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» Vol. 6, Issue 2, 2005 «



Title: A Special Case of Selberg's Integral
Author: Andrea Anderson, University of Wisconsin-Eau Claire Author Bio    
Abstract: In this paper we evaluate a difficult integral that arose in the study of a class of differential operators. Although we later discovered the integral could be evaluated using published formulae, we present our own interesting proof. Our computation of the integral uses algebraic techniques to solve a problem that at first seems to be strictly analytic.
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Title: The Membership Problem for Ideals in Z[X]
Author: Carlos E. Arreche, Princeton University Author Bio    
Abstract: There exists a feasible procedure to decide whether or not an arbitrary polynomial belongs to a given ideal in Z[x] if the ideal's minimal basis is known. However, when this is not the case there is no feasible procedure to decide whether or not an arbitrary polynomial belongs to a given ideal. There already exists an effective procedure to find an ideal's minimal basis, but it depends on solving the membership problem for the ideal (i.e. the problem of deciding whether an arbitrary polynomial belongs to the ideal). Therefore, we develop a modification of the existing algorithm to find an ideal's minimal basis so that there is no need to solve the membership problem to carry it out, and then we use this minimal basis to solve the membership problem for this ideal.
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Title: Portfolio Optimization: MAD vs. Markowitz
Authors: Beth Bower,College of William and Mary
Pamela Wentz, Millersville University
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Author Bio    
Abstract: We look at investment portfolio optimization. We create portfolios consisting of five stocks and a six-month bond by randomly selecting the stocks from the S&P 500. We take the data from July 1, 2004 to December 31, 2004 and use the Markowitz minimum variance model as well as the Mean Absolute Deviation model to determine the allocation of funds to each asset in each of the portfolios. We then compare the returns of the portfolios from January 3, 2005 to June 30, 2005 using a series of parametric and non-parametric tests.
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Title: Non-Constant Stable Solutions to Reaction-Diffusion Equations in Star-Shaped Domains
Authors: Greg Drugan, University of Texas at Austin Author Bio    
Abstract: In the following we will discuss some known results on the behavior of solutions to reaction-diffusion equations. We will be concerned with the stability of steady-state solutions in different classes of domains. A result in Matano states that for convex domains, every non-constant stable steady-state solution to the reaction-diffusion equation is unstable. As an application of a theorem in Matano, we show that this result for convex domains does not generalize to the larger class of star-shaped domains.
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Title: The Riemann Surface of The Logarithm Constructed in a Geometrical Framework
Authors: Nikolaos Katzourakis, University of Athens Author Bio    
Abstract: I present a geometrical method that produces the fundamental holomorphic surface of the complex logarithm (classically obtained via analytic continuation) without any tools concerning the complex structure or the Covering Spaces theory. The only tools employed are elementary notions of (real) Differential Geometry and ordinary convergence of surface sequences.
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Title: Good Sequences, Bijections and Permutations
Author: Igor Kortchemski, Lycée Louis-Le-Grand in Paris Author Bio    
Abstract: In the present paper we study general properties of good sequences by means of a powerful and beautiful tool of combinatorics - the method of bijective proofs. A good sequence is a sequence of positive integers k=1,2,... such that the element k occurs before the last occurrence of k+1. We construct two bijections between the set of good sequences of fixed length and the set of permutations of the same length. This allows us to count good sequences as well as to calculate generating functions of statistics on good sequences. We study avoiding patterns on good sequences and discuss their relation with Eulerian polynomials. Finally, we describe particular interesting properties of permutations, again using bijections.
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Title: A New Computation of the Codimension Sequence of the Grassmann Algebra
Authors: Joel Louwsma, University of Michigan
Adilson Eduardo Presoto, Univ. of Sao Carlos, Brazil
Alan Tarr,Pomona College
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Abstract: Krakowski and Regev found a basis of polynomial identities satisfied by the Grassmann algebra over a field of characteristic $0$ and described the exact structure of these relations in terms of the symmetric group. Using this, they found an upper bound for the the codimension sequence of the $T$-ideal of polynomial identities of the Grassmann algebra. Working with certain matrices, they found the same lower bound, thus determining the codimension sequence exactly. In this paper, we compute the codimension sequence of the Grassmann algebra directly from these matrices, thus obtaining a proof of the codimension result of Krakowski and Regev using only combinatorics and linear algebra. We also obtain a corollary from our proof.
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Title: A Modified Lotka-Volterra Competition Model with a Non-Linear Relationship Between Species
Author: Austin Taylor, University of Alabama
Amy Crizer, James Madison University
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Author Bio    
Abstract: In this article, we consider a modified Lotka-Volterra competition model, which incorporates a non-linear relationship representing the interaction between species. We study the qualitative properties of this new system and compare them to the qualitative properties of the classical Lotka-Volterra equations and obtain results suggesting that the modified model is a better representation of some biological situations.
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Title: Application of the Mathematical Model of Tumor-Immune Interactions for IL-2 Adoptive Immunotherapy to Studies on Patients with Metastatic Melanoma or Renal Cell Cancer
Author: Asad Usman, University of Michigan
Chris Cunningham, University of Michigan
Author Bio    
Author Bio    
Abstract: Recent developments in Adoptive Immunotherapy for cancer management have lead clinicians to employ these techniques in hospital settings. Much data has been produced that indicates the effectiveness of introducing enhanced and expanded immune systems into cancer hosts. In this retrospective study we take another look at the Kirschner mathematical model for immune-tumor interactions in light of data presented by Rosenburg on patients with Metastatic Melanoma or Renal Cell Cancer.
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Title: Expansion into Partial Fractions
Author: Noe Ricardo Arellano Velazquez, University of Michoacan Author Bio    
Abstract: There are several methods of calculating the unknown coefficients appearing in the partial fraction expansion of a fraction of two polynomials. This paper shows how to derive a method based on standard results from the theory of complex variables, namely, the Cauchy-Goursat theorem and the Cauchy integral formula.
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Title: An Identity of Derangements
Author: Le Anh Vinh, University of New South Wales Author Bio    
Abstract: In this paper, we present a new identity for derangements, and as a corollary, we have a combinatorial proof of the irreducibility of the standard representation of symmetric groups
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