Mathematical Moments From The American Mathematical Society

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 w

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 w

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. Linea

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 Milosevic 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 Milosevic's claim in its entirety. Guatemalan disappearances. Here, statistics is being used to extract information from over 80 million Nation

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 Milosevic 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 Milosevic's claim in its entirety. Guatemalan disappearances. Here, statistics is being used to extract information from over 80 million Nation

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 aren.t complete because, for example, some cases aren.t reported. Armed with reliable models, mathematicians help public health officials battle the complex, rapidly changing world of modern disease. Today.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

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 aren.t complete because, for example, some cases aren.t reported. Armed with reliable models, mathematicians help public health officials battle the complex, rapidly changing world of modern disease. Today.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

What.s in store for our climate and us? It.s an extraordinarily complex question 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. It.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 drastically 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 averag

What.s in store for our climate and us? It.s an extraordinarily complex question 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. It.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 drastically 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 averag

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., A.s donor to B, B.s donor to C, and C.s donor to A). The problem is that the possible number of lo

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 Matt.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 con

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 Beatles. "A Hard Day.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 doesn.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 - McCartney.s or Le

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 world.s population. Many other animals exhibit group dynamics; some organisms involved are small while their groups are huge, so researchers. 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 a

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 Archimedes. work was done using X-ray fluorescence. A painting, forged in the 1940s by one of the book.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 Archimedes. treatise The Method of Mechanical Theorems. The entire process of uncovering this

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 Archimedes. work was done using X-ray fluorescence. A painting, forged in the 1940s by one of the book.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 Archimedes. treatise The Method of Mechanical Theorems. The entire process of uncovering this

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 cent

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 cent

The racing team is just as important to a car.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 I.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 emissio

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 earth.s crust, even though the substances are otherwise dissimilar. One major difference between the two is the drastic changes in permeability of sea ice,

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 earth.s crust, even though the substances are otherwise dissimilar. One major difference between the two is the drastic changes in permeability of sea ice,