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Nesl: A Nested DataParallel Language
, 1990
"... This report describes Nesl, a stronglytyped, applicative, dataparallel language. Nesl is intended to be used as a portable interface for programming a variety of parallel and vector supercomputers, and as a basis for teaching parallel algorithms. Parallelism is supplied through a simple set of dat ..."
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Cited by 134 (4 self)
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This report describes Nesl, a stronglytyped, applicative, dataparallel language. Nesl is intended to be used as a portable interface for programming a variety of parallel and vector supercomputers, and as a basis for teaching parallel algorithms. Parallelism is supplied through a simple set of dataparallel constructs based on vectors, including a mechanism for applying any function over the elements of a vector in parallel, and a broad set of parallel functions that manipulate vectors. Nesl fully supports nested vectors and nested parallelismthe ability to take a parallel function and then apply it over multiple instances in parallel. Nested parallelism is important for implementing algorithms with complex and dynamically changing data structures, such as required in many graph or sparse matrix algorithms. Nesl also provides a mechanism for calculating the asymptotic running time for a program on various parallel machine models, including the parallel random access machine (PRAM...
Minimum diameters of plane integral point sets
"... ABSTRACT. Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets P, which are sets of n points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its ..."
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Cited by 12 (11 self)
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ABSTRACT. Since ancient times mathematicians consider geometrical objects with integral side lengths. We consider plane integral point sets P, which are sets of n points in the plane with pairwise integral distances where not all the points are collinear. The largest occurring distance is called its diameter. Naturally the question about the minimum possible diameter d(2, n) of a plane integral point set consisting of n points arises. We give some new exact values and describe stateoftheart algorithms to obtain them. It turns out that plane integral point sets with minimum diameter consist very likely of subsets with many collinear points. For this special kind of point sets we prove a lower bound for d(2, n) achieving the known upper bound nc2 log log n up to a constant in the exponent. A famous question of Erdős asks for plane integral point sets with no 3 points on a line and no 4 points on a circle. Here, we talk of point sets in general position and denote the corresponding minimum diameter by ˙ d(2, n). Recently ˙ d(2, 7) = 22 270 could be determined via an exhaustive search. 1.
A note on ErdösDiophantine graphs and Diophantine carpets
 Math. Balkanica
, 2007
"... ABSTRACT. We give an effective construction for ErdösDiophantine graphs and characterize the chromatic number of Diophantine carpets. 1. ..."
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Cited by 6 (5 self)
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ABSTRACT. We give an effective construction for ErdösDiophantine graphs and characterize the chromatic number of Diophantine carpets. 1.
Low regularity local wellposedness of the derivative nonlinear Schrdinger equation with periodic initial data
 SIAM J. Math. Anal
"... Abstract. The Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition is considered. Local wellposedness for data u0 in the space b Hs r (T), defined by the norms ‖u0 ‖ bH s r (T) = ‖〈ξ〉s bu0‖ ℓ r ′ ξ is shown in the parameter range s ≥ 1 4, 2> r>. The pro ..."
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Cited by 5 (1 self)
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Abstract. The Cauchy problem for the derivative nonlinear Schrödinger equation with periodic boundary condition is considered. Local wellposedness for data u0 in the space b Hs r (T), defined by the norms ‖u0 ‖ bH s r (T) = ‖〈ξ〉s bu0‖ ℓ r ′ ξ is shown in the parameter range s ≥ 1 4, 2> r>. The proof is based on an 2 3 adaptation of the gauge transform to the periodic setting and an appropriate variant of the Fourier restriction norm method. 1. Introduction and
Distribution of generalized Fermat prime numbers
 Math. Comp
, 1999
"... Abstract. Numbers of the form Fb,n = b2n +1 are called Generalized Fermat Numbers (GFN). A computational method for testing the probable primality of a GFN is described which is as fast as testing a number of the form 2m −1. The theoretical distributions of GFN primes, for fixed n, are derived and c ..."
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Cited by 5 (3 self)
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Abstract. Numbers of the form Fb,n = b2n +1 are called Generalized Fermat Numbers (GFN). A computational method for testing the probable primality of a GFN is described which is as fast as testing a number of the form 2m −1. The theoretical distributions of GFN primes, for fixed n, are derived and compared to the actual distributions. The predictions are surprisingly accurate and can be used to support Bateman and Horn’s quantitative form of “Hypothesis H” of Schinzel and Sierpiński. A list of the current largest known GFN primes is included. 1.
Seven consecutive primes in arithmetic progression
 Math.Comp
, 1997
"... Abstract. It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. In 1967, the first such sequence of 6 consecutive primes in arithmetic progression was found. Searching for 7 consecutive primes in arithmetic progression is difficult because it ..."
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Cited by 2 (0 self)
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Abstract. It is conjectured that there exist arbitrarily long sequences of consecutive primes in arithmetic progression. In 1967, the first such sequence of 6 consecutive primes in arithmetic progression was found. Searching for 7 consecutive primes in arithmetic progression is difficult because it is necessary that a prescribed set of at least 1254 numbers between the first and last prime all be composite. This article describes the search theory and methods, and lists the only known example of 7 consecutive primes in arithmetic progression. 1.
BILINEAR SPACETIME ESTIMATES FOR LINEARISED KPTYPE EQUATIONS ON THE THREEDIMENSIONAL TORUS WITH APPLICATIONS
, 901
"... Abstract. A bilinear estimate in terms of Bourgain spaces associated with a linearised KadomtsevPetviashvilitype equation on the threedimensional torus is shown. As a consequence, time localized linear and bilinear space time estimates for this equation are obtained. Applications to the local and ..."
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Abstract. A bilinear estimate in terms of Bourgain spaces associated with a linearised KadomtsevPetviashvilitype equation on the threedimensional torus is shown. As a consequence, time localized linear and bilinear space time estimates for this equation are obtained. Applications to the local and global wellposedness of dispersion generalised KPII equations are discussed. Especially it is proved that the periodic boundary value problem for the original KPII equation is locally wellposed for data in the anisotropic Sobolev spaces Hs xHε y(T3), if s ≥ 1 and ε> 0. 2 1. Introduction and