Category: Blogs on News

Modeling Crime

A recent article from Math in the News titled Preventing Crime with Math discusses the work of University of California-Santa Clara Professor of Mathematics George Mohler. Mohler has applied an algorithm used in modeling seismic activity to crime in Los Angeles. The model developed by Mohler models burglaries and gang violence.

According to the article, crime is similar to seismic activity in that it is initially difficult to predict, but is usually followed by smaller “afterschocks”. The model uses the algorithm to compute which houses are likely to be struck by a burglar given their initial crime. This model has been considered so effective that the Santa Cruz Police Department is working with Mohler to adapt the model for their use.


Equal Understanding

According to an article from Science News titled, Students’ Understanding of the Equal Sign Not Equal, Professor Says, one of the reasons students in the United States are so far behind their foreign peers in mathematics is that they have a lesser understanding of the equal sign. Research conducted at Texas A & M university suggests that roughly 70% of students in middle grades in the U.S. have misconceptions about what the equal sign means. This is a staggeringly large proportion considering that students in some Asian countries have almost no misconceptions about the equal sign.

A large part of the problem is believed to come from students memorizing procedures rather than learning how to interpret the symbols. An example given in the article is 4+3+2=__+2 where students are asked to fill in the blank. Many U.S. middle grade students would solve this by recognizing that 4+3+2=9, and thus filling in the blank with 9. The source of this problem is thought to be the textbooks that students use. In comparing textbooks from China with those from the U.S., researchers noticed that the Chinese texts offered examples explaining things such as the equal sign, while those in the U.S. did not.

Mandelbrot on Markets

In my last post on Benoit Mandelbrot’s final TED talk, I mentioned that he had begun his career in finance. The article ‘Economist’ Obituary Recalls Mandelbrot’s Scrutiny of Financial Markets goes into more detail about this. The article discusses the obituary written in the Economist for Mandelbrot (he died on October 14, 2010).

Benoit Mandelbrot believed that fractals could be used to model financial markets. The Gaussian (normal) distribution had been being used to model financial markets, and if Mandelbrot was correct, the Gaussian distribution would not be an effective means of modeling. While it has not been proven that fractals can be used to model financial markets, it is now widely accepted that such markets are not Gaussian.

Women in Math

There are more men than women with mathematically intensive careers as a result of the choices made by women. This is according to a recent article posted on Math in the News titled Most Women Say No to Working in STEM. The article states that rather than discrimination keeping women from having Science/Technology/Engineering/Mathematics (STEM) jobs, many women simply opt-out of working in a mathematically intensive field. According to a study, while about half of the math undergraduates are women, only about 27% of those with PhD’s in math are female, and fewer still are professors. The article gave a possible reason for this.

Aside from simply choosing to study something outside of the STEM fields, many women still choose to stay home and raise a family rather than pursue their career. As a result, young female math professors are less likely to work towards tenure than their male counterparts. Furthermore, women are more likely to relocate to accommodate their partner’s work. The article concludes that perhaps more should be done by universities to support women (and men) playing a care-taking role.

Sportswriter Gregg Doyel in his recent column, Numbers Don’t Lie: Sabermetrics Should Win AL Cy Young, explains why Felix Hernandez (and sabermetrics itself) should win the American League Cy Young Award. For those who don’t know, Felix Hernandez is a pitcher for the Seattle Mariners, who despite having the best stats in the American League (AL) (2.27 ERA and 232 strikeouts), had an underwhelming record this season of 13-12. Sabermetrics is the study of baseball using baseball statistics and objective evidence. The Cy Young Award is given to the best pitchers in baseball (one from the American League and one from the National League).

Doyel argues that Hernandez will win the AL Cy Young because he is the best pitcher in the league, a fact backed up by the mathematics of sabermetrics. In his opinion, it is not fair for a pitcher to lose the award because of factors beyond their control, such as playing on a team with very little offense (sorry Mariners). A pitcher cannot win a game if his team does not score runs.

I hope that Doyel is right. An award reserved for the best pitcher in baseball should go to the best pitcher in baseball, regardless of the success of their team as a whole. With that being said, go Mariners!

Math in Economics

For many people who don’t go past an introductory ECON course, the relevance of mathematics in the field of economics may not be immediately obvious. In fact, I would assert that it isn’t until graduate level work that one would really be able to appreciate the amount of math that is used in economics. I say this because, as a double major in Mathematics and Economics,  I have been surprised at how little math has been used in the ECON courses I have taken. Even in Mathematical Economics (ECON 345 here at PLU), the math covered was only basic concepts from Multivariate Calculus and Linear Algebra. Luckily, Paul Krugman, a professor of Economics and International Affairs at Princeton University, has described (very concisely) one of the ways he uses mathematics in economics in his New York Times article Mathematics and Economics.

In his article, Krugman explains that he uses mathematics in his economic research to help define and describe a situation. He writes, “In the economic geography stuff, for example, I started with some vague ideas; it wasn’t until I’d managed to write down full models that the ideas came clear. After the math I was able to express most of those ideas in plain English, but it really took the math to get there, and you still can’t quite get it all without the equations.”

Krugman also pointed out the limitations of economics even with the use of mathematics. He explains that just because something can be modeled nicely does not mean that the model is correct. I fear that this is a shortcoming that many have fallen prey too. While mathematics is an incredibly useful tool in economics, they are two distinct fields, and treating models as fact in economics can have disastrous results.

Despite the limitations of mathematical economics, it is still a fascinating field. I’d be interested to learn more about the math going on behind the scenes in economics, as well as its applications.

The article The Math Behind 96 Billion Burger Choices tells of a burger restaurant where the options are seemingly endless. The Manhattan restaurant 4food offers 5 buns, 4 add-ons, 12 condiments, 7 cheeses, 4 slices (avocado, mushroom, etc.), 17 scoops (beans, nuts, mac & cheese, etc.), and 8 patties.

The article also delves into the mathematics of determining how many possible burger combinations there are given this vast array of toppings. Assuming one bun and as many as you want of each of the others within the restraints established by 4food, the number of possible combinations is derived as follows:

5 * (2^4) * (2^12) * (2^7) * (2^4) * 18 * 8 = 96,639,764,160.

Math Magic

Today I read the article, Colm Mulchay’s Fibonacci-Inspired Trick. This article described two card tricks that use math. The first trick uses the Fibonacci sequence and good shuffling skills.

The trick is done by having the first six numbers in the Fibonacci sequence [1, 2, 3, 5, 8, 13] (1=Ace and 13=King) at the top of the deck. Then the deck is shuffled thoroughly, ensuring that those same cards are on the top of the deck (even if they are in a different order). You have someone draw the top two cards and tell you their sum. Then you tell them what the two cards are. This is where the Fibonacci sequence comes into play. All of the possible sums of these cards are unique, and thus you know the value of each card in the pair. Tah-dah!

Sports Statistics in Politics

When I think about mathematics, one of the last things on my mind is politics. That’s why I was surprised when I read Statistics-Based Blog Debuts in NY Times. According to the article, 25-year-old statistician Nate Silver, who writes the blog, ‘FiveThirtyEight: Nate Silver’s Political Calculus‘, uses statistics to accurately predict the outcome of elections.

Using his knowledge of Sabermetrics and other baseball statistics, Silver takes polling data (weighting it based on its historical merit as well as sample size and recent polling) and ‘balances’ it against demographic data. The article does not give any specifics of the statistics beyond this, but I’m sure Silver’s methods can be learned about in greater detail elsewhere.

The really interesting part about all of this is that Silver’s predictions were astonishingly accurate. According to the article, “[During the 2008 election] he predicted the presidential winner in 49 states—and the winner of every Senate race.”

Who would have thought that using baseball statistics you could almost completely accurately predict the outcome of an election? I know I wouldn’t have.

The title of the article, Internet Has Been Mapped to Hyperbolic Space, says it all. But what does this mean?

Unlike Euclidean space, hyperbolic space does not require parallel lines to be paired uniquely. That is, given a line A and a point B which is not on A, infinitely many lines can exist that intersect B and are coplanar to A but do not intersect A. If that wasn’t clear enough (I’m not sure I even understand it, someone please correct me if I am wrong), just think of every point as being able to be a hyperbola.

Anyways, with the capability of mapping the internet to hyperbolic space, the article explains that glitches could, in theory, be entirely gotten rid of, while speed would be greatly increased.

I’m not sure what it means to “map the internet”, and I don’t really fully understand what hyperbolic space is, but this sounds like an interesting topic. Given the time and proper resources, I would like to learn more about this.