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Generalized FFTs  A Survey Of Some Recent Results
, 1995
"... In this paper we survey some recent work directed towards generalizing the fast Fourier transform (FFT). We work primarily from the point of view of group representation theory. In this setting the classical FFT can be viewed as a family of efficient algorithms for computing the Fourier transform of ..."
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Cited by 57 (7 self)
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In this paper we survey some recent work directed towards generalizing the fast Fourier transform (FFT). We work primarily from the point of view of group representation theory. In this setting the classical FFT can be viewed as a family of efficient algorithms for computing the Fourier transform of either a function defined on a finite abelian group, or a bandlimited function on a compact abelian group. We discuss generalizations of the FFT to arbitrary finite groups and compact Lie groups.
Some applications of generalized FFTs
 In Proceedings of DIMACS Workshop in Groups and Computation
, 1997
"... . Generalized FFTs are efficient algorithms for computing a Fourier transform of a function defined on finite group, or a bandlimited function defined on a compact group. The development of such algorithms has been accompanied and motivated by a growing number of both potential and realized applicat ..."
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Cited by 32 (5 self)
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. Generalized FFTs are efficient algorithms for computing a Fourier transform of a function defined on finite group, or a bandlimited function defined on a compact group. The development of such algorithms has been accompanied and motivated by a growing number of both potential and realized applications. This paper will attempt to survey some of these applications. Appendices include some more detailed examples. 1. A brief history The now "classical" Fast Fourier Transform (FFT) has a long and interesting history. Originally discovered by Gauss, and later made famous after being rediscovered by Cooley and Tukey [21], it may be viewed as an algorithm which efficiently computes the discrete Fourier transform or DFT. In between Gauss and CooleyTukey others developed special cases of the algorithm, usually motivated by the need to make efficient data analysis of one sort or another. To cite but a few examples, Gauss was interested in efficiently interpolating the orbits of asteroids [43...
Applications of the Generalized Fourier Transform in Numerical Linear Algebra
 Department of Information Technology, Uppsala University
"... Matrices equivariant under a group of permutation matrices are considered. Such matrices typically arise in numerical applications where the computational domain exhibits geometrical symmetries. In these cases, group representation theory provides a powerful tool for block diagonalizing the matrix v ..."
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Cited by 5 (1 self)
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Matrices equivariant under a group of permutation matrices are considered. Such matrices typically arise in numerical applications where the computational domain exhibits geometrical symmetries. In these cases, group representation theory provides a powerful tool for block diagonalizing the matrix via the Generalized Fourier Transform. This technique yields substantial computational savings in problems such as solving linear systems, computing eigenvalues and computing analytic matrix functions. The theory for applying the Generalized Fourier Transform is explained, building upon the familiar special (finite commutative) case of circulant matrices being diagonalized with the Discrete Fourier Transform. The classical convolution theorem and diagonalization results are generalized to the noncommutative case of block diagonalizing equivariant matrices. Our presentation stresses the connection between multiplication with an equivariant matrices and the application of a convolution. This approach highlights the role of the underlying mathematical structures such as the group algebra, and it also simplifies the application of fast Generalized Fourier Transforms. The theory is illustrated with a selection of numerical examples. Key words: Non commutative Fourier analysis, equivariant operators, block diagonalization. 1 Introduction.
Multilinear Algebra and Chess Endgames
 of No Chance: Combinatorial Games at MRSI
, 1996
"... Abstract. This article has three chief aims: (1) To show the wide utility of multilinear algebraic formalism for highperformance computing. (2) To describe an application of this formalism in the analysis of chess endgames, and results obtained thereby that would have been impossible to compute usi ..."
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Cited by 4 (0 self)
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Abstract. This article has three chief aims: (1) To show the wide utility of multilinear algebraic formalism for highperformance computing. (2) To describe an application of this formalism in the analysis of chess endgames, and results obtained thereby that would have been impossible to compute using earlier techniques, including a win requiring a record 243 moves. (3) To contribute to the study of the history of chess endgames, by focusing on the work of Friedrich Amelung (in particular his apparently lost analysis of certain sixpiece endgames) and that of Theodor Molien, one of the founders of modern group representation theory and the first person to have systematically numerically analyzed a pawnless endgame. 1.
A Memory Efficient Retrograde Algorithm and Its Application To Chinese Chess Endgames
 IN MORE GAMES OF NO CHANCE
, 2002
"... We present an improved, memory efficient retrograde algorithm we developed during our research on solving Chinese chess endgames. This domainindependent retrograde algorithm, along with a carefully designed domainspecific indexing function, has enabled us to solve many interesting Chinese chess e ..."
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Cited by 2 (0 self)
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We present an improved, memory efficient retrograde algorithm we developed during our research on solving Chinese chess endgames. This domainindependent retrograde algorithm, along with a carefully designed domainspecific indexing function, has enabled us to solve many interesting Chinese chess endgame on standard consumer class hardware.
Cayley graphs formed by conjugate generating sets of Sn
"... We investigate subsets of the symmetric group with structure similar to that of a graph. The “trees” of these subsets correspond to minimal conjugate generating sets of the symmetric group. There are two main theorems in this paper. The first is a characterization of minimal conjugate generating set ..."
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We investigate subsets of the symmetric group with structure similar to that of a graph. The “trees” of these subsets correspond to minimal conjugate generating sets of the symmetric group. There are two main theorems in this paper. The first is a characterization of minimal conjugate generating sets of Sn. The second is a generalization of a result due to Feng characterizing the automorphism groups of the Cayley graphs formed by certain generating sets composed of cycles. We compute the full automorphism groups subject to a weak condition and conjecture that the characterization still holds without the condition. We also present some computational results in relation to Hamiltonicity of Cayley graphs, including a generalization of the work on quasihamiltonicity by Gutin and Yeo to undirected graphs.
SOME RECENT APPLICATIONS OF SEMIRING THEORY
, 2005
"... A semiring is an algebraic structure, consisting of a nonempty set R on which we have defined two operations, addition (usually denoted by +) and multiplication (usually denoted by · or by concatenation) such that the following conditions hold: (1) Addition is associative and commutative and has a n ..."
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A semiring is an algebraic structure, consisting of a nonempty set R on which we have defined two operations, addition (usually denoted by +) and multiplication (usually denoted by · or by concatenation) such that the following conditions hold: (1) Addition is associative and commutative and has a neutral element. That is to say, a+(b+c) = (a+b)+c and a+b = b+a for all a,b,c ∈ R and there exists a special element of R, usually denoted by 0, such that a + 0 = 0 + a for all a ∈ R. It is very easy to prove that this element is unique. (2) Multiplication is associative and has a neutral element. That is to say, a(bc) = (ab)c for all a,b,c ∈ R and there exists a special element of R, usually denoted by 1, such that a1 = a = 1a for all a ∈ R. It is very easy to prove that this element too is unique. In order to avoid trivial cases, we will always assume that 1 = 0, thus insuring that R has at least two distinct
Games solved: Now and in the future
"... In this article we present an overview on the state of the art in games solved in the domain of twoperson zerosum games with perfect information. The results are summarized and some predictions for the near future are given. The aim of the article is to determine which game characteristics are pred ..."
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In this article we present an overview on the state of the art in games solved in the domain of twoperson zerosum games with perfect information. The results are summarized and some predictions for the near future are given. The aim of the article is to determine which game characteristics are predominant when the solution of a game is the main target. First, it is concluded that decision complexity is more important than statespace complexity as a determining factor. Second, we conclude that there is a tradeoff between knowledgebased methods and bruteforce methods. It is shown that knowledgebased methods are more appropriate for solving games with a low decision complexity, while bruteforce methods are more appropriate for solving games with a low statespace complexity. Third, we found that there is a clear correlation between the firstplayer’s initiative and the necessary effort to solve a game. In particular, threatspacebased search methods are sometimes able to exploit the initiative to prove a win. Finally, the most important results of the research involved, the development of new intelligent search methods, are described. © 2001 Published by