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ALMOSTSCHUR LEMMA
"... Schur’s lemma states that every Einstein manifold of dimension n ≥ 3 has constant scalar curvature. Here (M, g) is defined to be Einstein ..."
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Schur’s lemma states that every Einstein manifold of dimension n ≥ 3 has constant scalar curvature. Here (M, g) is defined to be Einstein
On the ring lemma
, 2008
"... The sharp ring lemma states that if n >= 3 cyclically tangent discs with pairwise disjoint interiors are externally tangent to and surround the unit disc, then no disc has a radius below cn = (F 2 n−1 +F2 n−2 −1)−1 – where Fk denotes the k th Fibonacci number – and that the lower bound is attaine ..."
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The sharp ring lemma states that if n >= 3 cyclically tangent discs with pairwise disjoint interiors are externally tangent to and surround the unit disc, then no disc has a radius below cn = (F 2 n−1 +F2 n−2 −1)−1 – where Fk denotes the k th Fibonacci number – and that the lower bound
Almost Optimal Lower Bounds for Small Depth Circuits
, 1986
"... We give improved lower bounds for the size of small depth circuits computing several functions. In particular we prove almost optimal lower bounds for the size of parity circuits. Further we show that there are functions computable in polynomial size and depth k but requires exponential size when ..."
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Cited by 277 (8 self)
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when the depth is restricted to k1. Our main lemma which is of independent interest states that by using a random restriction we can convert an AND of small ORs to an OR of small ANDs and conversely.
Graph removal lemmas
 SURVEYS IN COMBINATORICS
, 2013
"... The graph removal lemma states that any graph on n vertices with o(nv(H)) copies of a fixed graph H may be made Hfree by removing o(n²) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory and com ..."
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Cited by 9 (3 self)
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The graph removal lemma states that any graph on n vertices with o(nv(H)) copies of a fixed graph H may be made Hfree by removing o(n²) edges. Despite its innocent appearance, this lemma and its extensions have several important consequences in number theory, discrete geometry, graph theory
Around Sperner’s lemma
"... We consider a generalization of the classic Sperner lemma. This lemma states that every Sperner coloring of a triangulation of a simplex contains a fully colored simplex. We found a weaker assumption than Sperner’s coloring. It is also shown that the main theorem implies Tucker’s lemma and some othe ..."
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Cited by 2 (2 self)
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We consider a generalization of the classic Sperner lemma. This lemma states that every Sperner coloring of a triangulation of a simplex contains a fully colored simplex. We found a weaker assumption than Sperner’s coloring. It is also shown that the main theorem implies Tucker’s lemma and some
THE REGULARITY LEMMA AND GRAPH THEORY
"... Abstract. The Szemerédi Regularity Lemma states that any sufficiently large graph G can be partitioned into a bounded (independent of the size of the graph) number of regular, or “randomlooking, ” components. The resulting partition can be viewed as a regularity graph R. The Key Lemma shows that u ..."
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Abstract. The Szemerédi Regularity Lemma states that any sufficiently large graph G can be partitioned into a bounded (independent of the size of the graph) number of regular, or “randomlooking, ” components. The resulting partition can be viewed as a regularity graph R. The Key Lemma shows
Sperner’s Lemma
"... Summary. In this article we introduce and prove properties of simplicial complexes in real linear spaces which are necessary to formulate Sperner’s lemma. The lemma states that for a function f, which for an arbitrary vertex v of the barycentric subdivision B of simplex K assigns some vertex from a ..."
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Summary. In this article we introduce and prove properties of simplicial complexes in real linear spaces which are necessary to formulate Sperner’s lemma. The lemma states that for a function f, which for an arbitrary vertex v of the barycentric subdivision B of simplex K assigns some vertex from a
On Yao’s XOR lemma
 Electronic Colloquium on Computational Complexity
, 1995
"... Abstract. A fundamental lemma of Yao states that computational weakunpredictability of Boolean predicates is amplified when the results of several independent instances are XOR together. We survey two known proofs of Yao’s Lemma and present a third alternative proof. The third proof proceeds by firs ..."
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Cited by 65 (6 self)
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Abstract. A fundamental lemma of Yao states that computational weakunpredictability of Boolean predicates is amplified when the results of several independent instances are XOR together. We survey two known proofs of Yao’s Lemma and present a third alternative proof. The third proof proceeds
The Generalized XOR Lemma
"... Abstract The XOR Lemma states that a mapping is regular or balanced if and only if all the linear combinations of the component functions of the mapping are balanced Boolean functions. The main contribution of this paper is to extend the XOR Lemma to more general cases where a mapping may not be ne ..."
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Abstract The XOR Lemma states that a mapping is regular or balanced if and only if all the linear combinations of the component functions of the mapping are balanced Boolean functions. The main contribution of this paper is to extend the XOR Lemma to more general cases where a mapping may
Note The Generalized XOR Lemma
"... The XOR Lemma states that a mapping is regular or balanced if and only if all the linear combinations of the component functions of the mapping are balanced Boolean functions. The main contribution of this paper is to extend the XOR Lemma to more general cases where a mapping may not be necessarily ..."
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The XOR Lemma states that a mapping is regular or balanced if and only if all the linear combinations of the component functions of the mapping are balanced Boolean functions. The main contribution of this paper is to extend the XOR Lemma to more general cases where a mapping may not be necessarily
Results 1  10
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1,137