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SURC / YMC 2003
August 8-10
at
Abstractsof
Talks and Posters
Talk/Presentation by
(Canton, NY)‘‘The Shape of Space" Author of
Location:(The Ohio State University)
Date: 9:30 amTime:
Everyone is cordially invited !
Hitchcock Hall 131
Sat., August 9 , 2003th
Sponsored by the NSF/VIGRE Program of the Department of Mathematics at OSU.
Recent satellite data are beginning to reveal the
introduce curved space, using physical
models and interactive 3D graphics to
build intuition and demonstrate some
surprising visual effects. We’ll see how
curvature and topology of the universe.
measurements of the cosmic microwave
background radiation are determining the curvature
computer games to introduce the concept
of the universe to unprecedented precision.
The second half of the presentation will use
of a finite, multiconnected universe, and
Abstract:
we will see how the same satellite data
suggest that the real universe may be
multiconnected. For undergraduates in
(graduate students and faculty also welcome).
The first half of this presentation will
mathematics, physics and astronomy.
Younes Abouyaala
Title of Poster Presentation: The Aggregates of George Cantor
Abstract:This is a poster presentation of my research project entitled" The Aggregates of GeorgeCantor" supported by the BMCC/CUNY Mathematics Department MSEIP StudentResearch Program. The project recalls the work done by the German mathematicianGeorge Cantor (1845-1918) in the field of set theory. It examines some of the basic ideasof set theory such as: finite and infinite sets, cardinal numbers, equivalency between sets,using a historical approach which includes proofs of set theorems. This approachemphasizes the outstanding accomplishment of George Cantor, providing an example ofhis mathematical creativity.
Jonas Azzam
Title of Talk: Stabilization of the Beam Equation
Abstract:We start with a beam equation with boundary control, and a continuous time feedbackthat exponentially stabilizes the beam. We consider a sample-and-hold version of thefeedback, with sampling time tau. We consider the following question: Does thesample-and-hold version also stabilize the beam if tau >0 is small enough? We show thatif there is some damping in the beam, the answer is yes.
Melissa Banister
Title of Talk: On Factorization Properties of Congruence Monoids
Abstract:This summer, the REU at Trinity University has examined various properties ofcongruence monoids; in particular, we have studied the factorization properties ofcongruence monoids, and have sought to find necessary and sufficient conditions forwhen a singular monoid is half-factorial.
Nick Bauer, Paul Carmany, and Peter Landry
Title of Talk: Progress and Discoveries in Relative Difference Sets
Abstract:Difference sets, relative difference sets, and some basic important results relating to thesetopics will be introduced. Then, we will describe progress towards a multiplier theoremfor relative difference sets. Finally, we will present two non-abelian (8,8,8,1) relativedifference sets that we have discovered.
Prerna Bihani
Title of talk: Inverse Problem for Upper Banach Density
Abstract:The research project has been about applying methods of nonstandard
analysis to additive number theory. As the faculty PI Renling Jin recentlydeveloped a general scheme with the help of nonstandard methods for deriv-ing theorems about Banach density parallel to every existing theorem aboutShnirelman density or lower density (see R. Jin, ”Nonstandard methods forupper Banach density problems,” ’The Journal of Number Theory,’ 91 (2001),20–38), there are many interesting concrete questions about Banach densityremaining to be solved using the scheme. For an infinite set A of naturalnumbers, one can measure its size by various densities. Several importanttheorems in recent decades indicate an inverse phenomenon, according towhich if the size of A+A is small, then the set A must have some rigid struc-ture. However, all these theorems are either about Shnirelman density, lowerdensity, or finite sets. With the help of nonstandard methods, the projectinvestigators have clarified the structure of a set A when the Banach densityof A+A is small.
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Suzanne Borgschulte
Title of Talk: Panting and Resonance in Canine Respiration
Abstract:Past studies have claimed that dogs pant at the resonant frequency of their chest-lungsystems. This is said to be necessary in order to breathe most efficiently so as not togenerate more heat than is dissipated. It can be shown analytically that under reasonableassumptions the resonant frequency of a dog's chest-lung system scales inversely with theradius of the lung. Using this conclusion, we investigate the phenomenon of pantingthrough modeling, numerical simulation, and analysis of published data, in order todetermine the validity of the resonant frequency hypothesis.
Joseph Breen
Title of Talk: Discrete-Time Modeling of Time-Dependent Queueing Systems
Abstract:The presentation will describe the basic ideas of Markov chains, and of homogeneous andinhomogeneous queueing systems. The use of a discrete-time model to obtainapproximate results for inhomogeneous queues with continuous service time distributionswill then be explained, and its computer implementation will be described.
Nick Bauer, Paul Carmany, and Peter Landry
Title of Talk: Progress and Discoveries in Relative Difference Sets
Abstract:Difference sets, relative difference sets, and some basic important results relating to thesetopics will be introduced. Then, we will describe progress towards a multiplier theoremfor relative difference sets. Finally, we will present two non-abelian (8,8,8,1) relativedifference sets that we have discovered.
Jonathan Chaika
Title of Talk: Factorization of Congruence Monoids
Abstract:This talk will deal with showing when Congruence Monoids i.e. a monoid in which all ofthe elements are congruent to a multiplicatively closed subset of Z mod nZ (with 1thrown in). This talk will deal with when these monoids are factorial, half-factorial, andcongruence half-fractorial. It will also discuss the elasticity of congruence monoids andwhen they are fully elastic. The particular congruence monoids being focused on for thistalk will be singular and semi-regular congruence monoids, i.e. they contain elements thatare not in the group of multiplicative units of Z mod nZ.
Lauren M. Childs
Title of Poster Presentation: Activation Level-Dependent Mutation Rates in the AffinityMaturation of B-cells
Abstract:During a specific immune response, classes of B-cells are activated and then migrate tothe germinal centers of lymph nodes. Within these germinal centers, the activated B-cellsundergo affinity maturation, a process in which the correspondence between therecognition sites of B-cells and the invading pathogens dramatically increases. For theoptimal response to an infection, affinity maturation, part of the germinal center reaction(GCR), must occur quickly. Current hypotheses indicate that the RNA-editor Activation-induced deaminase (AID) initiates the processes that alter the recognition sites of B-cells.It is known that AID facilitates both somatic hypermutation (SHM), the method primarilyresponsible for modifications of the recognition sites of B-Cells during the GCR, andclass-switching recombination (CSR), another method responsible for alterations in therecognition sites. AID is thought to invoke a double-stand break in a DNA moleculewithin a break-repair pathway that is essential to SHM and CSR. This project used amodel to compare affinity maturation with and without activation, where the doublestrand-break initiated by AID was considered activation. The purpose was to determinewhether AID influenced affinity maturation through its activation. In accordance withthe biological necessity of AID, the AID-activated simulations revealed a shorter,biologically-preferable time for B-cells to reach the optimal level of affinity for thepathogen.
Romain Coulibaly
Title of Poster Presentation: Quantum Planes, Trains, and Automobiles
Abstract:The quantum plane is a noncommutative ring generated by two variables x and y subjectto the relation yx=qxy where q is an arbitrary real number. In the classical commutativecase, q=1. Elements of the quantum plane are called polynomials. A polynomical of theform ax^2 + bxy + cy^2 is called a quadratic form. Necessary and sufficient conditionsfor a quadratic form to be irreducible are provided. Every prime polynomial isirreducible, but the value of q plays a key role in determining if irreducible polynomialsare prime. Detailed descriptions are given for the quadratic forms x^2 + cy^2 and x^2 +bxy + y^2. Graphs indicate values of q, b, and c where these quadratic forms arereducible (in red), irreducible (in yellow), or prime (in green).
Gary L. Crum
Title of Talk: Triangle inequalities on the 2-sphere
Abstract:The triangle inequalities in the Euclidean plane R^2 are a well-known part of analysis. IfA, B and C are the vertices of a geodesic triangle in R^2, and d(X,Y) denotes the distancefrom X to Y, then d(A,C) <= d(A,B) + d(B,C). This inequality is a necessary andsufficient condition for existence of a triangle in the plane with the given side lengths.Here we investigate abstract triangles embedded in the 2-sphere S^2 with edges are alonggeodesic arcs, where triangle sides are allowed to wrap around the sphere. We thus allowangles greater than 2*Pi.
These abstract triangles appear in the study of topology and metrics, in locally sphericalstructures with three conical singularities. Preliminary results show the solution space forthese abstract triangles is connected, but not convex and not star-shaped. The inequalitiescan be derived by using dual triangles with side lengths satisfying the regular inequalitiesa<=b+c, b<=a+c, c<=a+b, with the additional condition that a+b+c<2*Pi. The solutionspace is a union of tetrahedrons, each in a quadrants of side length Pi, and the regions indifferent quadrants are related by composition of reflections.
Thomas Davis
Title of Talk: Optimal Design of a Gaupillaud-Type Layered Media
Abstract:In this work we identify all the optimal designs that provide the smallest stress amplitudein a three-layered elastic strip subjected to transient loading. We accomplish this byusing explicit formulas for the stress, previously obtained using the method ofcharacteristics.
We then demonstrate that the derived analytical results are successfully supported bycomputational experiments using Maple and Matlab software. We conclude by makingcomparative observations with the previously obtained results for the two-layer case.
Adele Douglin
Title of Poster Presentation: Numerical Integration of the Levy Motion Driven SDE withDrift Boundaries for Motile Microbes
Abstract:A random walk solution was constructed for a Levy-motion driven, stochastic differentialequation (SDE) with drift and subject to alpha-stable sticky boundaries. The modelrepresents motile bacteria that swim in the aqueous phase but that stick to the walls forsome finite period of time when contact is made. The numerical algorithm wasimplemented in C++. Maple Software was used to graph the movements of the microbesand ensure that they followed a Levy motion. The code evaluates the random walks andrecords when and where each microbe sticks to the walls of the flow tube, and for whatduration it stays adsorbed. This information is very useful because the microbes are morelikely to transfer genetic information when they are adsorbed. The code is currentlybeing modified to compute first passage time distributions that will the user to see howlong it takes each microbe to pass a certain point in the flow field.
Inessa Epstein
Title of Talk: On two classes of graphs associated with zero divisor semigroups
Abstract:To each commutative zero divisor semigroup S, we associate a simple graph G(S). Thevertices of the graph are the nonzero elements of S and there is an edge between twovertices a,b if and only if ab = 0. We examine the relationship between the semigrouptheoretic properties and the graph theoretic properties of G(S). In particular, we willpresent results pertaining to properties of semigroups associated with the class completer-partite graphs and determine which graphs in the class of triangulated planar graphs aregraphs of semigroups.
Mihaela S. Facaianu
Title of Talk: The Electoral College and the Proportional Plan: A New Analysis Usingthe Shapley-Shubik Power Index
Abstract:This paper analyzes the Electoral College and the Proportional Plan in order to determinewhich method is better for selecting the United States president from an individual voter'sstandpoint. This analysis is conducted using voting theory elements, the Shapley-Shubikpower index, and election data from three different elections years: 1984, 1992, and2000. The two voting systems are set up as weighted voting games in which the 50 statesand the District of Columbia are the players. Using a C++ computer program, eachstate's Shapley-Shubik index is calculated. The same algorithm is used to compute therelative power of an individual in each state. Simulations are created by introducingrandomness in the data for the three election years. The data indicated different resultsbased on the election year. For a landslide election, the proportional plan seems to besimilar if not slightly better than the current system. For a close election, the proportionalplan appears to even out the power discrepancies between the states.
Jennifer Ferrell
Title of Talk: The Use of Statistical Methods and Data Mining in Applied Mathematical
Abstract:Until the development of formalism, mathematical theory was related to real applications.The purpose of this research project is to use different statistical models to examinerelationships in the data of a clinical trial. The data include demographic variables,treatment variables, clinical measurements, and quality of life data. We will use differenttechniques, including linear models, tests for categorical data, and kernel densityestimation. By using these techniques, we will show the effectiveness of the placement ofpacemakers and implantable cardioverter defibrillator (ICD) on patients‚ health. Alternateplacing of ICDs and Pacemakers on certain places of the heart may proof more beneficialto patients than first thought. Through analyzing the data, we will be able to determine ifthe overall quality of life improves with the alternate placement. Linear Models and Chi-Square tests will aide in this analysis. Data mining techniques such as neural networkanalysis an d C.5 rule induction will also be used. This application of mathematicalprocesses illustrates the importance of mathematics in numerous fields.
Kerri Fletcher
Title of Poster Presentation: A Quantitative Study of Student Retention
Abstract:As a part of a summer research program at Pacific Lutheran University, we conducted aquantitative study of student retention at PLU using past eight years of student entry,graduation, and withdrawal data. Approaches for our study include hypothesis testing,regression analysis, as well as risk assessment. Our findings will be of interest to bothresearchers and university administrators.
Amanda Geiser
Title of Talk: Classifying Semigroups Associated to Refinements of a Star Graph
Abstract:A zero divisor semigroup is a semigroup with zero where every element is a zero divisor.To each commutative zero divisor semigroup S, we associate a simple graph Gamma(S).The vertices of the graph are the nonzero zero divisors of S and there is an edge betweentwo vertices a and b if and only if ab = 0. Via examination of the relationship betweenthe semigroup theoretic properties and graph theoretic properties of Gamma(S), weclassify those semigroups where Gamma(S) is a star graph or the refinement of a stargraph.
Amy Heaton
Title of Talk: Fluid Transport Through Sea Ice
Abstract:The sea ice that covers the polar regions of the world is a composite material of pure icewith incorporated brine and air inclusions. Fluid transport through sea ice is offundamental importance in air-sea-ice interactions, in supplying nutrients to sea ice algae,in desalination processes, in sea ice production and decay, and in thermal transportthrough sea ice. However, until now, little has been understood theoretically about theeffective fluid transport properties of sea ice. Current work using continuum percolationtheory has yielded striking classification of the critical exponent k characterizing thebehavior of fluid permeability of sea ice near its critical temperature. For experimentsand processes involving macroscopic transport on geophysical length scales, k takes theuniversal lattice value of 2.0. While, for mesoscopic transport on smaller scales relevantto local biological processes, non-universal continuum values with k>2 can be obtained.Indeed, Arctic field experiments and laboratory centrifuge experiments yield matchingresults.
Devin Henson
Title of Talk: Group Divisible Designs block size 4 and with small number of groups
Abstract:Constructing group divisible designs where the number of groups is less than the blocksize is a challenging problem. Recently Fu and Rogers have completed the existenceproblem for block size three. Very little is known for block size four except the work byHurd and Sarvate. We will review this work and present the new result we have obtainedso far including the complete results on some special GDDS, so called even, odd andmixed GDDS with block size four and three groups.
Aaron Johnson
Title of Poster Presentation: Ground Truth Seismic Events in Tanzania and Ethiopia
Abstract:Determining accurate seismic locations with representative uncertainty estimates is offundamental importance to ground-based nuclear explosion monitoring. In this project,we are developing a catalog of reference events (ground-truth) in the northeast Africanarea where reference event coverage is exceptionally poor due to the limited stationcoverage by historic networks. The results of this project will enable the seismicmonitoring community to enhance their operational capability to monitor for nuclear testsin North Africa and the Middle East by increasing their ability to accurately locate andidentify seismic events in these regions.
These events can be located using regression or inverse techniques however the depths ofevents are hard to constrain. To fix the depth more accurately the orientation of the faultneeds to be determined and then modeling of the seismograms by solving the waveequation at different depths can constrain the depth more accurately.
Buddy Lagani
Title of Talk: Solving the Diophantine Equation
Abstract:Given an integer n, we may derive the solutions of the Diophantine equation frac{1}{x}+ \\frac{1}{y} = \\frac{1}{n} by simply examining the divisors of n^{2}. We will alsodiscuss methods of solving the general Diophantine equation sum_{i=1}^{k}frac{1}{x_{i}} = \\frac{1}{n} for its nontrivial solutions.
Nick Bauer, Paul Carmany, and Peter Landry
Title of Talk: Progress and Discoveries in Relative Difference Sets
Abstract:Difference sets, relative difference sets, and some basic important results relating to thesetopics will be introduced. Then, we will describe progress towards a multiplier theoremfor relative difference sets. Finally, we will present two non-abelian (8,8,8,1) relativedifference sets that we have discovered.
Catherine Lichten
Title of Poster Presentation: An Application of the Geometric Heat Equation for RapidEdge Detection
Abstract:In recent years, the application of differential equations to image processing hasundergone significant growth, leading to a number of results related to image analysis,processing and comparison, and computer vision. Our goal is to develop an edgedetection approach capable of providing large capture range and dealing with images ofdifferent sizes. We begin with a convex, smooth, closed curve, C(t,p) = C[x(t,p), y(t,p)],where t parameterizes the family of curves and p parameterizes each particular curve.We base our model on the geometric heat equation, dC/dt= KN, where N denotes thenormal vector and K is curvature. If N is the inward normal, the curve shrinks and if it isthe outward normal, the curve enlarges in the direction of the normal vector. A penaltyfunction is used in our model to detect the presence of objects, control the curve‚sevolvement, and segment the image if more than one object is encountered. On the basisof the model, we developed an algorithm applying finite differences. UsingMathematica, we created a tool capable of detecting the edges of multiple objects locatedin an image. Our algorithm is advantageous in that it has low run-time, large capturerange, and requires no prior knowledge of the image. Also, it can detect one or multipleobjects, with varying size and position in the image (although there are certain objectrelations that the model cannot yet resolve). So far, the method has been applied to blackand white images but could be adapted to gray level images as well. Research isunderway to analyze the algorithm's stability and to improve the model so that the toolfully detects all boundary concavities, object holes, and object relations.
William Meyerson
Title of Talk: Half-Factorial Properties of Congruence Monoids
Abstract:The classification of half-factorial congruence monoids has been somewhat known for awhile in the regular case, but semi-regular and singular congruence monoids haveresisted classification until very recently. I seek to help resolve this problem byclassifying half-factorial singular and semiregular congruence monoids.
Erin Militzer
Title of Talk: Two Families of Randomly Decomposable Graphs
Abstract:A graph G, is randomly H-decomposable if any subgraph isomorphic to H is part of anH-decomposition. The set of all randomly H-decomposable graphs is denoted by RD(H).We examine RD(H) where H is one of the following: (1) H = K_{m}P_{e}, a graphconstructed by identifying a vertex of the complete graph K_{m} with an end of the pathP_{e} or (2) H = H_{0} + P_{1} where RD(H_{0}) is known.
Lee Mitchell
Title of Poster Presentation: Multiple Objective Simulated Annealing applied to SeismicWaveform modeling in the Baikal Rift.
Abstract:The Baikal Rift region is geologically complicated. The rift, an archean craton, and 6kmof sediment in the Baikal Lake influence seismic paths. This structure requires modelingof seismic waveforms that will focus on finding an optimal solution efficiently. Thisresearch has two components, solving the wave equation using a discrete wavenumberintegration technique and a discussion of Multiple Objective Optimization and SimulatedAnnealing.
The eventual goal is to model six events recorded from along the Rift axis at Talaya(TLY) in Russia at distances ranging from 400-1300 km using such parameters as phasetiming and amplitude as well as a point-to-point comparison of the wave shape.
Michael H. Moriarty
Title of Talk: Cages of degree k are k-edge-connected
Abstract:This talk determines the edge-connectivity of cages, regular graphs of specified girth withminimum order. I show that cages of degree k are k-edge-connected.
Adenrele Oloye
Title of Poster Presentation: Introduction To Turing Machines
Abstract:The purpose of this project is to design a Turing machine that verifies set containment,and to provide awareness about the workings of a Turing machine. The researcher willdefine and describe the workings of Turing machines. The researcher will also design aTuring machine that generates a particular output (010010100101001) and a Turingmachine that verifies set containment such that, given two finite, non-empty sets A andB, the verification of A being a subset of B can be carried out with the Turing machine.
Lindsey Olson (joined by Kerri Fletcher)
Title of Poster Presentation: A quantitative study of student retention
Abstract:As a part of summer undergraduate research program at Pacific Lutheran University, weconduct a quantitative study of student retention at PLU using past eight years of studententry, graduation, and withdrawal data. Approaches for our study include hypothesistesting, regression analysis, as well as risk assessment. Our findings will be of interest toboth researchers and university administrators.
Maria Osorio
Title of Poster Presentation: Modeling Bacterial Conjugation
Abstract:Bacterial conjugation, part of three types of horizontal gene transfer, occurs via an extra-chromosomal DNA molecule called a plasmid through direct cellular contact between adonor and a recipient. Ecologically, the spread of genetic information by engineeredmicroorganisms (GEM), as well as the transfer and distribution of natural plasmids hasgenerated interest in mathematically modeling conjugation. Mathematical modeling ofmicrobial populations not only serves to reflect a general pattern, but also quantifiesthrough its constants, a system's characteristics. Thus, by continuously improving and/ormodifying the models used to parallel observed conjugation data, we ameliorate themethods for accounting attributions to conjugation. The data used here fit an alternateduse of a simple modified mass action model and an additive Fermi-logistic equation. Weuse a differential form of the logistic or Verhulst equation, to model conjugationaccounting for both the growth and decay of populations. We used nonlinear regressionsto find parameters, such as growth and decay rates. We expect to improve upon theprevious models of the data used by providing one equation as opposed to the two aboveto determine the conjugation rate constant.
Ryan Ottman
Title of Talk: A Conjecture on Intrinsically Linked Graphs
Abstract:I will discuss the conjecture mentioned in Adam's "The Knot Book" that removing anyvertex from an intrinsically knotted graph results in an intrinsically linked graph. Then Iwill discuss intrinsic linking with an unused vertex and suggest ways to prove theconjecture or special cases of the conjecture.
Jeffrey Overbey
Title of Poster Presentation: On the Keyspace of the Hill Cipher
Abstract:In its most general form, the Hill cipher's keyspace consists of all matrices of a givendimension that are invertible over Z_m. We present a formula for the number of suchmatrices, outlining a proof that uses only undergraduate mathematics. We also comparethis result with the total number of matrices and the number of involutory matrices for agiven dimension and modulus, identifying the effects of change in dimension andmodulus on the order of the keyspace.
Candice Price
Title of Talk: Coloring Invariant and Determinants in Knot Theory
Abstract:I have studied the Coloring Invariant and matrices associated with Knot theory. I havelooked at different conjectures associated with the Coloring Invariant and a specialcategory of knots called pretzel knots which include: 1) The determinant of a (m, n)pretzel is |n +m|. 2) A (n, m) pretzel is a link when both m and n are even or odd.Otherwise the pretzel is a link. And, 3) the determinant is divisible by 2 if the pretzel is alink.
Alex Rand
Title of Talk: Extinction Dynamics Due to an Invading Predator
Abstract:In many ecosystems, invasion of a new predator has led to the extirpation or localextinction of native species. The Rosenzweig-MacArthur predator-prey model withHolling Type II predation on multiple prey species is used in this study to classify suchextinction mechanisms by an invading predator. Prey species are assumed to have widelyvaried timescales in reproduction, which makes the classification possible in terms ofsingular orbits. Detailed analyses on singular orbits are carried out for models of one,two, and three prey species and the results are then generalized to n prey species. Theresults can be used to predict which prey species will be driven to extinction and in whatorder these extinctions will occur.
Jason M. Richwine
Title of Talk: Shapley/Owen Voting Power Analysis for Electoral College
Abstract:I have been studying the statistical methods of George Rabinowitz and Stuart MacDonaldin their 1986 American Political Science Review article "The Power of the States in theU. S. Presidential Elections". Their paper uses presidential election data from 1944 to1980 to determine the most powerful states in the Electoral College in terms of how oftenthey are "pivotal" to the election outcome. Massachusetts, for instance, has a largenumber of electoral votes but has little power because it is so reliably Democratic. I havebeen redoing their calculations using the much newer data since 1980. A lot hasobviously changed in the twenty years that provide new data for my study. 1980 is nowviewed as the affirmation of a political realignment that had begun with the presidentialcampaigns of Richard Nixon. Conservative southern states, having been solidlyDemocratic for a hundred years, have now moved to the Republican Party. Additionally,northern liberal Republicans have become a dying breed. These movements polarizedthe American electorate, because both parties lost much of their ideological balance.Such a political shift will have a profound impact on the relative power of states in theElectoral College; this makes updating Rabinowitz and MacDonald's study all the morenecessary. I have not completely finished the analysis as of yet, but I am on schedule tocomplete the project by the end of July. I expect the power of states in the ElectoralCollege to have substantially changed since 1980.
This study has required statistics and game theory to analyze the data. It uses Owen'smodification of the Shapley value to account for which states are more likely to bepivotal in electoral voting. A principal components analysis is used to describe thelikelihood of certain states being pivotal.
Christopher Scheper
Title of Poster Presentation: N-Dimensional Medians and Convex Functions
Abstract:The classical definition of a median in R^1 is defined in the following way: Given a setS, where S={x1,x2,∑,xk} and x1<x2<∑<xk, the median is the middle term. The idea ofa middle term does not work well in more than one dimension, for the median will not bepreserved through coordinate changes. The median can be described as the point z thatminimizes the following function, f(z) = |z-x1| + |z-x2| +∑+|z-xk|. Using this definition,the median will be preserved through any kind of coordinate changes, translational, androtational motion. This definition applies to all spaces R^n.
The goal of the project is to construct an algorithm that will find the median given arandom number of points in R^n, where n is randomly generated also. Because theminimizing function is comprised of convex functions, we can exploit this in order toprove that the function f gives a unique median.
Martin J. Senica
Title of Poster Presentation: Bifurcations of the Henon Map: Routes to Chaos
Abstract:Many simple physical systems exhibit chaotic behavior. A simplification of a weatherprediction model due to Lorenz is the Henon map f(x,y)=(1-ax^2+y, bx). This mappinghas been devised by the theoretical astronomer Michael Henon to illuminate themicrostructure of chaotic attractors and is the subject of intensive research. Theimportance of the characteristics of the Henon map is that it leads to a betterunderstanding of chaotic behavior.
Our methodology was that we investigated the existence and transformation of thechaotic attractor of the Henon mapping f(x,y)=(1-ax^2+y, bx), as 0<a<2 and |b|<1, withnumerical methods.
The classical parameter values are a=1.4 and b=0.3. For these values, the mapping iscontracting the area and has a trapping region, so it exhibits an attractor. However, for allvalues |b|<1, and for a wide range of values of a, the mapping is still contracting the area,and still has a trapping region. The global dependence on (a,b) is largely unexplored inexisting literature. We investigated the whole set of parameter values (a,b) that generatechaotic attractors. Our results are shown through bifurcation diagrams.
We developed computer programs using Maple software to explore the dependence ofdynamics on the parameters. We introduced a new exploration tool, which is aMandelbrot-type set of parameter values (a,b) that generate chaotic attractors. Thisparameter-set has a fairly regular boundary, except for some portions of it, which arefractals. Various regions of this parameter set yield various classes of attractors. We alsodeveloped three-dimensional bifurcation diagrams that show the dependence of theHenon attractor on both parameter values, which allow a better understanding of doublecrises and their loci.
Blerta Shtylla
Title of Talk: Knots, ortho-projection matrices and Jones polynomials.
Abstract:We present an algorithm that converts an alternating knot diagram into an ortho-projection matrix over the two-element field, and we explain how to obtain the Jonespolynomial of the knot from the matrix. We give examples of ortho-projection matricesthat do not arise from knot diagrams, and examine the Jones polynomials that correspondto these matrices.
Sarah Srodulski
Title of Talk: The Effects of a Constant Alternative Food Source on Ecosystems WithOne Predator and One Prey
Abstract:In this presentation, long term behaviors of a predator-prey model in which the predatorhas a varying amount of one main prey and a constant amount of alternative prey areconsidered. In the case of the 1950's invasion of the brown tree snake Boiga Irregularison Guam, the varying prey is the native population of birds and reptiles and thedomesticated animals are the constant alternative food source. The model of thissituation is loosely based on the devastation of indigenous species on Guam. Throughsingular orbit analysis, conditions can be given to determine which of these long-termoutcomes, either coexistence or extinction of one or more species, will occur for a giveninitial condition for the equation.
David Stroup
Title of Talk: Using the Desktop Hypercube to Analyze Spatially Distributed QueueingModels
Abstract:The Desktop Hypercube allows an individual to analyze and redesign patrol zones ofdifferent emergency vehicles. The name hypercube comes from the state space which,unlike the conventional stochastic flow diagram, can be mapped onto the vertices of anN-dimensional cube.
Wendy Wang
Title of Talk: Minimum Rank of Positive Semi-Definite Matrices with a PrescribedGraph
Abstract:A complex nxn matrix A = [aij] is said to be combinatorially symmetric if for i ≠j, aij ≠ 0 implies aji ≠ 0. We associate a simple graph G to acombinatorially symmetric matrix A such that V(G) = {1, 2, ∑, n} and join vertices i andj if and only if aij ≠ 0. The graph is independent of the diagonal entries of A.Define P(G) to be the class of all positive semi-definite matrices associated with a givengraph G. Denote #(G) = min {rank A | A Є P(G) } the minimum rank of G.Results about the minimum rank of certain classes of graphs and related topics will bepresented in this talk.
Alex Yuffa
Title of Talk: A New Formulation of the Integral Equation Method for ElectromagneticScattering
Abstract:We have developed a new formulation of the integral equation method by choosingelectric field and its normal derivative as the boundary unknowns. Our formulationprovides a significant computational advantage over the standard Stratton-Chuformulation. Because our impedance matrix has three diagonal blocks and is 80% sparse.We used our new formulation to model the behavior of light emanating from an NSOMfiber tip as a test of our formalism.
