OFF ON A TANGENT
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
November 5, 2003 Vol. 2, No. 5


Here's what is coming up in the next fortnight
Hope REU students will give the next colloquium

The students who participated in the 2003 Hope College Mathematics REU program will give presentations relating to their research tomorrow, November 6 at 4:00 p.m. in VWF 104.  Information and application procedures for the 2004 REU program will also be included. 

Brandon Alleman and Michael Cortez talk is titled, "Take Me Out to/of the Ballgame."  They looked at when should a person leave a baseball game in order to maximize his/her enjoyment?  How does this decision depend on the score of the game?  Assuming a modified logistic rate of departure from the stadium, and a constant maximum exit rate from the parking lot, they found optimal strategies for leaving games of various scores. (By the end of the talk, you will know whether you should have left early.)

James Boerkoel's talk is, "When Good Matrices Go Bad."  You have probably experienced the frustration of completing a math problem, and yet the answer is nowhere remotely close to the correct answer.  What if you were told that it might not be your fault; it may be the fault of the problem itself.  These “ill-posed” problems need to be fixed so that results are reliable.  In our research, we examine a discrete method for regularizing ill-posed Volterra problems.  We quantify how much better the condition number of the original ill-conditioned matrix is when using this method in a number of cases.  We also provide numerical evidence of the improved condition number in all cases.

Andrew Wells will present, "Counting Quadratic Forms of Rank 1 and 2."  Quadratic forms are polynomials in several variables where each term has degree 2.  The set of all quadratic forms in n variables corresponds bijectively to the set of all n x n symmetric matrices, and one can define the rank of a quadratic form as the rank of its associated matrix.  This talk deals with vector subspaces of quadratic forms (equivalently, with vector subspaces of the space of all symmetric matrices) with special attention given to counting forms (or matrices) of rank less than or equal to 2.  Included is a brief introduction to matrices and rank, along with the results of our research in the 2, 3, and 4 variable cases.


Next week's colloquium will be presented by Matt Boelkins

Grand Valley mathematics professor Matt Boelkins will present "Who's the Greatest Polynomial of Them All?" for next week's colloquium.  The talk is scheduled for Thursday, November 13 at  4:00 p.m. in VWF 104.

In his talk, Professor Boelkins will show that in some applied settings, like numerical analysis, mathematicians are interested in making a polynomial as "small" as possible.  This problem ultimately leads to the famous Chebyshev polynomials.  He will explore some of the amazing properties of these polynomials, as well as how, in one sense, a monic, degree n Chebyshev polynomial is the "smallest" polynomial one can find. 

It is also natural to ask, "How can we make a polynomial big?"  After implementing some natural restrictions, he will look at  how the "size" of the polynomial is a function of n variables, specifically the location of its n roots.  His ultimate goal will be to prove an upper bound for the maximum absolute value of members of a large class of polynomial functions. Along the way he will provide information about the recently-discovered Polynomial Root-Dragging Theorem, see how to "grow" a polynomial, and find that a family of polynomials almost as famous as the Chebyshev polynomials arises in a beautiful way.

This talk is largely devoted to the results of an undergraduate research project from Summer 2003. The big ideas are accessible to any student who has completed calculus I.

Fourteen students compete in the MATH Challenge

Fourteen Hope students competed on six teams in the Michigan Autumn Take-Home (MATH) Challenge this past weekend.  Heidi Libner, Daniela Banu, Stefan Coltisor, Keven Lin, Stephen Minnich, Rachel Lindner Emily Walsh, Megan Vivian, Andy Jarosz, Betsy Carlson, Tasuku Nishino, Giao Tran, James Boerkoel, and Brandon Alleman represented Hope College in this team event.  In groups of two or three, these students spent Saturday morning working on ten interesting mathematical problems.  We will be looking forward to finding out their results in the near future.
 

The problem of the fortnight   

The Fibonacci numbers are defined by:

f1 = f2 = 1;  fn = fn-1 + fn-2 .

Show that if k divides a single Fibonacci number, then it will divide infinitely many of them.

Drop your solution in the Problem of the Fortnight slot outside Dr. Pearson’s office (VWF 212) by 3:00 p.m. on Friday, November 14.


Problem solvers of the fortnight

We had quite a number of students who picked up the Halloween spirit and found that the number of goblins and werewolves were 55 and 65 respectively. Correct solutions were received from: Tim Angeli, Michael Banducci, Michael Cortez, Martha Graham, Ben Onken, Summer Pickhover, Nick Sumner and Matt Westveer. A panel of goblins and werewolves determined that the blue ribbon prize goes to Matt Westveer this enchanted fortnight. All solvers of the problem of the fortnight should pick up their treats from Dr. Pearson. No costume required.


Surfing the Web for interesting numbers

A famous story in mathematics goes as follows.  The remarkable young Indian mathematician Ramanujan came to England to discuss his work with G.H. Hardy.  However, Ramanujan's health was not good and he landed in a hospital. Hardy went to visit him and arrived in a taxi with plate number 1729.  Hardy remarked that he thought 1729 was rather a dull number.  Ramanujan immediately got very excited and told Hardy that, far from dull, 1729 was a very interesting number.  It is the smallest positive integer which can be written as the sum of two cubes in two different ways,  for example 1729 = 13 + 123 and 1729 = 93 + 103.  

Since 1729 is interesting, one might wonder what other numbers are interesting as well.  Did you know that 6 is the smallest perfect number?  Or that 1900 is the largest palindrome in Roman numerals and that 4096 is the smallest number with 13 divisors?  If you are interested in the interesting qualities of these and other numbers, check out http://www.stetson.edu/~efriedma/numbers.html. This site, maintained by Erich Friedman of Stetson University, relates interesting qualities of hundreds of numbers.


Not to propagate stereotypes, but . . .

In light of the recent string theory special on the PBS show Nova these past two weeks, the following amusing anecdote, found in H. Eves' Mathematical Circles Squared (71), seems timely:  "It has been told that when Dr. Einstein received his first check as a member of the Institute of Advanced Study at Princeton, New Jersey, he used the back of the check for some mathematical figuring and then in a short time lost the scrap of paper.  After that, his stipend checks were turned over directly to Mrs. Einstein for safer keeping."


"Obvious" is the most dangerous word in mathematics.  Eric Temple Bell (1883-1960)