Mathematical Moments from the American Mathematical Society cover image

Mathematical Moments from the American Mathematical Society

Latest episodes

undefined
Jun 15, 2012 • 4min

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 girl.s (or boy.s) best friend.
undefined
Jun 15, 2012 • 4min

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 can.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 models. 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 doesn.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.
undefined
Oct 5, 2011 • 4min

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 today.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 that.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.
undefined
Oct 5, 2011 • 4min

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 today.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 that.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.
undefined
Oct 5, 2011 • 4min

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 doesn.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.
undefined
Oct 5, 2011 • 4min

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 doesn.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.
undefined
Oct 5, 2011 • 4min

Keeping the beat - Part 2

The heart.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 heart.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 heart.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.
undefined
Oct 5, 2011 • 4min

Keeping the beat - Part 1

The heart.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 heart.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 heart.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.
undefined
Jul 12, 2011 • 4min

Sustaining the Supply Chain - Part 2

It.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.
undefined
Jul 12, 2011 • 6min

Sustaining the Supply Chain - Part 1

It.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.

The AI-powered Podcast Player

Save insights by tapping your headphones, chat with episodes, discover the best highlights - and more!
App store bannerPlay store banner
Get the app