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The particel swarm: Explosion, stability, and convergence in a multidimensional complex space
 IEEE TRANSACTIONS ON EVOLUTIONARY COMPUTION
"... The particle swarm is an algorithm for finding optimal regions of complex search spaces through interaction of individuals in a population of particles. Though the algorithm, which is based on a metaphor of social interaction, has been shown to perform well, researchers have not adequately explained ..."
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Cited by 852 (10 self)
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in discrete time (the algebraic view), then progresses to the view of it in continuous time (the analytical view). A 5dimensional depiction is developed, which completely describes the system. These analyses lead to a generalized model of the algorithm, containing a set of coefficients to control the system
A variant of the hypergraph removal lemma
, 2006
"... Abstract. Recent work of Gowers [10] and Nagle, Rödl, Schacht, and Skokan [15], [19], [20] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [26] and FurstenbergKatznelson [7] concerning onedimensional and multidimensional arithmetic progressions respecti ..."
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Cited by 75 (7 self)
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Abstract. Recent work of Gowers [10] and Nagle, Rödl, Schacht, and Skokan [15], [19], [20] has established a hypergraph removal lemma, which in turn implies some results of Szemerédi [26] and FurstenbergKatznelson [7] concerning onedimensional and multidimensional arithmetic progressions
(1.1) ∥ e(n·) ∥. p
, 2002
"... This paper is concerned with the majorant property of various randomly generated subsets of [1,N]. More precisely, suppose AN ⊂ [1,N] is a sequence of sets so that AN  ≍ Nρ for some fixed 0 < ρ < 1 as N → ∞. For example, one can take AN to be the squares, cubes, etc., or (multidimensional) ..."
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This paper is concerned with the majorant property of various randomly generated subsets of [1,N]. More precisely, suppose AN ⊂ [1,N] is a sequence of sets so that AN  ≍ Nρ for some fixed 0 < ρ < 1 as N → ∞. For example, one can take AN to be the squares, cubes, etc., or (multidimensional
The primes contain arbitrarily long arithmetic progressions
 Ann. of Math
"... Abstract. We prove that there are arbitrarily long arithmetic progressions of primes. ..."
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Cited by 268 (31 self)
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Abstract. We prove that there are arbitrarily long arithmetic progressions of primes.
Linear Algebra Operators for GPU Implementation of Numerical Algorithms
 ACM Transactions on Graphics
, 2003
"... In this work, the emphasis is on the development of strategies to realize techniques of numerical computing on the graphics chip. In particular, the focus is on the acceleration of techniques for solving sets of algebraic equations as they occur in numerical simulation. We introduce a framework for ..."
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Cited by 324 (9 self)
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direct solvers for sparse matrices, and by applying these solvers to multidimensional finite difference equations, i.e. the 2D wave equation and the incompressible NavierStokes equations.
Cultivating competence, selfefficacy, and intrinsic interest through proximal selfmotivation.
 Journal of Personality and Social Psychology,
, 1981
"... Abstract: The present experiment tested the hypothesis that selfmotivation through proximal goal setting serves as an effective mechanism for cultivating competencies, selfpercepts of efficacy, and intrinsic interest. Children who exhibited gross deficits and disinterest in mathematical tasks pur ..."
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Cited by 295 (6 self)
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pursued a program of selfdirected learning under conditions involving either proximal subgoals, distal goals, or no goals. Results of the multifaceted assessment provide support for the superiority of proximal selfinfluence. Under proximal subgoals, children progressed rapidly in selfdirected learning
Discrepancy of Cartesian Products of Arithmetic Progressions
, 2004
"... We determine the combinatorial discrepancy of the hypergraph of cartesian products of d arithmetic progressions in the [N ] lattice ([N]={0, 1,...,N 1}). The study of such higher dimensional arithmetic progressions is motivated by a multidimensional version of van der Waerden's theorem ..."
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Cited by 8 (4 self)
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We determine the combinatorial discrepancy of the hypergraph of cartesian products of d arithmetic progressions in the [N ] lattice ([N]={0, 1,...,N 1}). The study of such higher dimensional arithmetic progressions is motivated by a multidimensional version of van der Waerden
A NEW PROOF OF SZEMERÉDI’S THEOREM FOR ARITHMETIC PROGRESSIONS OF LENGTH FOUR
 GAFA, GEOMETRIC AND FUNCTIONAL ANALYSIS
, 1998
"... ..."
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
Results 1  10
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1,624