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» Vol. 13, Issue 2, 2012 «



Title: Improving on the Range Rule of Thumb
Author: Alfredo Ramirez, Tennessee Tech University
Charles Cox, Unjiversity of Tennessee Martin
Author Bio    
Author Bio    
Abstract: In manufacturing it is useful to have a quick estimate of the standard deviation. This is often done with the range rule of thumb: σ ~ (sample range)/4. This rule works well when the data comes from a normal distribution and the sample size is around 30, but fails miserably for other distributions and sample sizes. Through the use of Monte Carlo simulations we suggest new rules of thumb for the normal distribution, uniform distribution, and exponential distribution which are dependent on sample size. We then seek to verify these empirical results theoretically.
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Title: The Structure of the Tutte-Grothendieck Ring of Ribbon Graphs
Author: Daniel C Thompson, MIT Author Bio    
Abstract: W. H. Tutte's 1947 paper on a ring generated by graphs satisfying a contraction-deletion relation is extended to ribbon graphs. This ring of ribbon graphs is a polynomial ring on an infinite set of one-vertex ribbon graphs.
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Title: An Overview of Complex Hadamard Cubes
Author: Ben Lantz, Northern Arizona University
Michael Zowada, Northern Arizona University
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Author Bio    
Abstract: Hadamard matrices have been studied by many authors, but higher-dimensional generalizations of Hadamard matrices are new and still relatively unexplored. This paper will present an overview of Hadamard matrices and their generalizations. In particular we will explore Walsh functions and Hadamard matrices. We will also extend Yang's Product Construction to create complex 3-D Hadamard cubes.
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Title: The Cayley-Hamilton Theorem via the Z-Transform
Author: Casey Tsai, Louisiana State University Author Bio    
Abstract: The Z-transform is usually defined and developed in a typical course on Difference Equations. We extend the transform to matrix valued sequences. A couple of key observations leads to a rather novel and simple proof of the Cayley-Hamilton theorem.
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Title: Rational Function Decomposition of Polynomials
Author: Steven Carter, Birmingham-Southern College Author Bio    
Abstract: We determine conditions under which an arbitrary polynomial can be expressed as the composition of two rational functions, generalizing the work of J. Rickards on the decomposition into two polynomials. We show that a polynomial can be expressed non-trivially as a composition of two rational functions if and only if it can be so decomposed into two polynomials.
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Title: The Mathematical Modeling Behind Duchenne Muscular Dystrophy
Author: William A Berrigan, Villanova University Author Bio    
Abstract: We look at a mathematical model for the role of the immune system in Duchenne Muscular Dystrophy. A linear stability analysis is used on a set of differential equations to determine stable and unstable states. These states are a basis for investigation into possible therapeutic treatments.
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Title: Fractals And The Weierstrass-Mandelbrot Function
Author: Anthony Zaleski, New Jersey Institute of Technology Author Bio    
Abstract: The Weierstrass-Mandelbrot (W-M) function was first used as an example of a real function which is continuous everywhere but differentiable nowhere. Later, its graph became a common example of a fractal curve. Here, we first review some basic ideas from measure theory and fractal geometry, focusing on the Hausdorff, box counting, packing, and similarity dimensions. Then we apply these to the W-M function. We show how to compute the box-counting dimension of its graph, and discuss previous attempts at proving the not yet completely resolved conjecture of the equality of its Hausdorff and box counting dimensions. We also consider a surface generalization of the W-M function, compute its box dimension, and discuss its Hausdorff dimension.
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Title: Constructing Mobius Transformations with Spheres
Author: Rob Siliciano, Princeton University Author Bio    
Abstract: Every Mobius transformation can be constructed by stereographic projection of the complex plane onto a sphere, followed by a rigid motion of the sphere and projection back onto the plane, illustrated in the video Mobius Transformations Revealed. In this article we show that, for a given Mobius transformation and sphere, this representation is unique.
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Title: On the Extension of Complex Numbers
Author: Nicholas Gauguin Houghton-Larsen, University of Copenhagen Author Bio    
Abstract: This paper proposes an extension of the complex numbers, adding further imaginary units and preserving the idea of the product as a geometric construction. These `supercomplex numbers', denoted S, are studied, and it is found that the algebra contains both new and old phenomena. It is established that equal-dimensional subspaces of S containing R are isomorphic under algebraic operations, whereby a symmetry within the space of imaginary units is illuminated. Certain equations are studied, and also a connection to special relativity is set up and explored. Finally, abstraction leads to the notion of a `generalised supercomplex algebra'; both the supercomplex numbers and the quaternions are found to be such algebras.
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Title: Establishing a Metric in Max-Plus Geometry
Author: Uri Carl, Yeshiva University
Kevin W. O'Neill, Harvey Mudd College
Nicholas Ryder, Rice University
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Author Bio    
Author Bio    
Abstract: Using the characterization of the segments in the max-plus semimodule Rnmax, provided by Nitica and Singer, we find a class of metrics on the finite part of Rnmax. One of them is the Euclidean length of the max-plus segment connecting two points. This metric is not quasi-convex. There is exactly one other metric in our class that does possess this property. Each metric in our class is associated with a weighting function, which is concave and non-decreasing.
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Title: Relations in the Dyer-Laslof Algebra for Morava E-theory
Author: Louis Atsaves, MIT
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Abstract: In this paper, we prove that a map u between two polynomial rings, each with an associated Adem relation, is injective. We prove injectivity of u, by first finding formulas for elements within each ring polynomial, and then by computing the map with our associated formulas. After having computed the mapping of u, we then use our computations to show that the kernel of $u$ only contains the zero vector, which proves that the map u is injective. Then having proved that the map u is injective, we then use it to find a basis for u*, the dual map of u.
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