Department of Mathematics and Statistics: 2006-2007 Colloquia

List of talks

Jan. 23
 name, affiliation
 Title
Jan. 30
 Artur Elezi,
American University, Washington DC.
                  Enumerative Geometry of Algebraic Curves                        
Feb. 1
 Dayanand Naik
Old Dominion Univ. 		Canonical Correspondence Analysis
Feb. 6
 James Sochacki
James Madison University
 Newton, Maclaurin, Picard and Pade:
Four Horsemen of Differential Equations (continued).
Feb. 13 ---------
  
Feb. 20
 Dorin Drignei
Oakland University
 Computer Experiments with Finite-Difference Codes: a Fast Statistical Surrogate Approach
Feb. 27
Spring break
Mar. 6
 Qin Shao
University of Toledo
 Periodic time series data analysis
Mar. 12 
Richard Aron,
Kent State University
 
Linearity in non-linear situations
Mar. 20
 Berry Tyrret
Oakland University
 Some Recent (and not so recent) Results in Metric Fixed Point Theory
Mar. 27
 Nart Shawash
Oakland University
 Generalizing Cayley Graphs Generated by Transposition Trees
Apr. 3
 Bo-Nan Jiang
Oakland University
 The Least-Squares Meshfree Collocation Method
Apr. 12
 Patrick Pan
Saginaw Valley State University
 Reflexivities of Transformations
Apr. 17
 Takis Sakkalis
University of Athens
    Proper Polynomial Maps and the Fundamental Theorem of Algebra

Abstracts:

A. Elezi: Given 4 lines in a compact, three dimensional space, how many lines meet them? Questions of this type are the focus of enumerative geometry of algebraic curves. We will briefly tour the history of the this subject and explain how recent ideas from physics are providing a new framework for formulating and answering some classic problems/conjectures in the field.

D. Naik:  Abstract: Canonical correspondence analysis is a multivariate method that has been widely used by ecologists.  In this talk we will briefly review this method and present it as a variation of canonical correlation analysis.  We will discuss some mathematical details about canonical correspondence analysis using the work of Olkin and Tate (1961, Annals of Mathematical Statistics, 32, 448-465).  Extensions of this method to use in the context of longitudinally observed data  will also be discussed.

J. Sochacki: In a previous talk I showed the relationship between Picard iteration for solving differential equations and Maclaurin Polynomials. I will give a review of that on some of  the differential equations Isaac Newton gave us. Maclaurin polynomials are easy to work with, but are not always computationally feasible. Pade showed a technique that depends on Maclaurin Polynomials that can be more computationally feasible. I will present a method for generalizing Pade’s method.  I will demonstrate these two methods on several famous differential equations, including Newton's planetary motion problem and the double pendulum with animations for visualization. I will also discuss how this is related to Hilbert’s 16th and 17th problems. These methods are quite easy and can be understood by undergraduates.

D. Drignei: Applications in fields ranging from weather and climate, to biology and industry, increasingly use computer experiments as substitutes for physical experiments in cases where the latter are difficult or impossible to perform. A computer experiment consists of several runs of a computer model (or code) for the purpose of better understanding the impact of various input conditions on the outcome of the experiment. One practical difficulty in computer experiments is that computer model runs may require prohibitive amounts of computational resources. A recent approach uses statistical approximations as computationally faster surrogates for such computer models. This talk presents computationally efficient statistical methodology for a widely used class of computer models, called finite difference solvers of differential equations.

R. Aron:  In surprisingly many instances, one encounters not only one function exhibiting ``odd'' behavior, but large vector spaces of such functions. In this expository talk, we will describe a number of occurrences of this situation.

B. Tyrret: Consider a set C in a Banach space and a mapping T from the set C into itself.  What conditions do you need to impose on C and T in order to ensure that T has a fixed point in C?  Questions of this type have been intensively studied since 1912 when L.E.J. Brouwer proved his famous fixed point theorem that continuous self-maps of compact, convex sets in finite-dimensional spaces have fixed points.  In this talk, we consider progress on this question when C is a closed, bounded, convex subsets of a Banach space and the mapping T is nonexpansive.

N. Shawash: For the purposes of parallel computing Star-Generated Graphs S<sub>n</sub> were introduced as an attractive alternative to Hypercubes Q<sub>n</sub>. However S<sub>n</sub> suffers from the problem of having ``too many'' n! vertices. Two alternative generalizations were given as a remedy to this problem, Arrangement graph A<sub>n,k</sub>, and (n,k)-Star or S<sub>n,k</sub>. In this talk we will consider a new decomposition method for A<sub>n,k</sub> and S<sub>n,k</sub>, discus the quotient/lifting relation between them, and generalize transposition-tree generated graphs in the same manner as both A<sub>n,k</sub> and S<sub>n,k</sub> generalize S<sub>n</sub>.

Bon-Nan Jiang:  A least-squares meshfree collocation method is based on the first-order differential equations to simultaneously obtain the dual variables (stresses) and the primary variables (displacements) with the same accuracy.  The moving least-squares approximation is employed to construct the shape functions.  The summation of squared residuals of the differential equations at nodal points is minimized.  The present method does not require any mesh and additional evaluation points, and thus is truly meshfree method.  Unlike other finite point methods, in the present method no stabilization terms are needed, and the resulting stiffness matrix is always symmetric positive definite.  Numerical examples show that the proposed method possesses an optimal rate of convergence.

P. Pan:  One of the fundamental problems in mathematics is to distinguish different objects.  This is a common theme shared by many (all?) versions of reflexivity problems:  More specifically, given a set S and a set of “tools” T, can one distinguish objects outside S from members of S using T?
Various results of the above nature-some well known and some definitely not—will be discussed.  For this talk, we will restrict S to special sets of linear transformations over a vector space V.  Special cases include V being a finite-dimensional vector space; a Banach or Hilbert space; or an algebra, in which case S could also be a set of deriviations or automorphisms.

T. Sakkalis:  Polynomials are, perhaps, one of the most simple and useful class of functions that appear in mathematics.  Apart from their theoretical interest, they have enjoyed considerable attention over the last two decades in applied mathematics areas such as Computer Algebra, Computational Algebraic Geometry as well as in disciplines like Computer Science, Engineering and many others.  There are two basic reasons for that:

·   Most functions can be approximated by polynomials, and
·   They are rather easy to use in a computer code.

Thus, they serve as good substitutes for functions that are difficult to deal with.  In this talk we will look at polynomials from the proper function viewpoint and will provide an elementary proof of the Fundamental Theorem of Algebra that is partially based on the second derivative test of functions of two variables.



Fall 2006
Sept. 21 1500h
SEB372
 Eddie Cheng,
Oakland University
 Wheel-type inequalities for stable (multi-)set polytopes
Oct. 5
 Ejaz Ahmed
University of Windsor
 Robust Estimation of Process Capability Indices
Oct. 12
 Robert P. Kaufman
University of Illinois at Urbana-Champaign and
University of Michigan, Ann Arbor
 The Brouwer Degree in Euclidean Spaces – First Steps
Oct. 19

Oct. 26
 Laszlo Liptak, 
Oakland University 		Fault Resiliency of Interconnection Networks
Nov. 2
 Tanush Shaska,
Oakland University 		Vanishing theta nulls for algebraic curves with automorphisms. 
Nov. 9
 David Yang Gao
(Virginia Tech)
 Canonical Duality and Triality: A Potentially Powerful Method for Solving
Nonlinear Variational/Optimization Problems 
Nov. 16

Nov. 23.

Nov. 30

Dec. 7
 F. Cakoni
(University of Delaware)
 Qualitative Approaches in Inverse Scattering Theory


  
 
Abstracts:


Eddie Cheng:  A stable set in a graph is a set of nonadjacent vertices. If a vertex is allowed to be included more than once subject to certain restrictions, then it is a stable multi-set. The objective is to find one with the largest size or one with the largest weight if weights are assigned to the vertices. This problem is NP-hard. The most popular method is to cast it as an integer linear program. Currently, the state-of-the-art approach to an integer linear programming problem (ILP) is to apply a branch-cut-price algorithm. Such an algorithm has three components: branch, cut and price. There are cuts available for general ILP. However, for problems that are formulated from a combinatorial problem, specific cuts dedicated to a specific problem need to be developed. In this talk, I will talk about these strong cuts for the stable set problems and their corresponding separation problems. I will digress frequently during the talk.


Ejaz Ahmed: Process capability may be defined as a comparison of the consumer’s demand to the natural variability of the process.  Given in the form of indices, Cp and Cpm are two such measures of process capability.  The main assumption underlying these indices is the normality of the process under investigation.  The normality assumption about the process population enables the practitioners to convert the values of these indices into process yield, which is within the specification limits.

However, the assumption of normality is not realistic for many real applications.  On the other hand, a moderate departure from normality can be attributed to the presence of outliers.  In this talk, we suggest robust estimators of Cp and Cpm, based on Gini’s mean difference.  The asymptotic distributions of the suggested statistics are derived.  The relative performance of the proposed estimators with the benchmark estimators is assessed theoretically as well as numerically.  In addition to asymptotic confidence intervals, two kinds of bootstrap confidence intervals are presented.

Robert P. Kaufman:  The Brouwer degree (after L.E.J. Brouwer) is defined for continuous maps of domains in Rn into Rn.  All of the details can be presented for n=1 and, for n=2, the main idea will be familiar from complex variables.  For higher dimensions, the use of differential forms ("tensor analysis") seems most natural.

The Brouwer degree leads to important theorems about the nature of Euclidean spaces:  the Fixed Point Theorem and Dimension Theorems.  The main ideas should be accessible after a course in advanced calculus.  Time permitting, a few words about the degree in infinitely many dimensions will be given.

Laszlo Liptak:



Tanush Shaska:
Let $X_g$ be an irreducible, smooth, projective curve of genus $g \geq 3$, defined over the complex field. We denote by $M_g$ the coarse moduli space of smooth curves of genus $g$ and by $Aut (X_g)$ the automorphism group of $X_g$. Each group $G \le \Aut(X_g)$ acts faithfully on the $g$-dimensional vector space of holomorphic differential forms on $X_g$.

The locus of curves in $M_g$ with fixed automorphism group consists of finitely many components; to determine their number requires mapping class group action on generating systems.  We denote by $M_g (G, \sigma)$ be the sublocus in $\M_g$ of all the genus $g$ curves $\X$ with  $G  \hookrightarrow \Aut(\X)$ and signature $\sigma$. The main goal of this talk is to describe  the loci $\M_g (G, \sigma) $ in terms of the theta nulls for any given $g$, $G$, and $\sigma$.

Fioralba Cakoni:
Inverse scattering theory has been a particularly active area in applied mathematics for the past twenty five years. The aim of research in this field has been to not only detect but also to identify unknown objects through the use of acoustic, electromagnetic or elastic waves.  Mathematically, such problems lead to nonlinear and severely ill-posed equations. Until a few years ago, essentially all existing algorithms for target identification were based on either a weak scattering approximation or on the use of nonlinear optimization techniques. In recent years alternative methods for imaging have been developed which avoid incorrect model assumptions inherent in weak scattering approximations and,  as oppose to nonlinear optimization techniques, do not require a priori information. Such methods come under the general title of qualitative methods in inverse scattering theory. This lecture will provide a basic  introduction of qualitative methods  together with numerical examples showing the practicality 
of these approaches to the problem of target identification and finally  discuss some related open problems and opportunities.


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