Stories tagged Math

Aug
24
2009

Philadelphia Phillies' second-baseman Eric Brunlett made an amazing and unassisted game-ending triple play against the New York Mets. The extremely rare play came in the ninth inning with the Mets trailing by two runs but threatening with runners on first and second base and no outs. Both base runners were stealing on a 2-2 pitch when the Mets batter hit a line drive right to Brunlett, who caught the ball for the first out, stepped on second base for the second out, then tagged out the runner from first. It's only the 15th unassisted triple play in major league history, and only the second to end a game. The poor Mets. The odds of this happening must be astronomical, but I'll let someone else figure that out.

Jun
08
2009

We've all heard statistics about how boys are better than girls when it comes to math. Especially the kinds of advanced math it takes to find solutions to complex problems, to win important prizes and to invent world-changing technologies. According to some people, you can blame this gender gap on basic biology. Female brains are smaller than male brains, which means males are just naturally smarter, and really, what else do you need to know?

It's easy to believe assumptions and stereotypes about girls and math when you look around classrooms where advanced technical subjects are taught. Fewer women fill the seats, and in the top math and science positions, men outnumber women by a dramatic margin. It seems like no matter how many women prove that female brains can be every bit as good at math and science, we still hear that women are just not cut out for crunching numbers.

In other words, if you're a girl and you like math, you should probably quit now, because you will never be as good as the boys. And if you're a girl and you just don't understand math, it's okay, you won't need math anyway. Like Barbie says, Math Is Hard! Let's go shopping!

Or not? Could it be that fewer women excel in math and science fields because they have fewer opportunities? Or because everyone tells them they will do poorly, so they never really try? Is biology to blame for the math-science gender gap, or is culture the culprit?

A new study published in the Proceedings of the National Academy of Sciences says girls are not born to be bad at math. Instead, the authors say, the gender gap stems from cultural inequalities that put girls and women at an unfair disadvantage. Fewer educational and professional opportunities, negative stereotypes, and classroom or workplace dynamics all hinder the potential of girls and women to excel in math.

The authors of this study came to their conclusion by comparing data on gender inequality and math scores from around the world. They found that in countries where men and women had more equal opportunities, women did much better at math. If you're wondering where the United States stands, in 2007 the US was number 32 on the World Economic Forum's Global Gender Gap Report. Not the greatest place to be, but in the US it appears as though the math-science gender gap is narrowing. According to a recent article in the New York Times, there are now more opportunities than ever for women in science and math. If only they paid as well!

May
01
2009

A research group led by Dirk Brockmann at Northwestern University has created a computer model that predicts the spread of the 2009 H1N1 influenza virus in the US. (It uses a complex set of mathematical equations to describe the movement of people and virus.)

How can you track and predict the movement of something so small?: Follow the money, of course! (This is a colorized negative stained transmission electron micrograph (TEM) showing some of the ultrastructural morphology of the A/CA/4/09 swine flu virus. Got that? Good.
How can you track and predict the movement of something so small?: Follow the money, of course! (This is a colorized negative stained transmission electron micrograph (TEM) showing some of the ultrastructural morphology of the A/CA/4/09 swine flu virus. Got that? Good.Courtesy CDC/C.S. Goldsmith and A. Balish

(Brockmann was a guest on Minnesota Public Radio's Midmorning show today, and you can listen to it online.)

The good news is that, based on what we know now, and assuming that no one takes any preventive measures, we could expect to see some 1,700 cases of swine flu in the next four weeks. Because of the preventive measures being taken wherever a suspected case of H1N1 flu has popped up, we should actually see fewer cases. (You can see Brockmann's models here.) That's lousy if you're one of the folks who picks up the virus, but not a devastating number of cases. Of course, this is a rapidly developing, fluid situation, and things may change. Still, tools like Brockmann's model help to ensure that emergency supplies and other resources get to the places likely to need them most before they're needed.

Professor's Computer Simulations Show Worst-Case Swine Flu Scenario from Northwestern News on Vimeo.

Don't have faith in computer models? Well, a second research group at Indiana University is using another model, with different equations, and getting very similar results. That's a pretty good indication that the predictions are reliable.

You might remember Brockmann from a 2006 study that used data from WheresGeorge.com, a site that allows users to enter the serial numbers from their dollar bills in order to see where they go, to predict the probability of a given bill remaining within a 10km radius over time. That gave him a very good picture of human mobility, reflecting daily commuting traffic, intermediate traffic, and long-distance air travel, all of which help to model how a disease could spread.

Feb
28
2009

The hounds of spring are on winter’s traces: It appears that winter and summer temperatures are yoked together.  (This photo is for you, Thor.  I'm a cat person myself. Title explained here.)
The hounds of spring are on winter’s traces: It appears that winter and summer temperatures are yoked together. (This photo is for you, Thor. I'm a cat person myself. Title explained here.)Courtesy sbpoet

Last week, Candace asked whether there’s a connection between winter temperatures and summer temperatures. She noted that the winter of 2007-08 was pretty cold by recent standards, and the following summer was cool as well. Is there something going on here?

Liza searched the Web but couldn’t find anything definitive. I (after pooh-poohing the idea that this has been a warm winter – if you want to pay my heating bill, you’re welcome to it!) decided to crunch the numbers.

First, I went to this site. It records monthly average temperatures going back to December 1978.

Actually, it records temperature anomalies – whether the observed temperature in a given month is higher or lower than the average. They use the 20 years of 1979-1998 as their baseline. A reading of 1.00 means the temperature was 1 degree Centigrade (1.8 °F) warmer than expected. (One degree may not sound like much, until you realize it means 1 degree of every minute of every hour of every day. It quickly adds up to a lot of heat.) A reading of -1.00 means it was 1 degree cooler.

Using temperature anomalies is good for this exercise, as it removes the effects of global warming. Because global temperatures rose during the period under study, a “warm” winter in the late ‘70s might be considered only “average” today. In fact, that’s exactly what happened last winter. It was almost perfectly average by historical standards, but because recent winters have been so much warmer, it felt cold to us.

Getting back to Candace’s question: does a warm winter predict a warm summer? To answer this question, we have to calculate my all-time favorite statistical formula, the coefficient of correlation!

It sounds like a mouthful, but it’s a pretty easy concept to grasp. The coefficient of correlation measures how tightly two sets of numbers go together. For example, if you surveyed 100 people, and asked each one what year they were born, and how old they were, you would find that every single person born in 1990 was the exact same age. The first number (year of birth) and the second number (age) are linked together 100%.

OTOH, if you asked those people for the last digit in their telephone number, you would find no relationship whatsoever. A person born in 1990 is just as likely to have a phone number ending in 9 as ending in any other number, and the same goes for people born in every year.

Calculating the coefficient of correlation (or “coco,” as I affectionately call her), requires wading through a truly horrific battery of equations all to arrive at a number between 0 and 1. A coefficient of 1 means the two sets of numbers are perfectly synched together; a coefficient of 0 means there is no connection whatsoever.

So, I went back to the temperature data. First, I defined “winter” the same way the weather bureau does: December, January and February, the three coldest months of the year. (None of that solstice-equinox nonsense here!) I defined “summer” as the three warmest months: June, July and August, again following weather bureau standards. Using the Northern Hemisphere Land figures (sorry, they didn’t have anything Minnesota-specific), I came up with an average anomaly for every winter and every summer. I crammed the numbers into the formula, turned the crank, and came up with a coefficient of…

(drum roll, please)

0.71

OK, now what does that mean?

Well, in general, a score below 0.30 is considered inconclusive. It’s too close to zero—the “relationship” could just be random. A score between 0.30 and 0.50 is generally considered moderate—there’s a connection there, but it’s somewhat weak. A score over 0.50 is generally considered strong—there’s definitely something important going on there.

(This is especially true in highly complex systems, like weather, where a lot of different factors can affect your results. In a very simple system, you’d probably want a result much closer to 1.)

It all boils down to this: we can be more than 99% certain that, yes, there is a connection between a warm winter and a warm summer, or a cold winter and a cool summer. How much of a connection? For that, we need another figure, the coefficient of determination.

This one is much easier. Just square the coefficient of correlation. 0.71 squared yields 0.5041. That means 50% of the variability in summer temperatures is determined by the winter temperatures.

And “variability” is the key. Like I said, weather is an extremely complex system. Lots of things can affect the temperature for a day, a week, even a season. The fact that this winter is warmer than last winter does not guarantee that this coming summer will be warmer than last summer. (For example, the winter of 2003-04 was one of the warmest on record, but the following summer was one of the coolest in the study period.)

What this number does mean is, that of all the factors that will affect next summer’s temperatures, half of them seem to be connected to winter temperatures. And this winter was warmer than last winter.

Just for fun, I also ran the calculations the other way, to see if a warm summer predicts a warm winter. The coefficient of correlation was 0.54, and the coefficient of determination was 0.29. So, again, there is a connection, but it seems to b a good deal weaker.

A word of caution: one thing statisticians like to say is “correlation is not causation.” Partly because it’s fun to say, but mostly because it’s true. Just because two things are correlated does not mean one causes the other. We have not proven that warm winters cause warm summers. It could be that winter temps and summer temps are both boosted by some other factor – El Nino, perhaps. All we can say is that there is some sort of connection going on, and that it probably wouldn’t hurt to lay in some tanning cream now.

Jan
06
2009

Year of Science Logo
Year of Science LogoCourtesy COPUS
The Coalition on the Public Understanding of Science (COPUS) kicked off Year of Science 2009 (YoS2009) -- a national, yearlong, grassroots celebration--this week in Boston at the annual meeting of the Society for Integrative and Comparative Biology. COPUS, which represents more than 500 organizations, is celebrating how science works, who scientists are, and why science matters.

YoS2009 participants—museums, federal agencies, K–12 schools, universities, scientific societies, and nonprofit and for-profit organizations from all 50 states and 13 countries—will host events in celebration of YoS2009. Regionally connected YoS2009 participants are bringing science to their local communities in innovative ways. To learn about YoS2009 events near you click here.

A special web site will help the general public learn more about this yearlong, national event. Highlights from the dynamic YoS2009 Web site include the integration of components from the newly launched Understanding Science web site, Flat Stanley explorations of science, the opportunity to name a new species of jellyfish or adopt a species for the Encyclopedia of Life, and a contest to build the most scientific pizza.

All of these events and activities foster innovative new partnerships that will bring science and the public closer together locally, regionally, and nationally—all in a growing celebration of science!

Nov
02
2008

So, I open up my web browser this weekend to check the news, and I see the following three polls, all on the same page:

  • Rasmussen: Obama up by 5 points
  • Gallup: Obama up by 10 point
  • Zogby: McCain up by 1 point

These can’t all be right, can they?

Actually, they can. Or, at least, they can all be properly conducted, and just lead to wildly different results.

The only way to get a perfect result is to interview everyone in the country. (In fact, that’s exactly what we do on Election Day.) But that takes so much time and money that no individual pollster can do it. Instead, they interview several hundred people, maybe a couple thousand, and from there extrapolate what the country as a whole will do.

Now, mathematically, you can do this. You just can’t be sure of your answer. Here are a few of the reasons why.

Margin of error

Most opinion polls will state the margin of error. For example, they may say that that Candidate X is ahead by, say, 5 points, with a margin of error of plus-or-minus 3 points. Meaning, the real answer could be as high as 8 points or as low as 2 points.

(Sometimes, the margin of error is actually larger than the result. The poll shows Candidate X leading by 2 points, but with a margin of error of 4 points. Meaning, he could be ahead by 6, or he could actually be behind by 2! This seems to have happened a lot this year.)

A range of a few percentage points, when applied to a country with over 100 million voters, can lead to some pretty huge differences.

Confidence interval

In addition to reporting a margin of error, polls also report a confidence interval, usually 90% or 95%. This means that, according to the laws of mathematics, there is a 95% probability that the real result is the same as the poll result, within the margin of error.

But what about the other 5% or 10% of the time? Well, the folks reporting the numbers don’t like to tell you this, but, mathematically speaking, the poll can do everything right, and still be completely wrong, as much as 10% of the time.

There have been over 700 polls released this election season, and over 200 just in October. No doubt, many of the polls you have heard about fall into this category.

Weighting

In most elections, more women vote than men. If you conduct a survey and talk to 100 men and 100 women, you are going to have to give the women’s answers more weight to accurately reflect the Election Day results.

How much more weight? That depends. Do you think this election will be pretty much the same as previous years? Is there something happening this year that will make a lot more women come out to vote? Or, perhaps, something that will attract a lot more men?

The fact is, nobody knows. Weighting is just educated guesswork. And this year, it is more complicated than usual:

  • Black voters are expected to come out in record numbers to support Barack Obama. How many will actually vote? Nobody knows.
  • Young people generally do not vote as much as other groups. But many analysts expect more young people to vote this year. How many more? Nobody knows.
  • Democrats, having lost the last two elections, are likely to turn out in larger numbers. How much larger? Nobody knows.
  • New voters. There has been a great push to register new voters, many of them poor or minority members. These groups tend to vote Democratic. But, the NY Times reports that as many as 60% of those registrations may be fake. If you are a poll taker, you really have no idea how many new voters there actually are.
  • Likely voters. Every pollster ends up talking to some people who will not vote on Election Day. Different polls use different methods of figuring out who is likely to actually vote—based on whether they voted last time, how much interest they have in the election, or just taking the voter’s word for it.

The different weighting factors used by the different polls probably accounts for most of the variability we see in the results.

Human factors

Let’s face it – humans are complicated and sometimes uncooperative beings. There are lots of ways they can foul up a perfectly good poll.

  • Lying. People have a tendency to tell a pollster what they think he wants to hear. Maybe they just want to be nice; maybe they want to avoid an argument. This skewing has been found in many, many types of polls.
  • Refusal. In any poll, a certain number of people refuse to participate—they don’t want to be bothered, or they don’t want to talk politics with a stranger. If refusals are more likely to come from one party than the other, this can skew results.
  • Hard-to-reach folks. For years, pollsters called people on the telephone. But today more and more people have cell phones, or call screening, and are hard to reach. Again, if people with such gadgets are more likely to support one candidate or the other, it will skew the results. (One blogger has noticed that John McCain does better in polls conducted during the week than on the weekend, and speculates that's when McCain supporters are home.)
  • Bias. Poll takers are only human. They have their own thoughts and opinions. And while they take great pains to be neutral, those opinions sometimes come through in the questions asked or the way the answers are interpreted. Most of the major polling companies are based in big cities like New York, Los Angeles, Boston, or Washington, DC – all cities that are heavily Democratic. This may explain why, in the last two elections, the Democrats did better in the polls than they did on Election Day.

So, with all these problems, how can we figure out who is going to win the election? Well, never fear – there is one sure-fire way to find out the winner:

Read the newspaper Wednesday morning.

And don’t forget to vote!

Oct
14
2008

Garrett Lisi uses math to describe everything

E8: the largest and most complex of the exceptional simple Lie algebras
E8: the largest and most complex of the exceptional simple Lie algebrasCourtesy Claudio Rocchini
Surfer dude, Garrett Lisi lives in his van on a beach in Maui. Using a type of algebra he calls E8, Garrett has developed an exceptionally simple theory of everything -- a grand unified theory that explains all the elementary particles, as well as gravity. (link to pdf of paper found below)

Lisi describes how gravity, the standard model bosons, and three generations of fermions can be unified as parts of an E8 superconnection. This unified field theory attempts to describe all fundamental interactions that physicists have observed in nature, and stands as a possible theory of everything, unifying Albert Einstein's general relativity with the standard model of particle physics.

"I think the universe is pure geometry - basically, a beautiful shape twisting and dancing over space-time. Since E8 is perhaps the most beautiful structure in mathematics, it is very satisfying that nature appears to have chosen this geometry."

"This is an 'all or nothing' kind of theory -- meaning it's going to end up agreeing with and predicting damn near everything, or it's wrong. At this stage of development, it could go either way." Garrett Lisi

This video explains how mathematics predicts reality

Warning, even though I have a degree in physics education, the material presented was way over my head. I will watch it again though, because it does give me a glimpse of how mathematics can lead to understanding, perhaps even someday making possible something like electrogravity. Click this link if the video below does not work

Learn more
Garrett Lisi forum frequently asked personal questions
Garrett Lisi forum frequently asked questions about E8 and Theory of Everything
31 page paper (pdf) An exceptionally simple theory of everything

Oct
14
2008

Cats and dogs, living together: You need more proof?
Cats and dogs, living together: You need more proof?Courtesy darkmavis
Put a star on your calendars today, Buzzketeers, because science is taking a vacation! And I don’t know about y’all, but I’ll be celebrating by eating my own head and diving into a good book. Literally!

Some of you (the sassboxes) might point out that natural laws apparently still apply, and that scientists the world over continue to fiddle around with their science-tools to learn things about the world. But the day I start paying attention to scientists is the day I give up my lifelong dream of eating my own head.

No, I’m listening to the people who have a more direct influence on my day-to-day life: bookies.

See, it seems that a bunch of bookies are concerned that there’s some chance that an alien spaceship will be landing today in the American desert. And so I’m concerned too.

The idea… Nay, the fact of the impending landing came from the Australian psychic, Blossom Goodchild.

—A little side note: I can’t believe it! My name used to be Blossom Goodchild too! I just changed it to JGordon in junior high. I can’t believe we were both born “Blossom Goodchild”! Amazing!

Anyway, Blossom Goodchild, Aussie supernatural, delivers the secrets of love, light, and laughter by channeling “a native American spirit energy” by the name of White Cloud. I’m not entirely clear on why a native American spirit energy would go to Australia for channeling, but who am I to question the ways of the spirits?

Goodchild, presumably with the help of White Cloud, has started a wave of Internet-enthusiasm by predicting the imminent arrival of a massive space ship full of aliens (or “light beings”), which will supposedly be happening today!

The enterprising residents of Earth, not wanting to be caught with our pants down by light beings, have rushed to prepare… by betting on the arrival! Betting so much, in fact, that bookmakers have had to suspend further all wagers.

There has been no evidence of the coming aliens—no radio transmissions, no detected incoming spaceships, and no precedence—except for the word of an Australian good child, and her wandering spirit. And so it would make sense that the odds are set up against the landing. Yet human gamblers aren’t into odds (who ever won something by betting on a sure thing?) or evidence, and they stand to make a lot of money if (when) the ship arrives. See, we shoot from the hip, and we follow our guts, and the bookies know it, and they’re afraid to take any more bets on the spaceship.

So today’s the day, Buzzers. Try to do something impossible. You won’t be the only one.

Sep
27
2008

Fluid flow separation: The fluid flow becomes detached from the surface of the object, and instead takes the forms of eddies and vortices.
Fluid flow separation: The fluid flow becomes detached from the surface of the object, and instead takes the forms of eddies and vortices.Courtesy jaganath

Fluid flow separation explained with mathematics in 1904

In 1904, Ludwig Prandtl, considered the father of modern aerodynamics, derived the exact mathematical conditions for flow separation to occur, but only in two dimensions for steady flows.

Unsteady fluid flow in three dimensions explained with mathematics in 2008

A century later, George Haller, a visiting professor in the Department of Mechanical Engineering at MIT led a group that explained the mathematics behind unsteady separation in two dimensions. This month, his team reports completing the theory by extending it to three dimensions. Papers on the experiments and theory are being published in the Sept. 25 issue of the Journal of Fluid Mechanics and in the September issue of Physics of Fluids, respectively. Haller's coauthors are Amit Surana, now at United Technologies; MIT student Oliver Grunberg; and Gustaaf Jacobs, now on the faculty at San Diego State University.

Fluid mechanics theorists are excited

The equation will forever change the face of advanced fluid dynamics and will have a profound impact on many industries, including the aerospace and automotive industries. This quote from Daily Tech Review shows that this breakthough has theorists in fluid mechanics excited;

The new work -- if it survives the extensive peer review that is to come -- will likely go down as the greatest scientific advance of the decade. The research has already survived a strenuous initial round of peer review.

Equally important, this month Thomas Peacock, the Atlantic Richfield Career Development Associate Professor and his colleagues report important experimental work verifying the theory.

"This is the tip of the iceberg, but we've shown that this theory works," Peacock said.

Fluid dynamics makes a difference

Understanding how surfaces effect how an object flows through a fluid (including air) can make big differences in maximizing performance. Did the new swimsuits make a difference in breaking world records in Olympic swimming competition? How about the surfaces of baseballs, golf balls, and tennis balls? The effects on miles per gallon for autos and airplanes can save millions (billions?) of dollars.

Source: MIT News