2000 Chapter Activities
Seminars
Upcoming Undergraduate Talks, Spring 2001
Who: Ashlie Singer
Date:Wednesday, May 2 Time: 5:30
Location: Applied Math Conference Room
Discussion: I have been working on finding short pulse solutions to Maxwell's equations. These solutions describe the output of the fairly new ultrafast laser. Assuming that we have a short pulse solution leads to a simpler set of equations that describe the pulse. I incorporated the method of multiple scales, asymptotic expansion, Fourier analysis, and spectral methods(FFTs) in effort to find short pulse solutions to Maxwell's equations.
Upcoming Undergraduate Talks, Fall 2000
Who: Saverio Spagnolie and Mark Snyder
Date:Wednesday, October 11
Time: TBA
Location: Applied Math Conference Room
Discussion: Mark Snyder's Research: If a data set satisfies Benford's Law, then it's significant digits will be distributed according to a logarithmic density. Our work involves creating discrete one-dimensional dynamical systems that satisfy Benford's Law. We construct dynamical systems satisfying Benford's Law that are conjugate to other dynamical systems that have been studied intensely (like the logistic map). The dynamical systems we create inherit the measures of ergodicity from the systems they are conjugate to. We also generate dynamical systems that satisfy Benford's Law by solving the inverse Frobenius-Perron problem for a given density. We are able to calculate the Lyapunov exponents for these new dynamical systems. In addition to these exact results we use Monte Carlo methods to analyze other dynamical systems. Because of the many physical data sets satisfying Benford's Law (tax data, geophysical data, statistics from diffusion of gasses) we believe our work may have applications in mathematical modeling. Saverio Spagnolie's Research: We have studied recurrence in numerical studies of two Hamiltonian differential equations. We began our investigation by exploring the well known cubic non-linear Schrodinger equation (NLSE), iAt=Axx+A|A|^2. As a volume preserving equation, recurrence is predicted in numerical solutions to the NLSE by the Poincare' recurrence theorem; this was observed when the Hamiltonian structure was preserved numerically. The second equation investigated, iAt=Axx+A|A|, has been less studied than the NLSE, but similarities suggested comparable behavior between the two. In contrast to the aforementioned theorem, researchers from LANL have observed a tendency towards the formation and persistence of a large scale solitary wave in numerical studies of the equation. In our own studies of the equation, we have found that not all of the quantities preserved analytically by the equation are behaving in the same way numerically. Thus, we have yet to reach a point from which we might comment confidently about recurrence.
Who: Holly Lewis, Josh Wells, and Jeremy Horgan-Kobelski
Date: Wednesday, November 15 Time: 5 - 6:15 pm
Location: Applied Math Conference Room
Discussion: Jeremy Horgan-Kobelski's research: I will be discussing the Hansen-Patrick family of iterative root finding methods. This is a parameter dependent family of cubically convergent methods for finding the roots of a function f(z) = 0. I will be looking at this class of methods from a dynamical systems perspective and will present several numerical experiments performed on different classes of complex polynomials. Halley's method, one special case contained in the family of methods will be presented in detail for the polynomials z(z-1)(z-r). Halley's method is unique since it corresponds to the only finite parameter value of the Hansen-Patrick family for which the method is rational. Dynamics of Halley's method in the complex plane will be presented along with some analytical work. Holly Lewis and Josh Wells' research: We have been studying the set of fractal images called the Sierpinski relatives using linear algebra techniques. The 512 images in the set can be created using several techniques; the most common is the use of iterated function systems. However, for the purpose of linear algebraic analysis, we have chosen to implement (in Matlab) a variation of Peitgen's method, called the tiling algorithm. These images are generated for sizes that are powers of two, up to 256x256 pixels. During our analysis we have used several linear algebra decomposition methods, including the SVD. Related to the SVD is the "Eigenface" technique, which we have adapted. This method determines a basis for the set of images and is used to confirm that the number of unique images is 456. While exploring the singular value (SV) spectra for each image, we have found that for 56 of the images, the SVs can be created using an equation related to Pascal's triangle and the golden number, [1+sqrt(5)]/2. The non-standard Haar Wavelet Decomposition (HWD) has been used in conjunction with the SVD for the 56 images that are related to Pascal's triangle and the golden number. The averages and differences matrices, which result from the HWD, also have interesting SV properties. For example, their SVs can be created using another equation involving Pascal's triangle and the golden number. Currently, we are investigating the results when permuting the SVs of each image and then reconstructing using A_new = U*S_new*V'. Under this transformation, some of the images map to other images in the set, some of these map to fractal images that are not in the Sierpinski set, and others map to images that are unrecognizable. Of primary interest are the new fractal images, whose properties will be analyzed.
Who: Jillian Redfern and Mark Snyder
Date: Wednesday, March 22
Time: 5pm
Location: Applied Math Conference Room
Discussion: Jillian will discuss her ride on the "vomet comet" and
and Mark will talk about his research
Jillian's synopsis:
Jillian Redfern flew abroad the KC-135, aka the Vomit Comet, this month.
She is going to discuss her experiment briefly and then show a video
of the actual flight. She also experienced a hyperbaric chamber (25000
ft) while she was in Texas. So if you are curious about what it feels like
to be weightless, please attend. She will be willing to answer all of
your questions.
Mark's synopsis:
Benford's Law owes its discovery to the "Grubby Pages Hypothesis," a
nineteenth century observation that the beginning pages of logarithm books
were dirtier than the last few pages, implying that scientists referenced
the values toward the front of the books more frequently. This peculiarity
led to the hypothesis that the significant digits of many data sets
describing the physical world are not uniformly distributed, but
distributed in a way that favors smaller digits. Benford's Law takes this
hypothesis further by providing a probability mass function for
combinations of significant digits. While the law may be obscure, it is
well established that many common data sets like stock prices, tax data,
and census statistics are described by Benford's Law. Over the past few
months, I have worked to establish that many dynamical systems are
consistent with Benford's Law. Numerical evidence for this is overwhelming
and easy to generate. I also have some analytical results that are
convincing, but limited in scope.
Who: Saverio Spagnolie and Jonathan Peeters
Date: March 14
Time: 5:30pm
Location: Applied Math Conference Room
Discussion: Saverio and Jonathan will discuss their research projects
with Professor Segur and Professor Ablowitz respectivly
Saverio's synopsis:
One of the more important concepts used to understand
partial differential equations is that of conserved quantities. One
amazing result of preserving a particular conserved quantity is the
Poincare Recurrence theorem; this theorem ensures the recurrence of the
initial condition for any 'Hamiltonian' system in finite dimensions. I
have been studying the Nonlinear Schrodinger equation, a PDE with a
preserved Hamiltonian in time, in preparation for studying a far less
understood but similar nonlinear PDE. I will discuss recurrence,
Hamiltonian preservation, and my strategies for better understanding
recurrence patterns for certain initial conditions in NLS.
Jonathan's synopsis:
One of, if not the classical example of non-linear
behavior is deep water gravity waves. My research centered on finding an
efficient numerical technique to advance water waves in time. Combining
an algorithm developed by Dr. Fornberg with a scheme of finding higher
order time derivatives of the velocity of the fluid surface, we will
hopefully obtain a more efficient numerical method with higher accuracy in
time. This idea has somewhat stalled due to a couple of factors:
Fornberg's Fortran code, while manipulated to gain slight increases in
efficiency, is fairly unreadable and progress slowed to a crawl. However,
grad-student Jonathan Birge has translated the Fortran code into a
generally perspicuous MATLAB code, and I am now in the process of
understanding and implementing the higher order technique.
Who: Peter Fox and Elaine Spiller
Date: Monday, Febuary 7
Time: 5:30 PM
Location: Applied Math Conference Room
Discussion: Elaine will talk about the research she has done for Professor Ablowitz and Peter will talk about the programming he did last summer.
Elaine's synopsis:
This past summer I started a research project which involves numerically
modeling FWM in fiber optics. FWM is a sort of noise term which results when
two signals, i.e. data being sent on 2 different channels through the same
wire, collide. In an ideal system, FWM is not of concern because the term is
small and vanishes quickly. However, in a realistic model, one must include
damping (an inherent feature of optical fiber) and amplification. With these
factors included in the model, FWM does not vanish and in fact becomes quite
problematic.
Pete's synopsis:
This summer I worked with a team from the Applied Math Department, funded by
the Vigre Grant, to develop the Mathematical Visualization Toolkit. The
Mathematical Visualization Toolkit(MVT) is a software package, written in
Java, that provides Calc 3 and Diff Eq students several mathematical plotting
tools for compleating lab assignments and homework. The MVT runs as a Java
Applet, allowing students access to these
tools over the web.
Who: Anna Segurson
Date: Wednesday, Febuary 23
Time: 5:00 PM
Location: Applied Math Conference Room
Discussion: Anna will talk about the applied math computer lab (the Newton Lab), and give a brief introduction to the UNIX operating system.
Who: Jeremy Horgan-Kobelski and Bill Woessner
Date: Wednesday, March 1
Time: 5:00 PM
Location: Applied Math Conference Room
Discussion: Jeremy will discuss the research he is doing for Professor Curry and Bill will give us an introduction to complexity theory.
Bill's synopsis:
NP-Completeness: The Travelling Salesman with a Twist
I'll talk a little bit about NP-completeness and bring everyone up to
speed on exactly what an NP-complete problem is. Then I'll introduce the
classic example of the Travelling Salesman Problem and demonstrate its
NP-completeness. I'll discuss the classic solution to this problem and
give some computational results. Then I'll define computational
intractability and talk about human-NP interaction. Finally, I'll
conclude with a discussion of spectral bisection and how it might be used
to reduce this to a polynomial runtime problem.
Jeremy's synopsis:
I will be discusing the Hansen-Patrick family of root finding methods in
the complex plane. This is a one parameter family of methods which exhibit
different convergence behavior for different parameter values. I will be
discussing specifically the behavior of the method when applied to the
cyclotomic polynomials: Z^n - 1, and focus on the complicated bifurcation
that occurs at the parameter value 1/(n-1).
Professor Ted Vessey, from St. Olaf College, will be here November 18 and 19 and will give a seminar to undergraduates November 18.
Time: November 18, 5pm-6pm
Location: Applied Math Conference Room
Title of talk: Records are made to be broken
Come and enjoy the wisdom of a visting professor!
**There will be refreshments served**
Spring '99 Seminar:
Mushroom Classification
Undergraduate Data Analysis Constest: Mike Rempe and Leda Schwartz
You may be wondering what mushrooms have to do with Applied Math. Last Spring semester, two of our graduated Applied Math majors, Mike Rempe and Leda Schwartz scored first place in the annual Data Analysis Contest sponsered by USAFA. The goal of this contest was to classify mushroom based on 22 characteristics. Last Feburary, Mike and Leda went into detail about the results they obtained during this contest.
