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Convex Analysis
, 1970
"... In this book we aim to present, in a unified framework, a broad spectrum of mathematical theory that has grown in connection with the study of problems of optimization, equilibrium, control, and stability of linear and nonlinear systems. The title Variational Analysis reflects this breadth. For a lo ..."
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Cited by 5350 (67 self)
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In this book we aim to present, in a unified framework, a broad spectrum of mathematical theory that has grown in connection with the study of problems of optimization, equilibrium, control, and stability of linear and nonlinear systems. The title Variational Analysis reflects this breadth. For a long time, ‘variational ’ problems have been identified mostly with the ‘calculus of variations’. In that venerable subject, built around the minimization of integral functionals, constraints were relatively simple and much of the focus was on infinitedimensional function spaces. A major theme was the exploration of variations around a point, within the bounds imposed by the constraints, in order to help characterize solutions and portray them in terms of ‘variational principles’. Notions of perturbation, approximation and even generalized differentiability were extensively investigated. Variational theory progressed also to the study of socalled stationary points, critical points, and other indications of singularity that a point might have relative to its neighbors, especially in association with existence theorems for differential equations.
Surface Reconstruction by Voronoi Filtering
 Discrete and Computational Geometry
, 1998
"... We give a simple combinatorial algorithm that computes a piecewiselinear approximation of a smooth surface from a finite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled ..."
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Cited by 418 (15 self)
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We give a simple combinatorial algorithm that computes a piecewiselinear approximation of a smooth surface from a finite set of sample points. The algorithm uses Voronoi vertices to remove triangles from the Delaunay triangulation. We prove the algorithm correct by showing that for densely sampled surfaces, where density depends on "local feature size", the output is topologically valid and convergent (both pointwise and in surface normals) to the original surface. We describe an implementation of the algorithm and show example outputs. 1 Introduction The problem of reconstructing a surface from scattered sample points arises in many applications such as computer graphics, medical imaging, and cartography. In this paper we consider the specific reconstruction problem in which the input is a set of sample points S drawn from a smooth twodimensional manifold F embedded in three dimensions, and the desired output is a triangular mesh with vertex set equal to S that faithfully represen...
Efficient Implementation of Weighted ENO Schemes
, 1995
"... In this paper, we further analyze, test, modify and improve the high order WENO (weighted essentially nonoscillatory) finite difference schemes of Liu, Osher and Chan [9]. It was shown by Liu et al. that WENO schemes constructed from the r th order (in L¹ norm) ENO schemes are (r +1) th order accur ..."
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Cited by 415 (40 self)
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In this paper, we further analyze, test, modify and improve the high order WENO (weighted essentially nonoscillatory) finite difference schemes of Liu, Osher and Chan [9]. It was shown by Liu et al. that WENO schemes constructed from the r th order (in L¹ norm) ENO schemes are (r +1) th order accurate. We propose a new way of measuring the smoothness of a numerical solution, emulating the idea of minimizing the total variation of the approximation, which results in a 5th order WENO scheme for the case r = 3, instead of the 4th order with the original smoothness measurement by Liu et al. This 5 th order WENO scheme is as fast as the 4 th order WENO scheme of Liu et al. and, both schemes are about twice as fast as the 4th order ENO schemes on vector supercomputers and as fast on serial and parallel computers. For Euler systems of gas dynamics, we suggest to compute the weights from pressure and entropy instead of the characteristic values to simplify the costly characteristic procedure. The resulting WENO schemes are about twice as fast as the WENO schemes using the characteristic decompositions to compute weights, and work well for problems which donot contain strong shocks or strong reflected waves. We also prove that, for conservation laws with smooth solutions, all WENO schemes are convergent. Many numerical tests, including the 1D steady state nozzle flow problem and 2D shock entropy waveinteraction problem, are presented to demonstrate the remarkable capability of the WENO schemes, especially the WENO scheme using the new smoothness measurement, in resolving complicated shock and flow structures. We have also applied Yang's artificial compression method to the WENO schemes to sharpen contact discontinuities.
Natural language and natural selection
 Behavioral and Brain Sciences
, 1990
"... Pinker, S. & Bloom, P. (1990). Natural language and natural selection. Behavioral and Brain Sciences 13 ..."
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Cited by 373 (3 self)
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Pinker, S. & Bloom, P. (1990). Natural language and natural selection. Behavioral and Brain Sciences 13
Conformal deformation of a Riemannian metric to constant curvature
 J. Diff. Geome
, 1984
"... A wellknown open question in differential geometry is the question of whether a given compact Riemannian manifold is necessarily conformally equivalent to one of constant scalar curvature. This problem is known as the Yamabe problem because it was formulated by Yamabe [8] in 1960, While Yamabe&apos ..."
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Cited by 318 (0 self)
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A wellknown open question in differential geometry is the question of whether a given compact Riemannian manifold is necessarily conformally equivalent to one of constant scalar curvature. This problem is known as the Yamabe problem because it was formulated by Yamabe [8] in 1960, While Yamabe's paper claimed to solve the problem in the affirmative, it was found by N. Trudinger [6] in 1968 that Yamabe's paper was seriously incorrect. Trudinger was able to correct Yamabe's proof in case the scalar curvature is nonpositive. Progress was made on the case of positive scalar curvature by T. Aubin [1] in 1976. Aubin showed that if dim M> 6 and M is not conformally flat, then M can be conformally changed to constant scalar curvature. Up until this time, Aubin's method has given no information on the Yamabe problem in dimensions 3, 4, and 5. Moreover, his method exploits only the local geometry of M in a small neighborhood of a point, and hence could not be used on a conformally flat manifold where the Yamabe problem is clearly a global problem. Recently, a number of geometers have been interested in the conformally flat manifolds of positive scalar curvature where a solution of Yamabe's problem gives a conformally flat metric of constant scalar curvature, a metric of some geometric interest. Note that the class of conformally flat manifolds of positive scalar curvature is closed under the operation of connected sum, and hence contains connected sums of spherical space forms with copies of Sι X Sn~ι. In this paper we introduce a new global idea into the problem and we solve it in the affirmative in all remaining cases; that is, we assert the existence of a positive solution u on M of the equation
SPARSKIT: a basic tool kit for sparse matrix computations  Version 2
, 1994
"... . This paper presents the main features of a tool package for manipulating and working with sparse matrices. One of the goals of the package is to provide basic tools to facilitate exchange of software and data between researchers in sparse matrix computations. Our starting point is the Harwell/Boei ..."
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Cited by 317 (22 self)
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. This paper presents the main features of a tool package for manipulating and working with sparse matrices. One of the goals of the package is to provide basic tools to facilitate exchange of software and data between researchers in sparse matrix computations. Our starting point is the Harwell/Boeing collection of matrices for which we provide a number of tools. Among other things the package provides programs for converting data structures, printing simple statistics on a matrix, plotting a matrix profile, performing basic linear algebra operations with sparse matrices and so on. Work done partly at CSRD, university of Illinois and partly at RIACS (NASA Ames Research Center). Current address: Computer Science Dept., University of Minnesota, Minneapolis, MN 55455. This work was supported in part by the NAS Systems Division, via Cooperative Agreement NCC 2387 between NASA and the University Space Research Association (USRA) and in part by the Department of Energy under grant DEFG02...
Remodeling of yeast genome expression in response to environmental changes. Mol. Biol. Cell 12:323–337
, 2001
"... We used genomewide expression analysis to explore how gene expression in Saccharomyces cerevisiae is remodeled in response to various changes in extracellular environment, including changes in temperature, oxidation, nutrients, pH, and osmolarity. The results demonstrate that more than half of the ..."
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Cited by 236 (2 self)
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We used genomewide expression analysis to explore how gene expression in Saccharomyces cerevisiae is remodeled in response to various changes in extracellular environment, including changes in temperature, oxidation, nutrients, pH, and osmolarity. The results demonstrate that more than half of the genome is involved in various responses to environmental change and identify the global set of genes induced and repressed by each condition. These data implicate a substantial number of previously uncharacterized genes in these responses and reveal a signature common to environmental responses that involves �10 % of yeast genes. The results of expression analysis with MSN2/MSN4 mutants support the model that the Msn2/Msn4 activators induce the common response to environmental change. These results provide a global description of the transcriptional response to environmental change and extend our understanding of the role of
Combinatorial stochastic processes
"... This is a collection of expository articles about various topics at the interface between enumerative combinatorics and stochastic processes. These articles expand on a course of lectures given at the École d’Été de Probabilités de St. Flour in July 2002. The articles are called ’chapters ’ and numb ..."
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Cited by 219 (15 self)
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This is a collection of expository articles about various topics at the interface between enumerative combinatorics and stochastic processes. These articles expand on a course of lectures given at the École d’Été de Probabilités de St. Flour in July 2002. The articles are called ’chapters ’ and numbered according to the order of these chapters in a printed volume to appear in Springer Lecture Notes in Mathematics. Each chapter is fairly selfcontained, so readers with adequate background can start reading any chapter, with occasional consultation of earlier chapters as necessary. Following this Chapter 0, there are 10 chapters, each divided into sections. Most sections conclude with some Exercises. Those for which I don’t know solutions are called Problems. Acknowledgments Much of the research reviewed here was done jointly with David Aldous. Much credit is due to him, especially for the big picture of continuum approximations to large combinatorial structures. Thanks also to my other collaborators in this work, especially Jean Bertoin, Michael Camarri, Steven
The PERFECT Club Benchmarks: Effective Performance Evaluation of Supercomputers
 International Journal of Supercomputer Applications
, 1988
"... This report consists of two major portions. First is the presentation of a methodology for measuring the performance of supercomputers. This includes a set of thirteen Fortran programs that total well over 50,000 lines of source code. They represent applications in a number of areas of engineering a ..."
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Cited by 211 (4 self)
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This report consists of two major portions. First is the presentation of a methodology for measuring the performance of supercomputers. This includes a set of thirteen Fortran programs that total well over 50,000 lines of source code. They represent applications in a number of areas of engineering and scientific computing, and in many cases they represent codes that are currently used by a number of computational research and development groups. We also present the PERFECT Fortran standard which is simply a set of guidelines that allow portability to a number of types of machines. Furthermore, we present some performance measures and a methodology for recording and sharing results among a group of diverse users on different machines. The second portion of the paper presents some of the results we have obtained over the past year and a half. The results should not be used to compare machines, except in a very preliminary sense. Rather, the results are presented to show how the methodolo...
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