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Carolina Math Seminar (CMS)





The Citadel

Rigoberto Flórez

Phone: (843) 953 5034

Antara Mukherjee

USC Salkehatchie

Wei-Kai Lai

Fidele Ngwane

Columbia College

Virginia Johnson

Alexandru Gabriel Atim

MAA State Director of South Carolina. Francis Marion University

Nicole Marie Panza

Newberry College

Naser AlHasan

Lander University

Josie Ryan

University of South Carolina

Paula Vasquez

Ralph Howard

Scott Dunn

California University of Pennsylvania

Leandro Junes

USC Lancaster

Shemsi Irene Alhaddad

Jason Holt

Andrew Yingst



Fall Meeting at Newberry College
April 3, 2020

Trevor Olsen

Triangles and Squares on Circles

Let G be a a connected graph. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide asymptotic formulae for the maximum Wiener index of simple triangulations and quadrangulations with given connectivity, as the order increases, and make conjectures for the extremal triangulations and quadrangulations based on computational evidence. If r(v) denotes the arithmetic mean of the distances from v to all other vertices of G, then the remoteness of G is defined as the largest value of r(v) over all vertices v of G. We give sharp upper bounds on the remoteness of simple triangulations and quadrangulations of given order and connectivity.

Bo Li

Simultaneous Confidence Intervals for Differential Gene Isoform Expressions

In this talk, we describe the simultaneous confidence interval method in detecting differentially expressed gene isoforms based on the Poisson generalized linear models. The joint asymptotic distribution of multivariate pivotal quantities is derived. Since the sample size of RNA sequencing data is often small in practice, the large-sample approximation method becomes problematic. We propose the bootstrap method based on pivotal quantities as a robust alternative. We investigate the performance of the proposed method in detecting differentially expressed isoforms through Monte Carlo simulation. It shows that the proposed method controls the family-wise error rate for large-scale inference. It is observed that the validity of robustness of the bootstrap method holds even when mild overdispersion presents in RNA sequencing data. We apply the proposed method to a real RNA sequencing data for illustration.

Fall Meeting at Columbia College
Nov 8, 2019

Tyler Brown

Computing Unity in C[0,1]

Let C[0,1] denote the Banach space of all continuous functions on the unit interval with the usual supremum norm. A computable presentation of C[0,1] can be thought of as an infinite list of functions E whose span is dense in C[0,1] and such that there exists an algorithm that computes the max height of |f|, where f is any function in the rational linear span of functions in E. In 2014, A.G. Melnikov and K.M. Ng demonstrated the existence of a computable presentation P of C[0,1] in which the constant unit function 1 is not computable, i.e. there is no algorithm that can find a function in the rational linear span of vectors in P that, within arbitrary precision, approximates 1 with respect to the sup-norm. In this talk, via the use of Turing reductions, we determine the amount of extra "computational power" sufficient for computing 1 in any computable presentation of C[0,1].

Sher Chhetri

Parameter Estimation for Geometric Lѐvy Processes with Stochastic Volatility

In finance, various stochastic models are used to model the price movements of financial instruments. After Robert Merton’s (1976) seminal work, several jump-diffusion models for option pricing and risk management have been proposed. In this work, we add alpha-stable Lѐvy motion to the process related to dynamics of log-returns in the Black-Scholes model where the volatility is assumed to be constant. We use sample characteristic functions approach and study parameter estimation for discretely observed stochastic differential equations driven by Lѐvy noises. We also discuss the consistency and asymptotic behavior of the proposed estimators. Simulation results and applications to real data sets will be presented. We will also introduce a model where the volatility is not constant.

Inne Singgih

Antimagic orientations of graphs with large maximum degree

Given a digraph D with m arcs, a bijection τ: A(D)→ {1, 2,..., m} is an antimagic labeling of D if no two vertices in D have the same vertex-sum, where the vertex-sum of a vertex u in D under τ is the sum of labels of all arcs entering u minus the sum of labels of all arcs leaving u. We say (D, τ) is an antimagic orientation of a graph G if D is an orientation of G and τ is an antimagic labeling of D. Motivated by the conjecture of Hartsfield and Ringel from 1990 on antimagic labelings of graphs, Hefetz, Mütze, and Schwartz in 2010 initiated the study of antimagic orientations of graphs, and conjectured that every connected graph admits an antimagic orientation. This conjecture seems hard, and few related results are known. However, it has been verified to be true for regular graphs and biregular bipartite graphs. In this paper, we prove that every connected graph G on n≥ 9 vertices with maximum degree at least n-5 admits an antimagic orientation. Joint work with Donglei Yang, Josh Carlson, Andrew Owens, K. E. Perry, Zi-Xia Song, Fangfang Zhang and Xiaohong Zhang.

Lokendra Paudel

Kronecker function rings and Prüfer extension

Let R be a subring of S. The classical Kronecker function ring when R is a domain and S its quotient field associates to the domain R a Bezout domain (that is, a domain in which every finitely generated ideal is principal). Knebusch and Kaiser introduce a notion of star operation ★ on the commutative ring extension R S and generalize the concept of Kronecker function ring Kr1(★) from integral domains to commutative ring extensions. We introduce a new ring Kr2(★) which is a subring of Kr1(★) and we will show with an example that Kr1(★) ≠ Kr(★). We also discuss the properties of ring extensions using the star operation ★ and in particular, we focus on the case where R S is a Prüfer extension. Joint work with Simplice Tchamna (Georgia College & State University).

Trevor Olsen

Triangles and Squares on Circles

Let G be a a connected graph. The Wiener index of a connected graph is the sum of the distances between all unordered pairs of vertices. We provide asymptotic formulae for the maximum Wiener index of simple triangulations and quadrangulations with given connectivity, as the order increases, and make conjectures for the extremal triangulations and quadrangulations based on computational evidence. If r(v) denotes the arithmetic mean of the distances from v to all other vertices of G, then the remoteness of G is defined as the largest value of r(v) over all vertices v of G. We give sharp upper bounds on the remoteness of simple triangulations and quadrangulations of given order and connectivity.

Joshua Thompson

Poset Ramsey Numbers for Boolean Lattices

A subposet Q' of a poset Q is a copy of a poset P if there is a bijection f between elements of P and Q' such that xy in P iff f(x) ≤ f(y) in Q'. For posets P, P', let the poset Ramsey number R(P,P') be the smallest N such that no matter how the elements of the Boolean lattice QN are colored red and blue, there is a copy of P with all red elements or a copy of P' with all blue elements. Axenovich and Walzer introduced this concept in Order (2017), where they proved R(Q2, Qn) ≤ 2n + 2 and R(Qn, Qm) ≤ mn + n + m, where Qn is the Boolean lattice of dimension n. They later proved 2nR(Qn, Qn) ≤ n2 + 2n. Walzer later proved R(Qn, Qn) ≤ n2 + 1. We provide some improved bounds for R(Qn, Qm) for various n,mN. In particular, we prove that R(Qn, Qn) ≤ n2 - n + 2, R(Q2, Qn) ≤ (5/3)n + 2, and R(Q3, Qn) ≤ (37/16)n + 39/16. We also prove that R(Q2,Q3) = 5, and R(Qm, Qn) ≤ (m - 2 + (9m - 9)((2m - 3)(m + 1)))n + m + 3 for all nm ≥ 4.

Fall MAA State Dinner at Columbia College
Nov 8, 2019

Plenary speaker Alex Kasman

Solitons - At the Intersection of Algebraic-Geometry and Mathematical Physics

Algebraic-geometry grew out of the study of geometric objects defined by polynomial equations, like elliptic curves and Grassmannian manifolds. In the 20th century, it developed a reputation for being an especially arcane area of “pure” mathematics. That is why it may be surprising to learn that it has applications in mathematical physics, the subject that can accurately predict the behavior of waves and particles. In this talk, I will explain how it was discovered in the late 20th century that certain nonlinear partial differential equations used by physicists are actually just equations of algebraic-geometry “in disguise”. This discovery has far-reaching implications. The interplay between the two fields has allowed us to answer open questions in physics using algebraic-geometry and vice versa. And, because these equations have “soliton” solutions which are waves that behave like particles, this active area of research may also help resolve some of the enduring mysteries of quantum physics.

Spring Meeting at Lander University
March 29, 2019

Blair Matthew

Matrices in Hosoya Triangle

A triangular array where the entries are products of two Fibonacci numbers is Hosoya. The matrices within this triangle are of rank one (product of two vectors; located on the sides of the triangle). In this talk, we discuss properties and the behaviors of the eigenvalues, eigenvectors, characteristic polynomial, determinants, and their connection with graph theory. The non-zero eigenvalue is a combination of Lucas and Fibonacci numbers. In addition, these matrices are diagonalizable where the entries of the eigenvectors are points within the Hosoya Triangle. The components of the graphs (when matrices are seen mod 2) are complete graphs with loops and isolated vertices.

Katie Warnken and Charnae Wilson

Curious about Cantor

This presentation will discuss the Cantor Set and its properties. Georg Cantor studied this set in the nineteenth century, and we still use his ideas about different sizes of infinite today. This unusual set can be used in a variety of areas in Mathematics, including Real Analysis, Geometry, and topology. The properties that we will prove in this presentation are the length of the Cantor Set is zero and the cardinality of the Cantor Set is 20. This categorizes the Cantor Set as a large, yet also small, set. While this can be mind-boggling for some, it can also open our eyes to another world of mathematics, as it did for Georg Cantor and other mathematicians in the nineteenth century. We will also discuss applications of continuing fractions to the Cantor set, among other curiosities.

Ben Reid

Topological Data Analysis and Persistent Homology

Topological Data Analysis seeks to better understand and classify large data sets, including those in large dimensions, and extract useful topological properties to help analyze the data. Some applications of these techniques include medical image processing, machine learning, and signal analysis. One of these techniques is Persistent Homology, which attempts to recreate topological information by connecting points within a particular distance of each other. As this distance threshold increases, we can start to see which features are part of the overall "shape" of the data, and which might just be caused by noise in the data set. In this talk, we'll discuss the implementation of this method, look at some examples in low dimensions, and look at an application to the processing of images.

Duncan Wright

An Introduction to Quantum Mechanics through Random Walks

We will discuss the peculiarities of quantum mechanics motivated by the Stern-Gerlach experiment. This will motivate the definition of quantum random walks. We will show similarities and differences that arise when considering classical and quantum random walks.

Zhiyu Wang

Erdős-Szekeres theorem for cyclic permutations

We provide a cyclic permutation analogue of the Erdős-Szekeres theorem. In particular, we show that every cyclic permutation of length (k-1)(l-1)+2 has either an increasing cyclic sub-permutation of length k+1 or a decreasing cyclic sub-permutation of length l+1, and show that the result is tight. We also characterize all maximum-length cyclic permutations that do not have an increasing cyclic sub-permutation of length k+1 or a decreasing cyclic sub-permutation of length l+1. Joint work with Eva Czabarka.

Josiah Reiswig

Given a connected graph G=(V,E) and a vertex set S⊂ V, the Steiner distance d(S) of S is the size of a minumum spanning tree of S in G. For a connected graph G of order n and an integer k with 2≤ k ≤ n, the Steiner k-eccentricity of v of a vertex v in G is the maximum value of d(S) over all S⊂ V with |S|=k and v∈ S. The minimum Steiner k-eccentricity, sradk(G), is called the Steiner k-radius of G and the maximum Steriner k-eccentricity, sdiamk(G), is called the Steiner k-diameter of G. In 1990, Henning, Oellermann, and Swart showed that for any integer k≤ 2, there exists a graph G_k such that sdiam_k(G_k)=(2(k+1))/(2k-1)sradk(Gk) and proved that sdiam3(G)≤ 8/5srad3(G), and sdiam4(G)≤ 10/7srad3(G) for all connected graphs G. In this talk, we will show that for each k≤ 5, sdiamk(G)≤ (k+3)/(k+1)sradk(G) and show that this bound is tight via a construction.

Anthony Vasaturo

Invertibility and Positivity of Truncated Toeplitz Operators on Certain Model Subspaces of the Hardy Space

We study the degree to which both the invertibility and positivity of certain Truncated Toeplitz Operators are determined by the Berezin transforms of their symbols on model Subspaces of the Hardy Space corresponding to the inner functions u=z2 and u=z3.

Fall Meeting at USC Columbia
Nov 2, 2018

Adam Gregory

The Asymmetric Index of a Graph

A graph G is asymmetric if its automorphism group of vertices is trivial. Asymmetric graphs were introduced by Erdos and Renyi in 1963 where they measured the degree of asymmetry of an asymmetric graph. They proved that any asymmetric graph can be made non-asymmetric by removing some number r of edges and/or adding some number s of edges, and defined the degree of asymmetry of a graph to be the minimum value of r+s. In this paper, we define another property that gives how close a given non-asymmetric graph is to being asymmetric. We define the asymmetric index of a graph, G , denoted ai(G), to be the minimum of r+s in order to change G into an asymmetric graph.
We investigate the asymmetric index of both connected and disconnected graphs and obtain precise values for paths, cycles, certain circulant graphs, Cartesian products involving paths and cycles, and bounds for complete graphs, split graphs, and stars.

Risto Atanasov

Odd edge-colorability of subcubic graphs

An edge-coloring of a graph G is said to be odd if for each vertex v of G and each color c, the vertex v either uses the color c an odd number of times or does not use it at all. The minimum number of colors needed for an odd edge-coloring of G is the odd chromatic index χo(G). In this presentation, we consider loopless subcubic graphs, and give a complete characterization in terms of the value of their odd chromatic index. This is a joint work with Mirko Petrusevski and Riste Skerkovski

Keller Vandebogert

Compressed k-algebras and some structure theory

In this talk I'd like to give an overview of the idea of a compressed k-algebra and their importance as extremal objects in the world of algebras. In the case of projective dimension 3, it turns out that there is an elegant structure theory for such objects in terms of their "Tor-algebras" due to Avramov, Kustin, and Miller. We will then see how these objects may come to fit into the above structure theory, with some concrete results in a couple of special cases.

Patrick McFaddin

Geometric study of subfields of some non-commutative algebras

In algebraic geometry, one produces geometric objects by considering the collection of all prime ideals in a given commutative ring. This begs the question of whether a similar construction is possible for non-commutative rings. In this talk, I will discuss one method for building projective space from a matrix algebra. I will then provide a twisted version of this construction which produces more interesting geometric spaces, and shed some light on how one may use these objects to study subfields of a given algebra.

Todd Wittman

What Not to Eat – Machine learning and image processing in agriculture

Agricultural scientists work tirelessly to protect crops from damage caused by insects, weather, and pesticides. Due to the high volume of crops, it is difficult to examine each fruit and vegetable by hand to evaluate the damage and develop a response plan. In a joint project with the US Department of Agriculture Vegetable Lab, we are developing an artificial neural network (ANN) to examine images of crops and identify the extent and sources of damage. We will provide a brief and somewhat over-simplified introduction to machine learning algorithms and discuss some of the mathematics behind it. We will then discuss how machine learning can be used to help computers learn to see how like humans and its application to agricultural science. Joint work with Dr. Phillip Wadl, USDA Vegetable Laboratory.

Jesse Kass

How to count lines on a cubic surface arithmetically

A smooth cubic surface is the solution set to a suitably nondegenerate polynomial in 3 variables. While there any many different possibilities for such a surface, Salmon and Cayley proved a celebrated 19th century result that a smooth cubic surface over the complex numbers contains exactly 27 lines. By contrast, the count over the real numbers depends on the surface, and these possible counts were classified by Segre. A number of researchers have recently made the striking observation that Segre’s work shows a certain signed count is always 3. In my talk, I will explain how to extend this result to an arbitrary field. This is joint work with Kirsten Wickelgren.

Fall MAA State Dinner at USC Columbia
Nov 2, 2018

Plenary speaker Mohammad Ghomi

Durer’s unfolding problem for convex polyhedra

A well-known problem in geometry, which may be traced back to the Renaissance painter Albrecht Durer, is concerned with cutting a convex polyhedral surface along a collection of its edges so that it may be developed without overlaps into the plane. We show that this is always possible after an affine transformation of the surface. In particular, unfoldability of a convex polyhedron does not depend on its combinatorial structure. On the other hand, we also show that the unfoldability conjecture fails if it is generalized to allow cutting along the so called pseudo-edges: i.e., a network of distance minimizing geodesics connecting vertices of the polyhedron. Thus there exits substantial evidence both for and against a positive resolution of Durer’s problem..

Spring Meeting at the Citadel
March 30, 2018

Elizabeth Schuyler Spoehel

A Fibonacci identity

In this presentation I talk about the solution of Problem B-1218 from Fibonacci Quarterly. I will show the solution of this problem in detail and speak on the process. This problem has been submitted for publication to the same journal.

Hsin-Yun Ching

Properties of Fibonacci and Lucas Matrices

In this presentation, I will show the solution of a problem that I solved from Fibonacci Quarterly. The problem was to find the solution of a system of linear equations with Fibonacci coefficients. I solved this problem using the result of another problem -that I solved last year- from the same journal. I also used linear algebraic techniques like matrix block multiplication to solve the problem. I have submitted this solution to Fibonacci Quarterly for consideration to be published. In addition, I found two interesting results by replacing Fibonacci numbers with Lucas numbers -in the problems mentioned above-. I will present these novel results in my presentation as well.

Wei-Kai Lai

Some Inequalities with Sum of Cyclic Fractions

Cvetkovski introduced several inequalities involving sums of cyclic fractions with three variables in his book “Inequalities: Theorems, Techniques and Selected Problems”. By analyzing their pattern, we easily generalized these inequalities to the case when the exponents are random integers, using only basic algebra, AM-GM inequality, and mathematical induction. We then applied rearrangement inequality, Chebyshev’s sum inequality and a weighted Hölder’s inequality and proved the case when exponents are real numbers. In this talk, we will first introduce the pattern we found among these inequalities, and then we will introduce Cvetkovski’s method that led us to our first generalization. We will finally prove the second generalization using our own method.

Paul Young

The power of 2 dividing a generalized Fibonacci number

The Fibonacci sequence is the best-known and best-loved sequence in all of mathematics. It is a recursively defined sequence in which each term is the sum of the two previous terms, beginning with 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144,.... So much has been written about it that it has its own quarterly journal and its own international association.

Last year Diego Marques and Tamas Lengyel proved some theorems and made some conjectures about which power of 2 would divide a generalized Fibonacci number. Their generalization dealt with recursive sequences in which each term was the sum of the three (or four, or five) previous terms. One of their conjectures seemed rather strange to me, but I checked it on my computer and found it to be correct for the first three million terms of the sequence. In this talk I’ll discuss the status of this conjecture.

Garner Cochran

A New RNA Folding Model

I will speak about a model which generalizes the folding of the RNA molecule in biology, first introduced by Black, Drellich, and Tymoczko (2017+). RNA is represented by a word from the alphabet of nucleotides A, U, C, and G in which Watson-Crick bonds form between nucleotides A and U and between C and G. Sometimes RNA sequences will fold such that some base pairs are left unmatched. We wish to consider only the case where all the base pairs completely match up. I will answer the question of when a sequence will fold completely and will answer some questions about the different ways that a sequence can fold onto itself. I will conclude with some open problems in the area.

Inne Singgih

Vertex Magic Total Labelings of 2-Regular Graphs

A vertex magic total (VMT) labeling of a graph G=(V,E) is a bijection from the set of vertices and edges to the set of integers defined by λ:V ∪ E → {1,2, ... ,|V|+|E|} so that for every x ∈ V, w(x)=λ(x)+∑ xy∈ E λ(xy)=k, for some integer k. A VMT labeling is said to be a super VMT labeling if the vertices are labeled with the smallest possible integers, 1,2,..., |V|. In this talk we introduce a new method to expand some known VMT labelings of 2-regular graphs.

Hays Whitlatch

Graph Pressing Sequences on Mitochondrial Genomes

One can construct a useful metric on genome sequences by computing minimal-length sortings of (signed) permutations by reversals. Hannenhalli and Pevzner famously showed that such sorting sequences are essentially equivalent to a certain sequences of operations -"vertex pressing"- on bicolored graphs. We examine the matrix algebra over GF(2) that arises from the theory of such sequences, providing a collection of equivalent conditions for their existence and showing how linear algebra, poset theory, and group theory can be used to study them. We discuss enumeration, characterization, and recognition of uniquely pressable graphs (those with exactly one pressing sequence); relations on pressing sequences that have a surprisingly diverse set of characterizations; and some open problems. Joint work with Joshua Cooper.

Ben Reid

Introduction to Homotopy

Algebraic Topology is a field that uses techniques and structures from abstract algebra (groups, rings, etc) to study topological spaces. One of the major goals is to find certain invariants that help us classify topological spaces up to homeomorphism. One way to begin classifying spaces is to look at their homotopy groups, which arise from studying functions from spheres into the space. In particular, if two spaces have different homotopy groups, then they cannot be homeomorphic. In this talk, I’ll introduce the idea of a homotopy between maps, and how it gives rise to a sequence of groups for each space. We’ll look at some common topological spaces and compute some of these homotopy groups.

Fall Meeting at Benedict College
Nov 3, 2017

James Andrus

Multicast Routing using Delay Intervals for Collaborative and Competitive Applications

In our research we propose a remodel of the Delay and Delay-Variation Bounded Multicast Tree (DVBMT) problem. We refer to this as the Interval Multicast Subgraph (IMS) problem. IMS addresses the requirements of DVBMT with an interval of acceptable path weights, eliminating the need to optimize, as other techniques required. Our proposed algorithm, Interval Multicast Algorithm (IMA), addresses IMS. In this talk, I will present the IMS problem, the IMA algorithm, and techniques used in our research.

Hsin-Yun Ching

System of linear equations with Fibonacci coefficients

In this talk we show the solution of two open problems given in the journal Fibonacci Quarterly. The first problem we find the determinant of a n by n matrix with Fibonacci numbers as entries. If n is even, then the determinant is zero and 1 otherwise.

In the second problem we find the solutions of a system of equations that can be written using the previous matrix. For this solution we consider two cases because determinant is as mentioned earlier.

Rigoberto Florez

Characterization of the strong divisibility property for generalized Fibonacci polynomials

It is known that the greatest common divisor of two Fibonacci numbers is again a Fibonacci number. It is called the strong divisibility property. However, this property does not hold for every second order sequence. In this talk we discuss the generalized Fibonacci polynomials and classify them in two types depending on their Binet formula. We give a complete characterization for those polynomials that satisfy the strong divisibility property. We also give formulas to calculate the greatest common divisor of those polynomials that do not satisfy the strong divisibility property. Joint work with R. Higuita and A. Mukherjee.

James Hepburn

Budapest Semesters in Mathematics and Gilman Award Abstract and Talk Outline

Budapest Semesters in Mathematics is a prestigious study abroad program for American undergraduates and recent graduates to study topics typically not found in American undergraduate programs and distinctly Hungarian mathematical teaching methods. Credits are transferable from an American partner university and students get to experience one of the most beautiful capitals of Europe with low living costs and great camaraderie among like-minded, motivated peers. The Gilman Award is offered by the US State Department to promote better cultural, political, economic and civil societal relations between nations by funding study abroad programs for American undergraduates who receive the Pell Grant. It offers the financial opportunity to have the enriching experience of BSM and to learn about the fascinating Hungarian culture and language.

Alexander Wiedemann

Graph Theoretical Conditions for Equilibrium in Quantum Mechanical Systems

If a quantum mechanical system is coupled to a reservoir one can describe, under certain limiting conditions, the time evolution of the system by a quantum dynamical semigroup Tt. The generator of this semigroup can be cast using the GKLS master equation

=L(ρ)= - i[H,ρ]+(½)∑i,j=1N2-1 cij([FiFj*]+[Fiρ,Fj*])

where for us the importance will be that (cij) forms a complex positive (N2-1)* (N2-1) matrix. A question of obvious physical interest is to delimit those generators for which the corresponding semigroup has a unique equilibrium state for which every initial state tends to as t → ∞. Though conditions for this exist in the literature, not much is known without demanding certain technical assumptions (e.g., existence of a faithful invariant state). In this talk we develop the requisite graph theoretical knowledge needed to drop these assumptions and introduce a novel graph theoretic interpretation of the matrix (cij), which in turn can be used to derive necessary and sufficient conditions on the uniqueness of equilibrium in certain cases. Further, we describe sufficient algebraic/graph theoretic conditions for uniqueness of equilibrium, again without using the common technical assumptions.

Jonathan Zheng Sun

Threshold Changeable Secret Sharing Based on Chinese Remainder Theorem

Secret sharing is a fundamental problem in cyber security. This talk will present Shamir’s definition of the problem, Asmuth and Bloom’s solution based on the Chinese Remainder Theorem, and our recent work in extending it to work with changeable thresholds.

Rade Musulin

Kac's Chaos and Quantum Kac's Chaos

We define some terms in linear algebra which are analogous to terms in probability theory. For example, density matrices in linear algebra correspond to density functions in probability theory. The terms in linear algebra also belong to a field of mathematics called quantum probability theory. We are going to state a result in probability theory, and state an analogous result in quantum probability theory.

Wei-Kai Lai

A Journey of Inequalities

Inequalities can be useful and powerful tools in finding upper and lower bounds. However, this topic is a little understated in our Math courses. In this talk I will introduce several commonly used inequalities. Using a simple problem, I will then provide different solutions and show the audience how to apply these inequalities in the proofs.

Fall MAA State Dinner at Benedict College
Nov 3, 2017

Plenary speaker Sarah Greenwald

Geometry of the Earth and Universe

The quest to understand the precise geometry and shape of our universe began thousands of years ago, when mathematicians and astronomers used mathematical models to try and explain their observations. We'll explore historical and current theories related to the geometry of the earth and universe during an interactive talk.

Spring Meeting at Lander University
March 24, 2017

Carrie Finch

A light in the attic: Fermat, Abel & Wiles

The Norwegian Academy of Science and Letters has been awarding the Abel Prize since 2003. The list of Abel laureates includes mathematical giants such as Jean-Pierre Serre, Peter Lax, John thompson, John Tate, Endre Szemeredi, Pierre Deligne, and John Nash. In this talk, we focus on the mathematical achievements of the 2016 Abel laureate, Sir Andrew Wiles, and the long history that led to his proof of Fermat's Last Theorem.

Virginia Johnson

Areas of triangles and other polygons with vertices from various sequences

A triangle with vertices given by Fibonacci numbers as follows:

(Fn,Fn+k), (Fn+2k,Fn+3k), and (Fn+4k,Fn+5k)

has area

5(F4k Lk)/2 for k even and (F2k L3k)/2 for k odd.

We have extended this result to calculate the area of triangles with vertices using other sequences and from there to calculate the area of any n-gon with such vertices. This is joint work with Charles Cook of USC Sumter.

Rade Musulin

A Connection Between Mixing and Kac's Chaos

We will examine a connection between two notions of chaos in this talk. The First notion was introduced by Kac in 1956 while studying the integro-differential equation known as the Boltzmann equation. In an attempt to find the solution to this equation, Kac introduced a property which is now referred to as Kac's chaos. On the other hand, in ergodic theory, chaos usually refers to the mixing properties of a dynamical system. In this talk, we will study a relationship between Kac's chaos and mixing.

Erik Palmer

A Parallel Approach to Modeling Polymer Gel Dynamics

Stimuli-responsive polymer gels have many surprising non-Newtonian properties such as shear-thinning and shear-thickening. Their transient network structures respond to environmental stimuli such as pH, UV or temperature, making them ideally suited for a variety of applications. In this talk we introduce a non-linear elastic bead-spring model for characterizing polymer gel network dynamics. This approach leverages the parallel processing power of graphical processing units (GPUs) to overcome the mathematical and computational challenges that arise in this micro-macro scale design. Finally, we demonstrate the model’s ability to efficiently recreate measured data from single polymer gels, as well as capture the emergent behavior of their mixture.

Diana Delach

Carbonate Chemistry, or Why Natural Waters Are Not Neutral

Everyone knows that water is neutral and has a pH of 7; however, most natural water systems can fall anywhere from pH 6.5 to 8.5. The abiotic carbonate system buffers water in this broader range, which resists pH changes due to the activity of additional chemical species. Normal biotic activity creates a dynamic equilibrium in the same range, too, but anthropogenic influences and extreme climates can cause the pH to shift dramatically, causing ecological distress. This presentation will serve to explain how waters function naturally, as well as how climate change may influence aquatic systems in the near future.

Fall Meeting at the Citadel
December 5, 2016

Robinson Higuita

GCD Properties of Generalized Fibonacci Polynomials.

A sequence that satisfies the recurrence relation F0=0,F1=1, and Fn=xFn-1(x)+Fn-2(x) for n≥2 is called the Fibonacci polynomial. The Generalized Fibonacci Polynomial (GFP) is a natural generalization of the above mentioned sequence. It is known that the greatest common divisor of two Fibonacci numbers is a Fibonacci number. However, this property does not always hold for every GFP sequence. In this presentation I will provide a complete characterization of those polynomials that satisfy the Fibonacci gcd property. I will also present a characterization of polynomials that do not satisfy the Fibonacci gcd property. In particular, I will show that the polynomials that satisfy the Fibonacci gcd property are Fibonacci polynomials, Pell polynomials, Fermat polynomials, Chebyshev polynomials of second kind, Jacobsthal polynomials and one type of Morgan-Voyce polynomials, while the polynomials that do not satisfy the Fibonacci gcd property are: Lucas polynomials, Pell-Lucas polynomials, Fermat-Lucas polynomials, Chebyshev polynomials of first kind, Jacobsthal-Lucas polynomials and second type of Morgan-Voyce polynomials. These last set of polynomials partially satisfy the above-mentioned property

Fall Meeting at Columbia College
November 4, 2016

Ralph Howard

The value of pi in taxicab and related geometries.

There a a family of geometries, called Minkowski geometries, on the plane that generalize Euclidean geometry. Probably the best known of these geometries is taxicab geometry, where the length of a vector (a,b) is |a|+|b|. Each such geometry determines a value of pi, the ratio of the circumference of a circle to its diameter. We will out line a proof of a theorem of Golab that the value of pi for a geometry is always at least 3 and at most 4.

Nathan McAnally

A Series of Four Sums of a Fibonacci Number to the Fourth Power

The Fibonacci Numbers provide a unique sequence observable in many different areas of nature and applicable in a significant portion of theoretical mathematics. These numbers provide a definition for the aesthetically appealing golden ratio and can define the growth of a population. They can be heard in music and observed in the petals of a flower. Due to the numerous applications of this sequence in the physical world and theoretical mathematics, it is important to find identities related to this unique set of numbers.

Marcus Harbol

A Rational Fibonacci to the n Identity

We solve an open problem from the Fibonacci Quarterly, a sum of three fractions involving the sum of n-power Fibonacci numbers. Solving this problem, we found an even more interesting that gives rise to the Golden Ratio. In this talk we discuss both problems.

In 1965 Graham published a closed formula for the sequence of four sums of squares of Fibonacci numbers. Since then, as far as I know, there has been no other similar results for these type of natural questions. However, in 2015 the Fibonacci Quarterly proposed a problem, which was classified by the journal as an Advanced Problem related to Graham’s result. In the problem, instead of power two, the Fibonacci number was raised to power four. I found a proof for the proposed problem and also found that it gives rise to a potential future research problem. In this talk I discuss the proof of the problem described above. I submitted this problem for publication in the Fibonacci Quarterly.

Gregory Clark

Splitting Numbers of Integer Tiles

Janos Pach conjectured the following: there is a least integer N so that every covering of the plane by unit disks such that each point is covered at least N times has a two-coloring of the disks (say, by red and blue) where each point is covered by a red disk and a blue disk. It has been shown that Pach's conjecture is false! We explore questions similar to Pach's by restricting our attention to coverings of the integers by integer tiles. In particular, we show that for finite integer tiles a suitable N (which we define as the splitting number) always exists and provide an upper bound for it based on the size of the tile. Moreover, we provide a classification of splitting numbers for tiles of cardinality at most three. We conclude by presenting a connection between computing the splitting number of an integer tile and two-coloring a uniform hypergraph.

Paula Vasquez

Mathematical Modeling of Viscoelastic Materials

In a broad sense, one can divide fluids into Newtonian and non-Newtonian according to their response to flow. For example, honey –a Newtonian fluid- always flows, while mayonnaise –a non-Newtonian material- does not flow under moderate stresses. From a modeling point of view, all Newtonian fluids can be described by the well-known Navier-Stokes equations. In general, this set of equations works well on systems in which the flow does not alter the dynamics of individual constituents. In contrast, applied flows are capable of altering the local microstructure in non-Newtonian fluids so that there is not a unique system of equations capable of describing different materials, as is the case with Newtonian fluids. In this talk, we discuss a class of non-Newtonian fluids known as viscoelastic materials and highlight some modeling principles applied to this materials that are accessible to both undergraduate and graduate students.

James Brown

Modeling the Spread of the Zika Virus in South Carolina

With the passing of the 2016 Olympic Games, the Zika virus is a pressing topic for researchers and world leaders alike as it spreads throughout the Americas. The possibility of a rapid spread and its negative impact on pregnant mothers amplifies the need for modeling Zika. Developed by Kermack and McKendrick, compartmental models in epidemiology provide a mathematical infrastructure to model complex systems and to show how a disease spreads through them. More specifically, SIR models allow us to compute the theoretical number of people infected with a disease in a closed population over time. We apply the use of this model in order to show how the Zika virus may spread through South Carolina.

Candace Bethea

Can you make a knot not a knot?

Knot theory is a robust and interesting field of low dimensional topology that crosses over into many other subjects such as algebra, analysis, graph theory, and combinatorics. Knots can be classified by an invariant called the crossing number, but given a random knot it can be difficult, if not impossible, to discern the actual "complexity" of the knot. A natural question to ask is how much you have to deform a knot in order to obtain a simple closed path. There are many known approaches to answering this question. In this talk I will introduce the unknotting problem and give a survey of methods that have been employed to answer it, as well as methods that could be used as potential research topics for undergraduates.


Spring Meeting at Newberry College
March 18, 2018

Kendrick Hardison

Closing the Gap on Multifold Triple Systems

In part of a dissertation completed by N. Newman, he partially solved the problem of enclosing a triple system of TS(v, λ) in a triple system of TS(v + s, λ + m). Enclosings were found for all admissible values outside of a quadratic gap.  The research I am doing under his direction attempts to close the gap.

Jeremiah Bartz

Tropical Nets

Nets are certain configurations of points and lines in the complex projective plane which satisfy certain incidence relationships. In this talk, we use tropical geometry to "tropicalize" nets into tropical nets, their tropical counterparts. This new setting gives an alternative approach to attack classical problems involving nets.

Garner Cochran

The Gale-Shapely Algorithm and its Applications

In 2012, the Nobel Prize in Economics was given to Lloyd Shapely and Alvin Roth for their work on stable matchings and their applications to real world problems. Consider a group of men and a group of women, where each person has a preference list for all of the people they want to date. A stable matching is an arrangement of marriages where everyone is satisfied with their mate. We will explain the algorithmic proof of the existence of such an arrangement. We will talk about one algorithm used today to assign medical residents to hospitals, students to schools, and organs to transplantees. While difficulties and complications may arise in real world situation, we will speak of how the original algorithm can be adapted to solve some of these problems.

Antara Mukherjee

Introducing Students to Conjectures, Exploration and Visual Proofs using Experiments in Topology

In this presentation I will talk about some classical experiments in topology that my collaborator Dr. R. Flórez and I used to stimulate the curiosity of our freshman students in math classes. We designed some experiments where the student could ask questions, conjecture results and ultimately reconstruct some visual proofs which helped them gain better understanding of what a mathematical proof is. The classical experiments involved construction of topological objects like the Möbius band, projective plane band and the Klein bottle band, observing their properties, stating conjectures, verifying the conjectures and writing sketch of proofs. I will also discuss how the students discovered via experiments that altering topological objects by cutting do not preserve their hereditary properties. These experiments encouraged them to learn more about topology and other complex mathematical topics.

Wei-Kai Lai

The Digital Root of Power Towers

The digital root of a positive integer is defined by the unique single digit after repeatedly summing all its digits. And the power tower of a positive integer is defined by the iterated exponentiation of the integer. If the exponentiation is iterated for n times, it is called a power tower of order n. Using the technique of congruence, we analyze the digital root of a power tower and find that it remains a constant when the order exceeds a certain number.

Charles Cook

Higher order boustrophedon transforms for certain well-known sequences

A review of the boustrophedon transform is presented and transforms of several familiar sequences are obtained.  In addition higher transforms are also investigated.  Representations of the transform will be given in terms of members of the original sequence using the Euler Up-Down number coefficients. This is joint work with Michael Bacon.

Tien Chih

The Fundamental Morphism Theorem in the Category of Graphs

The Noether Isomorphism Theorem was a seminal result that exposed the connection between the internal structure of an algebraic object, and the homomorphisms between that objects and other objects. Some version of this result has been established for every type of algebraic structure and has become an indispensable part of our understanding of algebra: both for it's application to questions of isomorphisms and homomorphisms, but also highlighting the importance of structure preserving maps and the categorical perspective.

Relatively recent developments in graph theory now show that classical internal structure questions (such as graph coloring) may be phrased as a question of graph morphisms (strict morphism to a complete graph). The parallels between this and the Noether Isomorphism Theorem should be clear, and drawing analogous results should have similar repercussions for questions of graph isomorphisms and homomorphisms. In this talk, we present a generalization of the Noether Isomorphism Theorem to the categories of graphs, and give an application to a longstanding pair of graph isomorphism conjectures: the Reconstruction Conjectures.


Fall Meeting at Francis Marion University
Nov 6, 2015

John Risher

Two Problems Involving Radon’s Inequality

In 1913 Radon proved an inequality dealing with a sum of fractions. Since then many generalizations have been discussed. In this talk we will introduce some frequently used versions of Radon’s inequality. We will also introduce solutions of two problems in The College Mathematics Journal provided by us applying this useful inequality.

Ryan Brown & Talon Brown

Modeling Historic Outbreaks of the Bubonic Plague

Throughout history, the bubonic plague has periodically ravaged many areas in Europe and Asia. This project focuses on modeling localized historic outbreaks of the bubonic plague using a basic susceptible, infected, recovered (SIR) model. The SIR model is widely used within the field of epidemiology and, with proper parameterization, allows for modeling specific events. This presentation will discuss the implementation of the model and the attained results as well as limitations and challenges in proper parameterization.

Rachel Graves

Using Matrices to Derive Identities for Recursive Sequences

The matrix representation for various second and third order recursive sequences are squared and their eigen-equations are investigated. In the case of second order sequences, after completing the square, the Caley-Hamilton theorem is applied and the binomial expansion yields several binomial summation identities. In the case of the third order sequences the Caley-Hamilton theorem is not helpful but many similar binomial summation identities are still obtainable.

Risto Atanasov

Groups and Loops Partitioned by Subgroups

A set of subgroups of a group is said to be a partition if every nonidentity element belongs to one and only one subgroup in this set. The study of groups with partition dates back to a paper by Miller published in 1906. In this presentation we will talk about groups and loops that are partitioned by subgroups. We will also discuss finite p-groups such that a subset of their maximal subgroups form an equal quasi-partition

John Adams

Fibonacci and Lucas Numbers: Applications of Binet's Formula

Fibonacci's Numbers have intrigued many mathematicians over the years.  In this talk, I discuss two such mathematicians and how their formulas and discoveries may be used to express Fibonacci numbers.  I also discuss how these formulas are necessary to solve certain problems involving Fibonacci and Lucas numbers, including my solution to an open problem posed in Fibonacci Quarterly.

Mary Mulholland & Phillip Rouse

Modeling the Dengue Virus

This research focuses on modeling the dengue virus, a dangerous but still mysterious disease. The SIR model is used to mathematically describe the interactions between susceptible humans and mosquitoes along with their infected and recovered counterparts. Euler's method is implemented to simulate the spread of the dengue virus in both human and mosquito populations. Model results for human populations are be compared with data from recorded outbreaks of the dengue virus.

Joe Anderson.

An extremal problem on contractible edges in 3-connected graphs

An edge e in a 3-connected graph G is contractible if the contraction G=e is still 3-connected. The existence of contractible edges is a very useful induction tool. Let G be a simple 3-connected graph with at least five vertices. Wu proved that G has at most |V (G)|/5 vertices that are not incident to contractible edges. In this paper, we characterize all 3-connected graphs with exactly |V (G)|/5 vertices that are not incident to contractible edges. We show that all such graphs can be constructed from either a single vertex or a 3-edge-connected graph (multiple edges are allowed, but loops are not allowed) by a simple graph operation.

Spring Meeting at USC Salkehatchie
April 3, 2015

Anton Khristyuk

Magic Square and 15-Puzzle

In the world of mathematical puzzles, both Magic Square and 15-Puzzle are very popular and require different techniques. What would happen if these two puzzles were merged into one? In this talk, by analyzing an old problem, we will introduce the mathematics of these two puzzles, and discuss a solution of a special 15-Puzzle that also requires the knowledge of a Magic Square.

Kaige Lindberg

A closed forms for the summation of Lucas numbers

In this talk I will be talking about how I found the closed form for the summation of a Lucas number squared times the consecutive the Lucas number squared. I will discuss the identities and techniques I used. I will additionally talk about the generalization of the series. The summation was an open problem in the August, 2014 issue of the Fibonacci Quarterly.

Shane Latchman

An infinite Fibonacci Lucas series

We will compute one interesting infinite series involving Fibonacci and Lucas Numbers. This problem was submitted to be published in the Fibonacci Quarterly.

Eric Numfor

Optimal Control Applied in a Multi-group Immuno-epidemiological

The two key features in infectious diseases are the transmission between host and the immunological process at the individual host level. Understanding how the two features influence each other can be assisted through mathematical modeling. Linking components of the immune system with the compartments of the epidemic model leads to a two-scale model. On the other hand, optimal control can be used to design intervention strategies for the management of infectious diseases, and has been applied in decoupled immunological and epidemiological models of HIV.

We formulate an immuno-epidemiological model of coupled within-host model of ordinary differential equations and between-host model of ordinary and partial differential equations. Existence and uniqueness of solution to the epidemiological (between-host) model is established, and an explicit expression for the basic reproduction number of the multi-group coupled between-host model is derived. Steady state solutions are calculated and stability analysis of the disease-free and endemic equilibria is investigated. An optimal control problem with drug-treatment control (fusion and protease inhibitors) on the within-host system is formulated and analyzed. Numerical simulations based on the semi-implicit finite difference scheme and the forward-backward sweep iterative method are obtained.

Michael Tiemeyer

On z-Cycle Factorizations with Two Associate Classes

Let K = K(a,p12) be the multigraph with: the number of vertices in each part equal to a; the number of parts equal to p; the number of edges joining any two vertices of the same part equal to λ1; and the number of edges joining any two vertices of different parts equal to λ2. In this presentation, we give history and current progress regarding z-cycle factorizations of K.


Virginia Johnson

Catalan Numbers 

Catalan numbers have a fascination almost equal to Fibonacci numbers.  In this talk, we will give an overview of Catalan numbers with problems that are suitable for the undergraduate classroom.  After a few examples of proofs, one generalization of Catalan numbers is presented with suggestions for projects for students.


Tien Chih

Abstracted Primal-Dual Affine Programming

The classical study of linear (affine) programs, pioneered by George Dantzig and Albert Tucker, studies both the theory, and methods of solutions for a linear (affine) primal-dual maximization-minimization program, which may be described as follows:

Given A in ℜ mn, b y in ℜ m, c→ in ℜ n, d in ℜ, find x→ in ℜn such that Ax ≤ b, and x ≥ 0, that maximizes the affine functional f(x→) := c→.x→ - d; and find y→ in ℜm such that AT y→ ≥ c, and y ≥ 0, that minimizes the affine functional g(y→) := y→.b→ - d."

In this classical setting, there are several canonical results dealing with the primal-dual aspect of affine programming. These include: I: Tucker's Key Equation, II: Weak Duality Theorem, III: Convexity of Solutions, IV: Fundamental Theorem of Linear (Affine) Programming, V: Farkas' Lemma,VI: Complementary Slackness Theorem, VII Strong Duality Theorem, VIII Existence-Duality Theorem, IX: Simplex Algorithm.

We note that although the classical setting involves finite dimensional real vector spaces, moreover the classical viewpoint of these problems, the key results, and the solutions are extremely coordinate and basis dependent. However, these problems may be stated in much greater generality. We can define a function-theoretic, rather than coordinate-centric, view of these problem statements. Moreover, we may change the underlying ring, or abstract to a potentially infinite dimensional setting. Integer programming is a well known example of such a generalization. It is natural to ask then, which of the classical facts
hold in a general setting, and under what hypothesis would they hold?

We describe the various ways that one may generalize the statement of an affine program. Beginning with the most general case, we prove these facts using as few hypotheses as possible. Given each additional hypothesis, we prove all facts that may be proved in this setting, and provide counterexamples to the remaining facts, until we have successfully established all of our classical results.


Fall Meeting at Columbia College
October 24, 2014

Audrey Danielle Talley

Autonomous Adventures with the NAO Robot

During the summer of 2014, two students and I coded a NAO robot in ‘Python’  to perform three different autonomous tasks: design of winning strategies to games, design on human-like behavior motion planning for obstacle avoidance, and vision recognition applied to music sheets to identify and play songs.  These three tasks are a clear exponent on how basic mathematics help achieve very complex feats in artificial intelligence.  The design of strategies to win at games is done by generalizing root-finding techniques in Calculus.  The analysis of the different strategies is performed with techniques of statistics.  Motion planning is carried within the field of computational geometry, while the design of smooth paths is merely an application of interpolation.  Finally, the project related to vision recognition is done under the scope of image processing and analysis, which is in a set of basic application of multivariate Calculus.  This resulted in a game-winning, music-reading, obstacle-avoiding body of artificial intelligence.


Rachel Graves

Holder's Inequality and Fibonacci Sequences

In this talk, I will give a brief history of Fibonacci and Fibonacci numbers.  I will then explain the proof of a problem that I solved and is now published in the Fibonacci Quarterly.  I solved this problem by using Holder inequality which is a well known inequality in Functional Analysis.

Heather Smith

Zero Forcing: The Spread of Infection on a Graph

Even when we try to prevent it, sickness spreads quickly among friends. In this model, we say that a sick person will spread his disease to a friend if this is his only healthy friend. If we start with a network of people, some of whom are already sick, will everyone become sick after a sufficient amount of time? In the cases when the infection does spread throughout the graph, we establish a lower bound on the number of people who are initially sick. This lower bound is based upon the girth and minimum degree of the graph. 


Zibusiso Ndimande

A Fibonacci Numbers Identity

We will prove one interesting identity involving Fibonacci numbers. This problem was submitted to be published in the Fibonacci Quarterly


Francisco Blanco-Silva

Searching for the SS Central America

In the early 1980’s, the Columbus-America Discovery Group was formed with the intention of rescuing the shipwreck of the steamboat SS Central America. This ship sank somewhere off the coast of the Carolinas during the Gold Rush era in the 1850’s, with 3 tons of gold and valuable Geographic documentation.  In this talk we will discuss briefly (and very informally!) about some of the science that lead to the precise site of the treasure:  an intriguing combination of many fields including Geometry, Zoology, Statistics, and many other.


Scott Dunn

Arithmetic Progressions in the Polygonal Numbers

In this talk, we investigate arithmetic progressions in the polygonal numbers with a fixed number of sides. We first show that four-term arithmetic progressions cannot exist. We then describe explicitly how to find all three-term arithmetic progressions. Finally, we show that
not only are there infinitely many three-term arithmetic progressions, but that there are infinitely many three-term arithmetic progressions starting with an arbitrary polygonal number. Special attention is paid to the case of squares and triangular numbers with an emphasis on
undergraduate research possibilities.


Josie Ryan

Math is not alone: The undisciplined mind and a polymathic approach to life, the universe, and everything


Spring Meeting at The Citadel
March , 2015

Leandro Junes

Fibonacci Numbers and Non-Decreasing Dyck Paths.

The Fibonacci sequence has fascinated both amateurs and professional mathematicians for centuries, and it continues to charm us with its beauty, its abundant applications, and its ubiquitous habit of occurring in totally surprising and unrelated places. I will discuss in this talk how Fibonacci numbers appear in combinatorial objects called Non-Decreasing Dyck paths. In particular, the Fibonacci numbers helps us to count several statistics for Non-Decreasing Dyck paths.

Antara Mukherjee

Properties of the Hosoya Polynomial Triangle

The Hosoya triangle is a triangular arrangement of numbers similar to Pascal’s triangle where the entries are product of Fibonacci numbers. In this research we discuss the generalized Fibonacci polynomials, these polynomials have Fibonacci numbers as coefficients. Next we construct the Hosoya’s polynomial triangle which is a generalization of the Hosoya triangle where each entry is a product of two generalized Fibonacci polynomials. We show that some algebraic and geometric properties that occur in the Pascal triangle also hold in the new triangle. In particular we find closed formulas for the alternating sum of products of polynomial such as Fibonacci polynomial, Chebyshev polynomials, Morgan-Voyce polynomials, Lucas polynomials, Pell polynomials,  Fermat polynomials and other familiar sequences of polynomials.


Fall Meeting at The Citadel

David Forge

Bijections Between Affine Arrangements and Valued Graphs

I show new bijective proofs of previously known formulas for the number of regions of some deformations of the braid arrangement, by means of a bijection between the no-broken circuit sets of the corresponding integral gain graphs and some kinds of labelled binary trees. This leads to new bijective proofs for the Shi, Catalan, and similar hyperplane arrangements. It may lead to interesting new properties of the Linial arrangement.

This is joint work with Sylvie Corteel and Véronique Ventos.

Orsay Laboratoire de Recherche en Informatique(LRI)


Fall Meeting at Benedict College
October 11, 2013

Adela Vraciu

Degrees of relations in positive characteristic

We consider the polynomials x1^d1, ... , xn^dn, (x1 + ... + xn)^dn+1 in k[x1, ... ,xn]. We study the smallest degree of a non-Koszul relation in this polynomial. The answer depends on the characteristic of the field k


Andrew R. Kustin

Studying singularities by way of syzygies

Consider a parameterized curve in the projective plane. We investigate the singularities on the curve from the point of view of the relations on the homogeneous forms which parameterize the curve.


W. Garrett Mitchener

Simulating the Evolution of Regulatory Networks

The Utrecht Machine (UM) is a discrete abstraction of a biochemical gene regulatory network (GRN). Virtual organisms based on the UM can perform any computation, given sufficient resources. Such simulations combine ideas from molecular genetics, artificial life, and evolutionary dynamics to form a platform for studying how GRNs evolve to solve problems. I'll discuss the modeling process, explaining why I designed the UM the way I did, which biological details are included and which are left out. I'll discuss a case study in which selective breeding discovers agents that solve a data encoding problem, and its roots in a question about the evolution of linguistics.


Breeanne A. Baker

The k-Fixed-Endpoint Path Partition Problem on Trees and Block Graphs

The k-fixed-endpoint path partition problem is to determine the minimum number of vertex disjoint paths required to cover a graph such that every vertex in a given set T is an endpoint of a path. This problem is a generalization of the Hamiltonian path problem and is therefore NP-complete in general. When restricted to certain graph classes, the k-fixed-endpoint path partition problem becomes polynomial. Min-max theorems which characterize the k-fixed-endpoint path partition number for trees and block graphs are discussed.


Spring Meeting at USC Salkehatchie
March 22, 2013

Robbie Bacon

Rock Paper Scissors Lizard Spock

The classical Rock-Paper-Scissors game is revisited. We use payoff matrix and expected value to analyze the winning strategy of this game. We also examine several of its variations, like Rock-Paper-Scissors-Well and Rock-Paper-Scissors-Lizard-Spock. Other potential additional weapons are also discussed.


Fidele Ngwane

Trigonometrically-fitted Second Derivative Method for Oscillatory Problems

A continuous  Second Derivative Method (CSDM) whose coefficients depend on the frequency and stepsize is constructed using Trigonometric basis functions. Some discrete Second Derivative Methods are recovered from the CSDM as by-products and applied as a block  Second Derivative Algorithm (BSDA) to solve oscillatory initial value problems (IVPs). We discuss the stability properties of the BSDA and present numerical experiments to demonstrate the efficiency of the method.


Leandro Junes

Polygons in the Hosoya's Triangle

In this talk we discuss several GCD properties that generalize from Pascal triangle to Hosoyas triangle. In particular, we prove the GCD property for the Star of David and other polygons. We also give a criterion to determine whether a sequence of points in a polygon or in a rhombus have GCD equal to one.

Jeffrey Beyerl

Optimizing Balance in Video Games

Video games are a pervasive part of society and a growing multi-billion dollar industry. A current trend in modern video games is to give players multiple asymmetrical styles of play to choose from: that is, a customization of their gameplay that is distinctly different from another player’s gameplay. To encourage build diversity and game longevity these choices should not be easy to make. In this talk I will give some background motivation and a method for balancing these choices in action role-playing games.


Wei-Kai Lai

A Rearrangement Inequality on Ordered Tensor Products

In 1934, Hardy, Littlewood and Polya introduced a rearrangement inequality: the sum of the products of two real number sequences will reach its maximum if these two sequences are both in increasing (or decreasing) order, and will reach its minimum if these two sequences are in opposite order. With techniques introduced by Bu, Buskes, and Lai, we successfully create a similar version of the rearrangement inequality on ordered tensor products.



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.


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.