OFF ON A TANGENT
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
November 2, 2005 Vol. 4, No. 5
http://www.math.hope.edu/newsletter.html


Road Scholars: Hope students travel to Flint for the MUMC

A group of fourteen Hope students -- Shova KC, Liz Adenegan, Kim Harrison, David Visser, Daniela Banu, Pam Rexius, Martha Precup, Laura Schaedig, Jenny Birkenholz, Dan Halma, Koray Aya, Dan Emmendorfer, Jay Gibbs, and Brandon Alleman -- got up to hear the crack o' dawn, as the saying goes, this past Saturday.  The Hope contingent was the second largest, behind only UM-Flint, the host institution.

Now, fourteen college students waking up before 6:00 a.m. on a Saturday morning is worthy of headline news by itself.  But the reason for their premature departure from the Land of Snooze makes this story all the more remarkable.  These early risers bested the rooster on Saturday to drive to Flint for the MUMC, the Michigan Undergraduate Mathematics Conference.

Shova, Liz, Kim, David and Daniela all presented talks on research they conducted this summer.  And if you missed their road performance, you can catch them at home this Thursday, when Shova, Kim, David and Daniela will present an encore performance of their critically acclaimed talks in the weekly Mathematics Colloquium series.

Professor Ed Burger from Williams College started the conference with a bang, giving a fascinating talk titled "Conjugate Coupling."  In it he discussed why the continued fraction decomposition of a irrational quadratic number was in some sense a measure of its irrationality and how the continued fraction decomposition of such a number served as its "numerical DNA," as it were.  (For more on the subject Professor Burger's talk, please see the Amer. Math. Monthly 112 (2005), no. 4, 311--321.)  Three parallel sessions of student talks filled the rest of the morning and the part of the afternoon not given over to Hackenbush, a game of logic and strategy in which Shova, Kim and Koray competed.  Weary after a long day at the conference, the Hope crew returned to Holland in time to enjoy a terrific dinner at the Thai Palance on the Mathematics Department's dime. 

Students' reactions to the conference were overwhelmingly positive.  One conference-goer (who shall remain unnamed) told this reporter that even though it was a long day, she was surprised at how much fun she had.  Another (who shall also lurk in anonymity) confided that she was surprised at how much of the talks she was able to understand!  (Editor's note: she was well taught!)  Suffice it to say, a good time was had by all.

Next year Hope students won't have to roll out of bed quite so early.  Hope is hosting the MUMC next fall.




Tomorrow's colloquium features Hope research students
Tomorrow's colloquium will focus on undergraduate research in mathematics at Hope College.  There will be three 15-minute student presentations based on research done in Summer 2005.  Information will also be made available about research positions for Summer 2006.  The titles and abstracts appear below.

“Randomly Generated Triangles whose Vertices are Vertices of a Regular Polygon: Adventures in Area and Perimeter” by Shova KC

Abstract:
We generate triangles randomly by uniformly choosing a subset of three vertices from the vertices of a regular polygon. The expected area and perimeter based on the number of sides of the polygon is then determined.  We also determine the limit of these equations to compare with a classical result on triangles whose vertices are on a circle.

“Modeling Tri-trophic Interactions with Tall Fescue, Fall Armyworms, and Euplectrus Comstockii” by Kim Harrison

Abstract:
Differential equations have long been used to model the interactions between two populations, such as the Lotka-Volterra predator-prey equations. However, not as much has been done to model the interactions of three populations. We have constructed and analytically analyzed a mathematical model of three differential equations, giving us insight into the critical parameters for a system of tall fescue (Festuca arundinacea), fall armyworms (Spodoptera frugiperda), and a parasitoid wasp (Euplectrus comstockii).

“Presentations and Representations of Metacyclic Groups” by Daniela Banu

Abstract:
We explore the structure of metacyclic groups, particularly that of the dihedral and semidihedral groups, by constructing a geometric presentation for these groups and by examining its relationship to the linear representations of metacyclic groups. In particular, we are able to extract certain irreducible representations from these geometric presentations.


Next week's colloquium will focus on mathematical biology

Next week's colloquium it titled, "Using Mathematics to Gain Insight about the AIDS Virus" and will be presented by Prof. Janet Andersen.  Before the mid 1990s, it was believed that the AIDS virus was primarily dormant for the first 5 to 10 years in a HIV infected person.  However, improved lab techniques and mathematical models were able to determine that the AIDS virus was actually quite active throughout the infection.  These results dramatically changed the treatment protocol for HIV+ patients and the direction of research into this disease.  Dr. Andersen will describe this groundbreaking work accomplished by the collaboration of David Ho, currently Director of the Aaron Diamond AIDS Research Center, and Alan Perelson, Los Alamos National Laboratory.

Notice that this colloquium is scheduled for 11:00 instead of the usual 4:00.  So all of you that have 4:00 conflicts with our usual colloquium schedule, this one is for you!


Upper level mathematics courses offered for Spring 2006 announced

It is just about time to sign up for classes for next semester.  The following are the upper level courses being offered.

For more information about these courses you can talk to faculty members in the mathematics department or visit http://www.math.hope.edu/Math-courses-spring06.doc.


Opportunities for summer research in mathematical biology exist

As mentioned in the last newsletter, there will be other summer research opportunities in the mathematics department besides the REU.  Prof. Janet Andersen's research is in the area of mathematical biology. She currently has two research collaborations. One, with Dr. Tom Bultman, is in the area of ecology where they are looking at how plant defenses effect a tri-trophic system. The mathematics for this project is primarily differential equations and dynamical systems. The second project is with Dr. Leah Chase in the area of neuroscience where they are looking at ways to determine the kinetics of the individual steps of a cellular transport system. The mathematics for this project is primarily Bayesian statistics. If you are interested in either project, you should consider taking the Mathematical Biology course this spring. For the first project, it is also helpful if you have taken Math 232. For the second project, it is helpful if you have taken Math 310.  For more information, contact Prof. Andersen.


The MATH Challenge took place last weekend

 Fourteen Hope students competed on five teams in the Michigan Autumn Take-Home (MATH) Challenge on Saturday, October 29.  Megan Vivian, Abby Rockwood, Jordan Siemon, Julie Allerding, Amanda Allen, Brett Jager, Dave Visser, Jeff Shriner, Josh Warner, Jeff Ambrose, Keith Mulder, Ryan Weaver, Gabe Kalmbacher, and Nate Makowski  represented Hope College in this team event.  In groups of two or three, these students spent the morning working on ten interesting mathematical problems that involved bugs, a guy named Tom, and an elongated pentagonal orthocupolarotunda. 


The Mathematical Jeopardy results are in 

The Mathematical Jeopardy contest held a couple of weeks ago proved to be a success.  Twenty-nine people consisting of seven teams of students and one team of "non-mathematics department Hope faculty with mathematics degrees" competed.  The teams making it to the final round were The Differential Operators (consisting of Jon Moerdyk, Katie Heneveld, Katie Johnson, and Luc Leavenworth), The Strange Attractors (consisting of Chase Morris, Tim Boman, and Scott Peterson), and Complex (consisting of Kyle Williams, Ryan Weaver, Nathan Makowski, and Gabe Kambacher).

The winner of the final round was The Differential Operators with a score of 38 minus (e times pi).  No group managed to get the question for the final Jeopardy answer which was as follows. 
Congratulations to the winners and all those participating in the event.  If fact everyone was a winner since cider and doughnuts were served.


In the News: Mathematics helps to navigate the interplanetary superhighway

The spacecraft Genesis returned to Earth last April from what has been hailed as the most efficient space mission ever, thanks in part to mathematics.  Edward Belbruno, a Princeton mathematician, masterminded a space flight trajectory for Genesis that would minimize fuel consumption by working with gravity instead of against it.  From studying the mathematics underlying the gravitational interactions among the various planets Genesis would fly by, the optimal flight plan for the spacecraft was devised. 

The mathematics are subtle and age-old: near the beginning of the 20th century Poincare realized that the so-called three-body problem in mathematics was very sensitive to initial conditions.  Analyses of the differential equations governing the three-body problem, in part aided by computers unavailable to Poincare, have revealed a chaotic network of "tubes" through outer space, which researchers have begun to catalog in an atlas of such interplanetary superhighways for future space missions.  To read more about this fascinating development, please see http://www.sciencenews.org/articles/20050416/bob9.asp.    


Problem Solvers of the Fortnight

Congratulations to Benjamin Crumpler, James Daly, Maya Holtrop and Kristine Krcmar for bouncing the ant problem around in their heads and, after a minute or so, coming up with the right answer.

The key to this problem is this: the ants are always traveling with a speed of 1 m/min, regardless of which direction they're going, and so instead of thinking of the ants reversing directions when they collide, we can think of the ants just passing by one another.  Then the answer practically jumps off the meter stick at you: no ant can stay on the meter stick for more than 1 minute! 




Problem of the Fortnight

Suppose a straight stick is broken in two places.  The locations where the stick is to be broken are chosen randomly and the location of the second break does not depend on the location of the first.

What is the probability that the pieces will form a triangle?

Write your solution inside your favorite Pythagorean triple triangle, and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 p.m. on Friday, November 11.



 Got a Math Question?

Ask Elvis ...

... email him at elvis@hope.edu

Dear Friends,

I have a couple of questions to answer today.  Well that is not quite true. One is more of a comment and I, of course, comment back.  Keep those questions (or comments) coming.  I am eager to hear from you.  My email address is elvis@hope.edu.




Dear Elvis,
I went to the website that had the article on skiing and on that website they had changed it. Ah the powers of letter writing.
Benjamin

Dear Benjamin,
In the last newsletter, I wrote about a mistake in Skiing Magazine that described a 100% grade as being sheer vertical.  The online version of that article was changed to say, "A 100% grade would therefore be 45 degrees..."  However, it did not make the correction that said a 5.5% grade was 5.5 % of vertical.

A 5.5% grade is not 5.5% of vertical. In fact, in terms of the measure of the angle, it is not even true that it is 5.5% of a 45 degree angle. I think there is some theorem in mathematics that says not everything is linear.  This is one of those cases.   Since a grade is the same as the tangent of the angle of elevation and the arctangent of 0.055 is approximately 3.15 degrees, a 5.5% grade has an angle of elevation of about 3.15 degrees.  Therefore, a 5.5% grade would be about 3.15/45 of the way to 45 degrees (or about 7%.)
Elvis

P.S.  I think a corollary to the not everything is linear theorem is "not everything is as simple as it seems."

Dear Elvis,
A friend of mine at William and Mary saw your article and wanted to know, do all dogs know calculus or are only dogs of math professors so mathematically gifted?
Emily

Dear Emily,
While my mathematical abilities are exceptional, dogs not belonging to math professors also have certain mathematical skills.  If fact, other animals such as lobsters, birds, and (dare I say) cats also exhibit innate mathematical abilities.  A lot of these abilities are related to navigation.  You can find out more about these amazing animal acts in the book The Math Instinct written by Keith Devlin.  In fact, chapter two in this book is titled "Elvis: The Welsh Corgi who knows Calculus."  Need I say more?
Elvis


No human investigation can be called real science if it cannot be demonstrated mathematically.
da Vinci, Leonardo (1452-1519)