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An Analysis of SingleLayer Networks in Unsupervised Feature Learning
"... A great deal of research has focused on algorithms for learning features from unlabeled data. Indeed, much progress has been made on benchmark datasets like NORB and CIFAR by employing increasingly complex unsupervised learning algorithms and deep models. In this paper, however, we show that several ..."
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Cited by 223 (19 self)
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A great deal of research has focused on algorithms for learning features from unlabeled data. Indeed, much progress has been made on benchmark datasets like NORB and CIFAR by employing increasingly complex unsupervised learning algorithms and deep models. In this paper, however, we show
Integer sets containing no arithmetic progressions
 J. London Math. Soc
, 1987
"... lfh and k are positive integers there exists N(h, k) such that whenever N ^ N(h, k), and the integers 1,2,...,N are divided into h subsets, at least one must contain an arithmetic progression of length k. This is the famous theorem of van der Waerden [10], dating from 1927. The proof of this uses mu ..."
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Cited by 76 (0 self)
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lfh and k are positive integers there exists N(h, k) such that whenever N ^ N(h, k), and the integers 1,2,...,N are divided into h subsets, at least one must contain an arithmetic progression of length k. This is the famous theorem of van der Waerden [10], dating from 1927. The proof of this uses
Arithmetically defined dense subgroups of
, 2007
"... For every prime p and integer n � 3 we explicitly construct an abelian variety A/Fpn of dimension n such that for a suitable prime l the group of quasiisogenies of A/Fpn of lpower degree is canonically a dense subgroup of the nth Morava stabilizer group at p. We also give a variant of this result ..."
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For every prime p and integer n � 3 we explicitly construct an abelian variety A/Fpn of dimension n such that for a suitable prime l the group of quasiisogenies of A/Fpn of lpower degree is canonically a dense subgroup of the nth Morava stabilizer group at p. We also give a variant
ON ARITHMETIC STRUCTURES IN DENSE SETS OF INTEGERS
 DUKE MATHEMATICAL JOURNAL VOL. 114, NO. 2,
, 2002
"... We prove that if A ⊆ {1,..., N} has density at least (log log N) −c, where c is an absolute constant, then A contains a triple (a, a + d, a + 2d) with d = x 2 + y 2 for some integers x, y, not both zero. We combine methods of T. Gowers and A. Sárközy with an application of Selberg’s sieve. The resul ..."
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Cited by 24 (1 self)
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We prove that if A ⊆ {1,..., N} has density at least (log log N) −c, where c is an absolute constant, then A contains a triple (a, a + d, a + 2d) with d = x 2 + y 2 for some integers x, y, not both zero. We combine methods of T. Gowers and A. Sárközy with an application of Selberg’s sieve. The result may be regarded as a step toward establishing a fully quantitative version of the polynomial Szemerédi theorem of V. Bergelson and A. Leibman.
Arithmetic structures in random sets
, 2008
"... We extend two wellknown results in additive number theory, Sárközy’s theorem on square differences in dense sets and a theorem of Green on long arithmetic progressions in sumsets, to subsets of random sets of asymptotic density 0. Our proofs rely on a restrictiontype Fourier analytic argument of G ..."
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Cited by 8 (1 self)
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We extend two wellknown results in additive number theory, Sárközy’s theorem on square differences in dense sets and a theorem of Green on long arithmetic progressions in sumsets, to subsets of random sets of asymptotic density 0. Our proofs rely on a restrictiontype Fourier analytic argument
Zerofree regions for Dirichlet Lfunctions and the least prime in an arithmetic progression
 Proc. Lond. Math. Soc
, 1992
"... The classical theorem of Dirichlet states that any arithmetic progression a(mod q) in which a and q are relatively prime contains infinitely many prime numbers. A natural question to ask is then, how big is the first such prime, P (a, q) say? In one direction we have trivially ..."
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Cited by 73 (0 self)
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The classical theorem of Dirichlet states that any arithmetic progression a(mod q) in which a and q are relatively prime contains infinitely many prime numbers. A natural question to ask is then, how big is the first such prime, P (a, q) say? In one direction we have trivially
theorem for arithmetic progressions
 Contents ◭◭ ◭ ◮◮ ◮ Go Back Close Quit Page 24 of 24
, 1974
"... Let k be an integer> 1 and let I be an integer such that 1 Q 1 f k, (I, k) = 1. An asymptotic formula (valid for large x) is obtained for the product generalizing a familiar result of Mertens. 1. ..."
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Cited by 13 (2 self)
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Let k be an integer> 1 and let I be an integer such that 1 Q 1 f k, (I, k) = 1. An asymptotic formula (valid for large x) is obtained for the product generalizing a familiar result of Mertens. 1.
Results 11  20
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