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MULTIPLICITY OF INVARIANT ALGEBRAIC CURVES AND DARBOUX INTEGRABILITY
, 2000
"... We define four different kinds of multiplicity of an invariant algebraic curve for a given polynomial vector field and investigate their relationships. After taking a closer look at the singularities and at the line of infinity, we improve the Darboux theory of integrability using these new notion ..."
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Cited by 3 (1 self)
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We define four different kinds of multiplicity of an invariant algebraic curve for a given polynomial vector field and investigate their relationships. After taking a closer look at the singularities and at the line of infinity, we improve the Darboux theory of integrability using these new
QUADRATIC VECTOR FIELDS WITH INVARIANT ALGEBRAIC CURVE OF LARGE
, 904
"... Abstract. We construct a polynomial planar vector field of degree two with one invariant algebraic curves of large degree. We exhibit an explicit quadratic vector fields which invariant curves of degree nine and twelve and fifteen degree.. 1. ..."
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Abstract. We construct a polynomial planar vector field of degree two with one invariant algebraic curves of large degree. We exhibit an explicit quadratic vector fields which invariant curves of degree nine and twelve and fifteen degree.. 1.
Multiplicity of invariant algebraic curves in polynomial vector fields
 PACIFIC J. MATH
"... The aim of this paper is to introduce a concrete notion of multiplicity for invariant algebraic curves in polynomial vector fields. In fact, we give several natural definitions and show that they are all equivalent to our main definition, under some ‘generic ’ assumptions. In particular, we show th ..."
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Cited by 17 (2 self)
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The aim of this paper is to introduce a concrete notion of multiplicity for invariant algebraic curves in polynomial vector fields. In fact, we give several natural definitions and show that they are all equivalent to our main definition, under some ‘generic ’ assumptions. In particular, we show
A FAMILY OF QUADRATIC POLYNOMIAL DIFFERENTIAL SYSTEMS WITH INVARIANT ALGEBRAIC CURVES OF ARBITRARILY HIGH DEGREE WITHOUT RATIONAL FIRST INTEGRALS
, 2001
"... We give a class of quadratic systems without rational first integral which contains irreducible algebraic solutions of arbitrarily high degree. The construction gives a negative answer to a conjecture of Lins Neto and others. ..."
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Cited by 4 (2 self)
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We give a class of quadratic systems without rational first integral which contains irreducible algebraic solutions of arbitrarily high degree. The construction gives a negative answer to a conjecture of Lins Neto and others.
Guide to Elliptic Curve Cryptography
, 2004
"... Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves ..."
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Cited by 610 (18 self)
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Elliptic curves have been intensively studied in number theory and algebraic geometry for over 100 years and there is an enormous amount of literature on the subject. To quote the mathematician Serge Lang: It is possible to write endlessly on elliptic curves. (This is not a threat.) Elliptic curves
Homological Algebra of Mirror Symmetry
 in Proceedings of the International Congress of Mathematicians
, 1994
"... Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual Ca ..."
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Cited by 523 (3 self)
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Mirror Symmetry was discovered several years ago in string theory as a duality between families of 3dimensional CalabiYau manifolds (more precisely, complex algebraic manifolds possessing holomorphic volume elements without zeroes). The name comes from the symmetry among Hodge numbers. For dual
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 506 (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
Axiomatic quantum field theory in curved spacetime
, 2008
"... The usual formulations of quantum field theory in Minkowski spacetime make crucial use of features—such as Poincare invariance and the existence of a preferred vacuum state—that are very special to Minkowski spacetime. In order to generalize the formulation of quantum field theory to arbitrary globa ..."
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Cited by 689 (18 self)
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The usual formulations of quantum field theory in Minkowski spacetime make crucial use of features—such as Poincare invariance and the existence of a preferred vacuum state—that are very special to Minkowski spacetime. In order to generalize the formulation of quantum field theory to arbitrary
The use of the area under the ROC curve in the evaluation of machine learning algorithms
 PATTERN RECOGNITION
, 1997
"... In this paper we investigate the use of the area under the receiver operating characteristic (ROC) curve (AUC) as a performance measure for machine learning algorithms. As a case study we evaluate six machine learning algorithms (C4.5, Multiscale Classifier, Perceptron, Multilayer Perceptron, kNe ..."
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Cited by 685 (3 self)
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In this paper we investigate the use of the area under the receiver operating characteristic (ROC) curve (AUC) as a performance measure for machine learning algorithms. As a case study we evaluate six machine learning algorithms (C4.5, Multiscale Classifier, Perceptron, Multilayer Perceptron, k
Results 1  10
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14,079