| Section A – 10:00 - 11:10 |
| RKC 101 |
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| Critical Curves of Knot Energy |
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| Knot energy is potentially a very powerful tool in understanding and simplifying very complicated closed curves. Under an energy gradient flow, knots would simplify to a locally minimal energy configuration. We offer as a conjecture that for closed embedded curves in the plane, the only critical energy configuration is the round circle. We provide the beginnings of a proof using curve shortening flow, and some isoperimetric properties, as well as reason to suspect the proof will in fact hold and some possible approaches to complete it. |
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| Elementary |
| Prerequisites: |
| Author: Matthew Wise |
Vassar College |
| Advisor: John McCleary |
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| Finite Instances of Knot Quandles |
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| One classifying invariant of knots is an algebra known as quandles. Moreover, finite algebras, such as quandles, give rise to associated Constraint Satisfaction Problems. We will discuss finite instances of knot quandles in order to introduce our progress toward classifying knots by their underlying complexity classes. |
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| Elementary |
| Prerequisites: |
| Author: Peter Golbus |
Bard College |
| Advisor: Bob McGrail |
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| Toward Quandle Dichotomy |
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| Feder and Vardi (1993) discovered a strong correspondence between finite algebras and computational complexity through the constraint satisfaction problem (CSP). It allows a classification of algebras according to their complexity within NP. We focus on quandles, algebras that arise via knot theory. In particular, we demonstrate that all finite quandles that are not locally connected are NP-complete. Furthermore, we will present recent progress on the classification of locally connected quandles. |
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| Advanced |
| Prerequisites: |
| Author: Mona Merling |
Bard College |
| Advisor: Robert McGrail |
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| RKC 102 |
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| Data Mining Alogrithms |
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| Data mining is becoming a very important field. People have huge amount of data available, but we need to have an effective way of finding the pattern of our data in order to extract information. Starting from the basic k-means algorithm, our group experimented with different ways of clustering data. Finally, we came up with an adaptation to the existing spectral clustering technique. Moreover, we proposed the clustering aggregation technique that performs better than many of the existing algorithms. |
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| Elementary |
| Prerequisites: |
| Author: Dexin Zhou |
Bard College |
| Advisor: Lauren Rose |
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| Solving LaPlace’s Equation with Complex Polynomials Using Mathematica |
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| The theory of analytic functions found in the general topic of complex variables provides an exciting approach to approximating partial differential equations (PDE) of the LaPlace or Poisson type in two-dimensional spatial domains. Because analytic functions resolve into a set of conjugate two-dimensional real valued functions that exactly solve the LaPlace equation (and therefore given a particular solution, exactly solves the Poisson equation), they can readily be used to model two-dimensional potential problems in a wide variety of important applications including heat transport, corrosion, electrostatics, groundwater flow, among many others. With the advances in computer technology seen in the last decade, inexpensive computer programs are now available that provide considerable accuracy in the solution of matrix systems such as employed in the use of numerical methods, including the complex variable approach considered in the subject paper. As a result, computer programs such as Mathematica may be readily programmed to leverage the attractive features of analytic functions in modeling two-dimensional problems of potential problems. This result is significant because the traditional numerical techniques such as finite difference or finite element methods require the discretization of the two-dimensional domain into a myriad of nodes and finite elements, with the resulting numerical model still not exactly solving the governing PDE. In contrast, the complex variable technique presented exactly solves the governing PDE, and for a general two-dimensional spatial domain. A brief discussion of extending the presented modeling technique to three-dimensional problems is discussed. |
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| Elementary |
| Prerequisites: |
| Author: Cadets Addison Bohannon and Andrew Poler |
United States Military Academy |
| Advisor: Colonel Tom Kastner |
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| Dirichlet's Integral |
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| Using Lebesgue integration techniques, we prove the convergence of Dirichlet's integral and in the sequel we calculate its value. |
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| Advanced |
| Prerequisites: Advanced Calculus
Introductory Real Analysis(Lebesgue Integration) |
| Author: Stela Mihneva |
St. Francis College |
| Advisor: Fotios Paliogiannis, Ph.D. |
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| RKC 103 |
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| Mathematical Models of Reptile |
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| A well-known phenomenon among reptile species is temperature-dependent sex determination (TSD) in which the temperature of egg incubation determines the sex of the hatchlings. Based on previous work in (Murray, 2002) and (Woodward and Murray, 1993) we develop a delay differential equation (DDE) model describing the nesting habits of crocodilians; the delay accounts for some of the dependence of birth and death rates on the age of the population members. We are able to reasonably account for age structure in the population while finding a nonzero stable equilibrium. Stability of this equilibrium allows us to compare this to the biological data of Smith and Webb (1985, 1987); we obtain very strong agreement. Additionally, we solve our model numerically using a modified Runge-Kutta solver in Matlab and investigate the effects of environmental changes on the population. |
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| Elementary |
| Prerequisites: |
| Author: Michael Zawoiski |
City College |
| Advisor: Ethan Akin |
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| The Lion and Man Game in the P |
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| In this paper, we present a sufficient condition for a pursuer to capture an evader inside a convex environment with a single convex obstacle. This is a novel variant of the well-known lion and man game where a lion tries to capture a man with the same speed inside a circular arena. |
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| Elementary |
| Prerequisites: |
| Author: David Cerna |
Rensselaer Polytechnic Institute |
| Advisor: Volkan Isler |
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| Properties of Exponential Matrices |
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| In the project "Properties of Exponential Matrices" I will systematically examine several different properties of exponential matrices with the aide of technology (Matlab Software) and then provide proofs from theoretical considerations. My objective was to research if exponential matrices were commutable and what their practical applications could be.
I first examined composite transformations, which is the movement of a figure requiring two or more matrix multiplications in two dimensions. This application is similar to some of the most interesting breakthroughs at present in 3D Computer Graphics, especially when referring to modeling in the medical field.
I also scrutinized the non-commutative nature of exponential matrices and its cross-topic association to derivatives. Finally, perhaps the most interesting application is a culmination of solving ordinary simultaneous differential equations using matrices.
This project illuminates the amazing applications of exponential matrices in linear algebra, calculus, computer graphics and differential equations from elementary considerations. |
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| Elementary |
| Prerequisites: |
| Author: Lerone Bleasdille |
College of Technology of CUNY |
| Advisor: Satyanand Singh |
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| Section B – 2:00 - 3:10 |
| RKC 101 |
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| Generation of Megagraphs |
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| We discuss graphs whose vertices are graphs, known as megagraphs. More specifically we discuss graphs whose vertices are forests and a sensible method of defining edges between two megavertices of a megagraph. We also discuss looking at the megagraphs as partially ordered sets, an algorithm for computing these megagraphs, and consequences from the data gained from running the algorithm. |
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| Elementary |
| Prerequisites: |
| Author: Morgon Kanter |
Bard College |
| Advisor: Lauren Rose |
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| Self Contact in Fractal Trees |
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| Self-Contact in Three-Dimensional, Asymmetric Fractal Trees
Previous research on fractal trees has elaborated upon two dimensional symmetric and asymmetric fractal trees and three-dimensional symmetric fractal trees. I extend this research to study three-dimensional asymmetric fractal trees with three branches. Visualization and computational complexity leads to analysis of a finite set of canopy points to determine connectivity. Primary topics include connectedness of these fractal trees and the dimensions of connected fractal trees. |
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| Advanced |
| Prerequisites: |
| Author: Bailey Meeker |
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| Advisor: David Brown |
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| Subcubes of the hypercube, Qn |
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| The n-dimensional hypercube Qn is defined recursively, by Q1 = K2 and Qn = Qn-1 x K2. The second equation says that we get Qn by taking two copies of Qn-1 and rendering corresponding vertices adjacent. The two vertices of Q1 are labeled 0 and 1. As Q2 contains two copies of Q1, we label the vertices in the first copy 00 and 01, while the vertices in the second copy are labeled 10 and 11. Since Q3 consists of two copies of Q2, we attach a 0 to the labels of the vertices of the first copy of Q2 and a 1 to the labels of those of the second copy, in which case V(Q3) = {000, 001, 010, 011, 100, 101, 110, 111}. More generally, the vertex set of Qn consists of all binary strings of length n. The distance between x = (x1, x2, …, xn) and y = (y1, y2, …, yn) is |x1 – y1| + |x2 – y2| + … + |xn – yn|. We show that if d(x, y) = k < n, then there is a unique copy of Qk in Qn containing x and y. When d(x, y) = k < r < n, we find the number of copies of Qr in Qn containing x and y and obtain further results in this case. |
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| Elementary |
| Prerequisites: Elementary graph theory |
| Author: Anthony Delgado |
Purchase College (SUNY) |
| Advisor: Marty Lewinter |
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| RKC 102 |
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| A GENERALIZATION OF RECURSIVE INTEGER SEQUENCES OF ORDER 2 |
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Abstract(with LaTeX notation):
The algebraic structure of all general linear recursive sequences is developed utilizing an isomorphism between the golden ring, $Z[phi]$ and the ring $Z[Omega_alpha]$. The following results will then be obtained: (1) There is a ring extension from $Omega$ to $Omega_alpha$. (2) The general form of Period-1 matrices and Period-2 matrices using $Z[Omega_alpha]$ and vector spaces, (3) the determinant of all Period-1 matrices in $Omega_alpha$ is equal to 1, (4) the general form of powers of Period-1 and Period-2 matrices in the 2x2 and 4x4 case, (5) and the understanding of the consequences of multiplying n x m and m x r recursive matrices. (6) In addition, we will be able to generalize every $2k$ x $2k$ general recursive matrix forms a ring, and (7) every nxm recursive matrix that are elements of a complete orbit for a semigroup. (8) There exists only Period-1 and Period-2 orbits in $Omega_alpha.$ |
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| Elementary |
| Prerequisites: Abstract Algebra. |
| Author: Stephen Parry |
Elmira College |
| Advisor: Charlie Jacobson |
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| Iterating a Voronoi Diagram |
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| If you don't know what a Voronoi tessellation is, you should! From cell phones to crystalline structures, the Voronoi tessellation has found use in a multitude of varying disciplines. We'll be interested in talking about these objects in the framework of dynamical systems: for any finite bunch of points in the plane, there is a remarkably natural way of using the Voronoi tessellation to extract a new bunch of points. This map can be applied over and over again, yielding a sequence of evolving point sets. We shall define this novel system and present a number of original results which aim to explore just how this system changes over time. |
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| Elementary |
| Prerequisites: |
| Author: Sean Hart |
Vassar College |
| Advisor: Natalie Frank |
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| Voronoi Inverses |
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| We introduce an intuitive dynamical system that maps one point set to another using Voronoi tessellations. In most non-trivial cases, this system can be used to iterate forward an infinite number of times. But is it possible to reverse the mapping and iterate backwards from a given point set? I will be giving a brief background on this dynamical system before proceeding to answer that question: the question of the Voronoi inverse’s existence. In particular, I will be examining methods for finding Voronoi inverses on some basic point sets, which will lead to a useful property which allows one to quickly determine if a point set which lies entirely on its own convex hull has a Voronoi inverse. |
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| Elementary |
| Prerequisites: |
| Author: Richard Langford |
Vassar College |
| Advisor: Prof. Natalie Frank |
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| RKC 103 |
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| Mathematical Models of Reptile |
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| A well-known phenomenon among reptile species is temperature-dependent sex determination (TSD) in which the temperature of egg incubation determines the sex of the hatchlings. Based on previous work in (Murray, 2002) and (Woodward and Murray, 1993) we develop a delay differential equation (DDE) model describing the nesting habits of crocodilians; the delay accounts for some of the dependence of birth and death rates on the age of the population members. We are able to reasonably account for age structure in the population while finding a nonzero stable equilibrium. Stability of this equilibrium allows us to compare this to the biological data of Smith and Webb (1985, 1987); we obtain very strong agreement. Additionally, we solve our model numerically using a modified Runge-Kutta solver in Matlab and investigate the effects of environmental changes on the population. |
| |
| Elementary |
| Prerequisites: |
| Author: Michael Zawoiski |
City College |
| Advisor: Ethan Akin |
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| The Lion and Man Game in the P |
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| In this paper, we present a sufficient condition for a pursuer to capture an evader inside a convex environment with a single convex obstacle. This is a novel variant of the well-known lion and man game where a lion tries to capture a man with the same speed inside a circular arena. |
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| Elementary |
| Prerequisites: |
| Author: David Cerna |
Rensselaer Polytechnic Institute |
| Advisor: Volkan Isler |
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| Evolved Art - planetary motion |
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| Evolutionary computation is a relatively new field that uses concepts from biological evolution: differential selection in reproduction and replacement, mutation and crossover; evolved art is a subfield that aims to make art from evolutionary computation. Evolutionary computation is usually applied to problems of optimization or modelling - we present an application that covers both. The n-body problem in physics is solving for the trajectory of point particles under the influence of only gravity in an isolated system. For a rotationally symmetric 8-body problem, we evolve initial values (mass, position, velocity) to minimize the propagation of roundoff error in a numerical approximation, and view the intricate pictures thus drawn. |
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| Elementary |
| Prerequisites: |
| Author: Jeffrey Tsang |
U of Guelph |
| Advisor: |
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