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| Presenter | Coauthors | Title & Abstract |
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Ethan Berkove Lafayette College |
Mike Brilleslyper United States Air Force Academy |
Graph labeling and social intervals We say that a labeled graph admits a coprime labeling when integers can be assigned to vertices in such a way that integers associated to vertices connected by an edge are coprime. In this talk we will discuss which complete bipartite graphs $K_{p,q}$ with $p+q=n$ admit coprime labelings by $\mathcal{A}$, where $\mathcal{A}$ is a set of $n$ consecutive integers starting at $k$. It turns out that there are labeling sets $\mathcal{A}$ which cannot coprime label any complete bipartite graph at all. We call these sets of labels "social intervals"; we will provide both a graph theoretic and number theoretic description of such sets. |
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Byungchul Cha Muhlenberg College |
Dong Han Kim Dongguk University (South Korea) |
Lagrange and Markov Spectra of Pythagorean triples Call $(p, q)$ a Pythagorean pair if $p$ and $q$ are positive integers such that $p^2 + q^2$ is a perfect square. Draw a line $\ell$ from the origin into the first quadrant of the $xy$-plane. Suppose we want $\ell$ to avoid all but finitely many Pythagorean pairs with as large a margin as possible. What is the greatest possible margin? What is the second greatest? In 2008, Romik used a certain ternary tree consisting of Pythagorean triples to define a dynamical system on the unit quarter circle. We will study a Lagrange spectrum arising from Romik's dynamical system. This provides a natural setting for intrinsic Diophantine approximation on the unit circle. Our result gives a complete answer to the questions posed above. In addition, we obtain an analogue in this context to a classical theorem on Lagrange and Markoff spectra, which was first proved by Markoff in 1879. |
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Joe Chaffee Kaiser Permanente |
John Asplund Dalton State College James Hammer Cedar Crest College |
$\gamma^\prime$-Realizability and Other Musings on Inverse Domination This talk will introduce and study $\gamma'$-realizable sequences. For a finite, simple graph $G$ containing no isolated vertices, $I \subseteq V(G)$ is said to be an inverse dominating set if $I$ dominates all of $G$ and $I$ is contained by the complement of some minimum dominating set $D$. Define a sequence of positive integers $(x_1, \ldots , x_n)$ to be $\gamma'$-realizable if there exists a graph $G$ having exactly $n$ distinct minimum dominating sets $D_1, \ldots, D_n$ where for each $i \in \{1, \ldots, n\}$, the minimum size of an inverse dominating set in $V(G) \setminus D_i$ is equal to $x_i$. In this work, we show which sequences having minimum entry $2$ or less are $\gamma'$-realizable. We then detail a few observations and results arising during our investigations that may prove useful in future research. |
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Ji Young Choi Shippensburg University |
Four Generalization of Binomial Coefficients We find four different generalizations of binomial coefficients, by counting nonnegative integers with a restriction on the length and the digit sum of their base-$b$ representations, for any integer $b>1$. We identify one generalization as the extended binomial coefficients, and we express each of the rest generalizations in terms of the extended binomial coefficients. We also express each generalization as a sum of binomial coefficients and we find an explicit formula for each generalization. |
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Lindsay Dever Bryn Mawr College |
Modular Forms in the Hyperbolic Plane Modular forms are an important tool in a broad range of areas, including number theory, representation theory, and quantum physics. One class of modular forms are Maass forms, real analytic functions on the complex upper half-plane that satisfy the modularity property $f\left(\frac{az+b}{cz+d}\right) = f(z)$ for all matrices $\begin{pmatrix} a & b \\ c & d \end{pmatrix}$ in the modular group $SL_2(\mathbb{Z})$ and are eigenfunctions of the hyperbolic Laplacian. In this expository talk, I will introduce hyperbolic geometry in the upper half plane and define modular forms in this context. |
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Eva Goedhart Lebanon Valley College |
Helen G. Grundman Bryn Mawr College, AMS |
Solving Families of Diophantine Equations of the Form $NX^2+2^L3^M=Y^N$ I will introduce Lehmer pairs, Lehmer numbers, and a well-known result of Bilu, Hanrot, and Voutier on defective Lehmer pairs. As an application, I will then use them in an outline of the proof that the family of Diophantine equations $NX^2+2^L3^M=Y^N$ has no integer solutions for any positive integer values of $L$, $M$, and $N>1$ with $\gcd(NX,Y)=1$. |
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Gary Gordon Lafayette College |
Liz McMahon Lafayette College |
Counting Sets in SET and Flipping Coins We connect fundamental questions about the number of sets sharing $k$ attributes in $n$-attribute Set to a coin-flipping game. This allows us to answer questions about Set by using Bernoulli trials, the binomial distribution and the central limit theorem. The connection is direct, but somewhat surprising. We also count the number of transformations that preserve the "weight" of a set. |
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Ronald J. Gould Emory University |
A Look at Saturated Graphs Given a graph $H$, we say that a graph $G$ is H-saturated if it does not contain $H$ as a subgraph but the addition of any edge $e\not\in E(G)$ results in at least one copy of $H$ as a subgraph. The study of saturated graphs has a long and deep history. The question of the minimum number of edges of an $H$-saturated graph on $n$ vertices, known as the saturation number and denoted $sat(n, H)$, has been addressed for many different types of graphs. The saturation number contrasts the classic question of the maximum number of edges possible in a graph $G$ on $n$ vertices that does not contain a copy of $H$, known as the Turán number (or extremal number) and denoted $ext(n, H)$. The saturation spectrum of the family of $H$-saturated graphs on $n$ vertices is the set of all possible sizes $(|E(G)|)$ of an $H$-saturated.We look at a number of recent results that determine the saturation spectrum for a variety of graphs. |
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Joshua Harrington Cedar Crest College |
Undergraduate Research in Sums of Polynomial Residues In 1801 Gauss proved that the sum of distinct quadratic residues in the ring $\mathbb{Z}_p$ is always 0. About one hundred years later, Stetson studied the sum of distinct triangular residues in $\mathbb{Z}_p$. In this talk we will explore the proofs of Gauss and Stetson and generalize their results to certain polynomial residues. This talk is primarily target to undergraduate students in hopes to inspire future undergraduate research. |
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Richard Kliman Cedar Crest College |
Challenges in predicting the joint site frequency spectrum of source and founder populations Recently invasive species are natural experiments for the study of rapid adaptive evolution driven by exposure to novel environments. Intuitively, alleles (i.e., alternative versions) of genes that confer a fitness advantage in the founder population should increase in frequency relative to the source population. However, because of the dynamics of new mutations (which are initially very rare), the neutral frequency spectrum of mutations is heavily skewed toward lower frequencies. If a population is founded by a small number of individuals, and subsequently rapidly increases in size, low-frequency alleles that survive the founder event tend to increase in frequency even without the influence of selection. Therefore, to infer the action of natural selection, we must account for the spectrum-shifting effect of the founder-flush event. I will present a basic model that combines coalescent theory, Pólya urn sampling, and computer simulations to predict the neutral joint site frequency spectrum for founder and source populations. I will then outline the major challenges that arise from violation of the assumptions of the strictly bifurcating coalescent model. |
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Christopher Krizan Kutztown University |
Spectrum of the Szlam Numbers of the Plane Suppose that the plane $\mathbb{R}^2$ is equipped with a translation invariant distance function $\rho$ and suppose that $d>0$. The distance graph $G_\rho(\mathbb{R}^2,d)$ is the graph with vertex set $\mathbb{R}^2$ with $u,v\in\mathbb{R}^2$ adjacent if and only if $\rho(u,v)=d$. A rather red coloring of $G$ is a coloring of $\mathbb{R}^2$ with red and blue such that no two points adjacent in $G$ are both blue. The Szlam number of $G$ is the minimum cardinality, over all rather red colorings of $G$, of $X\subseteq\mathbb{R}^2$ such that no translate of $X$ is all red. Fixing $d=1$, we exploit results of Johnson, Szlam, and Kloeckner to show that for every positive integer $n$ there exists $\rho$ such that the Szlam number of $G_\rho(\mathbb{R}^2,1)$ is $n$. |
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Brian Kronenthal Kutztown University |
Allison Ganger University of Nebraska - Lincoln Shannon Golden Colorado State University Felix Lazebnik University of Delaware Carter Lyons Nebraska Wesleyan University |
The girth of two-dimensional algebraically defined graphs The objects of interest in this talk are algebraically defined bipartite graphs, which are constructed as follows. Let $\mathbb{F}$ denote a field, and consider the bipartite graph whose partite sets $P$ and $L$ are copies of $\mathbb{F}^2$ such that $(p_1,p_2)\in P$ and $[\ell_1,\ell_2]\in L$ are adjacent if and only if $p_2+\ell_2=p_1\ell_1$. This graph has girth six, and of particular interest is identifying any polynomials $f\in\mathbb{F}[x,y]$ such that replacing $p_1\ell_1$ with $f(p_1,\ell_1)$ in the adjacency condition produces a girth six graph that is not isomorphic to the original. In addition to discussing some results related to this question, we will also explain the connection between algebraically defined graphs and incidence geometry, which partially motivates this line of inquiry. |
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Elsa Magness Bryn Mawr College |
Quadratic Reciprocity and The Goldwasser Micali Encryption Scheme In 1982, cryptographers Goldwasser and Micali identified two common failures of encryption schemes based on the notion of a "trapdoor" function. To fix these problems, they propose the idea of probabilistic encryption where an encryption scheme is based on a rule with a random outcome and provide an example of a scheme that uses the quadratic residue problem to encode information in a random way. This talk will summarize the quadratic residue problem and the Goldwasser Micali encryption scheme. |
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Nicholas Mayers Lehigh University |
Andrew Mayers Vincent College |
Integer Partition Statistics Arising From Seaweed Algebras Using the index theory of seaweed algebras, we explore various new integer partition statistics. We find relations to some well-known varieties of integer partitions as well as a surprising periodicity result. |
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Elizabeth McMahon Lafayette College |
Jordan Awan The Pennsylvania State University Claire Frechette The University of Minnesota Yumi Li |
Geometry, Combinatorics and the Game of SET The deck of cards for the game of SET is an excellent model for the finite affine geometry $AG(4,3)$ and provides an entry to surprisingly combinatorial structure theorems for that geometry. In this talk, we’ll explore that geometry and find a beautiful structure hiding within the collections of cards that have no sets (called maximal caps). There’s a connection to the outer automorphisms of $S_6$ as well, but you don’t need to know what those are to follow what we’ll do. |
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Michael Mossinghoff Davidson College |
Come On Down! Barker Sequences Through Number Theory and Combinatorics A Barker sequence is a finite sequence of integers $\{a_i\}$, each $\pm1$, for which $\sum_i a_i a_{i+k}$ is in $\{-1, 0, 1\}$ for every positive integer $k$. Very few Barker sequences are known, and it is widely conjectured that only finitely many exist. We will discuss the origins of this problem and how it is related to a number of other open questions in number theory, analysis, and combinatorics, concerning for instance Wieferich primes, flat polynomials, and circulant Hadamard matrices. We will also describe some recent progress on this problem, from algebraic, analytic, and computational points of view. In particular, we will describe some algorithms recently designed to identify values that cannot be discounted as potential winning values in the Barker sequence problem and its relatives. As the former The Price is Right game show host Bob Barker would say, "Come on down!" to hear about this topic at the intersection of many mathematical subjects! |
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Viorel Nitica West Chester University |
About Some Relatives of the Taxicab Number The taxicab number, $1729$, became well known due to a discussion between Hardy and Ramanujan. It is the smallest positive integer that can be written in two ways as a sum of two cubes: $1^3+12^3$ and $9^3+10^3$. The number $1729$ also has a less well known property: if we add its digits we obtain $19$; multiplying $19$ by $91$, the number obtained from $19$ reversing the order of its digits, we obtain again $1729$. It is not hard to show that the set of integers with this property is finite and equal to $\{1, 81, 1458, 1729\}$. In a conversation that I had with my colleague, Professor Shiv Gupta, Shiv asked if the second property can be generalized. One replaces the sum of the digits of an integer by the sum of the digits times an integer multiplier and then multiplies the product by the number obtained reversing the order of the digits in the product. The taxicab number becomes a particular example with multiplier $1$. A computer search produced a large number of examples with larger multiplier. There are $23$ integers less than $10000$ having this property; see sequence A305131 in the Online Encyclopedia of Integer Sequences. For example, $2268$ has multiplier $2$. The sum of the digits is $18$, one has $18\times 2=36$, and $36\times 63=2268$. One may replace the last product in the above procedure by a sum. A computer search showed that there are numbers that have the property for sums. There are $264$ integers less than $10000$ having the property; see sequence A305130 in the Online Encyclopedia of Integer Sequences. For example, $121212$ has multiplier $6734$. The sum of the digits is $9$, one has $9\times 6732=60606$, and $60606+60606=121212$. The talk is dedicated to the study of these properties. |
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Caitlin Owens Rowan University |
Garth Isaak Lehigh University |
Forbidden subgraphs for Hamiltonian problems on 2-trees It is known that 2-trees are Hamiltonian if and only if they are 1-tough. However, the analogous statement for Hamiltonian paths does not hold. We will define a family of 2-trees such that a 2-tree has a Hamiltonian path if and only if it does not contain any graph from that family as an induced graph. To define this family, we will examine a variation of the Hamiltonian path problem, 2HP, which is to determine whether or not it is possible to find a Hamiltonian path in a graph when both endpoints of the path are fixed. These results will be extended to the Hamiltonian path problem on 2-trees. |
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Matthew Prudente Alvernia University |
Garth Isaak Lehigh University |
Two-Player Graph Pebbling Given a graph $G$ with pebbles on the vertices, we define a pebbling move as removing two pebbles from a vertex $u$, placing one pebble on its neighbor $v$ and discarding the other pebble as a toll. The pebbling number $\pi(G)$ is the least number of pebbles needed so that every arrangement of $\pi(G)$ pebbles can place a pebble on every goal vertex $r$ through a sequence of pebbling moves. We introduce a new variation on graph pebbling called two-player pebbling. In this, players called Mover and Defender alternate moves, with the stipulation that Defender cannot reverse the previous move. Mover wins if they can place a pebble on the root and Defender wins if the goal vertex has no pebble on it and there are no pebbling moves left. We define $\eta(G)$, analogously, as the minimum number of pebbles such that given every configuration of the $\eta(G)$ pebbles and every root vertex $r$, Mover has a winning strategy. We investigate winning strategies and configurations for both players on a special class of diameter 2 graphs and variations of paths. |
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Kathleen Ryan DeSales University |
Vincent Coll Lehigh University Grant Fickes Kutztown University Dylan Green Trevecca Nazarene University Jonelle Hook Mount St. Mary's University Colton Magnant Clayton State University Karen McCready King's College Nathaniel Sauerberg Carleton College Jill Stifano Fairfield University |
How many steps does it take to get between the ends of a tootsie roll? An edge-colored graph is properly connected if there exists a properly colored path between every pair of vertices. The presence of this property allows us to extend the notions of distance and diameter in such edge-colored graphs $G$. To do so, we let the proper distance between two vertices in $G$ be the length of a shortest properly colored path between them, and we define the proper diameter of $G$ to be the maximum proper distance between any pair of vertices in $G$. During this talk, we explore the proper diameter values of various graph families, including a graph family that we affectionately nicknamed "tootsie roll graphs." During our explorations, we'll even discover how many steps it can take to get between the endpoints of a tootsie roll graph. |
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Barry Smith Lebanon Valley College |
Classifying indefinite quadratic forms with combinatorial necklaces Dirichlet showed that reduction of indefinite binary quadratic forms corresponds to expansion of a corresponding continued fraction. Through this construction, a combinatorial necklace over the alphabet $\mathbb{N}$ can be attached to each form class – the necklace just encodes the minimal period of the continued fraction. It has long been known that primitive classes correspond to primitive necklaces and that mirror symmetry in the necklace forces the corresponding class to have order dividing 4. We provide further results about how visual properties of the necklaces are reflected in the arithmetic of the corresponding classes. In particular (1) we discuss Markoff’s famous family of forms, which correspond to Christoffel necklaces, and (2) we examine certain two-parameter families of forms and see how those of small order in the class group can be identified by visual properties of their necklaces. |
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Wing Hong Tony Wong Kutztown University |
Jiao Xu West Virginia University |
A Two-Person Random Walk Game Alice and Bob take turns to toss a fair coin, which decides whether they collect $1$ or $2$ chips in that turn. The first player who accumulates at least $100$ chips wins the game. We determine the winning probability of Bob, and generalize the problem to more complicated versions. |
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Daniel White Bryn Mawr College |
Power Moments of Dirichlet L-functions Study of the distribution of zeroes of the Riemann-zeta function and Dirichlet $L$-functions in the critical strip is essential to understanding the distribution of primes. The Lindelöf hypothesis on maximal rate of growth of the Riemann-zeta function along the critical line asserts that $\zeta(1/2+it) \ll |t|^{\varepsilon}$ for every $\varepsilon > 0$. This statement turns out to be equivalent to $$\frac{1}{T} \int_0^T |\zeta(1/2+it)|^{2k} \, dt \ll T^{\varepsilon}$$ for all positive integers $k$. Considerable effort has been placed in understanding these power moments over the past century. Hardy and Littlewood, and later Ingham, determined the asymptotic rate of growth for $k=1$ and $2$, respectively. Despite the time that has passed since, no asymptotics have been proved for $k \ge 3$. In the realm of Dirichlet $L$-functions, natural analogues are power moments in the $q$-aspect, where $|L(1/2,\chi)|^{2k}$ is averaged over primitive characters $\chi$ modulo $q$. The situation $k=1$ is classically understood, but work due to Heath-Brown and Soundararajan was required to seal off $k=2$. Only recently was Young able to produce power-saving asymptotics for this case. In this expository talk, we will develop an indispensable tool to this pursuit, the so-called approximate functional equation for Dirichlet $L$-functions at the central point. From this, we will recover classical results and present some heuristics. The talk will conclude with a brief survey of recent and current research on higher moments with emphasis on strong bounds for the twelfth moment. |