Center for Nonlinear Analysis
CNA Home
People
Seminars
Publications
Workshops and Conferences
CNA Working Groups
CNA Comments Form
Summer Schools
Summer Undergraduate Institute
PIRE
Cooperation
Graduate Topics Courses
SIAM Chapter Seminar
Positions
Contact 
2012 Summer Undergraduate Applied Mathematics Institute
May 29  July 17, 2012
► Students
► Faculty
Projects:
 Combinatorics on Words, Hrothgar, Michael Druggan, Archit Kulkarni
Advisor: Irina Gheorghiciuc
Abstract: According to Irina the main points of the project:
 Familiarize yourselves with the notion of subword complexity of finite
and infinite words and its significance. Read several important papers on
the subject.
 Analyze in detail the Sturmian and de Bruijn words, which are the words
with the highest and lowest subword complexity.
The group expanded the known list of necessary conditions for a vector
to be a subword complexity sequence of a finite word. They worked on a conjecture about the asymptotic behavior of the number of distinct subword complexity sequences by Matt Green and Arash Enayati.
While certain recurrence patterns in the sequence have been observed and
proved, this work in still in progress.

A First look at boundary value problems, Emmalyne Wyatt, Jessica Jennings
Advisor: Michael Goldman
Abstract: In this project, E. Wyatt and J. Jennings studied some boundary value problems in one
dimension such $ u'' + u = g\;\; u(0) = u(1) = 0\;\;\;\; (1)$
They tried to understand the difference between this kind of problems and ODEs and then
studied (1) by two methods. First, they computed explicit solutions using Fourier methods
taking as an example the tent function
\[ g(x) =
\left\{
\begin{array}{ll}
x & \mbox{if $0 \leq x \leq \frac{1}{2}$ }\\
1 x & \mbox{if $ \frac{1}{2} \leq 1$} \end{array}
\right. \]
After that they proved also existence of solutions using functional analysis. For this they
learned about Hilbert spaces with emphasis on weak convergence, Riesz Theorem, orthonor
mal basis (and made the link with the Fourier method). They also learned about the space
of square integrable functions $L^2$ and the Sobolev space $H^1,$ getting familiar with the notion
of weak derivatives. They then derived a weak formulation of (1) and solved it using both
the Riesz representation Theorem and by proving existence of minimizers of the functional
\[ J(u) = \frac{1}{2} \int_0^1 [(u')^2 + u^2] {\mbox dx}  \int_0^1 (gu)
{\mbox dx} \]
by the Direct Method of the Calculus of Variations. Then they used both the Fourier method
and the abstract method to study the convergence when $\epsilon \rightarrow 0$ of the solutions
$u_{\epsilon}$ of the
problem $ u'' + u = g_{\epsilon}\;\; u(0) = u(1) = 0$
where $ g_{\epsilon}=g(frac{x}{\epsilon}) $
with $g$ a periodic function. They showed that $u_{\epsilon}$ strongly converges to the
solution of $ u'' + u = \bar{g}\;\; u(0) = u(1) = 0$
where $\bar{g}= \int_0^1 g\; dx.$
 Using Mathematics to Restore or Correct Images, Jieun Lee, Nathan Scavilla
Advisor: Michael Goldman
Abstract: In this project, J. Lee and N. Scavilla studied two methods for denoising images. They First
concentrated on the Tychonov method where a corrupted image $g$ is denoised by considering
the minimization problem
\[ {\mbox min}\int_{[0,1]^2} \bigtriangledown u^2
+\frac{\lambda}{2}\int_{[0,1]^2} (ug)^2.\;\;\; (1)\]
They derived the Euler Lagrange equation for this problem, which is
$\bigtriangleup u+\lambda(ug)=0$
and understood the associated (Neumann) boundary conditions. They then solved this equation both
using discrete differences and Fourier methods. They studied the effect of the parameter $\lambda$
on the solution and showed that when considering the one dimensional analog with
for example $g=\chi_{[\frac{1}{2}]}$ then the denoised signal is always smooth meaning that this
model can
not capture the edges. To overcome this problem, they considered the RudinOsherFatemi
model
\[ {\mbox min}\int_{[0,1]^2} \bigtriangledown u^2
+\frac{\lambda}{2}\int_{[0,1]^2} (ug)^2.\]
involving the total variation. They showed in one dimension that for some values of $\lambda$, no
smooth solution can exist and studied the definition and basic properties of functions of
bounded variation again in one dimension. They then focused on the discretized problem
\[ {\mbox min}\Sigma \sqrt{(u_{i+1,j}u_{i,j})^2 +(u_{i,j+1}u_{i,j})^2}
+\frac{\lambda}{2}\Sigma (u_{i,j}g_{i,j})^2.\]
Since this is a convex but nonsmooth problem, in order to derive the Euler Lagrange equation
of this problem, they studied some convex analysis with particular emphasis on the notions
of subgradient and convex conjugate. Using these tools, they were able to rewrite the Euler
Lagrange equation as
$u= g\pi_{\lambda K}(g)$
where $\pi_{\lambda K}(g)$ is the projection of $g$ on the convex set $\lambda
K=\{{\mbox div}p : p_{i,j}\leq \lambda.$ They
computed this projection by using a semiimplicit gradient descent algorithm
proposed by A. Chambolle. They ran a large number of numerical examples comparing the Tychonov and
the ROF models.
 Generalized Mersenne Numbers, Christian Rodriguez
Advisor: Gregory Johnson
Abstract: Large prime numbers are useful in cryptography because of the computational complexity of determining the prime decomposition of products of two large primes. Current efforts are primarily focused on Mersenne primes which are of the form 2^{n}  1 as the largest known primes are of this form. In this project we study potential primes of the form 2^{2n} ± 2^{n} + 1. We determined properties of such numbers necessary for them to be prime as well as efficient primality tests for those that meet these conditions. In addition, we investigated the question as to whether a potential prime of this form must be square free.
 Modeling Stock Pairs Using ContinuousTime Markov Chains, Caroline Miller, Utkarsh Narayan, Elizabeth Newman, Tamar Oostrom
Advisor: Chad Schafer
Abstract: This group worked to fit and test a continuous time Markov chain model to the price fluctuations of a pair of equities. Briefly stated, the objective was to use the limiting distribution of the resulting Markov chain to quantify the proportion of the time the two stocks exhibit similar trends. This could be useful for pairs trading strategies which rely upon finding pairs of stocks which are typically in sync with one another. This measure should be more robust than the standard "beta" measure in current use, as it is less sensitive to outliers. The group made significant progress towards this goal, by developing and testing methods for determining when a stock price process shifts between upward, downward, and stable periods. This required an understanding of fundamental
issues of statistical hypothesis testing. The group explored the properties of the continuous time Markov chain, and used these to determine optimal estimators for the parameters of the process; again, this allowed us to discuss fundamental statistical inference issues. The group also used the fit model to estimate the limiting distribution for the Markov chain. All of this work required significant amount of data processing and coding, which was done in R. In this way the project represented a comprehensive exploration of an interesting and challenging data analysis and model fitting problem.
