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» Vol. 13, Issue 1, 2012 «



Title: Towards Bidding Connect Four
Author: Kenny Goodfellow, Juniata College Author Bio    
Abstract: Allis has determined that the player who goes first in Connect Four always has a winning strategy. We consider the discrete bidding variation of the game instead of alternating turns. In discrete bidding, each player holds an integer number of chips, and the players bid for the next turn. Whoever wins the bid takes a turn and gives his chips to the other player; thus, the total number of chips stays constant. Introducing bidding to the game alters a player's strategy, as multiple moves in succession are now possible. Develin and Payne have completed an analysis of Tic-Tac-Toe using discrete bidding and have determined a winning strategy. We analyze bidding Connect Two on all board sizes and bidding Connect Three on a 3-by-3 board, which will give us insight into the strategy for Connect Four.
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Title: Convex Hulls and the Casas-Alvero Conjecture for the Complex Plane
Author: Thomas Polstra, Georgia State University Author Bio    
Abstract: It has been conjectured by Casas-Alvero that polynomials of degree n over fields of characteristic 0, share roots with each of its n-1 derivatives if and only if those polynomials have one root of degree n. In this paper, using the analytic theory of polynomials, an equivalent formulation of the Casas-Alvero Conjecture is established for polynomials over the complex plane t ogether with several special cases of it.
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Title: A Factorial Power Variation of Fermat's Equation
Author: Matthew J. Green, Towson University Author Bio    
Abstract: We consider a variant of Fermat's well-known equation xn+yn=zn. T his variant replaces the usual powers with the factorial powers defined by xn=x(x-1)...(x-(n-1)). For n=2 we characterize all possible integer solutions of the equation. For n=3 we show that there exist infinitely many non-trivial solutions to the equation. Finally we show there exists no maximum n for which xn+yn = zn has a non-trivial solution.
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Title: Invariants of Finite Groups Acting as Flag Automorphisms
Authors: Dennis Tseng, Massachusetts Institute of Technology Author Bio    
Abstract: Let K be a field and suppose that G is a finite group that acts faithfully on $(x1,...,xm) by automorphisms of the form g(xi)=ai(g)xi+bi(g), where ai(g),bi(g) \in K(x1,...,xi-1) for all g \in G and all i=1,...,m. As shown by Miyata, the fixed field K(x1,...,xi-1)G is purely transcendental over K and admits a transcendence basis {\phi1,...,\phim}, where \phii is in K(x1,...,xi-1) [xi]G and has minimal positive degree di in xi. We determine exactly the degree di of each invariant \phii as a polynomial in xi and show the relation d1 ... dm=|G|. As an application, we compute a generic polynomial for the dihedral group D8 of order 16 in characteristic 2.
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Title: Structure and Randomness of the Discrete Lambert Map
Authors: JingJing Chen, Pomona College
Mark Lotts, Randolph-Macon College
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Author Bio    
Abstract: We investigate the structure and cryptographic applications of the Discrete Lambert Map (DLM), the mapping x --> xgx mod p, for p a prime and some fixed g \in (Z/pZ)* The mapping is closely related to the Discrete Log Problem, but has received far less attention since it is considered to be a more complicated map that is likely even harder to invert. However, this mapping is quite important because it underlies the security of the ElGamal Digital Signature Scheme. Using functional graphs induced by this mapping, we were able to find non-random properties that could potentially be used to exploit the ElGamal DSS.
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Title: Tiling with Penalties and Isoperimetry with Density
Authors: Yifei Li, Berea College
Michael Mara, Williams College
Isamar Rosa Plata, University of Puerto Rico
Elena Wikner, Williams College
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Abstract: We prove optimality of tilings of the flat torus by regular hexagons, squares, and equilateral triangles when minimizing weighted combinations of perimeter and number of vertices. We similarly show optimality of certain tilings of the 3-torus by polyhedra from among a selected candidate pool when minimizing w eighted combinations of interface area, edge length, and number of vertices. Finally, we provide n umerical evidence for the Log Convex Density Conjecture.
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Title: Recent Developments in Perfect Bricks with Dimension Higher than 2 x 2
Authors: Brooke Fox, Northern Arizona University Author Bio    
Abstract: A numerical semigroup S is a set of nonnegative integers such that S contains 0, S is closed under addition, and the complement of S is finite. This paper considers pairs (S,I) of a given numerical semigroup S and corresponding relative ideal I such that \mu(I)\mu(S-I) = \mu(I+(S-I)), where \mu denotes the size of the minimal generating set and S-I is the dual of I in S. We will present recent results in the research of such pairs (perfect bricks) with \mu(I) > 2 and \mu(S-I) > 2. We will also show the existence of an infinite family of perfect bricks.
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Title: Cold Positions of the Restricted Wythoff's Game
Authors: Ryoma Aoki, Hyogo Prefectural Akashi Kita Senior High School
Junpei Sawada, Hyogo Prefectural Akashi Kita Senior High School
Yuki Miyake, Hyogo Prefectural Akashi Kita Senior High School
Hiroaki Fujiwara, Hyogo Prefectural Akashi Kita Senior High School
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Abstract: Wythoff's game is a kind of 2-pile Nim game, which admits taking the same number of stones from both piles. It differs only a little from the 2-pile Nim game, but their winning strategies are quite different from each other. Amazingly the winning strategy of Wythoff's game is directly related to a real number, specifically the golden ratio. In this paper we add two restrictions to this game, and investigate the winning strategy of the revised game.
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Title: On the Number of 2-Player, 3-Strategy, Strictly Ordinal, Normal Form Games
Authors: Austin Williams, Portland State University Author Bio    
Abstract: The 2-player, 2-strategy, strictly ordinal, normal form games were originally studied by Anotol Rapoport and Melvyn Guyer in a paper entitled A Taxonomy of 2x2 Games. Their paper appeared in 1966 and included an exact count, an enumeration (that is, a complete listing), and a taxonomy of such games. Since then it has been known that there are 78 such games. If we allow each player access to one additional strategy, however, the number of games explodes to nearly two billion. In this paper we compute the exact number of 2-player, 3-strategy, strictly ordinal, normal form games.
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Title: Homology of Hom Complexes
Authors: Mychael Sanchez, New Mexico State University Author Bio    
Abstract: The hom complex Hom(G,K) is the order complex of the poset composed of the graph multihomomorphisms from G to K. We use homology to provide conditions under which the hom complex is not contractible and derive a lower bound on the rank of its homology groups.
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