OFF ON A TANGENT

A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics

September 22, 2004

Vol. 3, No. 2

Mathematics students again enjoy their summer research opportunities
 
A number of students spent last summer doing research in the mathematics department.  The accompanying picture shows Kyle Williams, Ryan Weaver, and Ryan Ter Louw making progress on their research this past June.

Dr. Pennings's students worked on modeling problems

Under the guidance of Prof. Pennings, Hope student Kyle Williams and Jocelyn Sikora from Carnegie Mellon looked at the question, "Why is the learning curve S-shaped?"  They developed a probabilistic model of neural behavior which does, in fact, predict (and perhaps explain) this behavior. 

Kyle Weaver and Ryan Ter Louw also worked on a modeling problem with Prof. Pennings.  They worked on the problem of optimizing a barge trip up a river.  They thought that as a captain of a barge, one would need to determine how fast to transport your barge up a river against current in order to minimize the cost of energy.  They found solutions to this question using various types of assumptions.

Dr. Andersen's students investigated mathematical biology

Working in the field of mathematical biology, Mike Nelsen, Mike Cortez, and Jennica Skoug worked with Dr. Tom Bultman (Biology) and Dr. Janet Andersen (mathematics) modeling tri-trophic interactions between tall fescue, fall army worm, and a parasitoid. They did both the experimental and the mathematical modeling. Steve Wirkus (CalPoly Pomona) and Erika Comacho (Loyola-Marymount) spent a week helping them with the modeling.  As mentioned in the last newsletter, Mike Cortez gave an award winning presentation of these results at MathFest in Providence, Rhode Island this past August.

Brandon Alleman and Bess Walker (computer science major from Purdue) worked with Dr. Leah Chase (Biology and Chemistry) and Dr. Janet Andersen doing electrophysiology on a cellular transport system. Brandon did both the experimental as well as the modeling work while Bess produced a computer program to help with the calculations. Brandon and Dr. Andersen presented this work at the Society of Mathematical Biology meeting in Ann Arbor in July.

Dr. Cinzori's students nursed ill-posed problems back to health

Dr. Cinzori had two groups of REU students this summer.  Two of these students, Andrew Craker from Notre Dame and Erin Wicker from Alma College, worked on two projects during the summer.  They began with a short project examining how linear algebra is used by practicing population biologists in their research.  The focus of this project was to determine the lengths of members of a certain class of piece-wise linear spirals.  Andrew and Erin cracked a case that had held Dr. Cinzori up for several months.  Their work resulted in them becoming co-authors on a paper that we recently submitted for publication.

Dr. Cinzori's other two students were David Levitt from Carnegie Mellon and Daniela Banu from Hope.  David and Daniela worked on extending the results that his 2003 REU students began.  In particular, they were examining the condition numbers of a certain class of matrices that arise in the regularization of ill-posed problems.  Dr. Cinzori and his collaborators had shown that a regularization method that they developed improved the condition of these matrices, but they never quantified the degree of improvement.  His 2003 research group quantified the conditioning effect in certain cases and made conjectures about others.  David and Daniela extended that work to more cases and verified some of the conjectures while opening still more questions.  They are currently working on putting together a manuscript to submit for publication.

Andrew Wells ...

In addition to the students working at Hope, Andrew Wells did mathematics research last summer at SUNY Potsdam.


It is time to think about competing in the fall mathematics competitions

Two mathematics competitions that take place each fall at Hope College are the MATH Challenge and the Putnam Exam.  Information about each of these follows.

The Michigan Autumn Take Home Challenge

The 2004 Michigan Autumn Take Home Challenge (or MATH Challenge) will take place on the morning of Saturday, November 6 this year.  Teams of two or three students take a three-hour exam consisting of ten interesting problems dealing with topics and concepts found in the undergraduate mathematics curriculum.  Each team takes the exam at their home campus under the supervision of a faculty advisor.  Each year 20-30 teams compete in this competition with teams from Hope regularly placing in the top three.  Last year, the team of  Daniela Banu, Stefan Coltisor, and Heidi Libner from Hope College won the event. 

For more information about this competition visit http://www.mcs.alma.edu/mathchallenge/.  If you are interested in competing, you need to sign up with Prof. John Stoughton before October 15.

The Putnam Exam

The William Lowell Putnam Mathematical Competition, administered by the Mathematical Association of America, is the most prestigious mathematical competition for undergraduates in the nation.  If you are interested in taking the 65th Annual Wm. Lowell Putnam Exam, you must sign up by Oct. 5 with Prof. Stoughton.   The date of the exam is Saturday, December 4, 2004. There is both a morning and an afternoon session of this exam; lunch will be provided by the mathematics department during the break.  For more information about the Putnam Exam visit http://math.scu.edu/putnam/. (As of today, this official Putman site hasn't been updated since last year, but still gives some useful information.)  For questions and solutions from past exams visit http://www.kalva.demon.co.uk/putnam.html. 


Colloquium: An Introduction to Simplicial Complexes
  

Thursday, September 23    VWF 104, 3:30 - 4:30 p.m. Julie Bergner from Notre Dame will introduce the idea of an n-simplex and various ways to describe them, accompanied by lots of pictures.  She will discuss how to put these simplices together to form a simplicial complex, and from there go on to talk about polyhedra and ways to build simplicial complexes from a graph.  Come to Julie's talk and learn about these interesting and important mathematical ideas . . . and find out whether a simplicial complex is complexly simple or simply complex!

   Tea time!  Now that's simple!

Take a little study break on Thursday and join us for tea at 3:00 in VWF 222 (the Reading Room) before Julie's colloquium talk.  We'll have some snacks, pop, and yup, you guessed it -- tea.  So grab your favorite mug, glass or sipping saucer and join us for some food and fellowship.  It'll be a great time to chat with Julie (and find out about that snowshoeing (mis)adventure), and visit with your professors and friends.



  Mathography: Henri Poincaré (1854 - 1912)

Henri Poincaré was one of the early developers of topology and one of the most influential mathematicians of recent times.  His ideas in algebraic topology governed investigations in the field for over 40 years.  He is probably most famous for the Poincaré conjecture, which has received its fair share of ink lately after the announcement that it had been proved by the Russian mathematician Dr. Grigori Perelman. 

The Clay Institute, which had taken out a million dollar bounty on the problem (who says math doesn't pay?), describes the Poincaré conjecture on its website as follows: "If we stretch a rubber band around the surface of an apple, then we can shrink it down to a point by moving it slowly, without tearing it and without allowing it to leave the surface. On the other hand, if we imagine that the same rubber band has somehow been stretched in the appropriate direction around a doughnut, then there is no way of shrinking it to a point without breaking either the rubber band or the doughnut. We say the surface of the apple is 'simply connected,' but that the surface of the doughnut is not. Poincaré, almost a hundred years ago, knew that a two dimensional sphere is essentially characterized by this property of simple connectivity, and asked the corresponding question for the three dimensional sphere (the set of points in four dimensional space at unit distance from the origin). This question turned out to be extraordinarily difficult, and mathematicians have been struggling with it ever since."  Until Perelman, that is.  Henri Poincaré was "a mathematician, geometer, philosopher, and man of letters, who was a kind of poet of the infinite, a kind of bard of science," as an address given at his funeral put it. 

To read more about the Poincaré conjecture, visit http://mathworld.wolfram.com/PoincareConjecture.html and http://www.claymath.org/millennium/Poincare_Conjecture/.  The BBC has a general interest article on the Perelman's proof of the Poincaré conjecture at http://news.bbc.co.uk/1/hi/sci/tech/3632908.stm.  To find out more about Poincaré himself, check out the nice biographical article of him at http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Poincare.html.

Got a math question?    Ask Elvis . . .                       . . . email him at elvis@hope.edu

Dear Elvis,

What in the world is a simplicial complex?  Did Julie make a deliberate mistake in the title of her talk, or was it only partially completed?  Maybe it's just a simply complicated topic, but it sounds like an oxymoron to me! 

                -- Clearly confused

To Clearly, dearly,

It sounds like an oxymoron to me, too.  Like simple calculus or domestic cat.  I know a lot of calculus, but I am almost totally sure I don't know anything about simplicial complexes.  So I asked around the department the other day when I was eating a green orange and feeling idly laborious, and I think I have a nearly complete picture of it now.  Here is my preliminary conclusion: A simplex is something like a line (a 1-simplex), a triangle (a 2-simplex) or a tetrahedron (a 3-simplex) -- you get the idea . . . it's clear as mud.  Well, mathematicians of a certain ilk (topologists) put these simplices together to form what they call a simplicial complex.  It's a way of building up complicated (I almost said complex!) surfaces out of simpler ones, and topologists use simplicial complexes to study all kinds of surfaces and spaces. 

When I was walking down the hall the other day, I noticed a very nice display about topology on the bulletin board.  Check it out sometime!  Or visit http://mathworld.wolfram.com/SimplicialComplex.html to find out more about simplicial complexes, or take a gander at http://www.math.hope.edu/pearson/topology.html to read a little more about topology.

            -- Simply yours,

                    Elvis
                   

Problem Solvers of the Fortnight

A record-setting 47 of you weighed in (sorry, couldn't resist) on the first Problem of the Fortnight -- a great way to start off the problem season!  The problem of determining which of the 8 coins was the bogus (heavier) one is a little more subtle than meets the eye.  It can actually be done in just two weighings.  First, weigh 3 coins on each side of the balance.  If they balance each other, then the heavier one must be one of the ones left over, and a second weighing will determine which.  If the pans containing 3 coins do not balance, take two from the heavier side; if they balance, the bogus one is the one left over, and if they don't, the scales will tell you which is heavier. 

Congratulations to Brandon Alleman, Daniela Banu, James Boerkoel, Mike Cortez, James Daly, David DeZwaan, Chris Johnson, Heidi Libner, Robert Lloyd, Joshua Morse, Trevor Shull and Nick Sumner for producing a balanced account (again, my apologies) of the solution!  The eleven of you who are on campus may stop by Dr. Pearson's office to claim your prizes; Brandon, since you're in Hungary on the Budapest Semester in Mathematics Program, you'll just have to be on the lookout for a carrier pigeon.

(The accompanying photo shows a collection of the artistic (i.e. winning) entries from the last Problem of the Fortnight.)


Problem of the Fortnight

Since so many of you enjoyed the first problem, here's another in a similar vein:

You have 50 coins, one of which is counterfeit and heavier than the other 49.  What is the minimum number of weighings needed, using a balance scale, to determine which coin is bogus, and how do you do it?

Affix your solution (containing your name and your math class, please) to the back of a buffalo nickel or a wheat penny and drop it in the Problem of the Fortnight slot outside Dr. Pearson's office by 3:00 p.m. on Friday, October 1.


Mathematics is the art of giving the same name to different things.
Jules Henri Poincaré  (1854-1912)