OFF ON A TANGENT
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
February 23, 2005 Vol. 3, No. 10
http://www.math.hope.edu/newsletter.html


Tomorrow's colloquium will be on path lengths Tomorrow's colloquium speaker is Professor Tom Scofield from Calvin College.  He will be talking on "Spiraling Sequences and their Path Lengths."

He states his problem as follows:  We begin with the four corners, (0,0), (1,0), (1,1) and (0,1) of a square in the plane, taken in that order. Then we generate a sequence of points by taking the average of two previous points, those 3 and 4 earlier in the sequence. So, our fifth point is [(0,0) + (1,0)]/2 = (1/2, 0), our 6th is [(1,0) + (1,1)]/2  = (1,1/2), etc. The points thus generated, connected sequentially by line segments, are depicted in the figure to the right.
 
It seems that the sequence must tend to a limit. How does one show this? Can we find the coordinates of that limit? Does the path traced out by these points constitute one of finite or infinite length? We will investigate these questions not just for the specific problem stated above, but for similarly-generated sequences starting with an n-gon.

Though the talk is self-contained, watch for Professor Aaron Cinzori's talk two weeks from now in which he and two of his summer 2004 REU students present their work on the structure one gets when beginning with an initial triangle instead of a square.  The ideas in this talk are broadly accessible, but it will be helpful if students have had calculus and some linear algebra.


Next week's colloquium will feature a professor from Wabash University
 Next week's colloquium speaker is Prof. Peter Thompson from Wabash College and the title of his talk is "The Almost-Binomial Distribution."  Binomial distributions are used to describe some basic probability experiments.  For example if you are rolling a die three times and counting the number of times a one shows up, this forms a binomial distribution.

In general a binomial distribution arises when we have a sequence of n independent trials, where trials result in either a success or failure, and the probability of a success on any given trial is p. When X denotes the number of successes, X has the binomial(n, p) distribution, with probability density function (pdf)  

    where    x = 0, 1, 2,…, n.

So, for example, if we roll a die three times, the number of ones we roll will have a binomial (n = 3, p = 1/6) distribution.

In this counting situation, n is a positive integer.  What happens if we generalize and allow n to be any positive number?  The new distribution we get is surprisingly simple and is very useful in approximation situations. Several undergraduate students have helped Prof. Thompson explore this area and in this talk he will give an overview of some of the work they have done.


Fall class schedule posted

The tentative fall schedule of mathematics classes is posted on Prof. Stephenson's door (VWF 210).  Your are invited to come check it out to see what is offered, when a certain class is offered, and who is teaching it.  It is not too early to start planning your fall classes.


The Academic Support Center is in need of tutors

The Academic Support Center is in need of tutors for mathematics this semester.  If you have completed Calculus II (with a B or better) and are interested in becoming a tutor for those taking classes at the Calculus II level and below, contact Mrs. Heisler in the Academic Support Center.  Her phone number is X7830 and her email address is jheisler@hope.edu.


Summer opportunities for research should be investigated soon

The time is running out on applying for summer research opportunities.  There are two different programs available in the mathematics department.  These are listed below.
Problem Solvers of the Fortnight

The Problem of the Fortnight in the last issue appears to have been a stumper!  We received nine submissions, but all of them involved a simplifying assumption that rendered their final answer not quite correct (though there were some good and insightful approximations!).  Some approximated the area around the silo with a semicircle and some with half an ellipse, but both approximations neglect the fact that the curve is not tangent to the circle along the silo.  Rather, it looks something like the figure shown at right (which isn't to scale and isn't even quite correct, but it conveys the idea).  To get an exact solution to the problem, consider the (approximately) triangular region cut out by Harry's rope as he moves some small distance ds along the outside curve and use the fact that this triangle is similar to the one inside the silo to solve the resulting integration problem.

As always, those who submit thoughtful solutions will receive credit for an extracurricular mathematics activity, and if the authors of submissions for the Great Goat Debate problem would like to submit another solution in light of the hints given above, we'd be happy to reward their efforts with a prize as well.


Problem of the Fortnight

This Problem of the Fortnight involves a little probability and comes from a Hope College professor who has devised an interesting system for collecting homework.  The question was originally send to Elvis to answer, but he thought it made a good problem of you to answer.  Professor Collectsumup writes:

"In each of my classes this semester, I begin by rolling a die to determine whether the homework assignment will be turned in. In my early morning session, if I roll a 1 or a 2, then the class turns in the assignment. In late morning session, things are more complicated; they wanted to turn in assignments more frequently than just 1 day in 3. In this class, there is an escalating chance that the assignment will be turned in for each day that it is not turned in. In particular, the first day of class there is a 1 in 6 chance that the assignment will be collected. If the assignment is not collected, then the next day there is a 2 in 6 chance. If it is still not collected, then the chance rises to 3 in 6, and so on. Whenever I do collect the assignment, the probability of collecting the assignment on the subsequent day drops back to 1 in 6 and then begins to rise again. My question is this: in the long term, what expected fraction of the total number of assignments will the late morning session turn in to be graded?"

Write your solution on the back of an old homework assignment that wasn't collected and drop it in the Problem of the Fortnight Slot outside Dr. Pearson's office (VWF 212) by 3:00 on Friday, March 4.  Solutions received by that time will have a probability p = 1 of being eligible for a prize.


Mathography: Shigefumi Mori

Born on February 23, 1951 in Nagoya, Japan, Shigefumi Mori was awarded the Fields Medal in 1990 for his work in algebraic geometry on classifying certain algebraic varieties.  Far from being a contradiction in terms, algebraic geometry is a well established area of modern mathematics that studies curves and surfaces (and their higher-dimensional analogues) defined by polynomial equations.  These curves and surfaces are called algebraic varieties.  Mori began his work on classifying algebraic varieties some twelve years before he was awarded the Fields Medal, which are awarded every four years (since 1936) to the most distinguished mathematicians aged 40 or under and are considered to be the highest professional honor a mathematician can attain.  In the same year that he won the Fields Medal, Mori was awarded the Cole Prize, currently awarded every three years by the American Mathematical Society for a notable contribution to algebra.  To read more about algebraic varieties, please visit http://mathworld.wolfram.com/AlgebraicVariety.html, and to find out more about our featured mathematician this fortnight, please visit http://www-gap.dcs.st-and.ac.uk/~history/Mathematicians/Mori.html. 


Got a Math Question?

Ask Elvis ...

... email him at elvis@hope.edu

Dear Friends,

You may have seen on Knowhope last week that I was out traveling over winter break.  Tim and I spent some time in Wisconsin trying to show some students there that I do indeed know calculus.  A nice reporter from the Wisconsin State Journal came by while I was working with the students at the Madison Area Technical College.  In case you missed it, his article can be found at http://www.madison.com/archives/read.php?ref=wsj:2005:02:15:403842:FRONT.

Also during break, someone sent me the following link to an interesting psychic website.  Go to http://www.cyberglass.net/flshstuff/mindreader.php and see if you can figure out the mathematics and the trick behind this nice "mind reading program."

I had three questions asked of me since the last issue.  I won't give the answer to one of them since that is being used as the Problem of the Fortnight.  We are still working on the solution to another one and should have an answer in the next issue.  The other one is answered below.




Dear Elvis,
Just how are the letters in an equation chosen? As an example: How did Einstein chose E = mc2?  Does each letter have an assigned value, which is why you would select it?  I have been watching the new program on TV called NUMB3RS and I don't know where he is getting the letters that he works with. Please explain.
Thanks, Scooter from Minnesota


Dear Scooter,
Starting with algebra in high school one learns that letters can be used to represent numbers.  In a high school algebra class much of the time is spent trying to find x.  The letter x is used to represent an arbitrary input (or independent variable) in some equation.  However, when the inputs are used to represent some actual variable like time, a letter more appropriate is chosen, like t.  

This is the case with Einstein's equation E = mc2.  The m is the input and represents mass.  The E is the output and that represents Energy.  Both of these are variables, meaning that one can put some value in for m to get out the appropriate value for E.  The c, however, is not a variable but is a constant.  It represents the speed of light.  This number doesn't change and is around 186,000 miles per second.  [Slightly faster than the speed at which a cat moves when it is shaken (not stirred) from an afternoon nap by the sound of my bark!]

You might wonder why the speed of light is represented by c and not s or l.  This is because Einstein based it on the word celeritas, the Latin word meaning speed.  [I think he originally was going to use l for lichtgeschwindigkeit, German for speed of light but had trouble spelling lichtgeschwindigkeit.]
Thanks for your question Scooter,
Elvis

I'm a great believer in luck, and I find the harder I work the more I have of it.
Thomas Jefferson