Mathematical Moments from the American Mathematical Society

Experts are adept at answering questions in their fields, but even the most knowledgeable authority can.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 manner.not at all elementary. Watson.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 problems.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.

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 it.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.

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 it.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.

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 ESPN.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 everyone.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.

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 ESPN.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 everyone.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.

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.

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.

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.

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.

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.