Pi Mu Epsilon is the National Mathematics Honorary Society of the United States and its purpose is to encourage scholarly mathematical activities. We congratulate the new members and current members. You can check out the organization at http://www.pme-math.org/.
The very idea of a “top 5″ list is kind of mathematical, isn’t it? Check out this list of the top 5 jobs from msnbc.com:
Four out of the five are Math/CS jobs: mathematician, actuary, statistician, and computer system analyst.
Do the Math/CS!
Here is a color image of Pascal’s triangle, with each number colored based on its value modulo 5. You can see how the pattern is self-similar like the Sierpinski gasket.
The following video of “Japanese” multiplication has been floating around the internet recently.
I thought it might make an interesting post to explain how it works. Every part of this trick matches exactly what you were taught to do in elementary school. Check out the following example:
The blue lines in the image above correspond to the digits of 123 and the red lines correspond to the digits of 321. The place where any two lines intersect is exactly where you were taught to put the product of the two corresponding digits. Of course, the product of two digits is the number of times their corresponding lines intersect.
If you think about it, it is easy to see that this method works for any numbers, but you wouldn’t want to multiply 987 × 789 this way! It is also not hard to see that the lines method will almost always be slower, although it might be easier for students to remember.
Here is a delightful article on why
would make a better fundamental constant than
I’ve wondered about this before, and I have to say, I agree with the author!
Hampden-Sydney’s first foray into the prestigious COMAP Mathematical Competition in Modeling was an unqualified success. All three teams did an outstanding job in the contest, and will receive certificates from COMAP recognizing their work. Team Beta did especially well, with team members Cameron Auker, Nathan Parr, and Doug Vermilya earning a coveted “Meritorious” award for their paper “A Simple Approach to Geographical Modeling: The Circle-Decay Overlay Model“. The Meritorious designation places Team Beta in the top 20% of all 2,254 teams that participated in the contest, alongside teams from schools such as Harvard, Duke, MIT, Cornell, and the U.S. Military Academy. As a result of this designation, Team Beta has been invited to give a talk on their solution at the next meeting of the Mathematical Association of America (MAA) next weekend at Virginia State University. In addition, H-SC’s Team Alpha – Matthew Carrington, Miguel Mogollon, and Tian Shihao – earned an “Honorable Mention” for their paper “Give Me a Bat, and I’ll Give You the Sweet Spot“, putting them in the top 44% of all teams.
These are truly outstanding results. With little more than a month of preparation, our teams more than held their own against schools like Davidson College, Colby College, Shippensburg University, not to mention some big-name schools like MIT and UC Berkeley. All the men who participated deserve credit for laying the foundation for what I hope will be a long and successful tradition at H-SC! If you see them, be sure to congratulate them on a job well-done!
The current issue of Science News has an interesting article on the use and misuse of statistics in science. There is a good discussion on the general lack of understanding of the concept of statistical significance within the scientific community. The article also touches on the problem of testing multiple hypotheses simultaneously, and on the philosophical underpinnings of the classical statistical approach (p-values, confidence intervals). It concludes with a discussion of Bayesian approach as a potential remedy for some of these problems, a position which I personally think has some merit.
On Monday evening Hampden-Sydney’s three COMAP teams successfully completed the arduous Mathematical Contest in Modeling (MCM), capping off a banner year in which H-SC competed in math contests at the regional, national, and international levels. H-SC’s Team Alpha, consisting of Matthew Carrington, Miguel Mogollon, and Tian Shihao, submitted their paper, entitled “Give Me a Bat, and I’ll Give You the Sweet Spot“, which analyzed the problem of modeling the “sweet spot” on a baseball bat.
Team Alpha’s paper used a moment of inertia approach to calculate the speed of the batted ball as a function of the point of contact between the ball and the bat. They were then able to identity the sweet spot as the point along the bat that resulted in the greatest batted ball speed.
Team Beta, made up of Cameron Auker, Nathan Parr, and Douglas Vermilya, analyzed the problem of predicting the next location where a serial criminal will strike. Their paper, “A Simple Approach to Geographical Modeling: The Circle-Decay Overlay Model“, used spatial distribution and probability distance strategies to predict the location of the next crime.
Team Beta tested their model on historical data from known serial killers such as David Berkowitz and Peter Sutcliffe. Their results were good, predicting the location of the criminal’s last crime with over 90% accuracy at the lowest level of spatial resolution, and 30% to 60% accuracy at the highest level of resolution.
Team Gamma, consisting of Paul Cottrell and Ke Shang, also analyzed the problem of explaining the sweet spot on a baseball bat. Their paper, “Mathematical Portraits of Baseball Bat Vibration“, analyzes the problem from the point of view of Euler-Bernoulli beam theory, the basic idea being that the sweet spot occurs at the “nodes” of the bat, where energy loss due to vibration is minimized. This approach is complicated by the non-constant cross-section of baseball bats, and the team developed a heuristic to address this situation.
Results from the COMAP competition should be in by early April. In the meantime, the Math/CS department congratulates all the H-SC students who have participated in the contests this year!
Hampden-Sydney COMAP’ers Tian Shihao, Paul Cottrell, Douglas Vermilya, Matthew Carrington, Miguel Mogollon, Ke Shang, Cameron Auker, and Nathan Parr have been hard at work on the 2010 COMAP MCM problems. Twenty four hours into the contest, the teams took a break to chow-down on sandwiches from Merk’s Place.
Team Beta is working on the Criminology problem. They are confident they will be able to predict the location of Miguel’s next crime…
Miguel, Matt and Tian are working on Problem A, on the physics of baseball.
Team Gamma is also working on Problem A:
Final solution papers are due Monday night at 6 PM. Please join the Math/CS department in wishing our HSC mathletes the best!
Three teams from Hampden-Sydney College are competing in the Consortium for Mathematics and Its Applications (COMAP) Mathematical Contest in Modeling (MCM):
- Team Alpha: Matthew Carrington, Miguel Mogollon, and Tian Shihao.
- Team Beta: Cameron Auker, Nathan Parr, and Douglas Vermilya.
- Team Gamma: Paul Cottrell and Ke Shang.
PROBLEM A: The Sweet Spot
Explain the “sweet spot” on a baseball bat.
Every hitter knows that there is a spot on the fat part of a baseball bat where maximum power is transferred to the ball when hit. Why isn’t this spot at the end of the bat? A simple explanation based on torque might seem to identify the end of the bat as the sweet spot, but this is known to be empirically incorrect. Develop a model that helps explain this empirical finding.
Some players believe that “corking” a bat (hollowing out a cylinder in the head of the bat and filling it with cork or rubber, then replacing a wood cap) enhances the “sweet spot” effect. Augment your model to confirm or deny this effect. Does this explain why Major League Baseball prohibits “corking”?
Does the material out of which the bat is constructed matter? That is, does this model predict different behavior for wood (usually ash) or metal (usually aluminum) bats? Is this why Major League Baseball prohibits metal bats?
PROBLEM B: Criminology
In 1981 Peter Sutcliffe was convicted of thirteen murders and subjecting a number of other people to vicious attacks. One of the methods used to narrow the search for Mr. Sutcliffe was to find a “center of mass” of the locations of the attacks. In the end, the suspect happened to live in the same town predicted by this technique. Since that time, a number of more sophisticated techniques have been developed to determine the “geographical profile” of a suspected serial criminal based on the locations of the crimes.
Your team has been asked by a local police agency to develop a method to aid in their investigations of serial criminals. The approach that you develop should make use of at least two different schemes to generate a geographical profile. You should develop a technique to combine the results of the different schemes and generate a useful prediction for law enforcement officers. The prediction should provide some kind of estimate or guidance about possible locations of the next crime based on the time and locations of the past crime scenes. If you make use of any other evidence in your estimate, you must provide specific details about how you incorporate the extra information. Your method should also provide some kind of estimate about how reliable the estimate will be in a given situation, including appropriate warnings.
In addition to the required one-page summary, your report should include an additional two-page executive summary. The executive summary should provide a broad overview of the potential issues. It should provide an overview of your approach and describe situations when it is an appropriate tool and situations in which it is not an appropriate tool. The executive summary will be read by a chief of police and should include technical details appropriate to the intended audience.