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» Vol. 16, Issue 2, 2015 «



Title: On Invariants for Spatial Graphs
Author: Elaina Aceves, California State University, Fresno
Jennifer Elder, California State University, Fresno
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Author Bio    
Abstract: We use combinatorial knot theory to construct invariants for spatial graphs. This is done by performing certain replacements at each vertex of a spatial graph diagram D , which results in a collection of knot and link diagrams in D. By applying known invariants for classical knots and links to the resulting collection, we obtain invariants for spatial graphs. We also show that for the case of undirected spatial graphs, the invariants we construct here satisfy a certain contraction-deletion recurrence relation.
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Title: Categorification of the Nonegative Rational Numbers
Author: Matteo Copelli, Department of Mathematics and Statistics, University of Ottawa Author Bio    
Abstract: In this document we describe a categorification of the semiring of natural numbers. We then use this result to construct a categorification of the semiring of nonnegative rational numbers.
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Title: A Graph Partitioning Model of Congressional Redistricting
Author: Shawn Doyle, Youngstown State University Author Bio    
Abstract: Redrawing congressional districts in the United States is a constitutionally required, yet politically controversial, task undertaken after each decennial census. Federal law requires contiguous, `relatively compact' congressional districts that maintain `approximately equal' population. Controversy is introduced when individual states redraw their districts, or redistrict, using partisan committees. States such as Ohio continue to redistrict with a committee appointed according to the current proportion of legislators' political parties to the whole. When political parties have majority power in redistricting committees, they can draw districts in a way that gives their party the best chance to keep its majority representation, a process called gerrymandering. Mathematical redistricting models seek an unbiased computational approach to the problem. Rather than trust partisan committees, mathematical modeling approaches rely upon well-defined methods in computational geometry, graph theory, game theory, and other fields. Here, we discuss two such approaches. The first, given as a background for comparison, constructs Voronoi diagrams to redistrict states into convex polygons, which are generally considered `compact'. We give greater emphasis to a new model that discretizes a state's population and partitions it into regions of approximately equal population. This model, our main focus, relies upon graph partitioning to achieve the desired result and uses census population data as the sole parameter in redistricting.
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Title: Transposing Noninvertible Polynomials
Author: Nathan Cordner, Brigham Young University Author Bio    
Abstract: Landau-Ginzburg mirror symmetry predicts isomorphisms between graded Frobenius algebras (denoted A and B ) that are constructed from a nondegenerate quasihomogeneous polynomial W and a related group of symmetries G . Duality between A and B models has been conjectured for particular choices of W and G . These conjectures have been proven in many instances where W is restricted to having the same number of monomials as variables (called \invertible). Some conjectures have been made regarding isomorphisms between A and B models when W is allowed to have more monomials than variables. In this paper we show these conjectures are false; that is, the conjectured isomorphisms do not exist. Insight into this problem will not only generate new results for Landau-Ginzburg mirror symmetry, but will also be interesting from a purely algebraic standpoint as a result about groups acting on graded algebras.
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Title: Linearization and Stability Analysis of Nonlinear Problems
Author: Robert Morgan, Wayne State University Author Bio    
Abstract: The focus of this paper is on the use of linearization techniques and linear differential equation theory to analyze nonlinear differential equations. Often, mathematical models of real-world phenomena are formulated in terms of systems of nonlinear differential equations, which can be difficult to solve explicitly. To overcome this barrier, we take a qualitative approach to the analysis of solutions to nonlinear systems by making phase portraits and using stability analysis. We demonstrate these techniques in the analysis of two systems of nonlinear differential equations. Both of these models are originally motivated by population models in biology when solutions are required to be non-negative, but the ODEs can be understood outside of this traditional scope of population models. In fact, allowing solutions for these equations to be negative provides some very interesting mathematical problems, and demonstrates the utility of the analysis techniques to be described in this article. We provide stability analysis, phase portraits, and numerical solutions for these models that characterize behaviors of solutions based only on the parameters used in the formulation of the systems. The first part of this paper gives a survey of standard linearization techniques in ODE theory. The second part of the paper presents applications of these techniques to particular systems of nonlinear ODEs, which includes some original results by extending the analysis to solutions lying anywhere in the plane, rather than only those in the first quadrant.
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Title: Bathtub and Unimodal Hazard Flexibility Classification of Parametric Lifetime Distributions
Author: Dana Lacey, North Central College
Anh Nguyen, Texas Christian University
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Author Bio    
Abstract: There are a number of bathtub and unimodal hazard shape parametric lifetime distributions available in the literature. Therefore, it is important to classify these distributions based on their hazard flexibility to facilitate their use in applications. For this purpose we use the Total Time on Test (TTT) transform plot with two different criteria: I. measure the slope at the inflection point on the scaled TTT transform curve; II. measure the slope at selected points from the constant hazard line on the scaled TTT transform curve. We confine our research to classify the flexibility of Weibull extensions and generalizations and also select one-shape and two-shape parameter lifetime distributions to exemplify the two criteria process.
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Title: The Ahlfors Lemma and Picard's Theorems
Author: Aleksander Simonic, University of Ljubljana, Slovenia Author Bio    
Abstract: The article introduces Ahlfors' generalization of Schwarz' lemma. With this powerful geometric tool of complex functions in one variable, we are able to prove some theorems concerning the size of images under holomorphic mappings, including celebrated Picard's theorems. The article concludes with a brief insight into the theory of Kobayashi hyperbolic complex manifolds. Although this survey paper does not contain any new results, it may be useful for the beginner in complex analysis to better understands the concept of hyperbolicity in complex geometry.
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Title: Tangent Line and Tangent Plane Approximations of Definite Integral
Author: Meghan Peer, Saginaw Valley State University Author Bio    
Abstract: Oftentimes, it becomes necessary to find approximate values for definite integrals, since the majority cannot be solved through direct computation. The methods of tangent line and tangent plane approximation can be derived as methods of integral approximation in two and three-dimensional spaces, respectively. Formulas are derived for both methods, and these formulas are compared with existing methods in terms of efficiency and error.
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Title: Two Poorly Measured Quantum Observables as a Complete Set of Commuting Observables
Author: Mark Olchanyi, Newton South High School Author Bio    
Abstract: In this article, we revisit the century-old question of the minimal set of observables needed to identify a quantum state: here, we replace the natural coincidences in their spectra by effective ones, induced by an imperfect measurement. We show that if the detection error is smaller than the mean level spacing, then two observables with Poisson spectra will suffice, no matter how large the system is. The primary target of our findings is the integrable (that is, exactly solvable) quantum systems whose spectra do obey the Poisson statistics. We also consider the implications of our findings for classical pattern recognition techniques.
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Title: Coarse embeddings of graphs into Hilbert space
Author: Dylan Bacon, University of Wisconsin - Stout
Michael Perlman, University of Notre Dame
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Author Bio    
Abstract: In this paper, we study coarse embeddings of graphs into Hilbert space. For a graph &Gamma expressible as an infinite union of coarsely embeddable subgraphs, &Gammai, we prove that if the nerve of the covering of &Gamma by the &Gammai is a tree and any nonempty intersections of the subgraphs have universally bounded diameter then &Gamma is coarsely embeddable into a Hilbert space.
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Title: The Discrete Lambert Map
Author: Caiyun Zhu, Mount Holyoke College, Department of Mathematics and Statistics
Anne Waldo, Mount Holyoke College, Department of Mathematics and Statistics
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Author Bio    
Abstract: The goal of this paper is to analyze the discrete Lambert problem (DWP) which is important for security and verification of the ElGamal digital signature scheme. We use p-adic methods (p-adic interpolation and Hensel's Lemma) to count the number of solutions of the DWP modulo powers of a prime. At the same time, we discover special patterns in the solutions.
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Title: Computing the elementary symmetric polynomials of the multiplier spectra of the maps z2+c
Author: Grayson Jorgenson, Florida Institute of Technology Author Bio    
Abstract: Let f be a complex quadratic rational map. The ith elementary symmetric polynomial of the formal n multiplier spectra of f is denoted &sigmai(n)(f). The values of these polynomials are invariant under conjugation by the projective linear group and are interesting to the study of the moduli space of quadratic rational maps. For every positive integer n and i in the appropriate range, &sigmai(n)(f) is in Z[&sigma1, &sigma2] where &sigma1, &sigma2 are &sigma1(1)(f, &sigma2(1)(f), respectively. Despite this, the &sigmai(n)(f) are difficult to compute. By restricting our focus to the family of quadratic polynomials z2+c, computations become simpler. We determine an upper bound for the degrees of the &sigmai(n) for the maps of the form z2+c by arguing in terms of the growth rates of their periodic points and corresponding multipliers. We also include computations of the forms of the &sigmai(n) for n =2,…,6 for these maps.
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Title: Apollonian Circle Packings and the Riemann Hypothesis
Author: Jason Hempstead, University of Washington Author Bio    
Abstract: In this paper, we describe how one can state the Riemann hypothesis in terms of a geometric problem about Apollonian circle packings. We use, as a black box, results of Zagier, and describe numerical experiments which were used in a recent paper by Athreya, Cobeli, and Zaharescu.
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Title: Bayesian Estimation in Autoregressive Models Using Reversible Jump Markov Chain Monte Carlo
Author: Nathan Rogers, University of Illinois at Chicago Author Bio    
Abstract: In most applications, there is uncertainty about the statistical model to be considered. In this paper, we consider a particular class of autoregressive time series models where the order of the model---which determines the dimension of parameter---is uncertain. A common approach for model selection is to balance model fit with model complexity using, say, an AIC criterion. However, such an approach provides no meaningful measure of uncertainty about the selected model. A Bayesian approach, on the other hand, which treats the model and model parameters as random variables, can directly accommodate model uncertainty. The challenge is that the Bayesian posterior distribution is supported on a union of spaces of different dimensions, which makes computation difficult. We review a reversible jump Markov chain Monte Carlo method for sampling from the posterior, and apply this method to provide a Bayesian analysis of simulated and real data.
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