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» Vol. 12, Issue 1, 2011 «



Title: Laplacians of Covering Complexes
Author: Richard Gustavson, Cornell University Author Bio    
Abstract: The Laplace operator on a simplicial complex encodes information about the adjacencies between simplices. A relationship between simplicial complexes does not always translate to a relationship between their Laplacians. In this paper we look at the case of covering complexes. A covering of a simplicial complex is built from many copies of simplices of the original complex, maintaining the adjacency relationships between simplices. We show that for dimension at least one, the Laplacian spectrum of a simplicial complex is contained inside the Laplacian spectrum of any of its covering complexes.
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Title: Proving n-Dimensional Linking in Complete n-Complexes in (2n+1)-Dimensional Space
Author: Megan Gregory, Massey University, Palmerston North, New Zealand, Author Bio    
Abstract: This paper proves that there is an intrinsic link in complete n-complexes on (2n+4)-vertices for n=1,2,3 using the method of Conway and Gordon from their 1983 paper. The argument uses the sum of the linking number mod 2 of each pair of disjoint n-spheres contained in the n-complex as an invariant. We show that crossing changes do not affect the value of this invariant. We assert that ambient isotopies and crossing changes suffice to change any specific embedding to any other specific embedding. To complete the proof the invariant is evaluated on a specific embedding. Conway and Gordon use a diagram to carry out the final step for a 3-dimensional example and we use a computer to do this in higher dimensions. Our code is written in MATLAB. Taniyama has a proof for higher dimensions that does not use a computer.
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Title: Discrete Logarithms on Elliptic Curves
Author: Aaron Blumenfeld, University of Rochester Author Bio    
Abstract: Cryptographic protocols often make use of the inherent hardness of the classical discrete logarithm problem, which is to solve gx ≈ y ( mod p ) for x. The hardness of this problem has been exploited in the Diffie-Hellman key exchange, as well as in cryptosystems such as ElGamal. There is a similar discrete logarithm problem on elliptic curves: solve kB = P for k. Therefore, Diffie-Hellman and ElGamal have been adapted for elliptic curves. There is an abundance of evidence suggesting that elliptic curve cryptography is even more secure, which means that we can obtain the same security with fewer bits. In this paper, we investigate the discrete logarithm for elliptic curves over Fp for p ≥ 5 by constructing a function and considering the induced functional graph and the implications for cryptography.
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Title: Zero-Divisor Graphs and Lattices of Finite Commutative Rings
Authors: Darrin Weber, Millikin University Author Bio    
Abstract: In this paper we consider, for a finite commutative ring R, the well-studied zero-divisor graph Γ(R) and the compressed zero-divisor graph Γc(R) of R and a newly-defined graphical structure --- the zero-divisor lattice Λ(R) of R. We give results which provide information when Γ(R) ≅ Γ(S), Γc(R) ≅ Γc(S), and Λ(R) ≅ Λ(S) for two finite commutative rings R and S. We also provide a theorem which says that Λ(R) is almost always connected.
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Title: A Mathematical Analysis of The Generalized Oval Track Puzzle
Authors: Samuel Kaufmann, Carnegie Mellon University Author Bio    
Abstract: The oval track puzzle (also known as Top Spin) is a game consisting of 20 numbered tiles in an oval shaped track. Also, there is a fixed window (the swapping window) of 4 tiles that reverses the order of the tiles within the window, leaving the other 16 tiles fixed. The object of the puzzle is to reorder the tiles into counting order using the mechanisms of the puzzle. Previously, conditions for both solvability and non-solvability for the generalized oval track puzzle with n total tiles and k tiles in the swapping window were shown. We will now prove tight asymptotic bounds on the number of swaps needed to solve any configuration of a puzzle with n total tiles and k tiles in the swapping window provided that n and k yield a solvable case to begin with. These bounds will be asymptotic because we will assume that n grows infinitely and k stays fixed.
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