OFF ON A TANGENT
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
November 16, 2005 Vol. 4, No. 6
http://www.math.hope.edu/newsletter.html


Tomorrow's colloquium will be presented by Iowa State professor
 The colloquium tomorrow is titled "The Integral Operator" and will be presented by Prof. Justin R. Peters from the Department of Mathematics at Iowa State University.  He discuss the integral operator
 
to gain insight into some of its properties from the viewpoint of operator theory.  Dr. Peters will also talk briefly about graduate school at the Iowa State. 

Don't forget that tea time will precede the colloquium at 3:30 p.m. in VWF 222.


A summer research opportunity

As mentioned in an earlier newsletter, the mathematics department has a number of areas of research in the Hope REU Program.  For the next four newsletters, we will highlight these areas.  First, up is Prof. Mark Pearson's project on algebra and topology.  His project description is as follows.

Every time you sit down to play with a Rubik's Cube, you are participating in a fundamental mathematical operation, regardless of whether you solve it: you are performing a group action on a set. Group actions are an important link between algebra and topology, and students in my research group will get a taste of both of these areas by investigating questions at the interface of algebra and topology. The exact nature of the project may be more algebraic or more topological depending on interest, but typically the problems we will explore will be algebraic questions whose answers shed light on questions in topology.

Participating REU students will receive a stipend of $2600 for the eight week program. Free apartment-style housing will be provided. Funds for travel as well as books and supplies are also available.  Visit http://www.math.hope.edu/reu.html for more information about the program.


Problem Solvers of the Fortnight

Congratulations to Trevor Bakker, Benjamin Crumpler, James Daly, Erica Dickinson, Brian McClellan, Stephanie Pasek, Mark Parrazzio and Kurt Pyle, all of whom correctly triangulated a solution to the previous problem.  If a stick is randomly broken in two places, where the location of the second break does not depend on the location of the first, there's a 25% chance of the pieces forming a triangle.

The solution is elegant.  We know that in order to form a triangle, the pieces must satisfy the triangle inequality, which says that the sum of the lengths of any two sides of the triangle must be greater than (or equal to) the length of the third side.  So, suppose we break a stick of length L into pieces of length x, y and L - x - y.  Then:
  1.  x < (L - x - y) + y = L - x
  2.  y < (L - x - y) + x = L - y
  3.  L - x - y < x + y
Simplifying we find (1) x < L/2, (2) y < L/2, and (3) y > -x + L/2.  Plotting these three inequalities gives a triangle of area L2/8 (shaded in the figure).  Since x + y < L, the large triangle, whose area is L2/2, represents all possible divisions of the stick into length L.  The breaks forming a triangle will therefore constitute 1/4 of all possible breaks.


Problem of the Fortnight

A bug starts from the origin on the plane and crawls one unit upwards to (0,1) after one minute.  During the second minute, it crawls two units to the right, ending at (2,1).  Then during the third minute, it crawls three units upward, arriving at (2,4).  It makes another right turn and crawls four units during the fourth minute.  From here it continues to crawl n units during minute n and then makes a 90-degree turn, either left or right.  The bug continues this until after 16 minutes, it finds itself back at the origin.  Its path does not intersect itself.  What is the smallest possible area of the 16-gon traced out by its path?

Cut a sheet of paper to replicate the minimal 16-gon of the problem, write your solution on it, and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Monday, November 28.  (Alternatively, since we received no copies of "A Bug's Life" for our earlier Problem of the Fortnight about the dance of a hundred ants, and since we have not yet seen the flick, problem solvers are invited to submit solutions on the back of "A Bug's Life" DVD; we request that problem solvers opting for this alternative format submit their solutions before Thanksgiving break so we can watch the movie in between naps and turkey sandwiches.)


Road Scholars: Hope students travel to Chicago for Pew Conference

Hope students Kenneth Kuper, Mitchell Plosz, Daniela Banu, Brenna Giaecherio and Lydia Hartsell were among the approximately 75 students from regional colleges and universities who participated in the Pew Midstates Undergraduate Research Symposium in the Physical Sciences and Mathematics at the University of Chicago on November 4 - 6. 

Ken, Lydia and Mitch presented posters summarizing research they had done over the summer.  Ken's poster detailed his findings on "Synthesis of Quinoline Intermediates toward more potent photochromic photooxidants"; Mitch's poster outlined his research on "Isolation of seed chemical defense compounds from tropical pioneer plants"; and Lydia's poster discussed her work on "Development of an interdisciplinary laboratory module for physical chemistry and biochemistry classes."  Despite an untimely bout of laryngitis, Brenna presented a talk on "Developing a luminosity model for radio pulsars," work she had done in a Hope College Physics REU this past summer.  And senior mathematics major Daniela Banu gave a talk on "Geometric presentations and linear representations of metacyclic groups," a topic she investigated in a Hope College Mathematics REU this past summer. 

Said Pew director and conference organizer Janet Andersen (Hope College mathematics department), "I continue to be impressed by the quality of the talks and posters presented at the Pew Symposia, both by Hope students as well as students from other institutions."


"Can you hear me now? . . .  Good."

The Verizon Wireless ads provoke two questions rather than one: not only, how much redundancy can I (the viewer) take from these commercials? but also -- and here's the mathematically more interesting question -- how much redundancy needs to be built into a signal in order to transmit information reliably?

Such questions are hardly pressing ones for cell phone users or designers anymore.  (Would anyone be particularly upset if they couldn't hear the man from the Verizon ads?)  But consider them from the vantage point of a space engineer: how can satellites transmit data hundreds of millions of miles without the messages being hopelessly distorted by noise?  The answer: redundancy.  I repeat: redundancy.

By duplicating parts of the information, the message can usually be recovered, even if it becomes somewhat distorted in transmission.  Over the years mathematicians and computer scientists have devised clever ways of building less redundancy into the code so that the messages are smaller in size.  But there's a theoretical limit to how small the messages with redundancy can be; it's called the Shannon limit in honor of mathematician Claude Shannon who discovered the limit in 1948 while working at Bell Labs. 

In the 1990s two French coding theorists shocked the information world with their announcement that they had come within a hair of the Shannon limit, using so-called "turbo codes" and "low-parity density-check" coding (known as LDPC coding in the biz.)  More recently, when the European Space Agency launched SMART-1 to the moon in 2003, these turbo codes were put to the test and transmitted data within a few percent of the Shannon limit.  To read more about what some have hailed as "the last big leap in coding," please see  http://www.sciencenews.org/articles/20051105/bob8.asp.



 Got a Math Question?

Ask Elvis ...

... email him at elvis@hope.edu

Dear Friends,

Did you hear that some of the math faculty will be singing the national anthem before each of the Hope basketball games this Saturday?  Professors Pennings, Pearson, and VanIwaarden (retired) from the mathematics department and Professor Pikaart from the chemistry department will be singing a cappella (I think that is Italian for without practicing). So if you can get to the games on Saturday, make sure you don't miss the Three Mathematicians and a Chemist sing the national anthem.

Hope's mathematical a cappella group reminds me of a similar group from Northwestern University. They call themselves the Klein Four Group and are becoming quite famous. (Not quite as famous as the dog that knows calculus, however.)  They recently released a CD titled Musical Fruitcake. It includes such songs as "Finite Simple Group," "Lemma," "Contradiction," and "Mathematical Paradise." As you can see, their songs are mathematically related and are quite humorous.

You can check out their website at http://www.kleinfour.com. From there, you can view some of their musical performances as well as other productions, buy their CD and other merchandise, view their bios, and get more information about this interesting group. While this week's colloquium speaker will give you some information about graduate school in mathematics, the Klein Four Group can show you that not all your time in graduate school need be spent studying.  (Speaking of websites, why isn't there a calculuselvis.com or something like that?  I did see that elvisthedog.com has already been taken.)

We do have a local connection to the Klein Four. Professor Pearson (who graduated from Northwestern) has a brother (named Paul who is a student at Northwestern) is roomates with Mike Johnson of the Klein Four.

I did not receive any letters in the past couple weeks.  Don't be shy, I will try to answer most anything you might send it.  My address is elvis@hope.edu.





Question:  What does the photo above show?

a)  A group of "the guys" gathering to watch NUMB3RS on the big screen in VWF 102.
b)  The staff for the Math Help Sessions.
c)  A group of aliens from planet Poindexter.
d)  One more option that you can use to substitute for colloquium credit.
e)  The new a cappella group, MATH!.

It is the mark of an educated mind to be able to entertain a thought without accepting it.
Aristotle