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11,204
The irreducibility of the space of curves of given genus
 Publ. Math. IHES
, 1969
"... Fix an algebraically closed field k. Let Mg be the moduli space of curves of genus g over k. The main result of this note is that Mg is irreducible for every k. Of course, whether or not M s is irreducible depends only on the characteristic of k. When the characteristic s o, we can assume that k ~ ..."
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Cited by 512 (2 self)
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Fix an algebraically closed field k. Let Mg be the moduli space of curves of genus g over k. The main result of this note is that Mg is irreducible for every k. Of course, whether or not M s is irreducible depends only on the characteristic of k. When the characteristic s o, we can assume that k ~ (1, and then the result is classical. A simple proof appears in EnriquesChisini [E, vol. 3, chap. 3], based on analyzing the totality of coverings of p1 of degree n, with a fixed number d of ordinary branch points. This method has been extended to char. p by William Fulton [F], using specializations from char. o to char. p provided that p> 2g qi. Unfortunately, attempts to extend this method to all p seem to get stuck on difficult questions of wild ramification. Nowadays, the Teichmtiller theory gives a thoroughly analytic but very profound insight into this irreducibility when kC. Our approach however is closest to Severi's incomplete proof ([Se], Anhang F; the error is on pp. 344345 and seems to be quite basic) and follows a suggestion of Grothendieck for using the result in char. o to deduce the result in char. p. The basis of both Severi's and Grothendieck's ideas is to construct families of curves X, some singular, with pa(X)=g, over nonsingular parameter spaces, which in some sense contain enough singular curves to link together any two components that Mg might have. The essential thing that makes this method work now is a recent " stable reduction theorem " for abelian varieties. This result was first proved independently in char. o by Grothendieck, using methods of etale cohomology (private correspondence with J. Tate), and by Mumford, applying the easy half of Theorem (2.5), to go from curves to abelian varieties (cf. [M2]). Grothendieck has recently strengthened his method so that it applies in all characteristics (SGA 7, ~968) 9 Mumford has also given a proof using theta functions in char. ~2. The result is this: Stable Reduction Theorem. Let R be a discrete valuation ring with quotient field K. Let A be an abelian variety over K. Then there exists a finite algebraic extension L of K such
Spectral AMGe (ρAMGe
 SIAM J. Sci. Comput
"... Abstract. Spectral AMGe (ρAMGe), is a new algebraic multigrid method for solving discretizations that arise in Ritztype finite element methods for partial differential equations. The method assumes access to the element stiffness matrices in order to lessen certain presumptions that can limit other ..."
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Cited by 38 (12 self)
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Abstract. Spectral AMGe (ρAMGe), is a new algebraic multigrid method for solving discretizations that arise in Ritztype finite element methods for partial differential equations. The method assumes access to the element stiffness matrices in order to lessen certain presumptions that can limit
OnLine Construction of Suffix Trees
, 1995
"... An online algorithm is presented for constructing the suffix tree for a given string in time linear in the length of the string. The new algorithm has the desirable property of processing the string symbol by symbol from left to right. It has always the suffix tree for the scanned part of the strin ..."
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Cited by 432 (2 self)
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An online algorithm is presented for constructing the suffix tree for a given string in time linear in the length of the string. The new algorithm has the desirable property of processing the string symbol by symbol from left to right. It has always the suffix tree for the scanned part of the string ready. The method is developed as a lineartime version of a very simple algorithm for (quadratic size) suffix tries. Regardless of its quadratic worstcase this latter algorithm can be a good practical method when the string is not too long. Another variation of this method is shown to give in a natural way the wellknown algorithms for constructing suffix automata (DAWGs).
Spectral AMGe (rhoAMGe)
 SIAM J. Sci. Comput
"... We introduce spectral AMGe (#AMGe), a new algebraic multigrid method for solving systems of algebraic equations that arise in Ritztype finite element discretizations of partial di#erential equations. The method requires access to the element sti#ness matrices, which enables accurate approximation o ..."
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We introduce spectral AMGe (#AMGe), a new algebraic multigrid method for solving systems of algebraic equations that arise in Ritztype finite element discretizations of partial di#erential equations. The method requires access to the element sti#ness matrices, which enables accurate approximation
A review of algebraic multigrid
, 2001
"... Since the early 1990s, there has been a strongly increasing demand for more efficient methods to solve large sparse, unstructured linear systems of equations. For practically relevant problem sizes, classical onelevel methods had already reached their limits and new hierarchical algorithms had to b ..."
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Cited by 345 (11 self)
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to be developed in order to allow an efficient solution of even larger problems. This paper gives a review of the first hierarchical and purely matrixbased approach, algebraic multigrid (AMG). AMG can directly be applied, for instance, to efficiently solve various types of elliptic partial differential equations
Algebraic Multigrid Based On Element Interpolation (AMGe)
, 1998
"... We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritztype finite element methods for partial differential equations. Assuming access to the element stiffness matrices, AMGe is based on the use of two local measures, which are derived from global meas ..."
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Cited by 102 (15 self)
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We introduce AMGe, an algebraic multigrid method for solving the discrete equations that arise in Ritztype finite element methods for partial differential equations. Assuming access to the element stiffness matrices, AMGe is based on the use of two local measures, which are derived from global
Discriminative Reranking for Natural Language Parsing
, 2005
"... This article considers approaches which rerank the output of an existing probabilistic parser. The base parser produces a set of candidate parses for each input sentence, with associated probabilities that define an initial ranking of these parses. A second model then attempts to improve upon this i ..."
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Cited by 327 (9 self)
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This article considers approaches which rerank the output of an existing probabilistic parser. The base parser produces a set of candidate parses for each input sentence, with associated probabilities that define an initial ranking of these parses. A second model then attempts to improve upon this initial ranking, using additional features of the tree as evidence. The strength of our approach is that it allows a tree to be represented as an arbitrary set of features, without concerns about how these features interact or overlap and without the need to define a derivation or a generative model which takes these features into account. We introduce a new method for the reranking task, based on the boosting approach to ranking problems described in Freund et al. (1998). We apply the boosting method to parsing the Wall Street Journal treebank. The method combined the loglikelihood under a baseline model (that of Collins [1999]) with evidence from an additional 500,000 features over parse trees that were not included in the original model. The new model achieved 89.75 % Fmeasure, a 13 % relative decrease in Fmeasure error over the baseline model’s score of 88.2%. The article also introduces a new algorithm for the boosting approach which takes advantage of the sparsity of the feature space in the parsing data. Experiments show significant efficiency gains for the new algorithm over the obvious implementation of the boosting approach. We argue that the method is an appealing alternative—in terms of both simplicity and efficiency—to work on feature selection methods within loglinear (maximumentropy) models. Although the experiments in this article are on natural language parsing (NLP), the approach should be applicable to many other NLP problems which are naturally framed as ranking tasks, for example, speech recognition, machine translation, or natural language generation.
On Generalizing the AMG Framework
 SIAM J. NUMER. ANAL
, 2003
"... We present a theory for algebraic multigrid (AMG) methods that allows for general smoothing processes and general coarsening approaches. The goal of the theory is to provide guidance in the development of new, more robust, AMG algorithms. In particular, we introduce several compatible relaxation met ..."
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Cited by 18 (3 self)
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We present a theory for algebraic multigrid (AMG) methods that allows for general smoothing processes and general coarsening approaches. The goal of the theory is to provide guidance in the development of new, more robust, AMG algorithms. In particular, we introduce several compatible relaxation
Results 1  10
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11,204