The Rose Mathematics Seminar - History
Overview
This page give a history of the talks in the Rose Mathematics Seminar, which was started in 1994-95 under the name of the Applied Math seminar. Some years later it was expanded to include all mathematics with a suitable name change. The speakers titles and abstracts are listed below with the later years first. The current year's schedule of talks in given on the on the current seminar page.
Speakers, Titles, and Abstracts
2007-08 (latest first)
- Topic: Generalizations of Niven Numbers
- Speaker: Robert Lemke Oliver, Rose Student
- Date: 14 May 2008
- Abstract: A Niven number is an integer that is divisible by the
sum of its base q digits. For example, 2008 is Niven both in base 3 and in base 5 (see abstract).
Several people have derived asymptotic formulae for the function N(x) that counts the number
of Niven numbers less than x. We proceed in a more general case, studying functions that act
only on the base q digits of an integer. An asymptotic formula for the counting function of these
generalized Niven numbers is known, but the question of divisibility by multiple functions is
still open. We present partial work toward acquiring an asymptotic formula in this case, as well
as conjectures based off of numerical evidence.
For more information see Announcement/Abstract/Paper
in PDF form
- Topic: A Generalization of the Fibonacci and Jacobsthal Sequences
- Speaker: Ian Rogers, Rose Student
- Date: 07 May 2008
- Abstract: Among the sequences of discrete mathematics, the Fibonacci
sequence is probably the most well-known. Turning up in myriad areas from geometry to graph theory,
seashells to the stock market, the Fibonacci numbers display an amazing number of interesting
properties. The Jacobsthal numbers, another well-known sequence, are defined by a different,
yet closely related, recurrence relation to that of the Fibonacci numbers. While slightly less
popular, the Jacobsthal numbers too display many desirable properties. In this talk, we will
describe a new class of generalizations of the Fibonacci and Jacobsthal numbers. We then look
at a few examples in which the Fibonacci and Jacobsthal numbers are known to occur, and expand
them to produce the new sequences. Finally, we show that many of the desirable properties of
the Fibonacci numbers still hold in the general case, and provide suggestions for further research
into this new family of sequences.
- Topic: Total Variation Image Restoration
- Speaker: Ely Spears, MIT Lincoln Labs - Rose Alum
- Date: 26 Mar 2008
- Abstract: One of the most widely studied areas of applied mathematics
is image processing. Image restoration, also called image inpainting, is one of the most prominent
uses for these mathematical techniques. In this talk, a particular procedure for restoring damaged
or corrupted images, called total variation, is discussed. Most of the material will be accessible
to students familiar with linear algebra. A brief description of numerical methods, in particular
the Fast Level Set Transform, is included.
- Topic: Modeling a Slice of French Bread
- Speaker: David Finn
- Date: 23 Jan 2008
- Abstract: Why does a slice of French or Italian bread have a
somewhat elliptical shape? In this talk, I will provide a heuristic model to describe the shape
based on treating dough as a liquid. Then from data from slices of bread, I will show that this
model provides a good description of a slice of bread.
- Topic: Introduction to the Life Table
- Speaker: Casimir G.Ksiazek III, Rose Student, Mathematics
- Date: 19 Dec 2007
- Abstract: Buying life insurance is a quite a common occurrence.
But how do people determine how much life insurance premiums should cost? Historically, actuaries
have used life tables to assist in pricing insurance and annuities. In this talk, the concept
of a life table will be introduced. In addition, examples will be given to show how from seemingly
simple data, quantities such as life expectancy and insurance premiums can be calculated. A knowledge
of probability is recommended, but not required. Anyone interested in actuarial science is strongly
encouraged to attend.
- Topic: Introduction to Infinity or Why Johnny Can't Add
- Speaker: Bill Butske
- Date: 07 Nov 2007
- Abstract: First I want emphasize that this talk is for anyone
who has wondered what mathematics has to say about the concept of infinity. In particular non-math
majors are encouraged to attend and the talk is aimed primarily at them. I'm going to talk about
infinity in two ways, first in counting, where we will see that there are two different kinds
of infinity (at least) and second in geometry where we know that parallel lines DO intersect,
namely at infinity. Of course this is a math talk and the underlying intent is to warp your mind.
- Topic: FETCHING WATER WITH MINIMUM RESIDUES: Generalization of
a problem from Die Hard 3
- Speaker: Herb Bailey, Emeritus Rose Math professor
- Date: 31 Oct 2007
- Abstract: Bruce Willis can disarm a bomb if he is able to get
exactly 4 gallons of water from a well using only a 3 gallon jug and a 5 gallon jug. This problem
dates back to the 13th century. A generalization of this problem is to determine all possible
integer gallons that can be obtained using an M gallon jug and an N gallon jug, with M < N.
We solve the generalized problem using some congruence results. It turns out that there are only
two distinct pouring sequences to get a given number of gallons. The shorter of the two can be
determined by solving a linear congruence equation. Short is good since Bruce has but 5 minutes
prior to detonation. Not to worry, no previous knowledge of number theory will be needed to enjoy
this talk.
- Topic: Blow-up Solutions to Differential Equations
- Speaker: Kurt Bryan
- Date: 24 Oct 2007
- Abstract: Nonlinear differential equations of the form u' = f(u)
where u=u(t) are common in applied mathematics. Usually t is time, u(t) is the amount of some "stuff" in
a system, and f(u) models the rate stuff is produced or destroyed, as a function of the amount
present. If the function f is positive and increasing (the stuff catalyzes its own production)
then solutions may grow to infinity in a finite time, a phenomena called "blow-up".
In this talk I'll start with the simple ODE above, then describe some recent progress in analyzing
blow-up phenomena for similar partial differential equations in which diffusion is present.
- Topic: Models for Emergent Behavior
- Speaker: Ely Spears, Rose Student, Mathematics
- Date: 17 Oct 2007
- Abstract: Emergent behavior is a division of biology that seeks
to understand and better explain phenomena that appear in group situations but not on an individual
basis. Fish schooling, bacterial growth properties, and bird flocking are just a few prominent
examples of this sort of behavior. The latter of these examples motivated summer research at
the City University of Hong Kong, in China. This introductory presentation will give the details
behind some popular mathematical models for bird flocking behavior. Additionally, numerical simulations
of these models will be discussed at length and the various model parameters will be explored.
The talk is such that students of any background are encouraged to attend.
- Topic: Generalized Niven Numbers
- Speaker: Robert Lemke-Oliver, Rose Student, Mathematics
- Date: 27 Sep 2007
- Abstract: A base-q Niven number is one which is divisible by
the sum of its digits. For example, 18 is a base 10 Niven number, since 9 divides 18. We will
be interested in simultaneous Niven numbers, numbers that are Niven in more than one base. Returning
to the example, 18 is also base 9 Niven, since 18 is 20 in base 9. Thus, 18 is a simultaneous
base 9 and base 10 Niven number. We are interested in counting the number of simultaneous Niven
numbers up to a point, x. One approach to this is to look at completely q-additive functions.
These functions essentially act on the digits of a number, so that f(124)=f(1)+f(2)+f(4). Note
that the sum of digits function is completely q-additive. If we can understand these generalized
Niven numbers, we can hopefully gain some information about the standard Niven numbers. In this
talk, we will prove an asymptotic formula for the number of generalized Niven numbers, and we
will present the work that has been done to relate this to Niven numbers.
2006-07 (latest first)
- Topic: Modeling Hysteresis (PART II): A load dependent hysteresis
model for a simple shape memory wire actuator.
- Speaker: Steve Galinaitis
- Date: 31 Jan 2007
- Abstract: To accurately position an object with an actuator that
exhibits load dependent hysteresis requires a hysteresis model that is capable of adjusting to
a change in load. In this talk we investigate the specific problem of modeling the hysteresis of
a simple shape memory alloy wire that is operated under changing tensile loads. A Preisach operator
that incorporates load dependent parameters in the Preisach density function is proposed as the
hysteresis model. In support of this selection, a relationship between the Preisach density function
and the wire’s thermal coefficient of expansion is established. It is then shown that the
load dependent Austenite-Martensite transition temperatures of the wire can be used to estimate
the parameters of the density function. Based on these findings a load dependent Preisach operator
is defined. To test this approach, a bivariate density function that incorporates two load dependent
parameters is substituted for the Preisach density function. Two load dependent linear estimators
are developed from experimental data and used to estimate the parameters of the density function.
These estimators and the load dependent Preisach operator are then used to estimate the length
of a SMA wire that is operated under several tensile loads. The estimates are compared to experimental
data and a discussion of the effectiveness of this approach is given.
- Topic: Modeling Hysteresis: A load dependent hysteresis model for
a simple shape memory wire actuator.
- Speaker: Steve Galinaitis
- Date: 24 Jan 2007
- Abstract: To accurately position an object with an actuator that
exhibits load dependent hysteresis requires a hysteresis model that is capable of adjusting to
a change in load. In this talk we investigate the specific problem of modeling the hysteresis of
a simple shape memory alloy wire that is operated under changing tensile loads. A Preisach operator
that incorporates load dependent parameters in the Preisach density function is proposed as the
hysteresis model. In support of this selection, a relationship between the Preisach density function
and the wire’s thermal coefficient of expansion is established. It is then shown that the
load dependent Austenite-Martensite transition temperatures of the wire can be used to estimate
the parameters of the density function. Based on these findings a load dependent Preisach operator
is defined. To test this approach, a bivariate density function that incorporates two load dependent
parameters is substituted for the Preisach density function. Two load dependent linear estimators
are developed from experimental data and used to estimate the parameters of the density function.
These estimators and the load dependent Preisach operator are then used to estimate the length
of a SMA wire that is operated under several tensile loads. The estimates are compared to experimental
data and a discussion of the effectiveness of this approach is given.
- Topic: Alignment of Protein Structures
- Speaker: Yosi Shibberu
- Date: 17 Jan 2007
- Abstract: Proteins play a key role in nearly all of the biochemical
processes of living organisms. A protein is a long molecular chain constructed from twenty types
of molecules called amino acids. Proteins produced by living organisms fold up into unique, tightly
packed, structures called folds. The particular sequence of amino acids in a protein's chain determines
its unique fold. The geometry of a protein’s fold largely determines the protein's specific
biological function.
Identifying the biological function of individual proteins is an important and challenging problem.
A better understanding of the evolution of protein folds will help us decipher the function of
individual proteins and will lead to major advances in biology and new treatments for many human
diseases.
The evolution of proteins is studied by making comparisons. Proteins are typically compared by
comparing their sequence of amino acids, by comparing the geometry of their folds, and more recently,
by comparing their expression profiles.
Fold-based comparisons of proteins is believed to be much more informative and robust than sequence
based comparisons. However, the problem of aligning protein folds is not as well understood as
the problem of aligning protein sequences. In this talk, we describe a new mathematical framework
for describing the geometry of protein folds. This mathematical framework may lead to a better
understanding of the fold alignment problem.
- Topics: Cookius Maximus by Robert Lemke Oliver and On the Minimum
Vector Rank of a Graph by Ian Rogers
- Speakers: Robert Lemke Oliver and Ian Rogers , Rose Students
- Date: 20 Dec 2006
- Abstracts:
Shape of a Cookie How can the shape of a sugar cookie be modeled mathematically?
It turns out that it’s a solution of a non-linear partial differential equation. In this
talk, we examine a simplified version of this “cookie equation” to find the highest
point on the cookie. Our eyes seem to be very good at locating it, but whatever process we’re
using turns out to be hard to explain mathematically. We will look in particular at convex regions,
which are known to have only one maximum.
Graphs Given a graph or multigraph G on n vertices, we associate a set of nonzero
complex vectors to the vertices of G in the following manner: If vertices i and j are not joined
then the corresponding vectors are orthogonal, and if i and j are connected by a single edge,
the associated vectors are not orthogonal. The rank of a vector representation is the maximum
number of linearly independent vectors in the representation. The minimum vector rank of G, mvr(G),
is the minimum rank among all vector representations of G. We present methods for determining
mvr(G) if G is among certain classes of graphs, including perfect graphs, complete graphs, and
cycles. Further, we present upper and lower bounds on mvr(G) for all multigraphs that contain
only multiedges, and provide two conjectures on the exact value of mvr(G) for a graph.
- Topic: Optimizing 4th-Order and 5th-Order Explicit Runge-Kutta
Formulas
- Speaker: Stephen Dupal, Rose Student
- Date: 13 Dec 2006
- Abstract: Differential equations have been solved numerically with
explicit Runge-Kutta methods for over a century. Runge-Kutta methods are used in the sciences as
well as mathematical software such as Matlab’s ode45 solver. Utilizing techniques in polynomial
theory based on Gröbner bases, it becomes more manageable to find Runge-Kutta formulas that
minimize higher-order truncation error. In this talk, we will discuss the connection between the
Runge-Kutta method and Gröbner bases, and we will present some of the results of exploring
the optimization of fourth- and fifth-order Runge-Kutta formulas. This presentation is based on
work done by Iowa State University’s summer 2006 Numerical Analysis REU group consisting
of Stephen Dupal (Rose-Hulman) and Michael Yoshizawa (Pomona College).
- Topics: Characterizing Holes in Wires and Plates Inverting the
Heat Equation: Tom Werne and
Characterizing Refinable Rational
Functions: Ely Spears
- Speakers: Thomas Werne and Ely Spears, Rose Students
- Date: 06 Dec 2006
- Abstracts:
Heat Equation: The heat equation is a classical partial differential
equation that can predict the temperature distribution on some domain subject to certain boundary
conditions. Motivated by the field of nondestructive testing, the equation turns out to be a useful
tool for characterizing defects in metallic plates. In this talk we will discuss solution methods
and results that show how to characterize certain defects in two dimensional regions using only
boundary data. The presentation is based on work done during the summer of 2006 by the Inverse
Problems REU group of Thomas Werne (Rose-Hulman) and Jay Preciado (The College of New Jersey) at
Rose-Hulman under Dr. Bryan (Rose-Hulman Mathematics Department).
Refinable Functions: A k-refinable function is a function f(x) that
can be re-written in terms of the function f(x). In recent decades, refinable functions
have become increasingly popular due to their desirable properties in many applications, such as
wavelet analysis. While the refinability properties of many popular classes of functions, such
as compactly supported splines, have been known for a while, rational functions had seemed to escaped
notice in terms of refinability. This talk is based on research investigating the refinability
of rational functions that took place at Texas A&M University during the summer. Preliminary
simplifications to the general problem are presented in a chronological collection of lemmas. A
complete characterization of refinable rational functions follows with an interesting connection
to an open problem in number theory.
- Topic: Knots, Braids, and an Application followed by Probability,
Electrical Circuits, and Rectangles
- Speaker: Jennifer Franko and Michael Bateman, Indiana University
- Date: 29 Nov 2006
- Abstract: This week we have two mathematics seminars on Wednesday
that may be of special interest to students. The seminars are 9th and 10th period in G221 on Wednesday
by two Graduate Students from the Mathematics Department of Indiana University. The first (during
all of 9th period) is by Jennifer Franko entitled “Knots, Braids, and an Application” which
concerns the application of topology to quantum computing, and the second (during the first half
of 10th period) is by Michael Bateman entitled “Probability, Electrical Circuits, and Rectangles”.
Following Michael Bateman’s talk, both graduate students will answer questions about graduate
school, the application process, what life is like as a graduate students, etc, so if you are considering
Graduate School in your future it might be worthwhile to attend.
Knots, Braids, and an Application: One method proposed to build quantum computers
is based on braid representations. In this talk, we will define the braid group and discuss the
connection between braids and knots. Any invertible matrix which satisfies the Yang Baxter Equation
can be used to obtain representations of the braid group, and we will study these types of representations
and as well as link invariants they might yield. Finally, we will mention how these representations
might be used in a topological model of quantum computation.
- Topic: Actuarial Mathematics
- Speaker: Nate Dorr, Rose Student
- Date: 08 Nov 2006
- Abstract: Actuarial Mathematics refers to the mathematics of the
insurance industry. Actuaries use probability and statistics in calculating premiums, determining
reserves, and modeling insurance products. In this talk, actuarial components of a whole life insurance
product will be covered. Life insurance, life annuities will be discussed and will lead to how
premiums are calculated. In addition, information about the actuarial profession will be presented
with time for questions at the end. Probability should be sufficient background for this talk.
- Topic: Geometry from Chemistry II - The Geometry of Nanotubes
- Speaker: Allen Broughton
- Date: 01 Nov 2006
- Abstract: Carbon nanotubes are an interesting but as of yet incompletely
understood part of nanotechnology, an area of science that has really grown up in just that last
15 years. From the mathematical perspective nanotubes have an interesting molecular structure based
on the hexagonal honeycomb structure of graphite. In this talk I will describe the geometry and
symmetries of nanotubes. There is an infinite family of such nanotubes, so describing the structure
takes some care. Multivariable calculus should provide plenty of background to make this talk accessible.
There will be a brief recap from lecture I of this series to motivate the atom labelling problem
- a graph theory problem - for nanotubes.
For more information see Announcement/Abstract/Paper
in PDF form
- Topic: Geometry from Chemistry I - Understanding Molecular Dynamics
of Bucky Balls
- Speaker: Allen Broughton
- Date: 25 Oct 2006
- Abstract: Buckminsterfullerene is a complex molecule consisting
of sixty carbon atoms is an arrangement like a soccer ball, and so the molecules are often called
bucky balls. Trying to understands the molecular dynamics of bucky balls leads to some interesting
problems in geometry, algebra and differential equations. In the talk, the theory will be described
in some detail for very simple objects such as triangular molecules such as water. We then will
examine the geometrical issues that come about in modeling the much more complex bucky balls. We
are only go to talk about classical dynamics as quantum mechanics add a level of complexity well
beyond an hour's talk This work is a collaboration with Dan Jelski of the chemistry department
and Guo-Ping Zhang of the ISU physics department. We do not have complete results at this stage,
in fact I'd like to describe some problems that could be tackled by undergraduates. I don't plan
to use much more beyond multi variable calculus, though understanding of differential equations
helps.
For more information see Announcement/Abstract/Paper
in PDF form
- Topic: How to Paint Your Way out of a Maze
- Speaker: Joshua Holden
- Date: 18 Oct 2006
- Abstract: Many people don't realize that what we now call "algorithm
design" actually dates back to the ancient Greeks! Of course, if you think about it, there's
always the "Euclidean Algorithm". A more dubious example might be Theseus's use of a
ball of string to solve the "Labyrinth Problem". (Google "Theseus, Labyrinth, string".)
Solutions to this problem got a lot less dubious after graph theory was invited, since a graph
turns out to be a good way of representing a maze mathematically. We will examine the classical
solutions to this problem, and then throw in a twist --- a Twisted Painting Machine that puts restrictions
on which paths we can take to explore the maze. Applications to sewing may also appear, depending
on the presence of audience interest and string.
- Topic: More Talks on the Numerical Range
- Speaker: Thomas Werne, Ted Lyman and Robert Lauer, Rose students
EE/MA, ME/MA , EE/MA
- Date: 27 Sep 2006
- Abstract:
First Talk
Title: Finding the Centroid of W(A)
Student: Thomas Werne
Abstract: The numerical range of a matrix A is a subset of the complex plane. One method of generating
this subset is to choose random vectors on the unit ball in complex hyperspace. The method of generating
these random vectors induces a probability density function on the numeric range. In this talk,
we examine these density functions and a possible connection with the centroid of the numerical
range and the spectrum of the matrix.
Second Talk
Title: Pre-Images of Points in the Numerical Range
Students: Ted Lyman (speaking) and Robert Lauer
Abstract: If A is an n x n matrix, the numerical range of A is the set of complex numbers W(A)
= (Ax,x), where x is a unit vector in Cn and (Ax,x) denotes the dot product between
Ax and x. Although W(A) appears simple, it has many intriguing properties. We give a brief overview
of some of these properties and take a look specifically at the connectedness of the pre-image
of points in W(A).
- Topic: Home on the (Numerical) Range
- Speaker: Dr. Roger Lautzenheiser
- Date: 20 Sep 2006
- Abstract: Like the eigenvalues, the numerical range of a matrix
is a subset of the complex plane. However, unlike the eigenvalues, the numerical range will not
be a finite set except when the matrix is a multiple of the identity. Indeed, the numerical range
of A is the singleton set {a} if and only if A = a I. In addition to containing the eigenvalues,
the numerical range has many interesting properties. In this talk we survey the history of the
numerical range, the relationships between the geometric properties of the numerical range and
the algebraic properties of the matrix, and perhaps most importantly, how the numerical range is
used as a research project in Linear Algebra 2.
2005-06 (latest first)
- Topic: Fuzzy Topological Spaces Part II (of II) - "Correct" Fuzzification of Topological Spaces: Functors and the General Tychonoff Theorem
- Speaker: Stephan Carlson*
- Date: 17 May 2006
- Abstract: In this second part of his presentation, the speaker will discuss Lowen's modified definition of a fuzzy topology on a set and its ramifications for the investigation of fuzzy topological spaces. Emphasis will be placed on the use of category theory as a test for a correct generalization of set-based topology and the success in proving a general theorem on products of compact fuzzy topological spaces. *Research on the results presented was completed during the presenter's 2004-2005 sabbatical leave.
For more information see Announcement/Abstract/Paper in PDF form
- Topic: Fuzzy Sets and Fuzzy Topologies: Early Ideas and Obstacles
- Speaker: Stephan Carlson
- Date: 10 May 2006
- Abstract: Fuzzy set theory and fuzzy logic were introduced in the 1960s by electrical engineers as tools for understanding and developing efficient control methods. Since fuzzy sets in a fixed set generalize subsets of the set, mathematicians – especially topologists – took on the challenge of generalizing existing set-based theories to the fuzzy set context. In this first part of his presentation, the speaker will survey the initial development of the field of fuzzy topology, which yielded some elegant results but also left some challenging gaps. The presentation will be intended for a general audience, in the sense that no previous background in either fuzzy set theory or topology will be necessary in order to comprehend basic ideas.
For more information see Announcement/Abstract/Paper in PDF form
- Topic: Bicycle Tracks on the Plane and the Sphere
- Speaker: David Finn
- Date: 29 Mar 2006
- Abstract: The title problem of the MAA book “Which way did the bicycle go? … and other intriguing mathematical mysteries” by Konhauser, Velleman and Wagon considers the following situation: Imagine a 20-foot wide mud patch through which a bicycle has just passed, with its front and rear tires leaving tracks as illustrated below. In which direction was the bicyclist traveling? This problem is motivated by the Sherlock Holmes mystery, The Priory School, in which the great detective encounters a pair of tire tracks in the mud and immediately deduces the direction the bicycle was going. This evidence then leads to the finding of a duke's son and the arrest of a murderer. In this talk, we will describe solutions to two variations of this problem on both the plane and the sphere in which a criminal could potentially fool the great detective as it is possible for an incredible bicyclist to create tracks for which it is impossible to determine which direction the bicycle went by only the geometry of the tracks. Moreover, an incredible bicyclist can also defeat the great detective by riding in such a way to leave only one track, possibly causing the detective into believing he is pursuing a unicyclist instead.
For more information see Announcement/Abstract/Paper in PDF form
- Topic: Breaking the MD5
- Speaker: Brandon Borkholder, Rose Student, Computer Science
- Date: 22 Mar 2006
- Abstract: The MD5 hash function and its family are security algorithms that have been used world-wide for nearly a decade. Just a few years after creation there were hints of weakness and now there are algorithms to crack it efficiently. How do these algorithms work? Is the MD5 completely broken? How can a potential hacker exploit this weakness to undermine the trust of those who use it?
- Topic: Investigating the Shape of a Cookie
- Speaker: Hari A. Ravindran, Rose Student, Mathematics
- Date: 15 Feb 2006
- Abstract: This is a continuation of the previous two talks on the Shape of a Cookie. The goal of the investigation is the establishment of an asymptotic expansion for the shape of a cookie with an elliptical base domain. The talk summarizes Hari’s work towards this goal over the past summer and during this academic year. This research was funded in part by a Joseph B. and Reba A. Weaver Undergraduate Research Award.
- Topic: Existence for a Heuristic Model for the Shape of a Cookie (Part II Cookie Series)
- Speaker: David Finn
- Date: 08 Feb 2006
- Abstract: Have you ever wondered why cookies are generically round? Well, I have. And, the reason involves some interesting mathematics: Calculus of Variations, Nonlinear Partial Differential Equations, and Differential Geometry (Sorry, cookie picture is too large to e-mail!) In this second of two talks on the shape of a cookie, I will first give an overview of the first talk developing a heuristic model for determining the shape of a cookie. Then, we will prove that this model can be solved mathematically and outline the method used to generate the numerical solutions presented. Finally, I will present some interesting questions that will be examined during the REU this summer. Homemade Cookies will be provided during the talk.
- Topic: Modeling the Shape of a Cookie
- Speaker: David Finn
- Date: 01 Feb 2006
- Abstract: Have you ever wondered why cookies are generically round? Well, I have. And, the reason involves some interesting mathematics: Calculus of Variations, Nonlinear Partial Differential Equations, and Differential Geometry. In this first of two talks on the shape of a cookie, I will overview a heuristic model for determining the shape of a cookie, and show under some physically reasonable assumptions that a cookie is generally round. Some interesting questions suggest themselves, when the generically round is stated in a mathematically precise language, and the physically reasonable assumptions are allowed not to hold. An investigation of some of aspects of this mathematical model for the shape of a cookie will examined during the REU this summer. Homemade Cookies will be provided during the talk.
- Topic: A Combinatoric Proof of the Chan-Robbins-Yuen Theorem
- Speaker: Daniel Litt, High School Student from Ohio
- Date: 25 Jan 2006
- Abstract:
For more information see Announcement/Abstract/Paper in PDF form
- Topic: Solving the Rubik's cube: An Introduction to Group/Graph Theory
- Speaker: William Butske
- Date(s): 02 Nov 2005, 09 Nov 2005
- Abstract: The Rubik's cube is one of the most concrete examples of a finite non-abelian group that one is likely to come across. If these terms don't mean anything to you, don't worry, they will by the end of the talk(s). We will see how group theory and graph theory can be used to solve fundamental problems about the Rubik's cube. For example, how many different positions are possible is the same as asking what is the order of the Rubik's group. How many moves are necessary to solve the worst possible scrambling (God's Algorithm) is a question about the diameter of the associated Cayley graph.
- Topic: Do Dogs Really Know Calculus?
- Speaker: Eric Reyes, Rose Student, Math/Econ Major
- Date: 26 Oct 2005
- Abstract: Least squares is a regression technique frequently used by engineers and scientists to gain insight into data generating processes. In 2003, Timothy Pennings of Hope College asked the question: “Do Dogs Know Calculus?” In an effort to see if his dog Elvis minimized the retrieval time when playing fetch, Professor Pennings collected data during a game of fetch on the beach. We take a second look at his data and statistical analyses. We show how a simple-looking problem can require intricate analysis. We use advanced methods, including weighted least squares, to detect and compensate for violations in the standard least squares assumptions. And, we seek to answer the question: Do Dogs Really Know Calculus?
- Topic: Nonparametric estimation of volatility models with serially dependent Innovations
- Speaker: Michael Levine, Purdue University
- Date: 20 Oct 2005
- Abstract: We are interested in modeling the time series process yt = ¾(xt)"(xt)) where "t = Á0"t¡1 + vt. This model is of interest as it provides a plausible linkage between risk and expected return of financial assets. Further, the model can serve as a vehicle for testing the martingale difference sequence hypothesis, which is typically uncritically adapted in financial time series models. When xt has a fixed design, we provide a novel nonparametric estimator of the variance function based on the difference approach and establish its limiting properties. When xt is strictly stationary on a strongly mixing base (hereby allowing for ARCH effects) the nonparametric variance function estimator by Fan and Yao (1998) can be applied and seems very promising. We propose a semiparametric estimator of Á0 that is pT-consistent, adaptive, and asymptotic normally distributed under very general conditions on xt.
- Topic: Finite Groups of Matrices with Integer Entries
- Speaker: James Kuzmanovich (joint work with Andrey Pavlichenk), Wake Forest University
- Date: 28 Sep 2005
- Abstract: Finite groups of nonsingular matrices with integer entries are some of the first groups seen in an undergraduate algebra course, since they only require knowledge of matrix multiplication and inversion. They nevertheless have many interesting properties and associated problems (some unsolved), and they have been the object of study by many famous mathematicians. Not much of this theory (or history) appears in undergraduate texts (and was not known by at least one algebraist – me), even though it is a good source of problems and projects. Indeed, this talk is a report on what Andrey and I learned as he wrote a term paper for my undergraduate algebra course and we followed it up with independent study. Most of this talk should be accessible to students who have had a linear algebra course. It will introduce ideas and concepts from many areas of mathematics, but prior knowledge will not be assumed.
2004-05 (latest first)
- Topic: An Introduction to Constructible Numbers
- Speaker: Kurtis Katinas, Rose Student, Mathematics
- Date: 18 May 2005
- Abstract: Around 2500 years ago, the ancient Greeks proposed a set of three geometry problems about constructing certain lengths with an unmarked straightedge and compass. These are trisecting an arbitrary angle, doubling the cube, and “squaring” the circle. It turns out that all three of these feats are impossible. Trisecting the angle was the first to be disproved, but the ancient Greeks were not the ones who did it. It wasn't until the 1800's, when all three were disproved. What is most surprising about these solutions was that they did not use any heavy geometry. Instead, they relied on field theory and number theory. This talk is aimed primarily at undergraduate students as a walkthrough of two of these proofs and some details on the proof of the third. No knowledge of field theory is required or assumed. Basic number theory and little geometry will be used, but not required either.
- Topic: New Goodness-of-Fit Tests
- Speaker: Dr. Mark Inlow, inlow@rose-hulman.edu
- Date: 11 May 2005
- Abstract: Goodness-of-fit tests are formal procedures for assessing the fit between a given model and the distribution of some quantity of interest. Using the moment-matching property of the exponential family of distributions, we derive new generalizations of the smooth goodness-of-fit test. (The exponential family of distributions encompasses many distributions including the normal, t, chi, exponential, gamma, beta, and Poisson families.) We compare the performance of our new tests with standard goodness-of-fit tests for the normal distribution.
- Topic: Computational Modeling with Partial Differential Equations
- Speaker: Chad Westfall, Wabash College
- Date: 13 Apr 2005
- Abstract: Partial differential equations (PDEs) are used in many areas of science to model the behavior of quantities that depend on several independent variables. In this talk we will look at the process of modeling physical phenomena with partial differential equations. Working through a simple example we will highlight the issues and challenges in the discretization and solver stages of the process.
- Topic: Batch Calculation of the Residues and Their Sensitivities, Or: How to compute almost any derivative using sum(prod([combnk(factors), dfactors.]))
- Speaker: Brad Burchett
- Date: 30 Mar 2005
- Abstract: In determining the time-domain response of linear time invariant systems, the inverse LaPlace transform technique using partial fraction expansions has both practical and historical significance. The values which constitute the numerators of the partial fraction expansion are commonly known as the residues. A recent application of interest is brute force computation of the quadratic cost function for optimal output feedback which can be facilitated by the Sylvester expansion. Sylvester's expansion requires computation of the system residues. Computation of the residues is typically accomplished by deconvolving the system transfer function and evaluating ratios of polynomials at a system pole. In this work, the first order form of the partial fraction expansion is investigated. A general matrix equation is derived for computation of the residues. This equation is generalized for cases involving repeated as well as distinct system poles. The sensitivities of the residues to changes in system parameters can then be computed by differentiating this matrix equation. Typical numerical results are presented.
- Topic: The $20,000,000,000 Eigenvector - Part II
- Speaker: Kurt Bryan, bryan@rose-hulman.edu
- Date: 26 Jan 2005
- Abstract: In the last talk I showed a key idea that lies behind how Google ranks the importance of each page in a web of interconnected pages. The problem boils down to computing an eigenvector of a certain n by n matrix, where n is the number of pages in the web. But Google currently indexes over 8 billion pages---how does one do linear algebra on matrices of that size? Gaussian elimination? If you believe that, I have got a bridge for sale. In part II we will look at how one can reasonably compute an eigenvector for these very large matrices, and I will address a few questions that were raised in the first talk.
- Topic: The $20,000,000,000 Eigenvector - Part I
- Speaker: Kurt Bryan, bryan@rose-hulman.edu
- Date: 19 Jan 2005
- Abstract: When Google went online in the last decade, one thing that set it apart from other search engines was that its search result listings always seemed to deliver the good stuff up front. With other search engines you often had to wade through screen after screen of links to unimportant web pages that just happened to match the search text. Part of the magic behind Google is its ability to quantitatively rate the importance of each page on the web and so rank the pages, then present to the user the more important pages first. In these two talks I will explain one popular approach to rating web page importance. It turns out to be a delightful application of standard linear algebra.
- Topic: Why Number Theorists Care About Elliptic Curves - Part II
- Speaker: Ken McMurdy, mcmurdy@rose-hulman.edu
- Date: 08 Dec 2004
- Abstract: Let E be an elliptic curve whose Weierstrass equation has rational coefficients. In the first installment of this talk, we defined an abelian group structure on E. We then showed how to compute the p-torsion subgroup, denoted E[p], which must always be isomorphic to two copies of the integers mod p. In Part II, we will show how a certain Galois group acts on E[p], resulting in a Galois representation into the group of invertible two-by-two matrices over the field Fp. This will all be done in great detail for the specific curve whose 3-torsion was worked out explicitly in Part I. Time permitting, I will then discuss an analogous construction of l-adic Galois representations, and connections with modular forms such as the Shimura-Taniyama-Weil Conjecture.
- Topic: Algebraic Cycles on Abelian Varieties
- Speaker: Reza Akhtar, Miami University of Ohio
- Date: 27 Oct 2004
- Abstract: The theory of algebraic cycles was initially developed with an view towards studying intersections on algebraic varieties. Since then, it has found many applications to K-theory, number theory, and most recently to the theory of motives. This talk will provide an introduction to algebraic cycles and abelian varieties, and will describe the interaction between the product structure on cycles and the group law on an abelian variety. Some recent results of the speaker in this area will also be discussed.
- Topic: Equivalence of Real Elliptic Curves - Part II - Birational Equivalence
- Speaker: Allen Broughton, brought@rose-hulman.edu
- Date: 13 Oct 2004
- Abstract: This second talk on real elliptic curves will complete the picture of birational equivalence of real elliptic curves by looking at the complex elliptic curve defined by the original curve. The complex curve is called a complexification of the real curve and the real curve is called a real form of the complex curve. The complex curve is a torus and it interesting to visualize the real forms as curves on the torus. We will spend most of the talk exploring the very interesting relationship among the real forms, mirror reflections on the torus, and the automorphisms of the complex curve.
Non-isomorphic real curves can have can have isomorphic complexifications. The main result we will show is that each complex elliptic curve defined by real equations has exactly two real forms which are birationally inequivalent. The most interesting part is that there is exactly one complex elliptic curve that has a real form with one component and another real form with two components.
We will not use any calculations more complex than high school algebra and nor any geometric concepts beyond what we cover in our multi-variable calculus course. The calculations are made quite easy by using the Weierstrass form discussed in the first talk. The first part of the talk will be a recap of the first talk in the context of complex elliptic curves. There will be lots of pictures.
For more information see Announcement/Abstract/Paper in PDF form
- Topic: Equivalence of Real Elliptic Curves - Part I - Linear Equivalence
- Speaker: Allen Broughton, brought@rose-hulman.edu
- Date: 06 Oct 2004
- Abstract: This is the first of several talks on elliptic curves given by Allen Broughton and Ken McMurdy. In the two talks by Allen Broughton a complete answer will be given to a question posed by Ken McMurdy during his job talk last spring.
What is the moduli space of real elliptic curves like?
Since then a complete answer has been worked out and it is surprisingly simple.
In the first talk a basic introduction to real elliptic curves will be given -- starting from definitions, smoothness, projective completion, the geometry of the group law, the geometry of tangents and inflection points and ending up with the notions of embedded linear equivalence, normal Weierstrass form, and linear classification. The main result is that there are two families of curves each depending on a single real parameter. Each curve in one family has one component and each curve in the other family has two components*. The talk does not use calculations more complex than high school algebra and the geometric concepts that we cover in our multi-variable calculus course (except a smidgen of topology at one point). There will be lots of pictures.
*Well that statement is almost true. The explanation of almost true will be given in the second talk, which will cover the complexifications of real elliptic curves, real forms of complex elliptic curves, the moduli space complex elliptic curves, and the automorphism groups of curves.
For more information see Announcement/Abstract/Paper in PDF form
- Topic: Fast Reconstruction of Cracks using Impedance Imaging
- Speaker: Dr. Kurt Bryan, bryan @rose-hulman.edu
- Date: 22 Sep 2004
- Abstract: This talk is based on the work done in our mathematics REU in the summers of 2002-2004, concerning some mathematical problems that arise in the non-destructive testing of materials. I will present an absurdly simple and fast algorithm to reconstruct linear cracks inside an object, by using electrical currents applied to the outer boundary of the object and then measuring the induced voltages on the outer boundary (or if you prefer to think in terms of heat, one applies a known heat source to the outer boundary and measures the resulting steady-state boundary temperatures). An insightful result by the 2003 group (extended by the 2004 students, using results from the 2002 group) turns this apparently hard problem into a DE I exercise!
2003-04 (latest first)
- Topic: The Celestial Sphere: Geometry and Astrolabes
- Speaker: Tanya Leise leise@rose-hulman.edu
- Date: April 14, 2004
- Abstract: In the first and second centuries BC, Greek thinkers took the Babylonian beginnings of astronomy, which included the zodiac, and incorporated their brilliant geometrical ideas to create a mathematical model of the heavens that was both useful and accurate. Ptolemy's Almagest (ca. 100-150 AD) marks the peak of the development of the Greek mathematical astronomy. This early astronomy viewed the heavens as a great rotating celestial sphere with a stationary Earth at its center. The stars were fixed to the celestial sphere, while the sun moved along the zodiac, making one full circle each year. We will survey some of the geometry used in developing coordinate systems on the celestial sphere and in projecting the sphere onto a plane to result in a working two-dimensional model of the heavens—the astrolabe. In order to visualize this sphere-to-plane stereographic projection, we will work some basic computations with astrolabes that I will provide to the audience, and compare the 2D astrolabe to a 3D celestial globe.
- Topic: The Joy of Zero Divisors (and possibly the horror if time permits )
- Speaker: Mike Axtell, Wabash College, axtellm@wabash.edu
- Date: March 31, 2004
- Abstract: The talk will focus on a beautiful and surprising result linking Abstract Algebra to Graph Theory. You need not know anything about Graph Theory (the speaker doesn't either). You need not know anything about Abstract Algebra - relevant ideas are basic and will be introduced. Warning: The speaker may use this opportunity to trash talk Rose prior to the ICMC (Indiana Collegiate Mathematics Competition) on Friday.
- Topic: Numerical ODE Solving for a Chaotic System
- Speaker: Brad Burchett, Rose-Hulman - Mech Eng, Bradley.T.Burchett@rose-hulman.edu
- Date: March 17, 2004
- Abstract: A simple non-linear dynamical system with chaotic properties is used to illustrate the advantages and limitations of Runge-Kutta (RK) based ODE solving. Herein we describe the course "Computer Applications in Engineering 2" (ME 323): how it fits in the ME curriculum, and course objectives. We quickly review the techniques of fixed and adaptive step fourth order RK (RK4). The definition of stability for non-linear autonomous systems is reviewed. We then present the physical system and its ODE representation. Results are shown for adaptive and fixed-step RK4 where the system stability boundary estimate visibly changes due to numerical inaccuracies.
- Topic: Probability Models in Genetics
- Speaker: Amanda Lynn Stephens, Rose student, stephanal@rose-rulman.edu
- Date: February 18, 2004
- Abstract: A discussion of probability models in genetics. Genetics models such as the Wright-Fisher and the Moran Model will be analyzed with Probability Modeling. The talk is based on an undergraduate research project by the speaker.
- Topic: Theme and Variations from Geometric Function Theory (3 talks)
- Speaker: Jerry Muir, muir@rose-rulman.edu
- Dates: January 28, 2004, February 4, 2004, February 11, 2004
- Abstracts of the talks:
I. Convex Mappings of the Unit Disk: The theory of univalent (one-to-one and analytic) functions of the unit disk in the complex plane has been an area of active research for almost a century. Bieberbach's conjecture, proposed in 1916 and proved by de Branges in 1985, that a univalent function defined on the unit disk of the form
must satisfy  for all n motivated a great deal of this research. In particular, many elegant results were proved for families of univalent functions that are defined by some geometric condition on the image of the function. Usually, there is no nice extension of results from one complex variable into higher dimensions, and this topic is no exception. Because of this, the geometric classes of functions are of special importance in that setting. In this, the first of three talks, we will consider univalent mappings of the unit disk whose image is a convex set in the plane. A sequence of appealing results will be given that draw upon some of the classical principles from Complex Analysis.
II: Some Examples and Obstacles in Higher Dimensional Geometric Function Theory: Having been introduced to some of the basic and elegant results of one variable geometric function theory, we turn our attention to the higher dimensional setting. Although natural to consider, this setting yields problems of much greater difficulty. Many of the simplest one variable results either have no reasonable extension or the extensions require difficult unintuitive arguments. We will introduce the basic ideas of function theory in higher dimensions, including all of the necessary definitions, and examine some situations where difficulties arise. This will include some counterexamples to natural generalizations of the one variable theory. We will conclude by considering different norms on the space C2 of two-dimensional complex vectors. The impact that changing norms has on the function theory is substantial. Recently developed constructions of convex mappings of the unit ball of C2 with certain non-Euclidean norms will be given.
III: Analysis of Convex Mappings of the Ball in Cn Onto Sets Containing a Line: In the last talk, we saw some instances in which elementary properties of convex mappings of the unit disk do not easily extend to a higher dimensional setting. Few examples of higher dimensional mappings are known, and those that are known fail to extend the familiar properties that some one-dimensional mappings posses. In this talk, we will focus on mappings F of the Euclidean ball B in Cn such that F(B) is a convex subset of Cn containing a line. These provide an interesting generalization of mappings of the unit disk onto strips and half-planes and may eventually be useful in the determination of the extreme points of the family of convex mappings.
- Topic: The Banach Fixed Point Theorem and Solvability of Integral Equations
- Speaker: Dan Abretske, Rose student, Daniel.A.Abretske@rose-rulman.edu
- Date: January 21, 2004
- Abstract: As part of my independent study last quarter I studied various solvability conditions that can be placed on both linear and non linear operators. As an extension of that course I will be discussing the Banach Fixed Point Theorem and the Geometric Series Theorem. I will then show how they can be applied to integral equations of the second kind.
- Topic: Black Box Linear Algebra
- Speaker: William Turner, Wabash College, turnerw@wabash.edu
- Date: November 12, 2003
- Abstract: In symbolic computation and its subfield of computer algebra, we desire algebraic methods to compute an exact solution to a problem, as opposed to the numerical approximations supplied in numerical analysis. In this talk, we introduce the black box model for symbolic linear algebra. We investigate Wiedemann's approach to solve a system of linear equations and compute the determinant and rank of a black box matrix.
- Topic: Inverse Electrocardiography
- Speaker: Lorraine Olson, Mech Eng, Lorraine.Olson@rose-rulman.edu
(Joint work with Robert Throne, Rose-Hulman Institute of Technology and John R. Windle, University of Nebraska Medical Center )
- Date: November 5, 2003
- Abstract: The heart is an electromechanical device. In its resting state, the heart is electrically polarized. For each heartbeat, a wave of “depolarization” travels through the heart muscle, causing the tissues to contract. If the electrical pathways in the heart malfunction, this leads to arrhythmias and poor blood flow. Hence, knowledge of the electrical patterns on the heart is extremely useful in diagnosing and correcting certain types of heart-conduction related defects. In recent years there have been a growing number of attempts at reconstructing surface potentials on the heart from minimally invasive remotely measured signals. Two basic approaches have been taken. In the oldest approach, body surface potentials are measured and used to estimate the potential patterns on the endocardium (outside surface of the heart). More recently, a probe which can be inflated within a heart chamber has been developed and is used to estimate potential patterns on the interior surface of the heart. Both of these estimation problems are “inverse problems”, and they are very sensitive to small errors in the measurements. We therefore need to use some form of “regularization”, or smoothing, to ensure that the answers we obtain are reasonable. The key question is how much smoothing to use, so that we obtain accurate answers. This talk will focus on the mathematical details behind the inverse electrocardiography problem for the inflated probe case: the governing equations, finite element methodology, regularization techniques, and methods for selecting the regularization parameters. We will also show preliminary results for the probe data.
- Topic: Small Cycles of the Discrete Logarithm (2 talks)
- Speaker: Joshua B. Holden, holden@rose-hulman.edu
- Dates: October 22, 2003 and October 29, 2003
- Abstract of the talks: Brizolis asked the question: does every prime p have a pair (g,h) such that h is a fixed point for the discrete logarithm with base g? In other words, is g^h congruent to h modulo p? We will extend this question to ask about not only fixed points but also two-cycles, and examine methods for estimating the number of such pairs given certain conditions on g and h. This problem has applications to cryptography, since one well-known cryptographically secure random number generator uses the idea of iterating the discrete logarithm and we hope that it does not fall into cycles too often!
- Topic: Fast Reconstruction of Internal Cracks with Thermal Imaging
- Speaker: Nic Trainor, Rose student, Nic.A.Trainor@rose-rulman.edu
- Dates: October 1, 2003 and October 8, 2003
- Abstract: The ability to characterize the interior of an object without damaging the object is an invaluable tool in industry. One useful technique of recent interest is “impedance imaging”, or equivalently, “steady-state thermal imaging”. The idea, in thermal terms, is to use temperature measurements on the boundary of an object---specifically, imposed thermal energy fluxes and measured boundary temperatures---to determine interior structure, for example, to find internal cracks or voids. In these two talks we'll discuss some new mathematical results on thermal imaging for cracks, obtained in Rose-Hulman's summer REU mathematics program. In the first talk we'll examine a new and very rapid approach to finding a single crack in the interior of an object, under the assumption that the crack blocks the flow of heat. In the second talk we'll discuss how to extend the procedure to the problem of finding several interior cracks, and look at the issue of what types of input fluxes provide optimal resolution and stability.
- Topic: Imaging the Inner Wall Profile of a Blast Furnace
- Speaker: Kurt Bryan, bryan@rose-rulman.edu
- Date: September 249,2003
- Abstract: A blast furnace is essentially a large vessel filled with molten material. It turns out that the inner wall of the furnace, which is in contact with the molten interior, can change shape over time, becoming either thinner due to the corrosive nature of the furnace interior, or the wall can become thicker due to the build up of deposits. Walls that become too thin are obviously dangerous, and it's also undesirable to have the walls become too thick.
It's obviously difficult to directly measure the profile of the inner wall when the furnace is in operation, so one would like a means of determining the profile indirectly, from the outside. One approach is to use thermal methods, by measuring the temperature and heat flux at positions on the outer wall and from this information infer the inner wall profile.
In this talk we'll consider a simple one-dimensional model of the situation, in which the furnace wall (or a cross section of it) is modeled as a thermally conductive bar, whose length changes slowly over time. We'll look at how one can use temperature and heat flow measurements at one end of the bar to determine the length of the bar at any time. This is work done during our summer REU program.
2002-03
- Topic: Approximate solutions to the Boussinesq equation
- Speaker: Aleksey Telyakovskiy
- Date: October 2, 2002
- Abstract : The Boussinesq equation is a nonlinear diffusion equation that models the behavior of groundwater in unconfined aquifers. Solutions of the Boussinesq equation are considered in many areas of hydrology. In case of zero initial conditions, solutions of the Boussinesq equation exhibit wetting fronts that propagate with finite speed. For certain types of initial-boundary value problems the Boussinesq equation can be reduced to boundary-value problems for an ordinary differential equation for a scaling function. In this talk we construct approximate closed-form solutions to the one-dimensional Boussinesq equation.
- Topic: An Inverse Problem Arising In Non-destructive Testing for Cracks
- Speaker: Kurt Bryan, bryan@rose-hulman.edu
- Date: October 9,2002
- Abstract : Consider some material object which may or may not have an internal "crack ". You want to find out if there is indeed such a crack, and if so, determine the location of the crack. The catch is that you must do it non-destructively---there's no point to cutting the thing in half only to find out it was good. Recently, two methods for imaging the interior of an object to find defects have been much investigated. The techniques use either heat or electrical energy to "see " inside objects, non-destructively. In this seminar I'll talk about mathematical research done with undergraduates in our REU program last summer, in which we extended some known theoretical and computational techniques for finding cracks in objects using thermal and electrical methods.
- Topic: The Distribution of the Kolmogorov-Smirnov Statistic for Exponential Populations with Estimated Parameters
- Speaker: Diane Evans, evans@rose-hulman.edu
- Date: October 23,2002
- Abstract: I will present the derivation of the distribution of the Kolmogorov-Smirnov, Cramer-von Mises, and Anderson-Darling test statistics in the case of exponential sampling when the parameters are unknown and estimated from sample data for n = 1 and n = 2 via maximum likelihood.
- Topic: Factoring Integers via Lenstra's Elliptic Curve Method
- Speaker: Noor Martin, noor.martin@rose-hulman.edu
- Date: October 30,2002
- Abstract: This talk examines a method for factoring integers based on the use of Elliptic Curves modulo some composite number n. Published by H. W. Lenstra in 1987, this method is a modification of Pollard's p-1 method for factoring integers. Background information on both Elliptic Curves and Pollard's p-1 method will be covered as well.
- Topic: Elliptic Curve Cryptography
- Speaker: Matthew Ford, matthew.ford@rose-hulman.edu
- Date: November 6,2002
- Abstract: Elliptic Curve Cryptography (ECC) provides an alternative method of public key cryptography. While RSA is based on the factorization of a composite number, ECC is based on the Elliptic Curve Discrete Log Problem. The difference in these problems makes ECC not vulnerable to some of the attacks against RSA. The current best known attack against ECC is an exponential time algorithm.
- Topic: The Combinatorics of Symmetric Functions
- Speaker: Thomas Langley, thomas.langely@rose-hulman.edu
- Date: November 13,2002
- Abstract: There is a remarkable connection between representations of the symmetric group and symmetric multivariable polynomials polynomials that are unchanged when the variables are permuted). This correspondence, in which characters of irreducible representations are mapped to Schur functions, allows the combinatorics of symmetric functions to be used to solve representation theoretic problems. This talk will provide an introduction to this complex and beautiful combinatorial world, introducing symmetric functions, tableaux, Schur functions, plethysm, and the Robinson-Schensted correspondence.
- Topic:Vanishing Cycles and Kaleidoscopic Quadrilateral Tilings
- Speaker: Allen Broughton, brought@rose-hulman.edu
- Date: December 11,2002
- Abstract: For the last 5 years the focus of the Rose-Hulman REU Tilings group has been hyperbolic, kaleidoscopic tilings of Riemann surfaces by triangles. A lot has been discovered about these objects including a complete classification up to genus 13. Last summer we pushed beyond triangles to consider quadrilateral tilings. On the plus side the group theory did get a bit simpler; on the minus side we lost rigidity. A surface constructed from triangles is rigid in the sense that there are no transformations that preserve both angles and area. This is not true in the quadrilateral case. The euclidean analog is that all triangles with congruent corresponding angles and the same area are congruent. However, there is a one-parameter family of mutually non-congruent rectangles with the same area. On hyperbolic surfaces the same holds true, but there is an interesting twist. As we vary the quadrilaterals through an infinite family of equiangular, equal area quadrilaterals some curves on the surface take on arbitrarily small lengths, and shrink to a point as we go to infinity. These are the so-called "vanishing cycles" studied in algebraic geometry. We will show how to identify the vanishing cycles in simple geometric terms. Much of the talk will be explaining the basic concepts in terms of small visual examples. Students Isabel Averill, Michael Burr, John Gregoire and Kathryn Zuhr all contributed to this project.
- Topic:Equations, Scramblings, and Random Walks in Finite Groups
- Speaker: Gary Sherman, gary.sherman@rose-hulman.edu
- Date: December 18, 2002
- Abstract: We prove (casually) that the probability of solving an equation in a (finite) group is just about the reciprocal of the cardinality of the groups derived subgroup. Our approach is to;
- view your favorite group equation, xy = yx, in terms of a permutation action,
- introduce a new class of permutations, so-called scramblings, which are combinatorially related to derangements
- spawn a natural random walk on the derived subgroup. Natural research questions suitable for Natural research questions suitable for undergraduates ensue
- Topic:de Casteljau's Algorithm in Hyperbolic Space
- Speaker: Alla Genkina, Rose CS major
- Date: January 22, 2003
- Abstract: Geometric Modeling can be defined as the application of mathematics to describe the shape and properties of physical or virtual objects. This application of mathematics extends to various industrial and graphical fields. Since most of the fields are computerized, the algorithms developed to describe objects in mathematical terms can be programmed and analyzed by computers. This presentation will describe de Casteljau's algorithm which is used to generate Bezier curves. The curves that are created can then be utilized to model various objects. The presentation will demonstrate the use of the algorithm in both Euclidean and Hyperbolic Space, but the main concentration will be on its application and use in Hyperbolic Space.
- Topic:Prescribing the curvature of level curves
- Speaker: Dave Finn, finn@rose-hulman.edu
- Date: January 29, 2003
- Abstract: Given a function u(x,y), it is a straight forward calculation in vector calculus to determine the curvature K[u] of the level curves of u. This curvature can be computed using the Hessian of u. In this talk, we consider the problem of prescribing the curvature of level curves of a function ,
Given a function k(x,y), is it possible
to find a function u(x,y) with K[u]=k
As a problem in nonlinear partial differential equations, this problem poses some interesting questions, starting with the nature of the equation, the correct boundary values to consider, the effect of the domain on solvability, the effect of the boundary values on solvability, and finally the existence of a solution.
- Topic: Why (and how!) we should all use group projects in all introductory statistics courses
- Speaker: Douglas Andrews, Wittenberg University
- Date: March 12, 2003
- Abstract: The overwhelming consensus emerging from the statistics education community over the past twenty years is for greater emphasis on exploratory data analysis, design, interpretation, and concepts, at the expense of probability, theory, recipes, and techniques. Moreover, education reform efforts in many fields highlight the benefits of active and collaborative learning pedagogies, as well as more authentic forms of assessment. Group data analysis projects -- in which students analyze data from simple observational studies and experiments of their own design -- can be an ideal way to implement these stat ed recommendations and realize these broader educational benefits in introductory statistics courses for all audiences. In this talk, I'll lay out some of the rationale for using such projects and give plenty of concrete advice for how to structure the experience.
- Topic: Automatic Differentiation of Algorithms (2 talks)
- Speaker: Jeffery Leader, leader@rose-hulman.edu
- Dates: March 26, 2003 and April 2, 2003
- Abstract: Many algorithms used in scientific computation require derivatives. Typically the function is provided--often in the form of a computer program--and the user must find or approximate its derivative. Automatic differentiation is a technique for automatically generating a program that produces those derivatives, by reading in the code of the program defining the function, considering its computational graph, and then finding its exact derivative. In Part I of this talk I will define the problem and outline its solution using this technique; in Part II of this talk I will discuss the two principal modes of automatic differentiation, forward and reverse, and how they use the computational graph to produce code for the required derivative(s).
- Topic: Homogenization: It's Not Just for Dairy Products
- Speaker: Kurt Bryan bryan@rose-hulman.edu
- Date: April 9, 2003
- Abstract: A material is homogeneous if its physical properties don't vary with position, at least at the physical scale of interest. But many (one could argue all) materials are not homogeneous at the microscopic level, but possess a structure with small-scale periodic or random variation. Indeed, composite materials are intentionally designed with such small-scale variations, in order to have certain desirable physical properties. In many cases one would like to predict the bulk or macroscopic physical properties of a composite material from the microscopic structure. Homogenization is a set of mathematical techniques for modeling a material with microscopic inhomogeneous structure as a macroscopically homogeneous material. In this talk I'll show one mathematical framework in which this is done, and illustrate with simple examples .
- Topic:Kaleidoscopic tilings on surfaces, this time with the groups (1st of 2 talk series)
- Speaker: Allen Broughton, brought@rose-hulman.edu
- Date: April 30, 2003 and May 7, 2003
- Abstract: In the past I have given several lectures on kaleidoscopic tilings by triangles and quadrilaterals on surfaces, and asserted in these talks that the tiling group completely determined the combinatorial and topological structure of a tiling. However, I have never really talked about the influence of the group theory! In this series of two talks I will give two examples of determining combinatorial and topological structure, by group computations. Each talk will focus on a problem I intend to give to REU students this summer. Thus, there will be no general theorems just problems statements with suggestions of attack, the talks will focus on developing the background to get to the problem statements. The first talk will include the necessary review of tilings and hyperbolic geometry. You don't need to know much about group theory or hyperbolic geometry.
First talk: Constructing a fundamental domain for kaleidoscopically tiled surfaces. We are all familiar with the process of creating a torus by identifying opposite sides of a euclidean rectangle. For higher genus surfaces of genus s > 1, a surface may be constructed by identifying sides of a hyperbolic 4s-gon. For a kaleidoscopically tiled surface can this be done so that the polygon is a "nice" collection of tiles? The group theory computation will be focus on relating the infinite tiling group on the hyperbolic plane to the finite tiling group on the surface.
Second talk: When are kaleidoscopic tilings separating? Every edge of a kaleidoscopic tiling generates a reflection of the surface to itself fixing the edge. In the case of a sphere the fixed point set (or mirror) of the reflection is a great circle which separates the sphere into two pieces. This is very misleading example, since for higher genus the mirror very rarely separates the surface. The question is: is there a fast way to determine this splitting property from the properties of the tiling group? The talk will present a method of attack using the group algebra of the talk. Again, no previous knowledge of group theory is assumed.
- Topic: Guessing Secrets
- Speaker: Jon Mastin - Rose CS major
- Date: May 21 , 2003
- Abstract: This talk will present a variation on the game “20 questions” which has arisen in the last few years in relation to internet security. In the two player game, one player holds two or more secrets (IP addresses) while the second player asks yes or no questions. If the first player must answer truthfully using one of his secrets, how much can the second player discover? We will discuss the answer to this question and discuss strategy from the point of view of both players.
2001-02
- Topic: Mathematical Phylogeny
- Speaker: Jeff Leader, leader@rose-hulman.edu
- Date: September 19 and 26, 2001
- Abstract (September 19): I will discuss how search engines use the singular value decomposition (SVD) to improve, and to score, the relevance of the results they return. This material will be needed for the second talk.
- Abstract (September 26): I will define the problem of mathematical phylogeny and the reconstruction of phylogenetic trees, then discuss current research being performed by Gary Stuart of ISU and myself that uses the ideas of the first talk to create such trees.
- Topic: Species Phylogenies from Whole Genomes using SVD
- Speaker: Gary Stuart (ISU), Gary.Stuart@isugw.indstate.edu
- Date: October 3, 2001
- Abstract : Following Jeff Leader's fine series of seminars intoducing SVD (Singular Value Decomposition) as a tool for generating biomolecular phylogenies, I will describe some of our very recent attempts to solve some very large problems using the same method. In particular, I will describe the generation of a tree summarizing the evolutionary relationships of 19 bacterial species. Unlike most trees, which result from the analysis of only one or a few genes or proteins, this tree is based on an exhaustive comparison of over 35,000 proteins predicted from whole genome sequence. Along the way, I plan to explore the "meaning" of the SVD relative to our application, and to present some of the (questionable?) assumptions upon which our method is based.
- Topic: The Mathematics of Financial Derivatives and Option Pricing
- Speaker: Kurt Bryan, bryan@rose-hulman.edu
- Date: October 17, 24, 31, 2001
- Abstract : Most people are familiar with traditional investments like stocks, bonds, and commodities. However, in the past few decades a huge market has arisen in the trading of "options" and other "financial derivatives", contracts in which payment is based on the value of some benchmark, e.g., the price of a given stock on a certain date. In short, the value of the contract is derived from the price of some underlying asset (hence the term "derivative').
As an example, suppose a contract is written in which I give you the option (but not the obligation) to buy from me one share of Microsoft stock for a guaranteed price of $50 on January 1, 2002 (today, October 15, it's selling for $57). This is an example of a European Call Option, in which you have the right to buy some asset at a guaranteed price sometime in the future. How much should you pay to enter into such an agreement? Surprisingly, there is a very quantitative strategy for determining the price of this option contract.
In these talks (3 or 4) we'll examine the problem of option pricing. We'll start by looking at some common options, then at basic probabilistic models for asset prices. Finally, we'll derive the celebrated Black-Scholes partial differential equation which shows how one can rationally determine option prices. This is work for which Robert Merton and Myron Scholes won the 1997 Nobel Prize in economics.
- Topic: Ideal Error-Correcting Codes: Reed-Solomon Decoding without Fourier Analysis
- Speaker: Matt Lepinski (Rose alumnus, MIT graduate student), lepinski@theory.lcs.mit.edu
- Date: December 19, 2001
- Abstract : An error-correcting code is a set of strings (called codewords) such that any two strings in the set differ in a large number of positions. Error-correcting codes are very useful in data transmission (where a noisy channel may corrupt some positions in the string). This is because many positions must be corrupted by the channel in order for the receiver to mistake one codeword for another.
This talk deals with the decoding problem for error-correcting codes. That is, given a string, how do we find the codeword that differs from the given string in the fewest number of positions. The codes considered in this talk will be the commonly used Reed-Solomon codes and the number theoretic Redundant Residue Number System codes. The talk will present a new algebraic framework for thinking about error-correcting codes and show how this framework allows us to use the same ideas to decode both Reed-Solomon codes and RRNS codes. These ideas are of particular interest because they can also be applied to decoding Algebraic Geometry codes which are asymptotically the best known codes. (Although a discussion of Algebraic Geometry codes is beyond the scope of this talk).
This talk assumes no prior knowledge of error-correcting codes. However, familiarity with polynomial algebra and finite fields is helpful in understanding some of the ideas in this talk. Most of the material in this talk comes from the work of Madhu Sudan, Venkatesan Guruswami and Amit Sahai.
Probably the most common Error-correcting codes in practice are the Reed-Solomon codes which are based on polynomials over finite fields. These codes are used by everyone from NASA to CD manufacturers.
- Topic: Optimizing College Enrollments Under Uncertainty
- Speaker: Concetta DePaolo, Indiana State University, sdcetta@befac.indstate.edu
- Date: January 16,2002
- Abstract : Each year an institution must decide which students to admit in order to accomplish its goals (e.g. quality, enrollment, etc.) while satisfying various capacity constraints. This presentation details a mathematical optimization model for this problem, which assumes that students are of different types that exhibit different (random) behavior. The presentation will describe the properties of the optimal solution, as well as an implementation and a heuristic algorithm that are both Excel-based. How the model is being used by Indiana State University to compare alternative admissions strategies and forecast the long-term effects of those strategies will also be touched upon.
- Topic: Automorphisms of Riemann Surfaces, Galois Groups, and Hecke Algebras
- Speaker: Allen Broughton, Rose-Hulman, brought@rose-hulman.edu
- Date: March 20 and 27, 2002
- Abstract : There is a classical and very well-understood connection between automorphism groups of compact Riemann surfaces and Galois groups of branched coverings of surfaces. In the first of this series of two talks we will introduce and explore this idea. In the second talk we will consider non-Galois coverings, and see how this situation can be partially captured by Hecke Algebras. These talks will highlight past and continuing work by students in the "tilings group" of the Rose-Hulman REU.
- Topic: The Theodorus Equations
- Speaker: Jeff Leader, Rose-Hulman, jeff.leader@rose-hulman.edu
- Date: May 1, 2002
- Abstract : The square-root spiral, or spiral of Theodorus, will be introduced, then generalized to a map on R^n with many strange attractors.
- Topic: Calculation of Bernoulli Numbers and Values of Zeta Functions
- Speaker: Josh Holden, Rose-Hulman, josh.holden@rose-hulman.edu
- Date: May 15, 2002
- Abstract : This talk will discuss some of the methods known to calculate Bernoulli numbers,
with an emphasis on asymptotic analysis of their running times. Definitions (and some motivation) will be provided. We will also discuss some more modern extensions of the Bernoulli number concept, and explore how and whether the methods for calculating Bernoulli numbers extend.
2000-01
- Topic: Statistics, Earwax, and the Bering Strait
- Speaker: Dr. Doug Wolf, Department of Statistics, Ohio State University
- Topic: Teaching statistics the EESEE way
- Speaker:Dr. Elizabeth Stasny, Department of Statistics, Ohio State University
- Date: September 12, 2000
- Abstract: This was a visit to recruit students into graduate statistics programs
- Topic: Singular Solutions to a Partial Differential Equation Arising in Corrosion Modeling
- Speaker: Kurt Bryan, bryan@rose-hulman.edu
- Date: September 20 and 27, October. 4, 2000
- Abstract for Sept. 20 and 27: I'll talk about some joint work with Michael Vogelius at Rutgers University, specifically a partial differential equation (PDE) that arises in the modeling of electrochemical systems. Although the PDE is linear, the boundary conditions contain an exponential type of nonlinearity. Under certain conditions the problem has a unique solution, but in other cases the boundary value problem has an infinite family of solutions with logarithmic singularities on the boundary of the domain. I'll show some numerical simulations, what we've been able to deduce about the nature of the solutions, and talk about what remains to be proved.
Abstract for Oct. 4: I'll discuss joint work with Lester Caudill at the University of Richmond, specifically a partial differential equation that arises in the modeling of heat flow through an object with an interior "crack" or flaw. The flaw is modeled as a discontinuity or jump in the temperature over the flaw, with a nonlinear relationship between the heat flux over the flaw and the temperature jump. I'll look at conditions under which the PDE has a unique solution, and discuss the inverse problem that motivates this: how to determine the location and nature of the interior flaw from boundary measurements.
- Topic: Cwatsets
- Speaker: Gary Sherman Gary.Sherman@rose-hulman.edu, and Dennis Lin, Rose student
- Date: October 18, 25, November 1, November 8, 2000
- Abstract: A cwatset is a subset of binary n-space that is closed (c) with (w) a (a) twist (t) For example, C = {000,110,101} is a cwatset because;
C + 000 = C,
C + 110 = {110,000,011} is C with the first two components of each element transposed,
C + 101 = {101,011,000} is C with the first and last components of each element transposed
That is, for each element c of C there exists a permutation, pi, of three symbols such that the coset C + c is just C with pi applied to the components of each element of C.
The theory of cwatsets has roots in statistics (a cwatset determines a confidence interval for the mean or median of a symmetric random variable) and blossoms in graph theory (each isomorphism class of simple graphs has a unique cwatset associated with it) and algebra (constructions, morphisms, representations). In this sequence of four talks we trace the development of the theory from the first cwatset sighting at Rose-Hulman in 1987 to the latest results on isomorphism classes of cwatsets while highlighting the contributions undergraduates have made to the theory
Talk 1: The statistical motivation for cwatsets, examples of cwatsets, and constructions of cwatsets.
Talk 2: The group theoretic ideas which bare the soul of the theory of cwatsets.
Talk 3: The connection between representation and isomorphism of cwatsets.
Talk 4: The determination of all cwatsets of order at most 23.
Talks 1,2 and 3 will be given by Gary Sherman and talk 4 will be given by Dennis Lin (a Rose student).
- Topic: Pi in the Mandelbrot set
- Speaker: Aaron Klebanoff
- Date: Dec. 6, 2000
- Abstract: The Mandelbrot set is arguably one of the most beautiful sets in mathematics. In 1991, Dave Boll discovered a surprising occurrence of the number pi while exploring a seemingly unrelated property of the Mandelbrot set. Boll’s finding is easy to describe and understand, and yet it is not widely known -- possibly because the result has never before been shown rigorously. In this presentation, I will provide the necessary background material to understand what the Mandelbrot set is and what Boll’s discovery was. I will then outline a proof of the result.
- Topic: Why Chaos Toys are Chaotic.
- Speaker: Aaron Klebanoff
- Date: Dec. 13 and 20, 2000
- Abstract for Dec. 13: The Horseshoe Map. The horseshoe map is a simple map of the unit square into itself that is the prototypical example for a chaotic map. I will define the map, explore its dynamics, and subsequently define what is meant by a chaotic dynamical system. Although this talk stands alone, it is preliminary material for next week's talk.
Abstract for Dec. 20: A Simple Chaotic Toy. I will describe a simple (chaotic) toy that my colleague and I developed, built, and analyzed. I'll outline a rigorous argument for showing that the toy (along with many executive-type "chaotic" desk toys) is chaotic by showing that it is well modeled by a system that is conjugate to the horseshoe map. I'll also show a picture of the real toy as well as some computer generated animations.
- Topic: A New Formula for Computing Frobenius Numbers in Three Variables.
- Speaker: Janet Trimm, Rose student
- Date: Jan 24, 2001
- Abstract : It is well known that if a and b are relatively prime positive integers, then the Frobenius number of a and b is equal to ab-a-b. Many authors have developed "explicit" formulas and algorithms for computing Frobenius numbers of relatively prime integers a1,a2, ... an when n>2. But these formulas and algorithms are clumsy and complicated even for n=3. In this paper, we will prove that there is surprisingly a nice formula that computes the Frobenius number of three positive integres a, b, and c where a and b are relatively prime.
- Topic: On the Probability that a Monic Integral Polynomial Is Irreducible
- Speaker: Timothy Kilbourn, Rose student
- Date: Jan 31, 2001
- Abstract : It is proved that if m is any positive integer, then the limiting value, as the prime-power q goes to infinity, of the probability that an m-th degree polynomial in F_q[X] is irreducible is 1/m. As a corollary, one obtains an identity which is indexed by the partitions of m and whose terms are unit fractions. Analogous probabilistic studies are carried out for various classes of integral polynomials, where the underlying notion of "probability" is defined in the spirit of "natural density," namely, as the limiting value, as n goes to infinity, of the usual combinatorial probability of irreduciblity in Q[X] (equivalently, Z[X]) for integral polynomials whose coefficients are bounded in absolute value by n. With this notion of "probability", it is shown that if m is between 2 and 6, then with probability 1, the random integral polynomial X^m+aX+b is irreducible; and if m is between 1 and 5, the same conclusion holds for the random monic integral m-th degree polynomial. Numerical evidence is presented in support of related conjectures.
- Topic: Mathematical Modeling with Categories
- Speaker: Ralph Wojtowicz
- Date: Feb 7, 2001
- Abstract : Every concept arises from the equation of unequal things. Just as it is certain that one leaf is never totally the same as another, so it is certain that the concept "leaf" is formed by arbitrarily discarding these individual differences and by forgetting the distinguishing aspects. ...What then is truth? A movable host of metaphors, metonymies, and; anthropomorphisms: in short, a sum of human relations which have been poetically and rhetorically intensified, transferred, and embellished, and which, after long usage, seem to a people to be fixed, canonical, and binding. Truths are illusions which we have forgotten are illusions... it is originally "language" which works on the construction of concepts, a labor taken over in later ages by "science".
--Friederich Nietzsche
"On Truth and Lies in a Nonmoral Sense" (1873)
A theory is a mathematical model for an aspect of nature. One good theory extracts and exaggerates some facets of truth. Another good theory may idealize other facets. A theory cannot duplicate nature, for if it did so in all respects, it would be isomorphic to nature itself and hence useless, a mere repetition of all the complexity which nature presents to us, that very complexity we frame theories to penetrate and set aside. With this sober and critical understanding of what a theory is, we need not see any philosophical conflict between two theories, one of which represents a gas as a plenum, the other as a numerous assembly of punctual masses. Models of either kind represent aspects of real gases; if they represent those properly, they should
entail many of the same conclusions, though of course not all.
---Clifford A. Truesdell and Robert G. Muncaster
"Fundamentals of Maxwell's Kinetic Theory of a Simple Monatomic Gas" (1980)
...in mathematical practice we must, more than in any other science, hold a given object quite precisely in order to construct, calculate, and deduce; yet we must also constantly transform it into other objects.
---F. William Lawvere
"Some thoughts on the Future of Category Theory" (1990)
Categories are abstract mathematical structures which may be viewed as the places where mathematical models live. A category consists of two sorts of things: objects and morphisms. Every morphism has source and target objects: Source ----" Target Each object has an identity morphism and there is an associative composition operation on adjacent pairs of morphisms. An example is the category having sets as objects and functions as morphisms.
The language of category theory is rich enough to describe diverse structures which arise in mathematical modeling and to express precise comparisons between models of different types. After discussing basic definitions and examples and giving a brief history of the theory, I will describe categories of sets and of stochastic matrices and a category having transition probabilities as morphisms. I will give examples of deterministic and stochastic, discrete-time dynamical systems and show how the former may be viewed as special cases of the latter. Certain constructions that can be made with sets (points, cartesian products, disjoint unions) have useful interpretations in other categories. I will also present an implementation of the category of stochastic matrices using Maple.
- Topic: Improving Solar Car Strategy
- Speakers: Brad Berron, Todd Goldfinger, Mike Ritter, Tom Schneider, Bill Stephen, Jerod Weinman, Rose students
- Date: Feb 14, 2001
- Abstract : The MA331 Mathematical Modeling class has been working for the past four weeks on modeling a few important aspects of the Rose-Hulman Solar Phantom VI solar car project, with the goal of improving race strategy for the upcoming 2001 American Solar Challenge. We have focused on two basic issues: calculating available power from sunlight and computing torque-power curves for various speeds and hill grades. The intensity of solar radiation changes over the course of a day, and depends on the current latitude, time of year, cloud conditions, and angle of the solar cell array on the car. The resulting power available versus time of day curve can then be used to help determine race strategy for that day (e.g., the maximum speed allowed by the available power for current road and weather conditions). To complement these calculations, we combined the efficiency curves provided by the engine manufacturer with the vast amounts of data compiled by past solar car runs to find torque-power curves for different constant speeds and hill grades. These models can use the GPS data supplied by the race coordinators, giving information like latitude and altitude along the racecourse, to help determine optimal race strategies.
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