Off on a Tangent A Fortnightly Electronic Newsletter from the Hope College Department of Mathematics
 October 28, 2009 Vol. 8, No. 4 http://www.math.hope.edu/newsletter.html

 Summer research is the focus of this week's colloquium

 Title: Undergraduate Research Presentations Speaker: Tarah Jensen and Jessalyn Boelkema Time: Thursday, October 29 at 4:00 p.m. Place: VWF 104

It's never too early to start thinking about summer!  Faculty in the department will be mentoring students in mathematics research this summer.  After December 1, descriptions of research projects will be found at http://sharp.hope.edu/.  If you are interested in applying for summer research in mathematics at Hope, please talk to Prof. Pennings (the program director) or any other mathematics professor.  Information will also be available at this week's colloquium in which undergraduate mathematics research students from last summer will give the following presentations about their work.

Taking Curvature to the Extreme

Tarah Jensen (Grand Valley State University senior)  Faculty Mentor: Stephanie Edwards (Hope College)

Abstract: Let F be a real polynomial of degree N. Then the curvature of F is defined to be

F"____

κ =  (1 + (F')2)3/2  .

Determining the maximum number of zeros of κ is an easy problem: since the zeros of κ are the zeros of F", the curvature of F is 0 at most N - 2 times. A much more intriguing problem is to determine the maximum number of relative extreme values for the function κ, or equivalently, determine the maximum number of zeros of κ'. In 2004, Edwards and Gordon showed that if all the zeros of F" are real, then  F has at most N - 1 points of extreme curvature. We use level curves and auxiliary functions to study the zeros of the derivatives of these functions. We provide a partial solution to this problem, showing that aF has at most  N - 1 points of extreme curvature, given that the real number a is smaller than a given bound. The conjecture that F has at most N - 1 points of extreme curvature remains open.

Geometric Spirals from Convex Combinations?

Jessalyn Boelkema (Hope College sophomore)  Faculty Mentor: Aaron Cinzori  (Hope College)

Abstract: Given a set of points P0, P1,…, Pn in the plane, we can connect the points (in order) by line segments.  We can then use a rule to generate further points and line segments and create a piece-wise linear spiral.  In this talk, we’ll describe the rule and provide necessary and sufficient conditions on the initial set of points to create a spiral whose segment lengths form a geometric series. We’ll also present and analyze some properties of such spirals.  Finally, we’ll show that if our initial set of points is a sub set of R3, rather than the plane, such spirals only exist if the original points were co-planar to begin with. The talk is appropriate for students who have studied some linear algebra.

 Next week's colloquium

 Title: Tales from the Crypt: The Mathematics of Secret Messages Speaker: Darin Stephenson Time: Thursday, November 5 at 4:00 p.m. Place: VWF 104

Abstract: Due to the growing usage of internet commerce, ATM and credit card transactions, and electronic correspondence, the need for secure communications has never been more evident.  However, the desire to transmit sensitive information from place to place privately dates back many centuries.  This need for security has resulted in the development of cryptology, a field that relies heavily on abstract algebra and number theory.

In this talk, we will discuss the basic goals involved in cryptology, as well as the mathematics involved in creating simple cryptosystems.  Our primary examples will include affine cryptosystems and simple examples of public key cryptosystems.  We will give historical information relating to the development of cryptology and indications as to what new directions the field has taken since the development of public key cryptography nearly 30 years ago.

 Integration Confrontation

The following students participated in the Integration Confrontation on Thursday, October 22:  Elly Earlywine, Jonathon Yarranton, Lauren Wilbur, Jessalyn Bolkema, Kyle McLellan, Greg Hubers, Ronald Radcliffe, Jonathan Wielenga, Marc Tory, Candace Gooden, Steven Vanhoven, Sara Lang, Danielle Yeadon, Andreanna Rosnik, Scott Declaire, Nathan Graber, Eric Hallquist, Daniel Simpson, Terra Fox, Kim Klask, Lydia Rau, Curtis Drozd, Sneha Goswami, and Christian McNally.

The team of Jessalyn Bolkema, Kyle McLellan, and Greg Hubers were undefeated in the initial qualifying rounds.  They, along with the teams of Elly Earlywine, Jonathon Yarranton, and Lauren Wilbur; Nathan Graber, Eric Hallquist, and Daniel Simpson; and Curtis Drozd, Sneha Goswami, and Christian McNally, advanced to the lightning round (which was shortened due to a power outage--that's the trouble with lightning).  Nathan, Eric, and Daniel were the only team to successfully calculate an integral in the lightning round, and they took the prize.

 Math Club News

Greetings math clubbers! First of all, we would like to thank you all for coming to our last pizza and board game event; it was a great success! We have decided to push our next meeting to this coming Tuesday (November 3rd) at 7 pm in VZN 298. We have pushed it back so we can get more t-shirt designs turned in. If you have a design in mind, you can either turn it in to Dr. Pearson or submit it on our moodle groups page. If you would like to be added to our group, please email me at
kimberly.klask@hope.edu, and I will make sure you are added. Hope to see you next week!

 Problem Solvers of the Fortnight

The previous Problem of the Fortnight asked for the minimum distance to Grandma's cabin, given that yours was two miles north of an east-west stream, hers was twelve miles east of yours and three miles north of the river, and you had to stop by the river en route to Grandma's in order to pick up water for her.  The minimum distance is 13 miles.

Congratulations to Jeff Minkus, Charlie Matrosic, Ben Hicks, Christine Gobrogge, Dan Waldo, Kyle Goins, Kelsey Ensz, Andrea Eddy, Kayla Lankheet, Zach Mitchell, Kyle Gibson, Ron Radcliffe, Zach Petroelje, Eric Hallquist, Jim Dratz, Eric Dulmes, Marissa Martz, Jessica Clouse, Tommy Waalkes, Kyle Alexander, Chas Sloan, Kylie Topliff, Cara Cannon, David Jenkins, Xisen Hou, Nolan Wiersma, Rebecca Danforth, Caitlin Roth, Rachel Jantz, Lute Olson, Robert Sjoholm, Marcus Bradstreet, Matt Koster, Kelly Petrasky, Ben Fineout, Scott DeClaire, and Brad Hekman.

 Problem of the Fortnight

A Cubic Polynomial (or cubic pumpkinomial)

For f(x) = x3 + 6x2 - 15x + k, the absolute maximum and absolute minimum values on the interval [-10,2] have the same absolute value.  Find the value of k.

Write your solution (not just the answer) on a bag full of Halloween candy and drop it off in the Problem of the Fortnight slot outside Dr. Pearson's office (VWF 212) by 3:00 on Friday, November 6.  As always, be sure to show all your work (answers without work will not be considered), and be sure to write your name, the name(s) of your professor(s), and your math course(s) -- e.g. Rush Inuit, Professor Sloan Cranky, Math 314.

The length of your education is less important than its breadth, and the length of your life is less important than its depth.

Marilyn Vos Savant

 Off on a Tangent