Summary
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| Friday, January 13, 3:00 - 4:00 p.m., MP3314 | Ram N. Mohapatra, Univ. of Central Florida | Some Extremal Problems in Polynomial and Rational Approximation |
| Friday, January 20, 3:00 - 4:00 p.m., MP3314 | Joy Darley, Georgia Southern University | Understanding Fractions as Numbers and the Ability to Perform Operations with Algebraic Expressions |
| Friday, January 27, 3:00 - 4:00 p.m., MP3314 | Frederic Mynard, Georgia Southern University | Differentiability as continuity |
| Friday, February 3 , 3:00 - 4:00 p.m., MP3314 | Francis Jordan, Georgia Southern University | Connected Partitions of Continua |
| Friday, February 10, 3:00 - 4:00 p.m., MP3314 | Grzegorz Michalski, Georgia Southern University | k-Independence in Vector Spaces over Finite Fields |
| Thursday, February 16, Public Lecture Series | Eric McDowell, Berry College | Counting |
| Friday, February 17, 3:00 - 4:00 p.m., MP3314 | Eric McDowell, Berry College | There and Back Again: An Introduction to the Study of Hyperspaces |
| Thursday, February 23, CLEC Lecture | Jimmie Lawson, Louisiana State University | Unraveling the Mysteries of Infinity |
| Friday, February 24, 3:00 - 4:00 p.m., MP3314 | Jimmie Lawson, Louisiana State University | Quasicontinuous functions and a generalized calculus |
| Friday, March 3, 3:00 - 4:00 p.m., MP3314 | Francois Ziegler, Georgia Southern University | Symplectic Geometry: Few Objects, *Many* Morphisms |
| Friday, March 10, 3:00 - 4:00 p.m., MP3314 | Robert Lund, Clemson University | A Random Walk Through Climate Change |
| Friday, March 17, 3:00 - 4:00 p.m., MP3314 | (--- Spring Break ---) | |
| Friday, March 24, 3:00 - 4:00 p.m., MP3314 | Xiezhang Li, Georgia Southern University | Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices -- application of perturbation theory for simple invariant subspaces |
| Friday, March 31, 3:00 - 4:00 p.m., MP3314 | Goran Lesaja, Georgia Southern University | A Class of Large- and Small-update Primal-dual Interior-point Algorithms for Linear Optimization Based on the New Barrier Function |
| Friday, April 7, 3:00- 4:00 p.m., MP3314 | Broderick Oluyede, Georgia Southern University | On Bounds and Approximating Weighted Distributions by Exponential Distributions. |
| Friday, April 14, 2006 Distinguished
Lecture
3:00 - 4:00 p.m. Room 1004, College of Information Technology Building |
Gene Golub, Stanford University | Numerical Methods for Rapid Computation of Page Rank |
| Friday, April 21, 3:00- 4:00 p.m., MP3314 | De Witt Sumners, Florida State University | DNA Topology |
| Saturday, April 22, 10:00 - 12:00
p.m. Third Public Lecture |
1. De Witt Sumners, Florida State University
2. Frederic Mynard, Georgia Southern University |
1. Calculating the secrets of life: mathematics in biology and
medicine
2. Everything you always wanted to know about topology but were afraid to ask |
| Friday, April 28, 3:00- 4:00 p.m., MP3314 | Homeira Pajoohesh, City College of New York | The relationship between speed of computer algorithm and category theory and topology |
Details of Spring 2006 lectures
1. Friday, January 13, 2006, 3:00 p.m. - 4:00 p.m. Room: MP 3314.
Speaker: Speaker: Ram N. Mohapatra, University of Central Florida (Host: Don Fausett)
Title: Some Extremal Problems in Polynomial and Rational Approximation.
Abstract: We shall discuss the extremal problems associated with the
bounds of the ratio of ||f'/f|| . Then we shall discuss Bernsteins's
inequalities for polynomials, rational functions and rational functions with a
prescribed zero. Extension of these results for weighted polynomials with
Hermite and Laguerre weights will also be discussed. Generalizations of the
latter results and some open problems will be mentioned.
2. Friday, January 20, 2006, 3:00 p.m. - 4:00 p.m. Room: MP 3314.
Speaker: Joy Darley, Georgia Southern University
Title: Understanding Fractions as Numbers and the Ability to Perform Operations with Algebraic Expressions
Abstract: In a twenty-six day design research study conducted in a ninth-grade beginning algebra classroom, activities were designed in order to lay the foundation for the use of fraction bars and number lines as quantitative models for the purpose of defining, renaming, adding, subtracting, multiplying and dividing fractions. Emphases were placed on building conceptual understanding prior to learning procedures, connecting conceptual knowledge to procedural knowledge, and the importance of the ability to abstract number operations to algebraic manipulations for future success in algebra. Prior to this study, the students were exposed to a curriculum that emphasized rote-memorization of procedures as opposed to conceptual understanding. The data sources utilized were pre-assessments, interviews, classroom discussions, performance assessments, and post-assessments.
Prior to instruction, many students did not see fractions as numbers. As a result, many of these students used incorrect algorithms when working with fraction operations and were predictably unsuccessful when working with algebraic operations. In contrast, those students who were able to recognize fractions as numbers were more likely to use correct algorithms and were more successful with both the fraction operations, and, consequently, the algebraic operations.
During instruction, students were required to show the location of a variety
of fractions on the number line given the position of zero and one. The
number line was effective in helping students understand fractions as
numbers. Likewise, students were asked to locate algebraic fractions on
the "algebra number line" given the position of zero and x. The "algebra
number line" was useful in bridging the gap from fractions to rational
expressions. As students became more skillful in visualizing fractions as
numbers, they became more proficient with fraction operations. Similarly,
as students became more skillful in seeing rational expressions as fractions,
they became more proficient with algebraic operations. After
participation, not only were students more successful with fraction and
algebraic operations but were also more apt to use algorithms when solving word
problems and were, therefore, more successful at solving the algebraic problems.
3. Friday, January 27, 2006, 3:00 p.m. - 4:00 p.m. Room: MP 3314.
Speaker: Frederic Mynard, Georgia Southern University
Title: Differentiability as continuity.
Abstract: We characterize differentiability of a map f:
R->R in terms of continuity of a canonically associated map
F. Both the local and global differentiabilities of f are investigated.
4. Friday, February 3, 2006, 3:00 p.m. - 4:00 p.m. Room: MP 3314.
Speaker: Francis Jordan, Georgia Southern University
Title: Connected Partitions of Continua
Abstract: By a continuum I mean a compact connected metric
space. Let X be a continuum and H be the space of nonempty compact subsets
of X. In the study arbitrary partitions of X into compact sets one finds that
the subsets of the partition that are connected in H seem to play an important
role. So, I dedicate this talk to connected partitions of continua. I was
surprised to find that for some continua every connected partition is actually
compact. I am far from characterizing continua with this property.
However, there is a characterization among graphs. I will discuss such
continua.
5. Friday, February 10, 2006, 3:00 p.m. - 4:00 p.m. Room: MP 3314.
Speaker: Grzegorz Michalski, Georgia Southern University
Title: k-Independence in Vector Spaces over Finite Fields
Abstract: For a positive integer k, a set of vectors is said to be
k-independent if every of its nonempty subsets that has no more than k elements
is linearly independent. In a previous talk two results were presented
concerning the maximal possible sizes of k-independent sets, for various k, in
the n-dimensional vector space over the two-element field. This time we consider
a vector space over any finite field, and obtain a generalization of one of the
earlier results.
6. Thursday, February 16, 2006. Public Lecture Series.
Speaker: Eric McDowell, Berry College. (Host: Francis Jordan)
Title: Counting
Abstract:
Once upon a time human beings had no concept of
number. They had no word or gesture to express the number of people in
their village or the number of nights between full moons.
A new mother of this era would have been unable to count the number of fingers on her newborn's right hand. However, she would have recognized that her baby's right hand had as many fingers as her own by touching each finger of her right hand to a corresponding finger of her baby's right hand. When each of her fingers touched exactly one of his and each of his fingers touched exactly one of hers, she knew that her collection of right-hand fingers was the same size as his collection of right-hand fingers.
The pairing of each element of one collection to exactly one element of another collection so that no elements are left over is called a one-to-one correspondence.
The goal of this talk is to demonstrate how the notion of one-to-one
correspondence can be used to "count" the members of very large sets.
Toward this end, we will encounter no-vacancy hotels that can accommodate
additional guests, different sized sets with the same "number" of elements, and
an infinite collection of different infinities.
7. Friday, February 17, 2006, 3:00 p.m. - 4:00 p.m. Room: MP 3314.
Speaker: Eric McDowell, Berry College. (Host: Francis Jordan)
Title: There and Back Again: An Introduction to the Study of Hyperspaces
Abstract: A metric on a set provides us with a way to measure distance between points. How can we use it to measure distance between sets?
The first part of this talk will offer several answers to this guiding
question. The answer that we settle upon will be used to define the notion
of a hyperspace of a set. Armed with this definition, we begin our
adventures in a remarkable world where arcs turn to n-cubes and nonseparating
plane continua admit fixed-point free maps. From there, our journeys will
take us back again to a hyperspace setting that looks very much like the space
where we began.
8. Thursday, February 23, 2006. Spring 2006 CLEC Lecture.
Speaker: Jimmie Lawson, Louisiana State University (Host: Frederic Mynard)
Title: Unraveling the Mysteries of Infinity
Abstract: Since ancient times philosophers, mathematicians, and
scientists have been fascinated and mystified by the concept of the infinite.
The first important insights into the mathematics of infinity arose in the
late nineteenth century in George Cantor's theory of infinite sets, which we
illustrate with a mathematical variant of the story of Hercules' fifth labor,
the cleansing of the Augean stables.
9. Friday, February 24, 2006, 3:00 p.m. - 4:00 p.m. Room: MP 3314.
Speaker: Jimmie Lawson, Louisiana State University (Host: Frederic Mynard)
Title: Quasicontinuous functions and a generalized calculus
Abstract: A subset of a topological space is said to be quasiopen if
has a dense interior and a function is quasicontinuous if the inverse image of
every open set is quasiopen. The topological theory of quasicontinuous
functions on compact metric spaces has seen considerable recent development,
motivated by connections with dynamical systems and viscosity solutions of
certain types of pde's. The latter is closely connected to the ability to
extend the differential calculus to the quasicontinuous setting. We trace
some of the recent developments of the topological theory of quasicontinuous
functions and sketch their generalized differential calculus.
10. Friday, March 3, 2006, 3:00 p.m. - 4:00 p.m. Room: MP 3314.
Speaker: Francois Ziegler, Georgia Southern University
Title: Symplectic Geometry: Few Objects, *Many* Morphisms
Abstract: Unlike riemannian manifolds (which abound but typically have no automorphisms), symplectic manifolds are rare birds with lots of symmetry. They are easily isomorphic, and their automorphism group is always infinite dimensional. Upon restriction to finite dimensional subgroups arises the key concept of a momentum map, and perhaps its most spectacular application: the complete classification (due to Kirillov, Kostant, and Souriau) of the homogeneous symplectic manifolds of any Lie group.
In this talk I will attempt a leisurely introduction to this theory, using
(again) as running examples R2n and the Cotangent Bundle of the Sky -
?a.k.a. the space of light rays, of interest in optics.
11. Friday, March 10, 2006, 3:00 p.m. - 4:00 p.m. Room: MP 3314.
Speaker: Robert Lund, Department of Mathematical Sciences Clemson University
Title: A Random Walk Through Climate Change
Abstract: This talk examines several statistical issues encountered in
temperature trend time series analyses. Regression models with
autocorrelated errors are introduced as the primary analysis vehicle.
These models are used to study the famous Hansen-Lebedeff series of global
temperatures and over 700 local temperature recording stations in the 48
contiguous United States. Issues of changepoints, periodicities,
regression response form, and regression memory error structure take prominent
roles. Spatial smoothings of temperature change rates in the United States
over the last 150 years show a cooling Southeast and a warming Northeast and
West.
12. Friday, March 24, 2006, 3:00 p.m. - 4:00 p.m. Room: MP 3314.
Speaker: Xiezhang Li, Georgia Southern University.
Title: Relative perturbation bounds for the eigenvalues of diagonalizable and singular matrices -- application of perturbation theory for simple invariant subspaces.
Abstract: Perturbation bounds for the relative error in the
eigenvalues of diagonalizable and singular matrices are derived by using
perturbation theory for simple invariant subspaces of a matrix and the group
inverse of a matrix. These upper bounds are supplements to the related
perturbation bounds for the eigenvalues of diagonalizable and nonsingular
matrices.
13. Friday, March 31, 2006, 3:00 p.m. - 4:00 p.m. Room: MP 3314.
Speaker: Goran Lesaja, Georgia Southern University
Title: A Class of Large- and Small-update Primal-dual Interior-point Algorithms for Linear Optimization Based on the New Barrier Function
Abstract: In the paper we present a class of polynomial primal-dual
interior-point algorithms for linear optimization based on a new kernel
function. This kernel function is not a self-regular function because its growth
term is increasing at the rate that is between linear and quadratic as opposed
to self-regular functions where it is increasing at least quadratically. Several
new arguments had to be developed regarding the new kernel function in order to
complete the complexity analysis of the algorithms. The obtained complexity
bounds match the best known complexity bounds obtained for these methods based
on self-regular functions.
14. Friday, April 7, 2006, 3:00 p.m. - 4:00 p.m. Room: MP 3314.
Speaker:Broderick Oluyede, Georgia Southern University
Title: On Bounds and Approximating Weighted Distributions by Exponential Distributions.
Abstract: Exponential Approximations to the class of weighted
residual and equilibrium distributions with monotone weight functions are
presented. These bounds and approximations are obtained for the class of life
distributions with increasing (decreasing) hazard rate and mean residual life
functions. Estimation of reliability measures from censored data as well as
Bayesian estimation of the length-biased exponential reliability function are
presented.
15. Friday, April 14, 3:00 - 4:00 p.m. 2006 Distinguished Lecture. Room 1004 (Auditorium), Information Technology Building.
Speaker: Gene Golub, Standord University (Host: Don Fausett)
Title: Numerical Methods for Rapid Computation of Page Rank
Abstract: We consider the problem of iteratively computing the
stationary distribution vector of large finite Markov chains. It is assumed that
the matrices involved are too large for a decompositional approach to be
effective, and matrix-vector products must be used. The problem is motivated by
Google's PageRank algorithm for large web databases. We consider an Arnoldi-type
restarted algorithm based on a combination of Arnoldi and the SVD. Connection of
the algorithm to other techniques such as the quadratic extrapolation method is
discussed, and the sensitivity of the PageRank problem is also addressed.
Numerical examples illustrate the performance and convergence behavior of the
algorithm. This is a Joint work with Chen Greif.
16. Friday, April 21, 2006, 3:00 p.m. - 4:00 p.m. Room: MP 3314.
Speaker: De Witt Sumners, Florida State University (Host: Frederic Mynard)
Title: DNA Topology
Abstract: Cellular DNA is a long, thread-like molecule with remarkably
complex topology. Enzymes which manipulate the geometry and topology of cellular
DNA perform many important cellular processes (including segregation of daughter
chromosomes, gene regulation, DNA repair, and generation of antibody diversity).
Some enzymes pass DNA through itself via enzyme-bridged transient breaks in the
DNA; other enzymes break the DNA apart and reconnect it to different ends. In
the topological approach to enzymology, circular DNA is incubated with an
enzyme, producing an enzyme signature in the form of DNA knots and links. By
observing the changes in DNA geometry (supercoiling) and topology (knotting and
linking) due to enzyme action, the enzyme binding and mechanism can often be
characterized. This expository lecture will discuss topological models for DNA
strand passage and exchange, and using the spectrum of DNA knots to infer
bacteriophage DNA packing in viral capsids.
17. Saturday, April 22, 2006, Public Lecture Series. Room: 1005 Information Technology Building.
Lecture 1 : De Witt Sumners, Florida State University (Host: Frederic Mynard)
Title: Calculating the secrets of life: mathematics in biology and medicine.
Abstract: The human body is an extremely complicated biological system. Spurred by spectacular recent progress, biology and medicine are experiencing an explosion of data. In order to convert this firehose of data into usable knowledge, mathematics and computation (both old and new) are needed to build models and navigational tools. This talk is intended for a general audience, and will briefly discuss a few applications to show the impact that mathematics can have in biology and medicine: in the cell (understanding how enzymes operate on DNA); in the heart (controlling fibrillation); and in the brain (understanding brain function).
Lecture 2: Frederic Mynard, Georgia Southern University
Title: Everything you always wanted to know about topology but were afraid to ask.
Abstract: Topology is NOT the study of terrain and terrain change as
people sometime believe. In other words, topology and
topography are two different things. So, what is topology? This is what
this talk will be about (hence, I'm not giving it away!). Topological ideas
underly a very large part of modern mathematics and are the foundation of many
notions and theories, even Calculus. But we don't need to talk about
sophisticate mathematics to understand what topology is about. Geometric
considerations easy to follow for a high school students will be sufficient for
a start. Then, a closer look at notions from Calculus will lead us to a deeper
understanding of topology and its more technical role in mathematics.
18. Friday, April 28, 2006, 3:00 p.m. - 4:00 p.m. Room: MP 3314.
Speakers: Homeira Pajoohesh, City College of New York (Host: Frederic Mynard)
Title: The relationship between speed of computer algorithm and category theory and topology
Abstract: Binary trees are very useful tools in computer science for
estimating the running time of comparison based algorithms, that is, algorithms
in which every action is ultimately based on a prior comparison between two
elements.
For two given algorithms A and B, where the decision tree of A is
more balanced than that of B, it is known that the
average and worst case
times of A will be better than those of B. Thus the most balanced and the most
imbalanced
binary trees play a major role. Here we define balance and
consider each comparison based algorithm as semilattices and characterize the
most balanced and the most imbalanced binary trees by topological and
categorical properties. Also we
define the composition of binary trees as a
commutative binary operation, *, such that for binary trees A and B, A*B is the
binary tree obtained by attaching a copy of B to each leaf of A. We show that
for the collection T of binary trees, (T, *) is a commutative po-monoid
and investigate its properties.
Please direct questions or comments regarding the
colloquium to Sze-Man
Ngai: mailto:ngai@gsu.mat.georgiasouthern.edu.
( Last updated: April 24, 2006.)