Carolina Math Seminar (CMS)

## Organizers

### Rigoberto Flórez

Phone: (843) 953 5034

Antara Mukherjee

### USC Lancaster

Jason Holt
jholt@mailbox.sc.edu

Andrew Yingst
yingst@mailbox.sc.edu

### USC Salkehatchie

Wei-Kai Lai
laiw@mailbox.sc.edu

Fidele Ngwane
ngwanef@mailbox.sc.edu

### Benedict College

Alexandru Gabriel Atim
atima@benedict.edu

### Columbia College

Virginia Johnson
vjohnson@columbiasc.edu

### Francis Marion University

Jeremiah Bartz
jbartz@fmarion.edu

### Lander University

Josie Ryan
pryan@lander.edu

California University of Pennsylvania

Leandro Junes
junes@calu.edu

 Fall Meeting at Citadel October 26, 2012 Rachel Hudson Cracking Codes In this talk I will discuss some basic aspects of cryptology. I will show how to generate a substitution cipher mathematically using a linear function, and also give an example of how to crack a substitution cipher when the encrypting function is unknown.   Matthew Ziemke An Introduction to Fractals and The Mandelbrot Set When we initially study geometry, we typically study objects such as lines, circles, and rectangles. Unfortunately these shapes rarely show up in nature. For example, what would you say is the shape of Mt. Everest? Or maybe the shape of the tree outside your window? One property Mt. Everest and the tree have in common is self-similarity, i.e., smaller sections of the shape are similar to the whole (a tree limb looks similar to a tree). We now designate shapes such as these as fractals. The Mandelbrot set is a compact subset of the complex plane with many interesting properties. The drive behind the definition of the Mandelbrot set was to have it be a "catalog" of a particular class of fractals but with the aid of computers in the early 1980's we soon realized the set seemed to be a fractal itself. In this seminar we will discuss some of the known properties of the Mandelbrot set along with its connection to fractals. László Székely Mixed orthogonal arrays and Sperner theory The well-known Bollobas-Lubbel-Yamamoto-Meshalkin inequality has been extended from Sperner families to M inequalities for M-part Sperner families, and recently to M choose k inequalities for k-dimensional M-part Sperner multi-families. It turns out that equality holds for all inequalities iff the Sperner multi-family is homogeneous and corresponds to a mixed orthogonal array. Mixed orthogonal arrays are used by statisticians to design experiments. Joint work with Harout Aydinian and Eva Czabarka.   Éva Czabarka Phylogenetic trees and Stirling numbers P.L. Erdos and L.A. Szekely provided a bijection between rooted semi-labeled trees and set partitions. This, with the asymptotic normality of the Stirling numbers of the second kind (Harper) translates into the asymptotic normality of rooted leaf-labeled trees with a fixed number of vertices and a variable number of internal vertices. Phylogenetic trees are rooted leaf-labeled trees where the only internal vertex that can have degree 2 is the root. We apply Harper's method and the Erd}os-Szekely bijection to obtain the asymptotic normality of phylogenetic trees in several sense. This is joint work with P.L. Erdos, V. Johnson, A. Kupczok and L.A. Szekely. Todd Wittman Variational Methods in Image Processing I will discuss how differential equations and the calculus of variations are used to solve problems in image processing. I will present the Rudin-Osher-Fatemi Total Variation (TV) denoising model and then discuss extensions of the model to problems in resolution enhancement.   Fall Meeting at USC Lancaster September 14, 2012 Emma Jane Riddle Two Applications of Mathematics to Business The first example shows how queuing theory can be used to analyze the waiting line at a drive-up teller window. The data for the problem includes the number of customers arriving per hour (arrival rate) and the number of customers the teller can serve in an hour (service rate). We will determine the average waiting time, the average length of the line, the utilization of the teller, and the idle time of the teller. Adding a second teller window would decrease waiting time but would also increase costs for the bank. The second example involves breakeven analysis. A business obtains revenue by selling goods or services. Every business pays fixed costs, such as rent, which must be paid even if no products are sold. Businesses also pay variable costs, such as materials and labor. Variable costs increase with the number of units produced. The breakeven point is the number of units that must be produced to pay fixed and variable costs, so that the business does not lose money. We will use algebra to find the formula for the breakeven point. If time permits, "what-if" analysis will be used to show the impact of pricing assumptions on the breakeven point. Andrew Yingst Measure, Category, Cardinality: Ubiquitous things that can't be found When a mathematician says that "almost everything" of a certain type has a certain property, she may be describing one of a few concepts. In this talk, aimed at the introductory level, we discuss the three most common interpretations of this phrase, we give examples of how these definitions lead to the knowledge that things may exist without the ability to construct any examples of them, and we'll touch on the surprising fact that almost every point might have some property in one sense, while almost every point does not have that same property in another. Allan Pangburn A New Algorithm for Maximum Flow Distribution Networks: The Modified Push Algorithm Abstract: Over the years researchers and programmers have created and revised algorithms to solve maximum flow network (MFN) problems. These problems contain: source node(s), transshipment (ordinary) nodes , sink node(s), and arcs with limited capacities. The objective is to send the maximum amount of flow from the source nodes, through the ordinary nodes to reach the termination nodes. A variation of MFN is a maximum flow distribution network (MFDN) problem. These problems contain the same nodes as MFN, but with additional nodes called distribution nodes. These nodes have only one ow entering and multiple flows leaving, but the leaving flows are proportion to the entering flow. In this presentation, we present a new method to determine an initial feasible flow by revising: Goldberg and Tarjan's algorithm of 1988, and Sheu, Ting, and Wang's algorithm of 2006. Major revisions include: defining a predetermined search order, resetting capacities on arcs, two formulas to lessen the amount of excess flow, and defining a new subgraph. Spring Meeting at USC Salkehatchie April 13, 2012 Joshua Smoak An Identity for Exradii In our talk, we will start with some basic properties of exradii of a triangle. We then will use a theorem introduced by Johnson (Modern Geometry, Dover, 2007) to prove an identity proposed in The American Mathematical Monthly Volume 118, No.8.     Kuo-Wei Yao Low Rank Matrix Approximation and Its Applications in Image Processing The aim of this paper is to find efficient methods to estimate approximation errors of low rank matrix approximations of a m × n matrix A Consider A ≈Ak formed by the first k singular values and their corresponding singular vectors where k is much smaller than the rank of A. We study the relation of the ratio of the kth largest and the largest singular values, and the change of the approximation error of Ak. We have applied our finding to applications in image processing to determine the rank of an approximation matrix that gives an acceptable image with the least storage possible. Results will be reported in the paper.   Virginia Johnson Counting gene trees Gene trees used in biology to describe the evolution of genetic material throughout different species. Internal nodes of the tree correspond to speciation or duplication events, and the leaves are labeled with the name of the species the gene comes from. Consequently, gene trees are leaf-labeled trees which ideally but not necessarily are rooted, the root is the only vertex that may have degree 2, and labels in the label set may be used multiple times or not at all (the latter corresponding to deletion events). Otter in 1949 has proved a formula on unlabeled trees that connects counts of rooted trees to corresponding counts of unrooted trees. We generalize this formula for semi-labeled graphs, and use this to provide ordinary generating functions for gene trees (binary or non-binary, rooted or unrooted) and leaf-labeled trees (where internal nodes may have degree 2 even if they are not the root). This is joint work with Éva Czabarka, Peter L Erdös, Virginia Johnson and Vincent Moulton.   Upasana Kashyap Picard group of dual operator algebras We discuss the Picard group of dual (weak*-closed) operator algebras. We prove that for a weak*-closed function algebra A, the weak Picard group Picw(A) is a semidirect product of the automorphism group of A, and subgroup of Picw(A) consisting of symmetric equivalence bimodules. In particular we show that the weak Picard group of space of bounded analytical functions is isomorphic to the group of conformal automorphisms of the disk.   Paul Nietert Use of novel biostatistics methods by researchers publishing in general/internal medicine journals Novel statistical methods are constantly being developed within the context of biomedical research; however, the characteristics of biostatistics methods that have been adopted into the field of general / internal medicine (GIM) is unclear.  This study highlights the statistical journal articles, the statistical journals, and the types of statistical methods that appear to be having the most direct impact on GIM research.  Descriptive techniques, including analyses of articles’ keywords and controlled vocabulary terms, were used to characterize the articles published in statistics and probability journals that were subsequently referenced within GIM journal articles during a recent 10-year period (2000-2009).  From the 45 statistics and probability journals of interest, a total of 597 unique articles were identified as being cited by 900 (out of a total of about 10,501) unique GIM journal articles.  The most frequently cited statistical topics included general/other statistical methods, followed by epidemiologic methods, randomized trials, generalized linear models, meta-analysis, and missing data.  I will briefly summarize these methods and what their uses.  As statisticians continue to develop and refine techniques, the promotion and adoption of these methods should also be addressed so that their efforts spent in developing the methods are not done in vain.   Fidele Ngwane Towers and Type I curves Type I curves   and Towers of functions are  very useful in coding theory.  We  will discuss Type I curves, Towers of function  and  we  will show how to put Towers of functions into Type I curves. All the Towers of functions considered here are asymptotically good towers.   Previous Meetings Spring Meeting at Benedict College Feb 10, 2012 Gurcan Comert Predicting System States in Transportation Networks Traffic systems often exhibit nonlinearities and sometimes abrupt changes due to various factors such as traffic accidents, inclement weather conditions, and demand surges. Developing robust adaptive models for real-time system state prediction is a significant challenge. This research develops strategies and models to predict traffic conditions under abrupt changes using Hidden Markov and Time Series Models. The data set used has incident databases that reveal the causes of sudden changes in traffic parameters (e.g., speed, occupancy, and flow). In the models, all of the significant characteristics of the changes (e.g., duration, amplitude) are statistically detected and analyzed.   Eva Czabarka Orthogonal arrays and transversals With H. Aydinian, K. Engel, P.L. Erdos and L.A. Szekely  we investigated a packing problem in M-dimensional grids, where bounds are given for the number of allowed entries in axis-parallel directions (i.e. in a 1-dimensional subgrid). This concept is motivated by error correcting codes and more-part Sperner theory, and it is closely connected to orthogonal arrays. We extend this concept from 1 to d-dimension: the bounds are given on the number of allowed entries in a d-dimensional subgrid. We prove that there are packing arrays that always reach the natural upper bound for their size, and prove some related extremal results.   László Zsilinszky Infinities – Stuff That Make People Go Nuts A gentle introduction to some questions involving the notion of infinity in Mathematics, and Set-Theory.   Danny Rorabaugh Collatz Generalized: An Expansion of the 3x+1 Problem The focus of this 2010 undergraduate research project is a generalization of the Collatz conjecture – an unsolved number theory problem involving the “3x+1 function” on the positive integers: if x is odd then C(x) = 3x + 1; if x is even then C(x) = x/2. The Collatz conjecture is that, given any positive integer x, the infinite sequence {x,C(x),C(C(x)),...} – the trajectory of x - contains the number 1. Although the conjecture has been proven for x up to at least 1017, it remains unproven for all positive integers. This project investigates the problem within a broader context of the following “Ax+B function”: if x is odd then C(x) = Ax + B; if x is even then C(x) = x/2. Under this wider scope, the project explores the relationships between A, B and x and whether a trajectory contains 1, loops without reaching 1, or is unbounded with no positive integer occurring twice. Understanding these relationships may help to shed light on why the trajectory of every positive integer under the 3x+1 function contains one, if such is in fact the case.   Fall Meeting at Citadel Oct 28 2011 Kristopher Liggins On Generalization of Fibonacci Numbers   Chris Rufty A Japanese Ladder Game, a simple version In MAA FOCUS newsmagazine Vol.31, No.3 June/July 2011, Dr. Dougherty and Dr. Vasquez introduced a puzzle, known as Japanese Ladder Game. It is known that every Japanese ladder will provide a 1-1 mapping, and for any sequence at the bottom of the ladder, one can always find a minimum solution. By studying a simpler version, we were able to use our method to prove some of the old results. In this talk, we will introduce the way to play the game, and the basic Math related to this game. Finally, we will give two simple proofs of our theorems.   Ralph Howard Integral Geometry with Applications to Geometric Inequalities We outline a few of the basic results in integral geometry, and their application to geometric inequalities such as the Bonnesen inequality (a refinement of the isoperimetric inequality) and Fray-Milnor inequality on the total curvature of knots. If time permits some recent work on knot energy will be given. The talk should be accessible to undergraduates with a knowledge of vector calculus.   Chuck Groetsch Inverse problems, von Neumann’s theorem, and stable approximate evaluation of unbounded operators The mathematical framework for many linear inverse problems in the mathematical sciences requires the application of an unbounded operator to a vector which is not in its domain – an impossible task! We present a brief introduction to a general approach, based on a classical theorem of von Neumann, for addressing this difficulty.   Oleg Smirnov Lie Algebras, Groups Triple Systems: the Truth In the talk we consider functorial connections between the categories of  the Lie algebras, Lie groups, Lie triple systems, and Symmetric Spaces. We present a full subcategory of graded Lie algebras which is equivalent to the category of triple systems and discuss a similar connection between Lie groups and symmetric spaces.   Mei Chen Eigenpairs of Adjacency Matrices of Balanced Signed Graphs In this talk, we present results on eigenvalues λ  and their associated eigenvectors x of an adjacency matrix A of a balanced signed graph. A graph G =(V,E) consists of a set V of vertices and a set E of edges between two adjoined vertices. A signed graph is a graph for which each edge is labeled with either + or -. A signed graph is said to be balanced if there are an even number of negative signs in each cycle (a simple closed path). Signed graphs were first introduced and studied by F. Harary to handle a problem in social psychology. It was shown by Harary in 1953 that a signed graph is balanced if and only if its vertex set V can be divided into two sets (either of which may be empty), X and Y, so that each edge between the sets is - and each within a set is +. Based on this fundamental theorem for balanced signed graphs, vertices of a balanced signed graph can be labeled in a way so that its adjacency matrix is well structured. Using this special structure, we find exactly all eigenvalues and their associated eigenvectors of the adjacency matrix A of a given balanced signed graph. We will present eigenpairs ( λ, x) of adjacency matrices of three types of balanced signed graphs: (1) graphs that are complete; (2) graphs with t vertices in X or in Y that are not connected; and (3) graphs that are bipartite Joint work with Spencer P. Hurd of The Citadel.   Previous Meetings abstracts Fall Meeting at USC Sumter September 16 2011   Shannon Michaels Areford A Relation Between Fibonacci Numbers and Lucas Numbers. In this talk I discuss some elementary properties of Fibonacci numbers. I will prove an identity that shows that the difference of two Fibonacci squares is a Lucas Number.   Thomas L. Fitzkee Topology Explains Why Automobile Sunshades Fold Oddly. The article uses topology and abstract algebra to examine "automatic folding" sunshades that coil up when not in use. From the authors' experience, it seems impossible simply to fold such a sunshade in half (i.e. coil it into exactly two loops). The object here is to figure out how many loops can appear in the coil and to understand why. Kevin Milans Subtrees with Few Labeled Paths. Consider a {0,1}-edge-labeling of the complete rooted ternary tree T of depth n. The edge labels along a path from the root to a leaf produce a bitstring of length n; such a bitstring is called a path label. For each complete binary subtree S of depth n, let L(S) be the set of path labels that occur along paths in S. We study the problem of finding a subtree S such that |L(S)| is small. The problem originated from a question in computability theory. This is joint work with Rod Downey, Noam Greenberg, and Carl Jockusch.   Jason Burns Extending Hadwiger's characterization theorem. In 1957, Hugo Hadwiger classified the continuous, rigid-motion invariant, finitely additive measures on convex sets in n-dimensional Euclidean space. Let me explain those buzzwords and persuade you this is important, then prove a 'baby' version and persuade you this is true, then talk about extensions to non-Euclidean space and persuade you there is more to be discovered. Leandro Junes Convolutions on the Geometry of Fibonacci numbers. We define a discrete convolution C using Fibonacci numbers that acts on the Hosoya’s triangle. We prove that this convolution give rise to some counting theorems. Those theorems are used to count some words’ patterns in formal languages. In particular, we have found that that those theorems can be used to count the weight of non-decreasing Dyck paths. Work in progress in collaboraton with Rigo Florez.   Abstracts, meeting at Lancaster April 15 2011 Speakers: Julia Smith (Student at USC Sumter) Inverse functions and ciphers A cipher is a method for creating secret messages. The purpose of using a cipher is to exchange information securely. Throughout history, many different methods have been created. I will discuss a cipher that depends on a linear function and the use of its inverse to decode a secret message. Basic modular arithmetic will be used throughout the talk.     Linyuan Lu A Fractional Analogue of Brooks' Theorem Abstract: Let Δ be the maximum degree of a connected graph G. Brooks' theorem states that the only connected graphs with chromatic number Δ+1 are complete graphs and odd cycles. Here we proved a fractional version of Brooks' theorem:  we classified all connected graphs G with the fractional chromatic number χf(G) ≥ Δ. (Joint work with Xing Peng)     Naima Naheed Convex Minorant of the Nonconvex Thomas-Fermi Energy Functional Mathematically rigorous versions of Thomas-Fermi theory and its generalizations were developed in the 1970s and 1980s by Lieb, Simon, Benilan, Brezis, Gisele and Jerome Goldstein and others. Later Phillippe Benilan, Gisele and Jerome Goldstein (BGG) incorporated Fermi-Amaldi correction into the Thomas-Fermi energy functional. As a result convexity is lost. The theory which will be presented here includes a convex minorant of the nonconvex Thomas-Fermi energy functional. The corresponding Euler-Lagrange equation will become a nonlinear elliptic system involving measures, which will be solved using the methods of BGG. Then the existence of a ground state Thomas-Femi density will be obtained using, among other things, topological degree theory.     Xing Peng The minimum number of monochromatic short progressions in Zn Abstract: For any n≥k≥3, let Mk(n) be the minimum number of monochromatic k-term progressions in any 2-coloring of Zn. We studied asymptotic bounds ofMk(n) for k=3,4 and large n.  For k=3, we show random colorings achieve the minimum number of monochromatic 3-term progressions.  For k=4, we construct a 2-coloring of Zn with few 4-term progressions than a random coloring has. Our upper bound and lower bound for M4(n) improve the previous result given by Wolf on M4(p) for prime p. (Joint work with Linyuan Lu)   Julian Buck The relationship between Topological Dynamics and C*-Algebras The classification program for C*-algebras is one of the main current branches of research in abstract functional analysis. C*-algebras that arise by looking at dynamical systems on topological spaces have provided especially good examples for the purpose of classification theory, where one can start with commutative C*-algebras (essentially, function spaces) and construct new noncommutative examples through a universal construction (the crossed product). This produces a sort of 3-tiered system: the base dynamical system, its associated function algebra, and the new crossed product C*-algebra. In this talk I will describe some of the interplay between the dynamical system and its crossed product, and how this is exploited to show the crossed product has a suitably nice structure for the classification program.         Abstracts, meeting at Benedict College Feb 11 2011 Speakers: Keyona James African-Americans in Mathematics In this talk we will discuss contributions made by African-American mathematicians and their involvement in the mathematical sciences. Not much is known, taught, or written about African-American mathematicians. Information lacks on their past and present contributions and on the qualitative and quantitative nature of their existence throughout mathematics.  In this talk we will provide some details about great mathematicians such as Benjamin Banneker, Elbert Frank Cox, Evelyn Boyd Granville, J. Ernest Wilkins, Jr., etc.     Peter Nyikos Discontinuities and smooth curves in n-space The following theorem, with various wordings, can be found in a number of calculus texts. Theorem 1. If the limit of a real-valued function on R2 exists at a point  p, then it will also be the limit along any smooth curve through  p. This theorem clearly extends to all n ≥2. The converse is true, but seems to be missing from all the standard calculus textbooks.  In fact, something stronger is true: Theorem 2.  If  f  is a real-valued function  defined in a deleted neighborhood of  p  in Rm, and the limit of  f  at  p  does not exist, then either: (1) there is a smooth curve through  p  on which the limit does not exist, or (2) there are two straight lines through   p on which the limits exist, but are unequal.     László Zsilinszky On some topological games The so-called Banach-Mazur game was first introduced and studied in the 1930's by Stanislaw Mazur and Stefan Banach,, who played it on the unit interval. Oxtoby generalized and studied the game in topological spaces, later Choquet  rediscovered it, and introduced a modification, which is now termed the strong Choquet game, to characterize complete metrizability in a metrizable setting. The talk will be an introduction to these games with a review of some interesting related results and examples.   Gurcan Comert Traffic parameter estimation from probe vehicles at signalized intersections Instrumented vehicle data (i.e., probe data) is becoming more important for real-time system parameter estimation in transportation networks. Probe data can be tracked anonymously and can report data on their locations, speeds, travel times, and arrival times as they perform their regular trips. This research develops analytical models for the real-time estimation of key traffic parameters (e.g., queue length, delay) at signalized intersections using the fundamental information (i.e., location, count, and time). For a single queue with Poisson arrivals, analytical models are developed to evaluate how error changes in estimation as percentage of probe vehicles in the traffic stream varies. The formulations presented assess the error in estimation for various scenarios of probe vehicle market penetration rates and congestion levels.   Ralph Howard The Geometry of Mirrors We use the principle of least action to show why light bounces off a mirror so that the angle of incidence equals  the angle of reflection.  One geometric consequence of this is  that when looking a mirror, one sees a left right reversed version of one's self.  Another is that light bouncing off of a corner will return parallel to its original path.  We also discuss how to construct the angles of a kaleidoscope so as to have non-overlaping images. There will be mirrors and kaleidoscopes for hands on experimenting     Abstracts, meeting USC Salkehatchie Nov 5 2010 Speakers: Fidele Ngwane Computing Integral Closures Monomial orderings  will be  discussed.  They  are  vital  in our polynomial  computations.   Integral  extensions  will  be   analyzed, in particular, type I integral extensions.  Integral   closures of type I  integral   extensions  have  great applications.  We will  present  a method  for computing integral  closures that is different  from   others.   Balaji Iyangar Multigrid Methods Iterative processes for solving the algebraic  equations  arising  from  discretizing  partial differential equations are stalling numerical processes, in which  the  error  has  relatively  small  changes  from one  iteration  to  the  next.  The  computer  grinds  very hard  for  very  small  or  slow  real  physical  effect with the use of too-fine discretization grids. In this case, in large parts of  the computational domain  the meshsize is  much  smaller  than  the  real  scale  of  solution changes.  Such  problems  can  be  overcome  by  the multigrid method, or more generally,  the Multi-Level Adaptive  Technique  (MLAT).  Stalling  numerical processes  are  usually  related  to  the  existence  of several  solution  components  with  different  scales, which conflict with each other. By using interactively several  scales  of discretization,  multigrid  techniques resolve such conflicts, avoiding stalling and also being computationally  efficient. Antara Mukherjee Isoperimetric Inequalities Using Varopoulos I will start by introducing Dehn functions and then show the compu tation of upper bounds of the second order Dehn functions for lattices of three-dimensional geometries, namely Nil and Sol. These upper bounds are obtained by using the Varopoulos transport argument on dual graphs. The idea is to reduce the original isoperimetric problem involving volume of three-dimensional balls and areas of their boundary spheres to a problem involving Varopoulos' notion of volume and boundary of nite domains in dual graphs. Upasana Kashyap: A Morita theorem for dual operator algebras. We consider some new variants of the notion of Morita equivalence appropriate to algebras of Hilbert space operators which are closed in the weak* -topology' (or equivalently, which are dual spaces and known as dual operator algebras), and we will describe how the earlier theory of strong Morita equivalence due to Blecher, Muhly, and Paulsen, transfers to this weak*-topology setting'. We will present our main theorem, that two dual operator algebras are weak*-Morita equivalent in our sense if and only if they have equivalent categories of dual operator modules. A key ingredient in the proof of our main theorem is W*-dilation, which connects the non-selfadjoint dual operator algebra with the W*-algebraic.     Abstracts, meeting USC Sumter September 17 2010 Speakers: Jason Holt: Non-Random Perturbations of the Anderson Hamiltonian and Cwikel-Lieb-Rozenblum Type Estimates We will consider the Anderson Hamiltonian H0 = ∆ + V (x; ω) where V is a random potential and \omega belongs to a probability space (ΩF; P). The main object of the present work is the perturbed operator H = H0 -W where W(x)≥ 0 decays at infinity. It is known that the spectrum of H below 0 is discrete consisting only of eigenvalues and that the total number N0(W) of eigenvalues below 0 is a random variable for which P{N0(W) < ∞ } = 1, or P{N0(W) < ∞ } = 0. We develop general conditions on V and W to guarantee P-a.s one case or the other and present several examples demonstrating the borderline decay in W. In particular, it will be shown that if V has a Bernoulli structure, then the borderline between finitely and infinitely many eigenvalues is obtained with a decay in W as O(c0 ln-2 |x| where c0 is a determined positive constant.     Joshua Cooper Tree reconstruction and a Waring-type problem on partitions Abstract: The line graph'' of a graph G is a new graph L(G) whose vertices are the edges of G, with a new edge in L(G) from e to f if e and f were incident in G. Graham's Tree Reconstruction Conjecture says that, if T is a tree (a connected, acyclic graph), then the sequence of sizes of the iterated line graphs of T uniquely determine T. That is, T can be reconstructed from | L (j) (G)|∞j=0 , where L (0) (G) = G and L (j+1) (G) = L(L (j) (G)). Call two trees equivalent if they yield the same sequence; we call the resulting equivalence classes Graham classes.'' Clearly, the conjecture is equivalent to the statement that the number of Graham classes of n-vertex trees is equal to the number of isomorphism classes of such trees, which is known to be about 2.955765n. We show that the number of Graham classes is at least superpolynomial in n (namely, exp(c log n 3/2)) by converting the question into the following Waring-type problem on partitions. For a partition λ = {λ_1,...,λ_k} of the integer n and a degree d polynomial f ∈ Z[x], define f(λ) = ∑ k j=1 f(λj). We show that the range of f(λ) over all partitions λ of n grows as Ω(n d-1). The proof employs a well-known family of solutions to the Prouhet-Tarry-Escott problem. Strong evidence suggests the conjecture that the size of the range is actually Θ(nd). Joint work with Bill Kay of USC     Kwan Lam: The formation of Turing pattern on networks of complete In this talk, we will discuss the formation of Turing pattern in networks of homogeneous coupled reactors with a focus on the a specific reaction diffusion model - Lengyel-Epstein kinetics. Special attention will be paid to the formation of bimodal pattern on the complete graph. We will use it as a building block to construct the linear and circular chains of the complete graphs with bimodal patterns.     Rigoberto Florez Some open questions In this talk I am going to discuss some open questions in number theory and combinatorics. If the time allows us, I discuss some potential research problems. Abstracts, meeting USC Lancaster April 23 2010 Speakers: Daniel Savu: On Sparse Approximation in Banach Spaces The sparse approximation problems ask for complete recovery of functions in a given space that are supported by few of the elements of a system of generators for the space or for approximate recovery that involves a limited number of generators. This is made in regard with redundant systems which offer convenience of representation as well as better rates of approximation. The redundancy raises, in turn, very difficult theoretical problems. We give answers to some of these problems in the very general setting of Banach spaces. The theoretical results complete the previous findings in greedy approximation in this setting and show, for the algorithms considered, the same general recovery properties as the ones known in the particular case of Hilbert spaces.   Moreover, we provide a novel idea of improvement of the geometry of the redundant systems by switching to a different setting than the standard Hilbert space. This improvement would translate in better recovery properties as we are able to prove the same efficiency of the greedy approach in the new setting. Wei-Kai Lai: The Radon-Nikodym Property for Positive Tensor Products of Banach Lattices In 1950’s, Grothendieck started the theory of projective and injective tensor products of Banach spaces. From the positivity perspective, Fremlin and Wittstock extended the theory to projective and injective tensor products of Banach lattices in 1972 and 1974 respectively. In this talk, we are going to discuss the Radon-Nikodym property for Fremlin and Wittstock’s versions of tensor product of Banach lattices. Wei-Tian Li: Lubell Function and Forbidden Subposets. Part II In 1928, Sperner proved that the size of a largest antichain in the Boolean lattice Bn is equal to n choose n/2. Since then, the results on largest sizes families not containing some specific posets were discovered gradually. The well-known inequality, the LYM-inequality, was individually used by Lubell, Yamamoto, and Meshalkin to reprove Sperner's Theorem. We will introduce the Lubell function, derived from LYM-inequality, and use it to estimate the maximal sizes of families do not contain some posets P. (This is a joint work with Jerrold R. Griggs and Lincoln Lu.) Yiting Yang: On the second order Randic index of trees Let G be a simple graph. The second Randic index of G is defined as 2R(G)=∑xyz 1/(dxdydz)1/2 where the summation runs over all paths xyz of length two, contained in G. It was first considered by chemists Randic, Kier and Hall in the study of branching properties of alkanes. One interesting problem on it is to find the maximum and minimum 2R value and its corresponding graphs among classes of graphs. In this talk, we will talk about the maximum and minimum 2R value on trees with fixed size. Abstracts, meeting USC Sumter February 19 2010 Speakers: Anthony Coyne Mathematics, Metaphysics, Morality Reflection on what Plato says about the place of the study of mathematics in the education of the guardians provides a sharp contrast with what is happening as USC approaches revising general education.  In particular, Plato does not advocate the study of mathematics because it leads to any of the learning outcomes identified by USC.  USC hopes students will be able to apply the methods of mathematics, statistics, or analytical reasoning to critically evaluate data, solve problems, and effectively communicate findings verbally and graphically. Plato hopes instead for  more important psychological, metaphysical, moral and political outcomes. Wei-Tian Li: Lubell Function and Forbidden Subposets In 1928, Sperner proved that the size of a largest antichain in the Boolean lattice Bn is equal to n choose n/2. Since then, the results on largest sizes families not containing some specific posets were discovered gradually. The well-known inequality, the LYM-inequality, was individually used by Lubell, Yamamoto, and Meshalkin to reprove Sperner's Theorem. We will introduce the Lubell function, derived from LYM-inequality, and use it to estimate the maximal sizes of families do not contain some posets P. (This is a joint work with Jerrold R. Griggs and Lincoln Lu.)   Wei-Kai Lai: Banach Space, Riesz Space, and Banach Lattice In many Functional Analysis books, we can find the following definitions: a Banach space is a complete normed linear space; a Riesz space is an ordered vector space with the lattice structure. Rephrasing them with earthly language, you will be able to measure the distance of two objects in a Banach space and you will be able to compare which one is bigger among two objects in a Riesz space. In this talk, I will introduce some basic examples of these two spaces, and together with their properties. Finally, I will introduce a special space with both structures, called Banach Lattice. Andrew Yingst: Partition Polynomials And Their Uses (part II) If a coin has probability x of coming up heads, and E is a set of possible outcomes of finitely many tosses of this coin, then the probability of event E is a polynomial in x, referred to as a partition polynomial.  We define and characterize these polynomials, and discuss some questions of dynamics on Cantor space that this characterization can be used to answer. Charlie Cook The “Magicness” of Powers of Some Magic Squares (This paper is in collaboration with Michael R. Bacon and Rebecca A. Hillman) Several Powers of a variety of additive magic squares are computed and conditions which guarantee that they are also magic are investigated. Abstracts, meeting USC Lancaster October 30 2009 Speakers:   Andrew Yingst: Partition Polynomials And Their Uses If a coin has probability x of coming up heads, and E is a set of possible outcomes of finitely many tosses of this coin, then the probability of event E is a polynomial in x, referred to as a partition polynomial.  We define and characterize these polynomials, and discuss some questions of dynamics on Cantor space that this characterization can be used to answer. Leandro Junes From Hyperplanes to Oriented Matroid Programs. :I will discuss the generalization of the simplex algorithm in Oriented Matroids. This will lead to an important class of Oriented Matroids called Euclidean Oriented Matroids. Rigoberto Florez Some topics in harmonic matroids A matroid is a combinatorial generalization of the linear independence concept. There are no linear relations, only dependent and independent sets. Many geometric properties extend to matroids. The harmonic conjugation extends from projective geometry to matroids. In this talk I am going to discuss some results in harmonic matroids Alexandru Gabriel Atim On balanced property of words over finite alphabet (part II) Sturmian words are infinite words over a binary alphabet that has exactly n+1 factors of length n for every n.  Morse-Heldun Theorem states that this equivalent to the set of all balanced aperiodic words. In this talk we will generalize the balanced property on words and then we will classify all words having this property. Abstracts, meeting USC Sumter September 18 2009 Speakers: Alexandru Gabriel Atim On balanced property of words over finite alphabet Sturmian words are infinite words over a binary alphabet that has exactly n+1 factors of length n for every n.  Morse-Heldun Theorem states that this equivalent to the set of all balanced aperiodic words. In this talk we will generalize the balanced property on words and then we will classify all words having this property.   Rebecca Hillman On Products of Fibonacci Numbers and Their Recurrence Relations Various products of Fibonacci numbers and their generalizations are investigated and recurrence relations for these products are obtained Jason Holt Estimates for the negative eigenvalues for a Random Schroedinger Operator with a decaying perturbation We will discuss a one dimensional random Schoedinger Operator with a decaying perturbation.  We will give sufficient condition for the existence of finitely many negative eigenvalues.  The model is closely related to the one dimensional motion of a quantum particle in a crystal lattice. Leandro Junes What is an oriented matroid? An oriented matroid is a combinatorial structure inspired by a hyperplane arrangement. In this talk I will give an informal definition of  oriented matroid from a Geometric view point. I will also discuss its interrelations with Graph Theory, Linear Programming and Topology. The views and opinions expressed in this page are strictly those of Rigoberto Florez. The contents of this page have not been reviewed or approved by the The Citadel.