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13,046
Closed measure zero sets
, 2003
"... We study the relationship between the σideal generated by closed measure zero sets and the ideals of null and meager sets. We show that the additivity of the ideal of closed measure zero sets is not bigger than covering for category. As a consequence we get that the additivity of the ideal of close ..."
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Cited by 16 (1 self)
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We study the relationship between the σideal generated by closed measure zero sets and the ideals of null and meager sets. We show that the additivity of the ideal of closed measure zero sets is not bigger than covering for category. As a consequence we get that the additivity of the ideal
Zero sets of univariate polynomials
, 2009
"... Let L be the zero set of a nonconstant monic polynomial with complex coefficients. In the context of constructive mathematics without countable choice, it may not be possible to construct an element of L. In this paper we introduce a notion of distance from a point to a subset, more general than the ..."
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Cited by 2 (2 self)
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Let L be the zero set of a nonconstant monic polynomial with complex coefficients. In the context of constructive mathematics without countable choice, it may not be possible to construct an element of L. In this paper we introduce a notion of distance from a point to a subset, more general than
THE ZERO SET OF CONFORMAL VECTOR FIELDS
, 2010
"... Abstract. We show that every connected component of the zero set of a conformal vector field on a Riemannian manifold is totally umbilical. ..."
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Abstract. We show that every connected component of the zero set of a conformal vector field on a Riemannian manifold is totally umbilical.
On the zero set of Gequivariant maps
, 2008
"... Let G be a finite group acting on vector spaces V and W and consider a smooth Gequivariant mapping f: V → W. This paper addresses the question of the zero set near a zero x of f with isotropy subgroup G. It is known from results of Bierstone and Field on Gtransversality theory that the zero set in ..."
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Let G be a finite group acting on vector spaces V and W and consider a smooth Gequivariant mapping f: V → W. This paper addresses the question of the zero set near a zero x of f with isotropy subgroup G. It is known from results of Bierstone and Field on Gtransversality theory that the zero set
On the optimality of the simple Bayesian classifier under zeroone loss
 MACHINE LEARNING
, 1997
"... The simple Bayesian classifier is known to be optimal when attributes are independent given the class, but the question of whether other sufficient conditions for its optimality exist has so far not been explored. Empirical results showing that it performs surprisingly well in many domains containin ..."
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Cited by 818 (27 self)
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containing clear attribute dependences suggest that the answer to this question may be positive. This article shows that, although the Bayesian classifier’s probability estimates are only optimal under quadratic loss if the independence assumption holds, the classifier itself can be optimal under zero
Zerosets of nearsymplectic forms
, 2008
"... We give elementary proofs of two ‘folklore ’ assertions about nearsymplectic forms on fourmanifolds: that any such form can be modified, by an evolutionary process taking place inside a finite set of balls, so as to have exactly n zerocircles, for any n ≥ 1; and that, on a closed manifold, the num ..."
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Cited by 5 (0 self)
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We give elementary proofs of two ‘folklore ’ assertions about nearsymplectic forms on fourmanifolds: that any such form can be modified, by an evolutionary process taking place inside a finite set of balls, so as to have exactly n zerocircles, for any n ≥ 1; and that, on a closed manifold
Zero sets of polynomials in several variables
"... Abstract. Let k, n ∈ IN, where n is odd. We show that there is an integer N = N(k, n) such that for every nhomogeneous polynomial P: IR N → IR there exists a linear subspace X ↩ → IR N, dimX = k such that P X ≡ 0. This quantitative estimate improves on previous work of Birch, et al, who studied th ..."
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Cited by 4 (1 self)
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Abstract. Let k, n ∈ IN, where n is odd. We show that there is an integer N = N(k, n) such that for every nhomogeneous polynomial P: IR N → IR there exists a linear subspace X ↩ → IR N, dimX = k such that P X ≡ 0. This quantitative estimate improves on previous work of Birch, et al, who studied this problem from an algebraic viewpoint. The topological method of proof presented here also allows us to obtain a partial solution to the GromovMilman problem (in dimension two) on an isometric version of a theorem of Dvoretzky. The present note was motivated by a question raised explicitly in [AGZ]: “Given k ∈ IN and an odd integer n, does there exist N ∈ IN such that for every nhomogeneous polynomial Q: IR N → IR there exists a kdimensional linear subspace X ↩ → IR N for which QX ≡ 0”. The authors gave a solution when n = 3, although as it turns out, a positive solution to a more general and essentially algebraic problem was obtained by Birch [B]. We are very grateful to the referee for pointing out that this problem has a long and distinguished history. In fact, progress on this type of question dates back to at least the 1950’s, with results mostly using algebraic techniques. For instance, Lewis [Le] showed exactly this type of result for algebraic number fields, in the cubic case. In a somewhat different direction, Schmidt [Sc] examined the number of common integer solutions, lying in a bounded domain, to systems of homogeneous polynomial equations. More recently, Wooley has found explicit bounds on the number of variables n = n(m, k) to ensure that every collection of m cubic polynomials in n variables vanishes on a subspace of dimension k (see, e.g., [W1, W2, W3]). This earlier work notwithstanding, we feel that our independent and rather topological solution has some advantages. For one thing, the estimate N ≥ (k + n) 3k that we obtain is incomparably better than the inductive methods of [B] would allow (leading to towering exponentials of fast increasing height). (On the other hand, it must be admitted that the
SOME POSSIBLE COVERS OF MEASURE ZERO SETS BY
"... of a measure zero set X (see [5]), that is, the rate of convergence of the series n In, for various sequences of open intervals 〈In: n ∈ N 〉 such ..."
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of a measure zero set X (see [5]), that is, the rate of convergence of the series n In, for various sequences of open intervals 〈In: n ∈ N 〉 such
Results 1  10
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13,046