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» Vol. 12, Issue 2, 2011 «



Title: The Poisson Integral Formula and Representations of SU(1,1)
Author: Ewain Gwynne, Northwestern University Author Bio    
Abstract: We present a new proof of the Poisson integral formula for harmonic functions using the methods of representation theory. In doing so, we exhibit the irreducible subspaces and unitary structure of a representation of the group SU(1,1) of 2 x 2 complex generalized special unitary matrices. Our arguments illustrate a technique that can be used to prove similar reproducing formulas in higher dimensions and for other classes of functions. Our paper should be accessible to readers with minimal knowledge of complex analysis.
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Title: A q = -1 Phenomenon for Pattern-Avoiding Permutations
Author: Xin Chen, Carleton College Author Bio    
Abstract: We give an instance of Stembridge's q = -1 phenomenon for pattern- avoiding permutations. In particular, we show that setting q = -1 in the generating function for 132-avoiding permutations with respect to the statistic rsg returns the number of 132-avoiding involutions.
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Title: The Exponential of a Quaternionic Matrix
Author: Casey Machen, University of Nevada, Reno Author Bio    
Abstract: The exponential map is important because it provides a map from the Lie algebra of a Lie group into the group itself. We focus on matrix groups over the quaternions and the exponential map from their Lie algebras into the groups. Since quaternionic multiplication is not commutative, the process of calculating the exponential of a matrix over the quaternions is more involved than the process of calculating the exponential of a matrix over the real or complex numbers. We develop processes by which this calculation may be reduced to a simpler problem, and provide some examples.
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Title: Liouville theorems in the Dual and Double Planes
Authors: Kyle DenHartigh, Calvin College
Rachel Flim, Calvin College
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Author Bio    
Abstract: Although the generalized complex planes have the same form as the complex plane, theorems and concepts that have been proven true for complex numbers cannot necessarily be extended to dual and double numbers. In this paper, we explore analytic functions of a dual and double variable and disprove two Liouville theorems in these cases. We also modify a domain coloring scheme in order to visualize analytic functions of a generalized complex variable.
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Title: On the Degree-Chromatic Polynomial of a Tree
Authors: Diego Cifuentes, Universidad de los Andes, Bogota, Colombia Author Bio    
Abstract: The degree chromatic polynomial Pm(G,k) of a graph G counts the number of k-colorings in which no vertex has m adjacent vertices of its same color. We prove Humpert and Martin's conjecture on the leading terms of the degree chromatic polynomial of a tree.
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Title: The Edge Slide Graph of the 3-cube
Authors: Lyndal Henden, Massey University, Palmerston North, New Zealand Author Bio    
Abstract: The goal of this paper is to study the spanning trees of the 3-cube by understanding their \emph{edge slide graph}. A spanning tree of a graph $G$ is a minimal set of edges that connects all vertices. An edge slide occurs in a spanning tree of the 3-cube when a single edge can be slid across a 2-dimensional face to form another spanning tree. The edge slide graph is the graph whose vertices are the spanning trees, with an edge between two vertices if the spanning trees are related by a single edge slide. This report completely determines the edge slide graph of the 3-cube. The edge slide graph of the 3-cube has twelve components isomorphic to the 4-cube, and three other components, mutually isomorphic, with 64 vertices each. The main result is to determine the structure of the three components that each have 64 vertices and we also describe their symmetries. Some partial results on the 4-cube are also provided.
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Title: A Family of Recompositions of the Penrose Aperiodic Protoset and its Dynamic Properties
Authors: Vivian Olsiewski Healey, University of Notre Dame Author Bio    
Abstract: This paper examines a recomposition of the rhombic Penrose aperiodic protoset due to Robert Ammann. We show that the three prototiles that result from the recomposition form an aperiodic protoset in their own right without adjacency rules and that every tiling admitted by this protoset (here called an Ammann tiling) is mutually locally derivable with a Penrose tiling. Although these Ammann tilings are not self-similar, an iteration process inspired by Penrose composition is defined on the set of Ammann tilings that produces a new Ammann tiling from an existing one, and the exact relationship to Penrose composition is examined. Furthermore, by characterizing each Ammann tiling based on a corresponding Penrose tiling and the location of the added vertex that defines the recomposition process, we show that repeated Ammann iteration proceeds to a limit for the local geometry.
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Title: A Beckman-Quarles type theorem for linear fractional transformations of the extended double plane
Authors: Joshua Keilman, Calvin College
Andrew Jullian Mis, Calvin College
Author Bio    
Author Bio    
Abstract: In this presentation, we consider the problem of characterizing maps that preserve pairs of right hyperbolas or lines in the extended double plane whose hyperbolic angle of intersection is zero. We consider two disjoint spaces of right hyperbolas and lines in the extended double plane $\mathscr{H}^+$ and $\mathscr{H}^-$ and prove that bijective mappings on the respective spaces that preserve tangency between pairs of hyperbolas or lines must be induced by a linear fractional transformation.
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Title: Conditions for Robust Principal Component Analysis
Authors: Michael Hornstein, Stanford University Author Bio    
Abstract: Principal Component Analysis (PCA) is the problem of finding a low-rank approximation to a matrix. It is a central problem in statistics, but it is sensitive to sparse errors with large magnitudes. Robust PCA addresses this problem by decomposing a matrix into the sum of a low-rank matrix and a sparse matrix, thereby separating out the sparse errors. This paper provides a background in robust PCA and investigates the conditions under which an optimization problem, Principal Component Pursuit (PCP), solves the robust PCA problem. Before introducing robust PCA, we discuss a related problem, sparse signal recovery (SSR), the problem of finding the sparsest solution to an underdetermined system of linear equations. The concepts used to solve SSR are analogous to the concepts used to solve robust PCA, so presenting the SSR problem gives insight into robust PCA. After analyzing robust PCA, we present the results of numerical experiments that test whether PCP can solve the robust PCA problem even if previously proven sufficient conditions are violated.
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Title: A Comparison of Lindelof-type Covering Properties of Topological Spaces
Authors: Petra Staynova, Oxford University Author Bio    
Abstract: Lindelof spaces are studied in any basic Topology course. However, there are other interesting covering properties with similar behaviour, such as almost Lindelof, weakly Lindelof, and quasi-Lindelof, that have been considered in various research papers. Here we present a comparison between the standard results on Lindelof spaces and analogous results for weakly and almost Lindelof spaces. Some theorems, similar to the published ones, will be proved. We also consider counterexamples, most of which have not been included in the standard Topological textbooks, that show the interrelations between those properties and various basic topological notions, such as separability, separation axioms, first countability, and others. Some new features of those examples will be noted in view of the present comparison. We also pose several open questions.
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Title: Exploring String Patterns with Trees
Authors: Deborah Sutton, Purdue University
Booi Yee Wong, Purdue University
Author Bio    
Author Bio    
Abstract: The combination of the fields of probability and combinatorics is currently an object of much research. However, not many undergraduates or lay people have the opportunity to see how these areas can work together. We present what we hope is an accessible introduction to the possibilities easily available to many more people through the use of many examples and understandable explanations. We introduce topics of generating functions and tree structures formed through both independent strings and suffixes, as well as how we can find correlation polynomials, expected values, second moments, and variances of the number of nodes in a tree using indicator functions. Then we show a higher order example that includes matrices as the basis for its generating functions. The study of this unique field has many applications in areas including data compression, computational biology with the human genome, and computer science with binary strings.
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