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Carolina Math Seminar (CMS)

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Organizers

The Citadel

Rigoberto Flórez

rigo.florez@citadel.edu
Phone: (843) 953 5034

Antara Mukherjee
antara.mukherjee@citadel.edu

USC Salkehatchie

Wei-Kai Lai
laiw@mailbox.sc.edu

Fidele Ngwane
ngwanef@mailbox.sc.edu

Benedict College

Alexandru Gabriel Atim
atima@benedict.edu

Columbia College

Virginia Johnson
vjohnson@columbiasc.edu

Francis Marion University

Jeremiah Bartz
jbartz@fmarion.edu

Newberry College

Naser AlHasan
Naser.AlHasan@newberry.edu

Tien Chih
Tien.Chih@newberry.edu

Lander University

Josie Ryan
pryan@lander.edu

Universtiy of South Carolina

Paula Vasquez
paula@math.sc.edu

Ralph Howard
howard@math.sc.edu

Scott Dunn
dunnsm@mailbox.sc.edu

California University of Pennsylvania

Leandro Junes
junes@calu.edu

USC Lancaster

Shemsi Irene Alhaddad
alhaddad@mailbox.sc.edu

Jason Holt
jholt@mailbox.sc.edu

Andrew Yingst
yingst@mailbox.sc.edu

 

 

Fall Meeting at Benedict College
Nov 3, 2017

James Andrus

Hsin-Yun Ching

James Hepburn

Antara 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.

  • Budapest Semesters in Mathematics independent status, started by academics like Lászlo Babai and indirectly by Erdős in late 1980s
  • Prestigious and rigorous, classes in topics not typically not taught to undergraduates, some effectively at graduate level, in many cases the Hungarian method
  • Hungarian method of teaching emphasizes individual problem solving and creativity
  • Method is also taught to those who wish to be educators; best found in BSME program and courses like “Proof and Method”
  • American credits, 4 credit hours, through St. Olaf’s
  • Beautiful city, low cost of living, like-minded and passionate, intelligent students

  • The Gilman Award: Through US State Department for Pell Grant recipients
  • To promote cultural understanding and good economic, political, and civil societal relations
  • Up to $4,000 for up to a year, and summers
  • Good stepping stone to higher fellowships and awards
  • Compensates for tuition difference for BSM

  • ReConnect Hungary: Birthright program for Americans and Canadians of Hungarian ancestry, in case anyone in audience is

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*])
dt

where for us the importance will be that (cij) forms a complex positive (N2-1)\times (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

Rade Musulin

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.

   
 
 

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