Stories tagged Math

Apr
17
2007

Friday nights at 9 pm.

"Numb3rs" is currently the most-watched program on Friday nights, attracting nearly 12 million viewers. Now in its third season, Numb3rs, along with the program's co-creators, Nick Falacci and Cheryl Heuton, will receive a National Science Board group Public Service Award for 2007 "for their contributions toward increasing scientific and mathematical literacy on a broad scale".

The annual Public Service Award recognizes individuals and organizations for their extraordinary contributions to increase public understanding of science. Recipients are chosen for their contributions to public service in areas such as: increasing the public's understanding of the scientific process and its communication; contributing to the development of broad science and engineering policy; promoting the engagement of scientists and engineers in public outreach; and fostering awareness of science and technology among broad segments of the population. NSF

Mathematical CSI

Cryptanalysis, probability theory, game theory, decision theory, principal components analysis, multivariate time series analysis and astrophysics are just some of the many disciplines employed in the series thus far. If you have not seen this show I recommend that you check it out.

Mar
23
2007


Math: Image courtesy Lodri.
I’m not good at math. Math and I are not friends. We’ll nod hello in the hallway, but we don’t hang out. So, when I read stories about researchers solving a century old math problem that when written out would cover the island of Manhattan, I am more than a little blown away. Seriously, can you even imagine a math problem that long? I can just hear Mr. Rambo, my 7th grade math teacher admonishing me to, “show your work!”

It took an 18-member international team four years to solve the theoretical puzzle known as the “Lie group E8”, which was discovered in 1887 and is the most complicated Lie group. What is a Lie group? Well, let’s ask Wikipedia:

In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie, is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. Lie groups represent the best developed theory of continuous symmetry of mathematical objects and structures. This makes Lie groups tools for nearly all parts of contemporary mathematics, as well as for modern theoretical physics, especially particle physics.

Since Lie groups are manifolds, they can be studied using differential calculus, in contrast with the case of more general topological groups. One of the key ideas in the theory of Lie groups, due to Sophus Lie, is to replace the global object, the group, with its local or linearized version, which Lie himself called an infinitesimal group and which has since become known as its Lie algebra.

Lie groups provide a natural framework to analyse continuous symmetries of differential equations (Picard-Vessiot theory), much in the same way as permutation groups are used in Galois theory to analyse discrete symmetries of algebraic equations.

I have tried, sitting here with my little brain, to make this explanation simpler, to make it more understandable. I think, perhaps, that the author of the Wikipedia article did that, and this is as simple as it gets.

"To say what precisely it is is something even many mathematicians can't understand," said Jeffrey Adams, the project's leader and a math professor at the University of Maryland.

So, I obviously have no chance.

But hey, I think this is cool none the less. It boggles my mind to think that there are problems like this to be solved – that seem impossible, and have remained unsolved for over 100 years – yet they are being solved today, and whose proof consists of more than 205 billion entries. Amazing.

Congratulations.

Mar
11
2007

That's how lotteries are often described. The odds of winning the top prize in the Mega Millions lottery is more than 600 times worse than your odds of getting hit by lightning. Yet people continue to play. Why?

Professor Lloyd Cohen suggests people aren't paying for the chance so much as they are paying for the dream. They enjoy fantasizing about winning, the same way people enjoy reading lifestyle magazines or watching movies of the rich and famous. And at one dollar a pop, a lottery ticket is not only cheaper than these other forms of entertainment, it actually has a chance -- no matter how infinitesimally tiny -- of actually paying off.

Mar
05
2007

It seems like every week there’s another medical breakthrough announced in the press – only later to fizzle when additional studies show it didn’t really hold up. Why are there so many false starts?

Dr. Peter Austin of the Institute for Clinical Evaluative Sciences in Toronto says it has to do with the way researchers use statistics. All statistical studies rely on “confidence intervals” – if an event has only a 5% chance of happening at random, then doctors can be 95% confident that it isn’t a random fluke. They assume they’ve discovered a real phenomenon, and start looking for a cause.

(For instance, a coin has about a 3% chance of landing heads five times in a row. If you had a coin that did that the first time you tried it, you’d have good reason to suspect something funny was going on, and conduct more tests.)

But Dr. Austin notes that many studies run multiple tests simultaneously. When you do that, the odds of at least one test giving an unusual result, just by chance, is very high. In our coin-flipping example, if you tested 100 identical coins by tossing each 5 times, it would be perfectly normal for at least one, and probably a few, to land all heads, without anything “funny” about them at all.

The point is, you have to look at the whole test, not just selected parts of it. And doctors – and journalists – need to be more careful when presenting the results of studies, so they don’t report false relationships.

Feb
28
2007

They call Economics “the dismal science” because it pays no attention to questions of right and wrong, good and evil, but only looks at supply and demand, profit and loss. But even with that limitation, it still helps illuminate certain moral precepts.

Take for instance the old adage “crime doesn't pay.” According to economist Steven Levitt (WARNING: 22-minute video, with occasional objectionable language), the worst job in America is drug dealer. Not only does it ruin lives; not only does it bring crime and violence that destroys entire neighborhoods; it simply doesn't pay well:

  • The average dealer earns $3.50 an hour – well under minimum wage.
  • There is little opportunity for advancement.
  • The death rate is extremely high. Not only is a dealer more likely to die on the job than a soldier on active duty; a dealer is seven times more likely to die in a given year than an inmate on death row!

Even without considering legality or morality, the math shows: dealing drugs is a pretty dumb way to make a living.

Feb
26
2007

Are life and death just a game?: Mathematics says they follow the same rules. Photo from the National Park Service.
Are life and death just a game?: Mathematics says they follow the same rules. Photo from the National Park Service.

Game theory is a branch of mathematics that attempts to explain how people make choices by weighing costs and benefits. It can be applied not just to games, but to all kinds of serious situations – business, politics, even war.

This report (abstract free; $ to download complete report) argues that even terrorists use classic game theory to maximize the impacts of their attacks:

We find that more educated and older suicide bombers are less likely to fail in their mission, and are more likely to cause increased casualties when they attack.

Knowing this, I wonder if anti-terrorist efforts are focusing more on those older, educated operatives, to minimize the threat of attack.

*(It's also the name of an '80s band, but that's neither here nor there.)

Jan
25
2007

The white male law professor -- an endangered species?: Photo from Rep. Jim Cooper, US House of Representatives
The white male law professor -- an endangered species?: Photo from Rep. Jim Cooper, US House of Representatives
The Science Museum of Minnesota is hosting a new exhibit Race: Are We So Different? A lot of racial topics are emotionally charged and hard to talk about. So, when we can find a scientific study, it gives us something a little more objective to discus.

This study shows that, over the last 14 years, white males have had a harder time getting jobs as law professors than minorities or female candidates:

Candidate Type Success Rate (%)
Minority Women 18.5
Minority Men 17.5
Non-Minority Women 15.0
Non-Minority Men 11.3

The study looks at success rate -- that is, what percentage of white candidates get hired, what percentage of black candidates get hired, etc. It does not look at what percentage of a law school's faculty is black, white, green, purple, etc.

The large amount of data in the study makes it pretty unlikely that this is a fluke -- the pattern has held steady for 13 of the past 14 years. It's hard to avoid drawing the conclusion that women and minorities enjoy an advantage in getting hired as law school teachers. Some people would say that's discrimination. Some would say it's justified to make up for the decades these groups were barred from the profession. Some would say it's necessary to give today's students a well-rounded education with many perspectives.

What do you say? Leave a comment.

Nov
30
2006

Digging dinosaurs in Utah: Courtesy Bureau of Land Management
Digging dinosaurs in Utah: Courtesy Bureau of Land Management

That’s how many types of dinosaurs remain to be discovered. According to Steve Wang, a statistician at Swarthmore College, Pennsylvania, and Peter Dodson, a palaeontologist at the University of Pennsylvania, Philadelphia, at least 70% of the dinosaurs that once existed have yet to be found. They arrived at this figure by taking the known dino discoveries and plugging them in to a mathematical model that has proven successful in extrapolating data.

The scientists estimate that about half of the missing dinos will probably never be found. They lived in upland areas where fossilization is rare. Or the rocks that held their bones have been destroyed by glaciers or other Earth processes. But that leaves some 700 types of dinosaurs yet to be discovered.

Dinosaur discovery has accelerated in recent years. Nearly half of all dinosaurs known today have been dug up in just the last 20 years. Countries like China and Argentina – long inaccessible to paleontologists – have been producing many new finds. But there are plenty of other countries, particularly in Africa, that have yet to be fully explored. Wang and Dodson figure that most of the remaining dinosaur discoveries should come to light in the next 100 to 140 years.

Nov
15
2006

Storm chasers know that puffy cumulus clouds often cause sudden rainstorms, while storms associated with stratus clouds form more slowly. Now physicists at England’s Open University have finally found an explanation.

They propose that neighboring water droplets in a stable stratus cloud don’t crash into each other because they’re all moving at about the same speed. But fast-forming, turbulent cumulous clouds contain water droplets moving at many different speeds. They crash into each other and form larger drops. As the turbulence grows, the drops grow quickly and fall as rain within a few minutes.

Cumulous cloud: These puffy clouds are associated with sudden rainstorms. Scientists are beginning to understand why.
Cumulous cloud: These puffy clouds are associated with sudden rainstorms. Scientists are beginning to understand why.

Sun and rain
Ever noticed the bright, moving lines on the bottom of a stream, bathtub, or swimming pool? They’re called caustics, and they’re caused when ripples on the water’s surface focus sunlight. (Caustics form whenever light rays are bent by a curved surface or object and then projected onto another surface.

Caustics have a characteristic shape. Physicists can graph the phenomenon mathematically, and the graph also describes other phenomena, such as particle motion or the movement of raindrops within a cumulus cloud.

Caustics: What do the rippling patterns on the bottom of a swimming pool and cumulous clouds have in common? (Photo by R. Motti)
Caustics: What do the rippling patterns on the bottom of a swimming pool and cumulous clouds have in common? (Photo by R. Motti)

Atmosphere to outer space
The researchers say their finding won’t have any impact on weather forecasting. But particle collisions in turbulent gases must have been involved in planet formation. Perhaps the same theory can be applied?

If you're at the museum on Saturday afternoon (11/18), the MakeIt team can help you play with caustics. Does bending mylar in a different direction produce a new pattern? Does using a different color flashlight or a brighter or dimmer light affect the design?

You can also play with caustics at home.