Friday, January 29, 2010

Famous mathematicians who became famous for something else

With nothing else to do, I was browsing through some old "starred" items on Google Reader, and found this blog by Dave Richeson.

It lists many interesting people, who I had no or little clue, were mathematicians. In particular Art Garfunkel (masters from Columbia and dropped out of the PhD program), Teri Hatcher (actress), Ahmed Chalabi (Iraqi deputy PM), Corazon Aquino (Phillippines premier), Michael Jordan (till his junior year), Ted Kaczynski (the unabomber, his PhD thesis in mathematics from the University of Michigan  won the Sumner B. Myers Prize for best dissertation of the year), Danica McKellar (better known as Winne Cooper from the Wonder Years, she is also the author of the best-selling books Math Doesn’t Suck, and Kiss My Math), and Lewis Caroll.

From the fictional world, James Moriarty, former Professor of Mathematics, author of The Dynamics of an Asteroid, whose essay on the binomial theorem is said to have had a continental vogue, became the leader of the most sinister criminal conspiracy in Victorian England. He has been called "the Napoleon of Crime." Sherlock Holmes's nemesis

I dug up one more myself.

Charlie Munger (Buffett's Berkshire partner) also studied mathematics at the University of Michigan, although he did not finish his degree.

Thursday, January 28, 2010

Brice Carnahan wins ASEE-CACHE award

Okay this is probably old news, but it is new to me.

I received the Winter 2009 CACHE newsletter in my mailbox today. I learned that Brice Carnahan had won the 2009 CACHE award for Excellence in Computing. Here is a pdf link to the citation, and previous award winners.

The news is interesting to me because I was in Brice's second-to-last numerical methods class as a graduate student at Michigan. I had never taken a numerical methods class before, and this class was taught with an amazing mix of analysis and practice. His authoritative textbook "Applied Numerical Methods" (with Jim Wilkes, another great retired professor at Michigan - I was in his last fluid mechanics class) has more than stood the test of time. In fact, I am teaching a numerical analysis class myself this term, and still use my copy of the old book more regularly than any other source.

Another reason I remember Brice very fondly is that he was the graduate coordinator at the time I was accepted into the PhD program. It is always heart-warming when nice self-effacing people get the awards they deserve.

Wednesday, January 27, 2010

"Unlocking" Firefox or Thunderbird sessions

Usually, I store my work on my shared local network. This has many obvious advantages, the primary being: I have access to all my files from everywhere, and not just my desktop machine. A peculiar problem that I run into very frequently, when I try to launch Firefox (esp. during lectures which take place in a different lecture hall) is a dialog box which says:

This happens because I usually leave Firefox running on my Desktop, when I go to the lecture hall. This source explains the problem:
Some applications (e.g. firefox, eclipse and more) use lock files to prevent users from opening two instances of the same application. A lock file is usually a simple file named "lock" or ".lock".

The application creates the lock file when it starts, and deletes it when it exits. When it crashes, it might forget to clean up the lock files, and when it is re-opened, and finds the lock files, it thinks there is already an instance running, and as a result, refuses to open.
On a linux box, the solution is relatively straightforward:

1. Go to the .mozilla/firefox, ./thunderbird, or ./mozilla-thunderbird directory
2. There is often a directory "X" which ends with ".default". cd into this directory.
3. Remove the "lock" file.
4. Of course, when you delete the lock, the session that the lock is protecting will probably crash.

Monday, January 25, 2010

Hackability of Passwords

Over last week, we heard that the great New York Times will slowly transition to a "pay for content" model for the heaviest readers. Nevertheless, here is a article on how many people set passwords that almost beg their accounts to be compromised.

The two most popular passwords are "123456" and "12346".
More disturbing, said Mr. Shulman, was that about 20 percent of people on the RockYou list picked from the same, relatively small pool of 5,000 passwords.

Thursday, January 21, 2010

Galileo's Genius: The Experiment That Never Was.

Many of us have this mental picture of Galileo standing on top of the Leaning Tower of Pisa and dropping a feather and stone to watch them fall on the ground simultaneously, thus upsetting the Aristotelian assertion that "heavier objects fall faster".

Of course, we know Galileo couldn't possibly have done that experiment, because if he actually had, the presence of air would have reaffirmed the Aristotelian world-view, and the feather would have fallen later.

So what exactly did Galileo do?

I haven't read his original work, but my PhD advisor - Ron Larson - told us this charming story during a class I took with him, and its stuck with me since.

Apparently Galileo did a thought experiment.

He used the idea of "reductio ad absurdum", which begins by assuming something that we wish to prove as false, and hitting a contradiction.

The proof is beautiful, and here is a rough outline.

Consider two objects, with the heavier object of mass M, and the lighter object of mass m. Let us assume that "heavier objects fall faster" - which is opposite of what we wish to prove.

Further consider a really tall tower. If you can think of an infinitely tall tower, that is even better.

Now we've assumed that mass M falls faster than mass m. Here's the beautiful construction: Consider tying the two masses with a piece of string - to create a new object of mass M + m, as shown below.

Let us drop this composite object from the top of our tall tower, and think about what may happen. Because of what we've assumed, the mass M is going to fall faster than m, and in a short while the composite object (that is still falling, remember) will have adopted a configuration that looks more like the following.

Let us now consider the tension inside the string (red line in my diagram). The big mass tugs on the small mass, while the small mass tugs back.

Thus, the small mass slows the big mass down.

Consequently, the composite object of mass (M+m) falls slower than the unfettered object of mass M. But the composite object is heavier, and should fall faster.

Reductio ad absurdum!

Saturday, January 16, 2010

The Economics of Speeding

I recently browsed a few interesting articles that tried to address the topic of this blog entry, but found some of the assumptions somewhat unrealistic. Here is my swipe at the same problem.

Before we begin to formulate our assumptions in detail, let us try to understand the problem qualitatively. When we speed, we reach our destination faster, which is desirable. However, at the same time, we risk getting a speeding ticket, making our trip more expensive. There is a secondary cost associated with high speeds, which is related to lower fuel efficiency. (There is also a safety issue, which is hard to account for properly, and is hence ignored.)

Thus, this is a risk-reward problem. The reward (faster arrival) needs to be traded off against the risk (speeding ticket + lower gas mileage).

Let us take an engineering approach, and try to put some numbers together. Let us say that the speed limit is 65 mph, and that our gas mileage decreases with excess speed (beyond the speed limit, thus at 75mph the excess speed is 75 - 65 = 10 mph) as shown in green on the graph below. The graph also plots (on the right y-axis, blue line) the cost of a speeding ticket at the different excess speeds if one were to get caught.

Next we need some way of estimating the probability of getting caught. Now this probability will depend on (a) the excess speed, and (b) the duration of the trip. Drive too fast, and the odds of getting caught increase. Drive over long distances, and the odds of lucking out in spite of speeding decrease - since there are now "more opportunities" (read cop cars on the journey) to catch you in the act. Based on some speeding ticket data, we assume a probability of the following form:

It basically suggests that if you speed less that 7 mph above the speed limit, it is very unlikely that you will get caught. However, around 10 mph above the speed limit, you chances increase quickly, and once you are about 15-17 mph above the speed limit, it is almost certain that you will get flagged over a journey of 1000 miles. Over a shorter journey, say 100 miles the chance of getting caught, even at very high speeds are capped at (100/1000) * probability, or 10% in this case. This is because, it is quite possible that there are no radar guns watching over a 100 mile stretch. Obviously, I think this is an underestimate - but I am after a conservative estimate, at least initially.

We further assume that gas costs $3 per gallon, and that the amount of time that a speeding ticket takes to administer is 15 minutes. The length of the trip is assumed to be 100 miles. This gives us the following expected time and costs, for various levels of over-speeding.

The blue line refers to the cost, and the green line refers to the time. As you can see, the total time required decreases with speed. The total cost increases, with a somewhat rapid increase around speeds in excess of 10 mph over the speed limit. This origin of this sharp increase is the sharp increase in the probability of getting caught in the earlier figure. In and above this region, you become very visibile to cops, if they are around - which increases you potential downside.

The risk-reward picture may best be visualized as:

This picture shows that if you drive slow (longer trip times) you spend less, and vice versa. The chart is strongly non-linear, which has two important lessons.

(1) If you are a low-risk person, then you probably want to drive 5-7 mph over the speed limit at most.
(2) If you are a high-risk person, then you are best off picking a speed somewhere along the sharp uprise.

Thursday, January 14, 2010

Amway and QuickStar followup

Since my recent article, I googled and found some interesting links on "the Business". Here is some resistance from some QuickStar folks.

A pdf of the book "Merchants of Deception", which is an insider's account, was once floating on the web. Unfortunately, it seems to have disappeared - probably due to copyright issues.

Wednesday, January 13, 2010

Two interesting science cartoons

1. Dimensional Analysis (xkcd)

2. Mathematical Proof (Abstruse Goose)

Tuesday, January 12, 2010

Rejecting a Null-Hypothesis under Uncertainty

Alexandre Borovik has a nice post on the fallacy of null hypothesis rejection. A brief overview of his post.

If the null hypothesis is correct, then this datum cannot occur.
It has, however, occurred.
Therefore, the null hypothesis is false.
Thus, for example,
If a person is a Martian, then he is not a member of Congress.
This person is a member of Congress.
Therefore, he is not a Martian.
So far, so good. Now, let throw in some uncertainty.
If the null hypothesis is correct, then these data are highly unlikely.
These data have occurred.
Therefore, the null hypothesis is highly unlikely.
While the above may seem reasonable, and is in fact quite widely misused even in academic literature, the following example highlights the inherent error.
If a person is an American, then he is highly unlikely to be a Congressman
This person is a member of Congress.
Therefore, it is highly unlikely that he is an American.

Sunday, January 10, 2010

Bayesian Analysis

I first encountered Bayes Theorem when I studied probability theory in high school, and until recently thought it was just another formula. Useful to solve puzzles, perhaps, but not the most elegant mathematical formula I've encountered.

Over the past year, I have been reading up on a lot of recent technical literature on Bayesian analysis, particularly as it pertains to solving important inverse problems using Markov chain Monte Carlo methods, and have been experiencing something like a muted, but prolonged epiphany.

Bayesian thinking is very under-emphasized in Math curricula (or maybe I just wasn't paying enough attention). This is particularly sorry since being able to tease apart statistics that get thrown at you has become a necessary tool in any survival kit. Here is a link to a great, gentle introduction for an interested, but uninitiated reader.

Saturday, January 2, 2010

Behavioral Finance Lectures

I have found behavioral finance to be an interesting connection between psychology, decision-making and economics. Wikipedia defines it as:
a separate branch of economic and financial analysis which applies scientific research on human and social, cognitive and emotional factors to better understand economic decisions by consumers, borrowers, investors, and how they affect market pricesreturns and the allocation of resources.
Here are a couple of lecture series on this subject:

1. Russel James at the University of Georgia (via Farnam Street, and Simolean Sense)
2. Sanjay Bakshi's "Behavioral Finance and Business Valuation" class at MDI.

They are both quite fascinating, and present a great synthesis which make them worth a look even if you are familiar with many of the ideas.