Off on a Tangent
|
A Fortnightly Electronic
Newsletter from the Hope
College Department of Mathematics
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Summer
research is the focus of this week's colloquium
|
Title:
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Undergraduate
Research Presentations
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Speaker:
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Tarah Jensen
and Jessalyn Boelkema |
Time:
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Thursday,
October 29 at
4:00 p.m.
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Place:
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VWF
104
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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.
Title:
|
Tales
from the Crypt: The Mathematics of Secret Messages |
Speaker:
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Darin Stephenson
|
Time:
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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.
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.
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