented many different ways. Compressed sensing relies on using the representation for the given class of signals that requires the fewest bits. Linear programming applied to that representation finds the most likely candidate fitting the particular low-information signal.

Mathematicians have proved that in all but the very rarest case that candidate-often constructed from less than a tiny fraction of the data traditionally collected-matches the original. The ability to locate and capture only the most important components without any loss of quality is so unexpected that even the mathematicians who discovered compressed sensing found it hard to believe.

For More Information: "Compressed Sensing Makes Every Pixel Count,"
What's Happening in the Mathematical Sciences, Vol. 7, Dana Mackenzie.…

## Getting at the Truth - Part 2

Mathematics has helped investigators in several major cases of human rights abuses and election fraud. Among them: The 2009 election in Iran. A mathematical result known as Benford's Law states that the leading digits of truly random numbers aren't distributed uniformly, as might be expected. Instead, smaller digits, such as 1's, appear much more frequently than larger digits, such as 9's. Benford's Law and other statistical tests have been applied to the 2009 election and suggest strongly that the final totals
are suspicious.

Ethnic cleansing. When Slobodan MiloÅ¡evicÂ´ went on trial, it was his contention that
the mass exodus of ethnic Albanians from Kosovo was due to NATO bombings and the activities of the Albanian Kosovo Liberation Army rather than anything he had ordered. A team collected data on the flow of refugees to test those hypotheses
and was able to refute MiloÅ¡evicÂ´'s claim in its entirety.

Guatemalan disappearances. Here, statistics is being used to extract information
from over 80 million National Police archive pages related to about 200,000 deaths and disappearances. Sampling techniques give investigators an accurate representation of the records without them having to read every page.

Families are getting long-sought after proof of what happened to their relatives, and investigators are uncovering patterns and motives behind the abductions and murders. Tragically, the people have disappeared. But because of this analysis, the
facts won't.

For More Information: Killings and Refugee Flow in Kosovo, March-June 1999, Ball et al., 2002.…

## Getting at the Truth - Part 1

Mathematics has helped investigators in several major cases of human rights abuses and election fraud. Among them: The 2009 election in Iran. A mathematical result known as Benford's Law states that the leading digits of truly random numbers aren't distributed uniformly, as might be expected. Instead, smaller digits, such as 1's, appear much more frequently than larger digits, such as 9's. Benford's Law and other statistical tests have been applied to the 2009 election and suggest strongly that the final totals
are suspicious.

Ethnic cleansing. When Slobodan MiloÅ¡evicÂ´ went on trial, it was his contention that
the mass exodus of ethnic Albanians from Kosovo was due to NATO bombings and the activities of the Albanian Kosovo Liberation Army rather than anything he had ordered. A team collected data on the flow of refugees to test those hypotheses
and was able to refute MiloÅ¡evicÂ´'s claim in its entirety.

Guatemalan disappearances. Here, statistics is being used to extract information
from over 80 million National Police archive pages related to about 200,000 deaths and disappearances. Sampling techniques give investigators an accurate representation of the records without them having to read every page.

Families are getting long-sought after proof of what happened to their relatives, and investigators are uncovering patterns and motives behind the abductions and murders. Tragically, the people have disappeared. But because of this analysis, the
facts won't.

For More Information: Killings and Refugee Flow in Kosovo, March-June 1999, Ball et al., 2002.…

## Resisting the Spread of Disease - Part 2

One of the most useful tools in analyzing the spread of disease is a system of
evolutionary equations that reflects the dynamics among three distinct categories
of a population: those susceptible (S) to a disease, those infected (I) with
it, and those recovered (R) from it. This SIR model is applicable to a range of
diseases, from smallpox to the flu. To predict the impact of a particular disease it
is crucial to determine certain parameters associated with it, such as the average
number of people that a typical infected person will infect. Researchers estimate
these parameters by applying statistical methods to gathered data, which arenâ€™t
complete because, for example, some cases arenâ€™t reported. Armed with reliable
models, mathematicians help public health officials battle the complex, rapidly
changing world of modern disease.

Todayâ€™s models are more sophisticated than those of even a few years ago. They
incorporate information such as contact periods that vary with age (young people
have contact with one another for a longer period of time than do adults from
different households), instead of assuming equal contact periods for everyone. The capacity to treat variability makes it possible to predict the effectiveness of targeted vaccination strategies to combat the flu, for instance. Some models now use graph theory and matrices to represent networks of social interactions, which are important
in understanding how far and how fast a given disease will spread.

For More Information: Mathematical Models in Population Biology and Epidemiology, Fred Brauer and Carlos Castillo-Chavez.…

## Resisting the Spread of Disease - Part 1

One of the most useful tools in analyzing the spread of disease is a system of
evolutionary equations that reflects the dynamics among three distinct categories
of a population: those susceptible (S) to a disease, those infected (I) with
it, and those recovered (R) from it. This SIR model is applicable to a range of
diseases, from smallpox to the flu. To predict the impact of a particular disease it
is crucial to determine certain parameters associated with it, such as the average
number of people that a typical infected person will infect. Researchers estimate
these parameters by applying statistical methods to gathered data, which arenâ€™t
complete because, for example, some cases arenâ€™t reported. Armed with reliable
models, mathematicians help public health officials battle the complex, rapidly
changing world of modern disease.

Todayâ€™s models are more sophisticated than those of even a few years ago. They
incorporate information such as contact periods that vary with age (young people
have contact with one another for a longer period of time than do adults from
different households), instead of assuming equal contact periods for everyone. The capacity to treat variability makes it possible to predict the effectiveness of targeted vaccination strategies to combat the flu, for instance. Some models now use graph theory and matrices to represent networks of social interactions, which are important
in understanding how far and how fast a given disease will spread.

For More Information: Mathematical Models in Population Biology and Epidemiology, Fred Brauer and Carlos Castillo-Chavez.…

## Predicting Climate - Part 2

Whatâ€™s in store for our climate and us? Itâ€™s an extraordinarily complex quesÂtion whose answer requires physics, chemistry, earth science, and mathematics (among other subjects) along with massive computing power. Mathematicians use partial differential equations to model the movement of the atmosphere; dynamical systems to describe the feedback between land, ocean, air, and ice; and statistics to quantify the uncertainty of current projections. Although there is some discrepancy among different climate forecasts, researchers all agree on the tremendous need for people to join this effort and create new approaches to help understand our climate.

Itâ€™s impossible to predict the weather even two weeks in advance, because almost identical sets of temperature, pressure, etc. can in just a few days result in drastiÂcally different weather. So how can anyone make a prediction about long-term climate? The answer is that climate is an average of weather conditions. In the same way that good predictions about the average height of 100 people can be made without knowing the height of any one person, forecasts of climate years into the future are feasible without being able to predict the conditions on a particular day. The challenge now is to gather more data and use subjects such as fluid dynamics and numerical methods to extend todayâ€™s 20-year projecÂtions forward to the next 100 years.

For More Information: Mathematics of Climate Change: A New Discipline for an Uncertain Century, Dana Mackenzie, 2007.…

## Predicting Climate - Part 1

Whatâ€™s in store for our climate and us? Itâ€™s an extraordinarily complex quesÂtion whose answer requires physics, chemistry, earth science, and mathematics (among other subjects) along with massive computing power. Mathematicians use partial differential equations to model the movement of the atmosphere; dynamical systems to describe the feedback between land, ocean, air, and ice; and statistics to quantify the uncertainty of current projections. Although there is some discrepancy among different climate forecasts, researchers all agree on the tremendous need for people to join this effort and create new approaches to help understand our climate.

Itâ€™s impossible to predict the weather even two weeks in advance, because almost identical sets of temperature, pressure, etc. can in just a few days result in drastiÂcally different weather. So how can anyone make a prediction about long-term climate? The answer is that climate is an average of weather conditions. In the same way that good predictions about the average height of 100 people can be made without knowing the height of any one person, forecasts of climate years into the future are feasible without being able to predict the conditions on a particular day. The challenge now is to gather more data and use subjects such as fluid dynamics and numerical methods to extend todayâ€™s 20-year projecÂtions forward to the next 100 years.

For More Information: Mathematics of Climate Change: A New Discipline for an Uncertain Century, Dana Mackenzie, 2007.…

## Matching Vital Needs - Increasing the number of live-donor kidney transplants

A person needing a kidney transplant may have a friend or relative who volunteers
to be a living donor, but whose kidney is incompatible, forcing the person to wait for a transplant from a deceased donor. In the U.S. alone, thousands of people die each year without ever finding a suitable kidney. A new technique applies graph theory to groups of incompatible patient-donor pairs to create the largest possible number of paired-donation exchanges. These exchanges, in which a donor paired with Patient A gives a kidney to Patient B while a donor paired with Patient B gives to Patient A, will dramatically increase transplants from living donors. Since transplantation is less expensive than dialysis, this mathematical algorithm, in addition to saving lives, will also save hundreds of millions of dollars annually.

Naturally there can be more transplants if matches along longer patient-donor cycles are considered (e.g., Aâ€™s donor to B, Bâ€™s donor to C, and Câ€™s donor to A). The problem is that the possible number of longer cycles grows so fast hundreds of millions of A >B>C>A matches in just 5000 donor-patient pairs that to search through all the possibilities is impossible. An ingenious use of random walks and integer programming now makes searching through all three-way matches feasible, even in a database large enough to include all incompatible patient-donor pairs.

For More Information:
Matchmaking for Kidneys, Dana Mackenzie, SIAM News, December 2008.
Image of suboptimal two-way matching (in purple) and an optimal matching (in green),
courtesy of Sommer Gentry.…

## Pulling Out (from) All the Stops - Visiting all of NY's subway stops in record time

With 468 stops served by 26 lines, the New York subway system can make visitors
feel lucky when they successfully negotiate one planned trip in a day. Yet
these two New Yorkers, Chris Solarz and Matt Ferrisi, took on the task of
breaking a world record by visiting every stop in the system in less than 24 hours.
They used mathematics, especially graph theory, to narrow down the possible
routes to a manageable number and subdivided the problem to find the best
routes in smaller groups of stations. Then they paired their mathematical work
with practice runs and crucial observations (the next-to-last car stops closest to
the stairs) to shatter the world record by more than two hours!

Although Chris and Mattâ€™s success may not have huge ramifications in other fields,
their work does have a lot in common with how people do modern mathematics
research

* They worked together, frequently using computers and often asking experts for
advice;

* They devoted considerable time and effort to meet their goal; and

* They continually refined their algorithm until arriving at a solution that was
nearly optimal.

Finally, they also experienced the same feeling that researchers do that despite all the hours and intense preparation, the project â€œfelt more like fun than work.

For More Information: Math whizzes shoot to set record for traversing subway system,â€ Sergey Kadinsky and Rich Schapiro, New York Daily News, January 22, 2009.

Photo by Elizabeth Ferrisi.

Map Â© New York Metropolitan Transit Authority.

The Mathematical Moments program promotes appreciation and understanding of the role mathematics plays in science, nature, technology, and human culture.
…

## Working It Out. Math solves a mystery about the opening of "A Hard Day's Night."

The music of most hit songs is pretty well known, but sometimes there are
mysteries. One question that remained unanswered for over forty years is: What
instrumentation and notes make up the opening chord of the Beatlesâ€™ "A Hard
Dayâ€™s Night"? Mathematician Jason Brown - a big Beatles fan - recently solved the
puzzle using his musical knowledge and discrete Fourier transforms, mathematical
transformations that help decompose signals into their basic parts.

These transformations simplify applications ranging from signal processing to multiplying large numbers, so that a researcher doesnâ€™t have to be "working like a dog" to get an answer.

Brown is also using mathematics, specifically graph theory, to discover who
wrote "In My Life," which both Lennon and McCartney claimed to have written.
In his graphs, chords are represented by points that are connected when one
chord immediately follows another. When all songs with known authorship
are diagrammed, Brown will see which collection of graphs - McCartneyâ€™s or
Lennonâ€™s - is a better fit for "In My Life." Although it may seem a bit counterintuitive
to use mathematics to learn more about a revolutionary band, these
analytical methods identify and uncover compositional principles inherent in some
of the best Beatlesâ€™ music. Thus itâ€™s completely natural and rewarding to apply
mathematics to the Fab 4

For More Information: Professor Uses Mathematics to Decode Beatles Tunes, "The Wall Street Journal", January 30, 2009..…

## Getting It Together

The collective motion of many groups of animals can be stunning. Flocks of birds
and schools of fish are able to remain cohesive, find food, and avoid predators without
leaders and without awareness of all but a few other members in their groups.
Research using vector analysis and statistics has led to the discovery of simple principles,
such as members maintaining a minimum distance between neighbors while
still aligning with them, which help explain shapes such as the one below.

Although collective motion by groups of animals is often beautiful, it can be costly
as well: Destructive locusts affect ten percent of the worldâ€™s population. Many other
animals exhibit group dynamics; some organisms involved are small while their
groups are huge, so researchersâ€™ models have to account for distances on vastly
different scales. The resulting equations then must be solved numerically, because of
the incredible number of animals represented. Conclusions from this research will
help manage destructive insects, such as locusts, as well as help speed the movement
of peopleâ€”ants rarely get stuck in traffic.

Photo by Jose Luis Gomez de Francisco.

For More Information: Swarm Theory, Peter Miller. National Geographic, July 2007.…

## Restoring Genius - Discovering lost works of Archimedes - Part 2

Archimedes was one of the most brilliant people ever, on a par with Einstein and Newton. Yet very little of what he wrote still exists because of the passage of time, and because many copies of his works were erased and the cleaned pages were used again. One of those written-over works (called a palimpsest) has resurfaced, and advanced digital imaging techniques using statistics and linear algebra have revealed his previously unknown discoveries in combinatorics and calculus. This leads to a question that would stump even Archimedes: How much further would mathematics and science have progressed had these discoveries not been erased?

One of the most dramatic revelations of Archimedesâ€™ work was done using X-ray fluorescence. A painting, forged in the 1940s by one of the bookâ€™s former owners, obscured the original text, but X-rays penetrated the painting and highlighted the iron in the ancient ink, revealing a page of Archimedesâ€™ treatise The Method of Mechanical Theorems. The entire process of uncovering this and his other ideas is made possible by modern mathematics and physics, which are built on his discoveries and techniques.

This completion of a circle of progress is entirely appropriate since one of Archimedesâ€™ accomplishments that wasnâ€™t lost is his approximation of pi.

For More Information: The Archimedes Codex, Reviel Netz and William Noel, 2007.…

## Restoring Genius - Discovering lost works of Archimedes - Part 1

Archimedes was one of the most brilliant people ever, on a par with Einstein and Newton. Yet very little of what he wrote still exists because of the passage of time, and because many copies of his works were erased and the cleaned pages were used again. One of those written-over works (called a palimpsest) has resurfaced, and advanced digital imaging techniques using statistics and linear algebra have revealed his previously unknown discoveries in combinatorics and calculus. This leads to a question that would stump even Archimedes: How much further would mathematics and science have progressed had these discoveries not been erased?

One of the most dramatic revelations of Archimedesâ€™ work was done using X-ray fluorescence. A painting, forged in the 1940s by one of the bookâ€™s former owners, obscured the original text, but X-rays penetrated the painting and highlighted the iron in the ancient ink, revealing a page of Archimedesâ€™ treatise The Method of Mechanical Theorems. The entire process of uncovering this and his other ideas is made possible by modern mathematics and physics, which are built on his discoveries and techniques.

This completion of a circle of progress is entirely appropriate since one of Archimedesâ€™ accomplishments that wasnâ€™t lost is his approximation of pi.

For More Information: The Archimedes Codex, Reviel Netz and William Noel, 2007.…

## Improving Stents - Part 2

Stents are expandable tubes that are inserted into blocked or damaged blood
vessels. They offer a practical way to treat coronary artery disease, repairing
vessels and keeping them open so that blood can flow freely. When stents
work, they are a great alternative to radical surgery, but they can deteriorate or
become dislodged. Mathematical models of blood vessels and stents are helping
to determine better shapes and materials for the tubes. These models are so
accurate that the FDA is considering requiring mathematical modeling in the
design of stents before any further testing is done, to reduce the need for expensive
experimentation.

Precise modeling of the entire human vascular system is far beyond the reach of
current computational power, so researchers focus their detailed models on small
subsections, which are coupled with simpler models of the rest of the system.
The Navier-Stokes equations are used to represent the flow of blood and its
interaction with vessel walls. A mathematical proof was the central part of recent
research that led to the abandonment of one type of stent and the design of
better ones. The goal now is to create better computational fluid-vessel models
and stent models to improve the treatment and prediction of coronary artery
disease the major cause of heart attacks.

For More Information: Design of Optimal Endoprostheses Using Mathematical Modeling,
Canic, Krajcer, and Lapin, Endovascular Today, May 2006.…

## Improving Stents - Part 1

Stents are expandable tubes that are inserted into blocked or damaged blood
vessels. They offer a practical way to treat coronary artery disease, repairing
vessels and keeping them open so that blood can flow freely. When stents
work, they are a great alternative to radical surgery, but they can deteriorate or
become dislodged. Mathematical models of blood vessels and stents are helping
to determine better shapes and materials for the tubes. These models are so
accurate that the FDA is considering requiring mathematical modeling in the
design of stents before any further testing is done, to reduce the need for expensive
experimentation.

Precise modeling of the entire human vascular system is far beyond the reach of
current computational power, so researchers focus their detailed models on small
subsections, which are coupled with simpler models of the rest of the system.
The Navier-Stokes equations are used to represent the flow of blood and its
interaction with vessel walls. A mathematical proof was the central part of recent
research that led to the abandonment of one type of stent and the design of
better ones. The goal now is to create better computational fluid-vessel models
and stent models to improve the treatment and prediction of coronary artery
disease the major cause of heart attacks.

For More Information: Design of Optimal Endoprostheses Using Mathematical Modeling,
Canic, Krajcer, and Lapin, Endovascular Today, May 2006.…

## Steering Towards Efficiency

The racing team is just as important to a carâ€™s finish as the driver is. With little to
separate competitors over hundreds of laps, teams search for any technological
edge that will propel them to Victory Lane. Of special use today is computational
fluid dynamics, which is used to predict airflow over a car, both alone and in relation
to other cars (for example, when drafting). Engineers also rely on more basic
subjects, such as calculus and geometry, to improve their cars. In fact, one racing
team engineer said of his calculus and physics teachers, the classes they taught to
this day were the most important classes Iâ€™ve ever taken.(1)

Mathematics helps the performance and efficiency of non-NASCAR vehicles, as
well. To improve engine performance, data must be collected and processed very
rapidly so that control devices can make adjustments to significant quantities such
as air/fuel ratios. Innovative sampling techniques make this real-time data collection
and processing possible. This makes for lower emissions and improved fuel
economy goals worthy of a checkered flag.

For More Information: The Physics of NASCAR, Diandra Leslie-Pelecky, 2008.…

## Hearing a Masterâ€™s Voice

The spools of wire below contain the only known live recording of the legendary folk singer Woody Guthrie. A mathematician, Kevin Short, was part of a team that used signal processing techniques associated with chaotic music compression to recapture the live performance, which was often completely unintelligible. The modern techniques employed, instead of resulting in a cold, digital output, actually retained the original concertâ€™s warmth and depth. As a result, Short and the team received a GrammyÂ© Award for their remarkable restoration of the recording.

To begin the restoration the wire had to be manually pulled through a playback device and converted to a digital format. Since the pulling speed wasnâ€™t constant there was distortion in the sound, frequently quite considerable. Algorithms corrected for the speed variations and reconfigured the sound waves to their original shape by using a background noise with a known frequency as a "clock." This clever correction also relied on sampling the sound selectively, and reconstructing and
resampling the music between samples. Mathematics did more than help recreate a performance lost for almost 60 years: These methods are used to digitize treasured tapes of audiophiles everywhere.

For More Information: "The Grammy in Mathematics," Julie J. Rehmeyer, Science News Online, February 9, 2008.…

## Going with the Floes - Part 1

Sea ice is one of the least understood components of our climate. Naturally its abundance or scarcity is a telling sign of climate change, but sea ice is also an important actor in change as well, insulating the ocean and reflecting sunlight. A branch of mathematics called percolation theory helps explain how salt water travels through sea ice, a process that is crucial both to the amount of sea ice present and to the microscopic communities that sustain polar ecosystems. By taking samples, doing on-site experiments, and then incorporating the data into models of porous materials, mathematicians are working to understand sea ice and help refine climate predictions.

Using probability, numerical analysis, and partial differential equations, researchers have recently shown that the permeability of sea ice is similar to that of some sedimentary rocks in the earthâ€™s crust, even though the substances are otherwise dissimilar. One major difference between the two is the drastic changes in permeability of sea ice, from total blockage to clear passage, that occur over a range of just a few degrees. This difference can have a major effect on measurements by satellite, which provide information on the extent and thickness of sea ice. Results about sea ice will not only make satellite measurements more reliable, but they can also be applied to descriptions of lung and bone porosity, and to understanding ice on other planets.

Image: Pancake ice in Antarctica, courtesy of Ken Golden.

For More Information: "Thermal evolution of permeability and microstructure in sea ice," K. M. Golden, et al., Geophysical Research Letters, August 28, 2007.…

## Going with the Floes - Part 2

Sea ice is one of the least understood components of our climate. Naturally its abundance or scarcity is a telling sign of climate change, but sea ice is also an important actor in change as well, insulating the ocean and reflecting sunlight. A branch of mathematics called percolation theory helps explain how salt water travels through sea ice, a process that is crucial both to the amount of sea ice present and to the microscopic communities that sustain polar ecosystems. By taking samples, doing on-site experiments, and then incorporating the data into models of porous materials, mathematicians are working to understand sea ice and help refine climate predictions.

Using probability, numerical analysis, and partial differential equations, researchers have recently shown that the permeability of sea ice is similar to that of some sedimentary rocks in the earthâ€™s crust, even though the substances are otherwise dissimilar. One major difference between the two is the drastic changes in permeability of sea ice, from total blockage to clear passage, that occur over a range of just a few degrees. This difference can have a major effect on measurements by satellite, which provide information on the extent and thickness of sea ice. Results about sea ice will not only make satellite measurements more reliable, but they can also be applied to descriptions of lung and bone porosity, and to understanding ice on other planets.

Image: Pancake ice in Antarctica, courtesy of Ken Golden.

For More Information: "Thermal evolution of permeability and microstructure in sea ice," K. M. Golden, et al., Geophysical Research Letters, August 28, 2007.…

## Going with the Floes - Part 3

Sea ice is one of the least understood components of our climate. Naturally its abundance or scarcity is a telling sign of climate change, but sea ice is also an important actor in change as well, insulating the ocean and reflecting sunlight. A branch of mathematics called percolation theory helps explain how salt water travels through sea ice, a process that is crucial both to the amount of sea ice present and to the microscopic communities that sustain polar ecosystems. By taking samples, doing on-site experiments, and then incorporating the data into models of porous materials, mathematicians are working to understand sea ice and help refine climate predictions.

Using probability, numerical analysis, and partial differential equations, researchers have recently shown that the permeability of sea ice is similar to that of some sedimentary rocks in the earthâ€™s crust, even though the substances are otherwise dissimilar. One major difference between the two is the drastic changes in permeability of sea ice, from total blockage to clear passage, that occur over a range of just a few degrees. This difference can have a major effect on measurements by satellite, which provide information on the extent and thickness of sea ice. Results about sea ice will not only make satellite measurements more reliable, but they can also be applied to descriptions of lung and bone porosity, and to understanding ice on other planets.

Image: Pancake ice in Antarctica, courtesy of Ken Golden.

For More Information: "Thermal evolution of permeability and microstructure in sea ice," K. M. Golden, et al., Geophysical Research Letters, August 28, 2007.…

## Going with the Floes - Part 4

Using probability, numerical analysis, and partial differential equations, researchers have recently shown that the permeability of sea ice is similar to that of some sedimentary rocks in the earthâ€™s crust, even though the substances are otherwise dissimilar. One major difference between the two is the drastic changes in permeability of sea ice, from total blockage to clear passage, that occur over a range of just a few degrees. This difference can have a major effect on measurements by satellite, which provide information on the extent and thickness of sea ice. Results about sea ice will not only make satellite measurements more reliable, but they can also be applied to descriptions of lung and bone porosity, and to understanding ice on other planets.

Image: Pancake ice in Antarctica, courtesy of Ken Golden.

For More Information: "Thermal evolution of permeability and microstructure in sea ice," K. M. Golden, et al., Geophysical Research Letters, August 28, 2007.…

## Bending It Like Bernoulli

The colored "strings" you see represent air flow around the soccer ball, with the dark blue streams behind the ball signifying a low-pressure wake. Computational fluid dynamics and wind tunnel experiments have shown that there is a transition point between smooth and turbulent flow at around 30 mph, which can dramatically change the path of a kick approaching the net as its speed decreases through the transition point. Players taking free-kicks need not be mathematicians
to score, but knowing the results obtained from mathematical facts can help players devise better strategies.

The behavior of a ball depends on its surface design as well as on how itâ€™s kicked. Topology, algebra, and geometry are all important to determine suitable shapes, and modeling helps determine desirable ones. The researchers studying soccer ball trajectories incorporate into their mathematical models not only the pattern of a new ball, but also details right down to the seams. Recently there was a radical change from the long-used pentagon-hexagon pattern to the adidas +TeamgeistTM. Yet the overall framework for the design process remains the same: to approximate a sphere, within less than two percent, using two-dimensional panels.

…

## Tripping the Light-Fantastic

Invisibility is no longer confined to fiction. In a recent experiment, microwaves
were bent around a cylinder and returned to their original trajectories, rendering
the cylinder almost invisible at those wavelengths. This doesn't mean that we're
ready for invisible humans (or spaceships), but by using Maxwell's equations, which
are partial differential equations fundamental to electromagnetics, mathematicians
have demonstrated that in some simple cases not seeing is believing, too.

Part of this successful demonstration of invisibility is due to metamaterials
electromagnetic materials that can be made to have highly unusual properties.
Another ingredient is a mathematical transformation that stretches a point into
a ball, "cloaking" whatever is inside. This transformation was discovered while
researchers were pondering how a tumor could escape detection. Their attempts
to improve visibility eventually led to the development of equations for invisibility.
A more recent transformation creates an optical "wormhole," which tricks
electromagnetic waves into behaving as if the topology of space has changed.
We'll finish with this:

For More Information: Metamaterial Electromagnetic Cloak at Microwave Frequencies, D. Schurig et al, Science, November 10, 2006.…

## Unearthing Power Lines

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 legislatorâ€™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.

For More Information: Porter, Mason A; Mucha, Peter J.; Newman, M. E. J.; and Warmbrand, Casey M., A Network Analysis of Committees in the United States House of Representatives, Proceedings of the National Academy of Sciences, Vol. 102 [2005], No. 20, pp. 7057-7062.…

## Making Votes Count

The outcome of elections that offer more than two alternatives but with no preference by a majority, is determined more by the voting procedure used than by the votes themselves. Mathematicians have shown that in such elections, illogical results are more likely than not. For example, the majority of this group want to go to a warm place, but the South Pole is the groupâ€™s plurality winner. So if these people choose their groupâ€™s vacation destination in the same way most elections are conducted, they will all go to the South Pole and six people will be disappointed, if not frostbitten.

Elections in which only the top preference of each voter is counted are equivalent to a school choosing its best student based only on the number of Aâ€™s earned. The inequity of such a situation has led to the development of other voting methods. In one method, points are assigned to choices, just as they are to grades. Using this procedure, these people will vacation in a warm place a more desirable conclusion for the group. Mathematicians study voting methods in hopes of finding equitable procedures, so that no one is unfairly left out in the cold.

For more information: Chaotic Elections: A Mathematician Looks at Voting, Donald Saari…

## Folding for Fun and Function

Origami paper-folding may not seem like a subject for mathematical investigation
or one with sophisticated applications, yet anyone who has tried to fold a
road map or wrap a present knows that origami is no trivial matter.
Mathematicians, computer scientists, and engineers have recently discovered that
this centuries-old subject can be used to solve many modern problems.The
methods of origami are now used to fold objects such as automobile air bags
and huge space telescopes efficiently, and may be related to how proteins fold.

Manufacturers often want to make a product out of a single piece of material.
The manufacturing problem then becomes one of deciding whether a shape
can be folded and if so, is there an efficient way to find a good fold? Thus,
many origami research problems have to do with algorithm complexity and
optimization theory. A testament to the diversity of origami, as well as the
power of mathematics, is its applicability to problems at the molecular level,
in manufacturing, and in outer space.

For More Information: http://db.uwaterloo.ca/~eddemain/papers/MapFolding/…

## Finding Fake Photos

Actually, they werenâ€™t caught together at all their images were put together with software. The shadows cast by the starsâ€™ faces give it away: The sun is coming from two different directions on the same beach! More elaborate digital doctoring is detected with mathematics. Calculus, linear algebra, and statistics are especially useful in determining when a portion of one image has been copied to another or when part of an image has been replaced.

Tampering with an image leaves statistical traces in the file. For example, if a person is removed from an image and replaced with part of the background, then two different parts of the resulting file will
be identical. The difficulty with exposing this type of alteration is that both the location of the replacement and its size are unknown beforehand. One successful algorithm finds these repetitions by first sorting small regions according to their digital color similarity, and then moving to larger regions that contain similar small ones. The algorithmâ€™s designer, a leading digital forensics expert, admits that image alterers generally stay a step ahead of detectors, but observes that forensic advances have made it much harder for them to escape notice. He adds that to catch fakers, At the end of the day you need math.(1)

For More Information: Can Digital Photos be Trusted?, Steve Casimiro, Popular Science, October 2005.

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1 It May Look Authentic; Hereâ€™s How to Tell It Isn't, Nicholas Wade, The New York Times, January 24, 2006.…

## Putting Music on the Map

Mathematics and music have long been closely associated. Now a recent mathematical
breakthrough uses topology (a generalization of geometry) to represent musical chords as points in a space called an orbifold, which twists and folds back on itself much like a MÃ¶bius 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.This 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.

For More Information: The Geometry of Musical Chords, Dmitri Tymoczko, Science, July 7, 2006.…

## Pinpointing 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.

For More Information: The Style of Numbers Behind a Number of Styles, Dan Rockmore, The Chronicle of Higher Education, June 9, 2006.…

## 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.

For More Information: A New Generation Hurricane Storm Surge Model for Southern Louisiana, by Joannes Westerink et al.…

## Targeting Tumors

Detection and treatment of cancer have progressed, but neither is as precise as
doctors would like. For example, tumors can change shape or location between
pre-operative diagnosis and treatment so that radiation is aimed at a target which
may have moved. Geometry, partial differential equations, and integer linear
programming are three areas of mathematics used to process data in real-time,
which allows doctors to inflict maximum damage to the tumor, with minimum
damage to healthy tissue.

One promising area of investigation is virotherapy: using viruses to destroy
cancerous cells. Researchers are using mathematical models to discover how to
use the viruses most beneficially.The models provide numerical outcomes for each
of the many possibilities, thereby eliminating unsuccessful approaches and identifying
candidates for further experimentation.Testing by simulation, which led to
the development of anti-HIV cocktails, means good medicine is developed faster
and cheaper than it can be by lab experiments and clinical trials alone.

For More Information: Treatment Planning for Brachytherapy, Eva Lee, et al,
Physics in Medicine and Biology, 1999.…

## Making Movies Come Alive

Many movie animation techniques are based on mathematics. Characters,
background, and motion are all created using software that combines pixels
into geometric shapes which are stored and manipulated using the mathematics
of computer graphics.

Software encodes features that are important to the eye, like position,
motion, color, and texture, into each pixel. The software uses vectors,
matrices, and polygonal approximations to curved surfaces to determine the
shade of each pixel. Each frame in a computer-generated film has over two
million pixels and can have over forty million polygons. The tremendous
number of calculations involved makes computers necessary, but without
mathematics the computers wouldn.t know what to calculate. Said one
animator, ". . . it.s all controlled by math . . . all those little X,Y.s, and Z.s that
you had in school - oh my gosh, suddenly they all apply."

For More Information:
Mathematics for Computer Graphics Applications, Michael E. Mortenson, 1999.…

# Mathematical Moments from the American Mathematical Society

The American Mathematical Society's Mathematical Moments program promotes appreciation and understanding of the role mathematics plays in science, nature, technology, and human culture. Listen to researchers talk about how they use math: from presenting realistic animation to beating cancer.