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Approximate Protein Structural Alignment in Polynomial Time
 Proc. Natl Acad. Sci. USA
, 2004
"... Alignment of protein structures is a fundamental task in computational molecular biology. Good structural alignments can help detect distant evolutionary relationships that are hard or impossible to discern from protein sequences alone. Here, we study the structural alignment problem as a family of ..."
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Cited by 38 (1 self)
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Alignment of protein structures is a fundamental task in computational molecular biology. Good structural alignments can help detect distant evolutionary relationships that are hard or impossible to discern from protein sequences alone. Here, we study the structural alignment problem as a family of optimization problems and develop an approximate polynomial time algorithm to solve them. For a commonly used scoring function, the algorithm runs in O(n ) time, for globular protein of length n, when we wish to detect all scores that are at most # distance away from the optimum. We argue that such approximate solutions are, in fact, of greater interest than exact ones, due to the noisy nature of experimentally determined protein coordinates. The measurement of similarity between a pair of protein structures used by the algorithm involves the Euclidean distance between the structures, after rigidly transforming them. We show that an alternative approach, which relies on internal distance matrices, must incorporate sophisticated geometric ingredients in order to both guarantee optimality and run in polynomial time. We use these observations to visualize the scoring function for several real instances of the problem. Our investigations yield new insights on the computational complexity of protein alignment under various scoring functions. These insights can be used in the design of new scoring functions for which the optimum can be approximated e#ciently, and perhaps in the development of e#cient algorithms for the multiple structural alignment problem.
Norming Sets and Spherical Cubature Formulas
, 1998
"... We investigate the construction of cubature formulas for the unit sphere in R^n that have almost equal weights. The corresponding knots are taken from equidistributed point sets on the sphere. The notion of norming sets in connection with the Markov inequality of spherical harmonics is used in ord ..."
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Cited by 8 (3 self)
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We investigate the construction of cubature formulas for the unit sphere in R^n that have almost equal weights. The corresponding knots are taken from equidistributed point sets on the sphere. The notion of norming sets in connection with the Markov inequality of spherical harmonics is used in order to provide a general result on uniformly stable cubature formulas. We also present some numerical evidence that there exist stable and almost equallyweighted cubature formulas, if the number of knots is slightly larger than required by the exactness conditions for spherical harmonics of a certain degree.
Galerkin Approximation for Elliptic PDEs on Spheres
 Journal of Approximation Theory
, 2004
"... We discuss a Galerkin approximation scheme for the elliptic partial differential equation −∆u + ω 2 u = f on S n ⊂ R n+1. Here ∆ is the LaplaceBeltrami operator on S n, ω is a nonzero constant and f belongs to C 2k−2 (S n), where k ≥ n/4 + 1, k is an integer. The shifts of a spherical basis functi ..."
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Cited by 3 (0 self)
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We discuss a Galerkin approximation scheme for the elliptic partial differential equation −∆u + ω 2 u = f on S n ⊂ R n+1. Here ∆ is the LaplaceBeltrami operator on S n, ω is a nonzero constant and f belongs to C 2k−2 (S n), where k ≥ n/4 + 1, k is an integer. The shifts of a spherical basis function φ with φ ∈ H τ (S n) and τ> 2k ≥ n/2 + 2 are used to construct an approximate solution. An H 1 (S n)error estimate is derived under the assumption that the exact solution u belongs to C 2k (S n). Key words: spherical basis function, Galerkin method
Fullerenes and Coordination Polyhedra versus HalfCubes Embeddings
, 1997
"... A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking fo ..."
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Cited by 2 (0 self)
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A fullerene F n is a 3regular (or cubic) polyhedral carbon molecule for which the n vertices  the carbons atoms  are arranged in 12 pentagons and ( n 2 \Gamma 10) hexagons. Only a finite number of fullerenes are expected to be, up to scale, isometrically embeddable into a hypercube. Looking for the list of such fullerenes, we first check the embeddability of all fullerenes F n for n ! 60 and of all preferable fullerenes C n for n ! 86 and their duals. Then, we consider some infinite families, including fullerenes with icosahedral symmetry, which describe virus capsids, onionlike metallic clusters and geodesic domes. Quasiembeddings and fullerene analogues are considered. We also present some results on chemically relevant polyhedra such as coordination polyhedra and cluster polyhedra. Finally we conjecture that the list of known embeddable fullerenes is complete and present its relevance to the Katsura model for vesicles cells. Contents 1 Introduction and Basic Properties 2 1...
A New Technique for Spherical Radiance Calculation
, 2002
"... This document describes a new technique for spherical radiance calculation. In a previous work, [Costa99] et al described a semiautomatic lighting design system in which the calculation of spherical radiance distributions dominated most of the processing time. In that system, the calculation of t ..."
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This document describes a new technique for spherical radiance calculation. In a previous work, [Costa99] et al described a semiautomatic lighting design system in which the calculation of spherical radiance distributions dominated most of the processing time. In that system, the calculation of the radiance that passes through a point in space is very important because it is used in the optimisation phase to search for the best lighting solution.
APPROXIMATION OF LINEAR PARTIAL DIFFERENTIAL EQUATIONS ON SPHERES
, 2003
"... The theory of interpolation and approximation of solutions to differential and integral equations on spheres has attracted considerable interest in recent years; it has also been applied fruitfully in fields such as physical geodesy, potential theory, oceanography, and meteorology. In this dissertat ..."
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The theory of interpolation and approximation of solutions to differential and integral equations on spheres has attracted considerable interest in recent years; it has also been applied fruitfully in fields such as physical geodesy, potential theory, oceanography, and meteorology. In this dissertation we study the approximation of linear partial differential equations on spheres, namely a class of elliptic partial differential equations and the heat equation on the unit sphere. The shifts of a spherical basis function are used to construct the approximate solution. In the elliptic case, both the finite element method and the collocation method are discussed. In the heat equation, only the collocation method is considered. Error estimates in the supremum norms and the Sobolev norms are obtained when certain regularity conditions are imposed on the spherical basis functions.