Spring Meeting at the Citadel
March 30, 2018
Elizabeth Schuyler Spoehel
A Fibonacci identity
In this presentation I talk about the solution of Problem B1218 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.
HsinYun 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.
WeiKai 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, AMGM 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 bestknown and bestloved 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 WatsonCrick 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 2Regular 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 2regular graphs.
Hays Whitlatch
Graph Pressing Sequences on Mitochondrial Genomes
One can construct a useful metric on genome sequences by computing minimallength 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 DelayVariation 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.
HsinYun 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 likeminded, 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 T_{t}. The generator of this semigroup can
be cast using the GKLS master equation
dρ

=L(ρ)=  i[H,ρ]+(½)∑_{i,j=1}^{N21 }c_{ij}([F_{i},ρ F_{j}^{*}]+[F_{i}ρ,F_{j}^{*}])

dt

where for us the importance will be that (c_{ij}) forms a complex positive (N^{2}1)* (N^{2}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 (c_{ij}), 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.
WeiKai 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 JeanPierre 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:
(F_{n},F_{n+k}), (F_{n+2k},F_{n+3k}), and (F_{n+4k},F_{n+5k})
has area
5(F^{4}_{k} L_{k})/2 for k even and (F^{2}_{k} L^{3}_{k})/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 ngon 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 integrodifferential 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
Stimuliresponsive polymer gels have many surprising nonNewtonian properties such as shearthinning and shearthickening. 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 nonlinear elastic beadspring 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 micromacro 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 F_{0}=0,F_{1}=1, and F_{n}=xF_{n1}(x)+F_{n2}(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 MorganVoyce polynomials, while the polynomials that do not satisfy the Fibonacci gcd property are: Lucas polynomials, PellLucas polynomials, FermatLucas polynomials, Chebyshev polynomials of first kind, JacobsthalLucas polynomials and second type of MorganVoyce polynomials. These last set of polynomials partially satisfy the abovementioned 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 npower 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 twocoloring 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 twocoloring a uniform hypergraph.
Paula Vasquez
Mathematical Modeling of Viscoelastic Materials
In a broad sense, one can divide fluids into Newtonian and nonNewtonian according to their response to flow. For example, honey –a Newtonian fluid always flows, while mayonnaise –a nonNewtonian material does not flow under moderate stresses. From a modeling point of view, all Newtonian fluids can be described by the wellknown NavierStokes 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 nonNewtonian 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 nonNewtonian 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 GaleShapely 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.
WeiKai 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 wellknown 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 UpDown 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 eigenequations are investigated. In the case of second order sequences, after completing the square, the CaleyHamilton theorem is applied and the binomial expansion yields several binomial summation identities. In the case of the third order sequences the CaleyHamilton 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 pgroups such that a subset of their maximal subgroups form an equal quasipartition
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 3connected graphs
An edge e in a 3connected graph G is contractible if the contraction G=e is still 3connected. The existence of contractible edges is a very useful induction tool.
Let G be a simple 3connected 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 3connected 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 3edgeconnected 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 15Puzzle
In the world of mathematical puzzles, both Magic Square and 15Puzzle 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 15Puzzle 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 Multigroup Immunoepidemiological
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 twoscale 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 immunoepidemiological model of coupled withinhost model of ordinary differential equations and betweenhost model of ordinary and partial differential equations. Existence and uniqueness of solution to the epidemiological (betweenhost) model is established, and an explicit expression for the basic reproduction number of the multigroup coupled betweenhost model is derived. Steady state solutions are calculated and stability analysis of the diseasefree and endemic equilibria is investigated. An optimal control problem with drugtreatment control (fusion and protease inhibitors) on the withinhost system is formulated and analyzed. Numerical simulations based on the semiimplicit finite difference scheme and the forwardbackward sweep iterative method are obtained.
Michael Tiemeyer
On zCycle Factorizations with Two Associate Classes
Let K = K(a,p;λ_{1},λ_{2}) 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 λ_{1}; and the number of edges joining any two vertices of different parts equal to λ_{2}. In this presentation, we give history and current progress regarding zcycle 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 PrimalDual 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) primaldual maximizationminimization 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 primaldual 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 ExistenceDuality 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 functiontheoretic, rather than coordinatecentric, 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 humanlike 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 rootfinding 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 gamewinning, musicreading, obstacleavoiding 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 BlancoSilva
Searching for the SS Central America
In the early 1980’s, the ColumbusAmerica 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 fourterm arithmetic progressions cannot exist. We then describe explicitly how to find all threeterm arithmetic progressions. Finally, we show that
not only are there infinitely many threeterm arithmetic progressions,
but that there are infinitely many threeterm 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 NonDecreasing 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 NonDecreasing Dyck paths. In particular, the Fibonacci numbers helps us to count several statistics for NonDecreasing 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, MorganVoyce 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 nobroken 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 x_{1}^d_{1}, ... , x_{n}^d_{n}, (x_{1} + ... + x_{n})^d_{n+1} in k[x_{1}, ... ,x_{n}]. We study the smallest degree of a nonKoszul 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 kFixedEndpoint Path Partition Problem on Trees and Block Graphs
The kfixedendpoint 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 NPcomplete in general. When restricted to certain graph classes, the kfixedendpoint path partition problem becomes polynomial. Minmax theorems which characterize the kfixedendpoint 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 RockPaperScissors 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 RockPaperScissorsWell and RockPaperScissorsLizardSpock. Other potential additional weapons are also discussed.
Fidele Ngwane
Trigonometricallyfitted 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 byproducts 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 multibillion 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 roleplaying games.
WeiKai 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.
