Off on a Tangent
A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
   April 9, 2008 Vol. 6, No. 13  
http://www.math.hope.edu/newsletter.html


Two Colloquium Opportunities Tomorrow

Title:
 Divine Action in the World
Speaker:
 Prof. Alvin Plantinga, Notre Dame
Time:
 Thursday, April 10 at 3:00 p.m.
Place:
 Schaap Science Center, Room 1000

  Abstract:  One of the points of tension between science and religion is the claim that modern science suggests or implies that God never acts specially in the world---that there are no miracles (Jesus' rising form the dead, for example), or any other kind of divine action in the world beyond creation and conservation.  Physics, so it is said, presupposes or implies determinism, and determinism implies that no miracles ever occur.  I will take a look at this claim and conclude that it doesn't hold water---either for Newtonian science, or for the more recent science of relativity theory and quantum mechanics.


Title:
 Quaternions and the Discovery of the Fourth Dimension
Speaker:
 Prof. Christopher Moseley, Calvin College
Time:
 Thursday, April 10 at 4:00 p.m.
Place:
 VWF 238

  Abstract:  We are familiar with the idea of time as a fourth dimension, as in Einstein's Special Theory of Relativity.  However, the first thorough mathematical exploration of the fourth dimension precedes Einstein by 60 years!  In this colloquium we will see how the mathematician William Hamilton “jumped….into a fourth dimension” (his words) as he discovered quaternions, 4-dimensional numbers that are used today in the control of spacecraft, quantum spin systems and the programming behind 3D graphics in computer games.
Two More Colloquium Opportunities This Semester

Title:
 Counting Distinct Sudoku Puzzles
Speaker:
 Hope Students Martha Precup and Laura Schaedig
Time:
 Tuesday, April 15 at 11:00 a.m.
Place:
 VWF 238

  Abstract:  The recent increase in popularity of Sudoku puzzles made us wonder: how many distinct Sudoku puzzles are there?  When will we run out?  Counting the number of distinct complete Sudoku puzzles involves a lot of interesting mathematics.  It brings together combinatorics and group theory.  We will discuss our efforts to count the number of distinct Sudoku puzzles and the mathematics behind this interesting counting problem, as well as some of obstacles we encountered.  Most of this talk should be accessible to mathematics students of all levels.
Title:

Paint by number: a visualization of complex functions
Speaker:
 Prof. Michael Bolt, Calvin College
Time:
 Thursday, April 24 at 4:00 p.m.
Place:
 VWF 238

  Abstract:  One challenge to understanding complex analysis is the difficulty one can have in forming an intuition for analytic functions.  Frank Farris suggested a new way to visualize complex functions.  The method is called domain coloring.  In this talk we present two implementations of domain coloring and we contrast it with the usual transformational approach.   We also use domain coloring to illustrate some of the deeper theorems in complex variables.


Students participate in the LMMC and Putnam

A number of Hope students just couldn't get enough of taking their regular exams recently and took part in regional and national competitions.  Martha Precup, Forrest Gordon, Joshua Kinder, and  Jessica Clouse took the Putnam Exam last December.  This exam, the most prestigious mathematical competition for undergraduates in the nation, is administered by the Mathematical Association of America.
 
Another mathematics competition that took place last Saturday, April 5, was the Lower Michigan Mathematics Competition (or LMMC).  This exam, held at Lawrence Tech in Southfield, consists of 10 interesting problems where teams of up to three students work together.  Ten Hope students (consisting of four teams) awoke very early last Saturday for the trip across the state.  Those representing Hope were Terra Fox, Aaron Silver, Luke Wendt, James Daly, Katie Johnson, Bryan McMahon, Thao Le, Josh Borycz, Garrison Benson, and Jon Boldt.


The Problem of the Fortnight

The final POF of the year is a winner!

A bin contains 25 balls: 10 red, 8 yellow, and 7 blue.  We draw three balls at random (without looking!) from the bin.  We will say that we win if we draw two balls of one color and another ball of a different color.

What is the probability of winning this particular game?

Tape your solution to a Rawlings Official Major League baseball and roll it into Dr. Pearson's office (VWF 212) by noon on Friday, April 18.  As always, be sure to write your name, the name(s) of your professor(s), and your math class(es) on your solution (e.g.  Ima Wynner, Prof. Van der Prob, Math 345).  Good luck, and have fun!
Problem Solvers of the Fortnight

The previous problem of the fortnight was:

Calculate the derivative with respect to x of the function



Congratulations to: Mark Gilmore, Felix Kikaya, Emily Bauss, Mark Becchetti, Luke Wendt, Ashley Grunenberg, Joel Mulber, Stephanie Pasek, Alayna Ruberg, Joel Blok, James Daly, Mark Panaggio, Nate Poel, Jeff Minkus, Laura Schaedig, Jonathan Winne, Tim Boman, Matt Glahn, Ryan Converse, Evan Ormiston, Kelsey Bos, Jared Wabeke, Chris Jardin, Chris Hall, Nicholas Rebhan, Jeanne Oxendine, Lucas Johnson, Josh Kinder, Collin Taylor, and Eric Lunderberg.  A solution is posted on the math bulletin board -- check it out!
Ask Elvis

Elvis,

Thank you very much for your answers to my previous questions.  (These appeared in Off on a Tangent v.6 n.8.)  Really you are very good.  I have another new problem. The problem is.....
We know that, if sin(A)=sin(B), then A=n(pi)+(-1)n(B).  One day a teacher asked me, why the (-1)n  is included?  I replied that because the sine function is periodic, the (-1)n is included.  But he said, "You are wrong. Since in the 1st & 2nd quadrants the value of sin(A) is positive, the (-1)n is included."  BUT I COULDN'T UNDERSTAND HIS WORDS. ELVIS, PLZ EXPLAIN THIS TO ME.
   
                                                                                                                                                                                                        Tanveer

Dear Tanveer,

I can certainly be empathetic with not understanding a person's words (particularly a teacher's).  I too have had these problems.  Humans can be very confusing at times.  Let me see if I can help you.

I would say that the (-1)n is included to help give the two different places that the sin(B) will give the same result in one period of the sine function.  In your answer for A [A=n(pi)+(-1)n(B)] we are assuming that n is any integer.  If we write out a few of these values, we might get a better picture of our answer.  For even values of n we get (2pi + B), (4pi + B), (6pi + B), and so on.  These are just accounting for the periodic nature of the sine function and giving answers for inputs in the sine function that are just one period (or 2pi) units apart.

However, the sine function will give the same result more often than once per period (as long as that result is not 1 or -1).  These are accounted for in your answer when n is odd.  Let's first look at the solution when n = 1 and hence A = pi - B.  In the figure of a unit circle that I have drawn below, we can see that when B is in the first quadrant, sin(B) gives the same result as sin(pi-B).  We can see this since the vertical sides of our two reference triangles (that represent sin(B) and sin(pi-B) have the same height.  For other even values of n we get (3pi - B), (5pi - B), and so on.  These again are just accounting for the periodic nature of the sine function and giving us answers for inputs that are one period apart.

Similar arguments (or drawings) can be given if B is in any of the other quadrant.  I hope this answers your question and you can understand a dog's words better than your teacher's.

Your pal,
Elvis

Dogs are the leaders of the planet.
If you see two life forms, one of them's making a poop, the other one's carrying it for him, who would you assume is in charge?

~ Jerry Seinfeld
Off on a Tangent