Results 1  10
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324
Expander Graphs and their Applications
, 2003
"... Contents 1 The Magical Mystery Tour 7 1.1 Some Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Hardness results for linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Error Correcting Codes . . . . . . . ..."
Abstract

Cited by 186 (5 self)
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Contents 1 The Magical Mystery Tour 7 1.1 Some Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.1 Hardness results for linear transformation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 1.1.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.1.3 Derandomizing Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 1.2 Magical Graphs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 1.2.1 A Super Concentrator with O(n) edges . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.2 Error Correcting Codes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.2.3 Derandomizing Random Algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13 1.3 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Landscapes and Their Correlation Functions
, 1996
"... Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive const ..."
Abstract

Cited by 89 (15 self)
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Fitness landscapes are an important concept in molecular evolution. Many important examples of landscapes in physics and combinatorial optimation, which are widely used as model landscapes in simulations of molecular evolution and adaptation, are "elementary", i.e., they are (up to an additive constant) eigenfuctions of a graph Laplacian. It is shown that elementary landscapes are characterized by their correlation functions. The correlation functions are in turn uniquely determined by the geometry of the underlying configuration space and the nearest neighbor correlation of the elementary landscape. Two types of correlation functions are investigated here: the correlation of a time series sampled along a random walk on the landscape and the correlation function with respect to a partition of the set of all vertex pairs.
Generic Properties of Combinatory Maps  Neutral Networks of RNA Secondary Structures
, 1995
"... Random graph theory is used to model relationships between sequences and secondary structures of RNA molecules. Sequences folding into identical structures form neutral networks which percolate sequence space if the fraction of neutral nearest neighbors exceeds a threshold value. The networks of any ..."
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Cited by 80 (36 self)
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Random graph theory is used to model relationships between sequences and secondary structures of RNA molecules. Sequences folding into identical structures form neutral networks which percolate sequence space if the fraction of neutral nearest neighbors exceeds a threshold value. The networks of any two different structures almost touch each other, and sequences folding into almost all "common" structures can be found in a small ball of an arbitrary location in sequence space. The results from random graph theory are compared with data obtained by folding large samples of RNA sequences. Differences are explained in terms of RNA molecular structures. 1.
On quantum algorithms for noncommutative hidden subgroups
, 2000
"... Quantum algorithms for factoring and finding discrete logarithms have previously been generalized to finding hidden subgroups of finite Abelian groups. This paper explores the possibility of extending this general viewpoint to finding hidden subgroups of noncommutative groups. We present a quantum ..."
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Cited by 75 (3 self)
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Quantum algorithms for factoring and finding discrete logarithms have previously been generalized to finding hidden subgroups of finite Abelian groups. This paper explores the possibility of extending this general viewpoint to finding hidden subgroups of noncommutative groups. We present a quantum algorithm for the special case of dihedral groups which determines the hidden subgroup in a linear number of calls to the input function. We also explore the difficulties of developing an algorithm to process the data to explicitly calculate a generating set for the subgroup. A general framework for the noncommutative hidden subgroup problem is discussed and we indicate future research directions.
Quantum mechanical algorithms for the nonabelian Hidden Subgroup Problem
 Proc. 33rd ACM Symposium on Theory of Computing
, 2001
"... We give a short exposition of new and known results on the “standard method” of identifying a hidden subgroup of a nonabelian group using a quantum computer. ..."
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Cited by 66 (7 self)
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We give a short exposition of new and known results on the “standard method” of identifying a hidden subgroup of a nonabelian group using a quantum computer.
Modularity of certain potentially BarsottiTate Galois representations
 J. Amer. Math. Soc
, 1999
"... Conjectures of Langlands, Fontaine and Mazur [22] predict that certain Galois representations ρ: Gal(Q/Q) → GL2(Qℓ) (where ℓ denotes a fixed prime) should arise from modular forms. This applies in particular to representations defined by the action of Gal(Q/Q) on the ℓadic Tate ..."
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Cited by 61 (6 self)
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Conjectures of Langlands, Fontaine and Mazur [22] predict that certain Galois representations ρ: Gal(Q/Q) → GL2(Qℓ) (where ℓ denotes a fixed prime) should arise from modular forms. This applies in particular to representations defined by the action of Gal(Q/Q) on the ℓadic Tate
A Reflective Symmetry Descriptor for 3D Models
 ALGORITHMICA
, 2004
"... Computing reflective symmetries of 2D and 3D shapes is a classical problem in computer vision and computational geometry. Most prior work has focused on finding the main axes of symmetry, or determining that none exists. In this paper we introduce a new reflective symmetry descriptor that represent ..."
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Cited by 60 (7 self)
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Computing reflective symmetries of 2D and 3D shapes is a classical problem in computer vision and computational geometry. Most prior work has focused on finding the main axes of symmetry, or determining that none exists. In this paper we introduce a new reflective symmetry descriptor that represents a measure of reflective symmetry for an arbitrary 3D model for all planes through the model’s center of mass (even if they are not planes of symmetry). The main benefits of this new shape descriptor are that it is defined over a canonical parameterization (the sphere) and describes global properties of a 3D shape. We show how to obtain a voxel grid from arbitrary 3D shapes and, using Fourier methods, we present an algorithm that computes the symmetry descriptor in O(N 4 log N) time for an N × N × N voxel grid and computes a multiresolution approximation in O(N 3 log N) time. In our initial experiments, we have found that the symmetry descriptor is insensitive to noise and stable under point sampling. We have also found that it performs well in shape matching tasks, providing a measure of shape similarity that is orthogonal to existing methods.
TwoDimensional Topological Quantum Field Theories And Frobenius Algebras
 J. Knot Theory Ramifications
, 1996
"... We characterize Frobenius algebras A as algebras having a comultiplication which is a map of Amodules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either ..."
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Cited by 59 (2 self)
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We characterize Frobenius algebras A as algebras having a comultiplication which is a map of Amodules. This characterization allows a simple demonstration of the compatibility of Frobenius algebra structure with direct sums. We then classify the indecomposable Frobenius algebras as being either "annihilator algebras"  algebras whose socle is a principal ideal  or field extensions. The relationship between twodimensional topological quantum field theories and Frobenius algebras is then formulated as an equivalence of categories. The proof hinges on our new characterization of Frobenius algebras. These results together provide a classification of the indecomposable twodimensional topological quantum field theories. Keywords: topological quantum field theory, frobenius algebra, twodimensional cobordism, category theory 1. Introduction Topological Quantum Field Theories (TQFT's) were first described axiomatically by Atiyah in [1]. Since then, much work has been done ...
Strong Uniform Times and Finite Random Walks
 ADVANCES IN APPLIED MATHEMATICS 8,6997 (1987)
, 1987
"... There are several techniques for obtaining bounds on the rate of convergence to the stationary distribution for Markov chains with strong symmetry properties, in particular random walks on finite groups. An elementary method, strong uniform times, is often effective. We prove such times always exist ..."
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Cited by 57 (7 self)
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There are several techniques for obtaining bounds on the rate of convergence to the stationary distribution for Markov chains with strong symmetry properties, in particular random walks on finite groups. An elementary method, strong uniform times, is often effective. We prove such times always exist, and relate this method to coupling and Fourier analysis.