Mathematics Symposium: "Mathematics Awareness Month at UNO"
Jay Pedersen
A Computer-Assisted Investigation of Numerical Ranges
This project contains computer investigations of the numerical
ranges of linear operators on certain Hilbert spaces. The focus is
on a class of operators called composition operators. The
numerical range of a linear operator has relevance to quantum
physics and is useful to engineers in determining the stability of
certain control systems. Interest in composition operators was
initially triggered by papers of theoretical mechanics.
Rebecca Pitz
Quantifying Degrees of Randomness in Word Rhythms of Literary Works
Rhythm is an intrinsic quality in literature. Intuitively, poems
are more rhythmic than novels, and novels are more rhythmic than
textbooks. By investigating word length and word order, we show
rhythm has a non-linear dynamic. To quantify the rhythm, we use
the fluctuation index and approximate entropy, which are both
effective in distinguishing the quantity of rhythm in various
types of literature.
Joseph Lee
Fractals in Nature - Understanding Their Structure
Fractal networks can be found in various natural
occurrences, as well as man-made structures.
Recent approaches have tried to explain this
phenomenon through optimization techniques.
Studying the fractal properties of these networks
can give a better understanding of how and why
these structures form. Comparing these structures
to other models points toward an economical reason
for their appearance - a limited amount of resources.
Birame Sene
Hoeffding-type probability inequalities for the
distribution of sums of possibly dependent random variables
In some practical problems, we must compute the probability distribution
of sums of random variables, which may or may not be independent.
In addition, these random variables may have different distributions. The
computation of the exact probability distribution is not always easy and
rarely of closed form. Hoeffding's Inequality and other such inequalities
may be applied, but often assume independent random variables. New
Hoeffding-type inequalities will be studied and new bounds will be derived.
Though these bounds are only implicit and not of closed form, they
are much easier to compute than the exact probability distribution.Both
analytical and numerical comparisons of these bounds are given in the
independent case.One case of dependence is also considered.
Chris Fischer
DNA Codes
DNA nanotechnology often requires collections of oligonucleotides
called DNA free energy gap codes. We will use new theoretical results
for designing new DNA codes. In this project we will study how to
design these codes to accomplish a desired amount of work within
an acceptable error rate. Using a probabilistic model of DNA code
and mathematical random coding theory methods, we will find DNA
code design parameters.