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1,703,070
log log k
"... Abstract. We study resonances of surfaces of revolution obtained by removing a disk from a cone and attaching a hyperbolic cusp in its place. These surfaces include ones with nontrapping geodesic flow (every maximally extended nonreflected geodesic is unbounded) and yet infinitely many long living ..."
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Abstract. We study resonances of surfaces of revolution obtained by removing a disk from a cone and attaching a hyperbolic cusp in its place. These surfaces include ones with nontrapping geodesic flow (every maximally extended nonreflected geodesic is unbounded) and yet infinitely many long living resonances (resonances with uniformly bounded imaginary part, i.e. decay rate). Let a < 0 < b, and let (X, g) be the surface of revolution X = R × S 1, g = dr 2 + f(r) 2 dθ 2, f(r) = 1 + ar if r ≤ 0, e−br if r> 0. Figure 1. The surface of revolution (X, g) for a + b = 0, a + b> 0, and a + b < 0. Let ∆g be the nonnegative Laplacian on (X, g). The resolvent (∆g − λ 2) −1 is holomorphic L 2 (X) → L 2 (X) for Im λ> 0. Poles of the continuation of its integral kernel from {Im λ> 0} to {Im λ ≤ 0, Re λ> b/2} are called resonances (see [Me, Zw3]). Theorem. The surface of revolution (X, g) has a sequence of resonances (λk)k≥k0 satisfying Re λk = πb
(log log N)2
"... §0. Statement and discussion of the argument This paper is a sequel to [B]. Our main result is an improvement of the density condition for a subset A ⊂ {1,...,N} to contain a nontrivial arithmetic progression of length 3. More specifically, we prove the following ..."
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§0. Statement and discussion of the argument This paper is a sequel to [B]. Our main result is an improvement of the density condition for a subset A ⊂ {1,...,N} to contain a nontrivial arithmetic progression of length 3. More specifically, we prove the following
Classical and new log logtheorems
, 710
"... We present a unified approach to celebrated log logtheorems of Carleman, Wolf, Levinson, Sjöberg, Matsaev on majorants of analytic functions. Moreover, we obtain stronger results by replacing original pointwise bounds with integral ones. The main ingredient is a complete description for radial proj ..."
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We present a unified approach to celebrated log logtheorems of Carleman, Wolf, Levinson, Sjöberg, Matsaev on majorants of analytic functions. Moreover, we obtain stronger results by replacing original pointwise bounds with integral ones. The main ingredient is a complete description for radial
The LogLog Term Frequency Distribution
, 2005
"... Though commonly used, the unigram is widely known as being a poor model of term frequency; it assumes that term occurrences are independent, whereas many words, especially topicoriented ones, tend to occur in bursts. Herein, we propose a model of term frequency that treats words independently, but ..."
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Though commonly used, the unigram is widely known as being a poor model of term frequency; it assumes that term occurrences are independent, whereas many words, especially topicoriented ones, tend to occur in bursts. Herein, we propose a model of term frequency that treats words independently, but allows for much higher variance in frequency values than does the unigram. Although it has valuable properties, and may be useful as a teaching tool, we are not able to find any applications that make a compelling case for its use. 1 The Unigram Model The unigram is a simple, commonly used, model of text. It assumes that each word occurrence is independent of all other word occurrences. There is one parameter per word, θi, � i θi = 1, which corresponds to that word’s rate of occurrence. For a document of length l, the chance that a unigram with parameters {θi} yields term frequencies {xi} is
References π, log π, log log π, log log log π,...
, 2008
"... Rough outlines of the project a) (Expanding a remark by S. Lang – [1]). Define K0 = Q. Inductively, for n ≥ 1, define Kn as the algebraic closure of the field generated over Kn−1 by the numbers e x, where x ranges over Kn−1. Let Ω+ be the union of Kn, n ≥ 0. Show that the numbers are algebraically i ..."
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Rough outlines of the project a) (Expanding a remark by S. Lang – [1]). Define K0 = Q. Inductively, for n ≥ 1, define Kn as the algebraic closure of the field generated over Kn−1 by the numbers e x, where x ranges over Kn−1. Let Ω+ be the union of Kn, n ≥ 0. Show that the numbers are algebraically independent over Ω+.
(poly(log log n), poly(log log n))Restricted Verifiers are Unlikely to Exist for Languages in NP
 in NP . Proc. of the 21th Mathematical Foundations of Computer Science
, 1996
"... . The aim of this paper is to present a proof of the equivalence of the equalities NP = PCP(log log n; 1) and P = NP. The proof is based on producing long pseudorandom bit strings through random walks on expander graphs. This technique also implies that for any language in NP there exists a res ..."
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Cited by 2 (2 self)
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. The aim of this paper is to present a proof of the equivalence of the equalities NP = PCP(log log n; 1) and P = NP. The proof is based on producing long pseudorandom bit strings through random walks on expander graphs. This technique also implies that for any language in NP there exists a
RUSPACE(log n) \subseteq DSPACE(log² n/log log n)
 THE 7TH ANNUAL INTERNATIONAL SYMPOSIUM ON ALGORITHMS AND COMPUTATION (ISAAC’96
, 1998
"... We present a deterministic algorithm running in space O , log n= log log n # solving the connectivity problem on strongly unambiguous graphs. In addition, we presentanO#log n# timebounded algorithm for this problem running on a parallel pointer machine. ..."
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We present a deterministic algorithm running in space O , log n= log log n # solving the connectivity problem on strongly unambiguous graphs. In addition, we presentanO#log n# timebounded algorithm for this problem running on a parallel pointer machine.
An optimal O(log log n) time parallel string matching algorithm
 SIAM J. COMPUT
, 1990
"... An optimal O(log log n) time parallel algorithm for string matching on CRCWPRAM is presented. It improves previous results of [G] and [V]. ..."
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Cited by 27 (11 self)
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An optimal O(log log n) time parallel algorithm for string matching on CRCWPRAM is presented. It improves previous results of [G] and [V].
A Log Log Law for Maximal Uniform Spacings
, 1982
"... this paper is to show that the constant c in (1.2) islog 2 ..."
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Cited by 8 (1 self)
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this paper is to show that the constant c in (1.2) islog 2
LogLog counting for the estimation of IP traffic
 IN &QUOT;PROCEEDINGS OF THE FOURTH COLLOQUIUM ON MATHEMATICS AND COMPUTER SCIENCE ALGORITHMS, TREES, COMBINATORICS AND PROBABILITIES
"... In this paper, we discuss the problem of estimating the number of “elephants ” in a stream of IP packets. First, the problem is formulated in the context of multisets. Next, we explore some of the theoretical space complexity of this problem, and it is shown that it cannot be solved with less than Ω ..."
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Cited by 3 (0 self)
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In this paper, we discuss the problem of estimating the number of “elephants ” in a stream of IP packets. First, the problem is formulated in the context of multisets. Next, we explore some of the theoretical space complexity of this problem, and it is shown that it cannot be solved with less than Ω(n) units of memory in general, n being the number of different elements in the multiset. Finally, we describe an algorithm, based on Durand & Flajolet’s LOGLOG algorithm coupled with a thinning of the packet stream, which returns an estimator of the number of elephants using a small amount of memory. This algorithm allows a good estimation for particular families of random multiset. The mean and variance of this estimator are computed. The algorithm is then tested on synthetic data.
Results 1  10
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