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Explicit points on the Legendre curve
"... ABSTRACT. We study the elliptic curve E given by y 2 = x(x + 1)(x + t) over the rational function field k(t) and its extensions Kd = k(µd, t 1/d). When k is finite of characteristic p and d = p f + 1, we write down explicit points on E and show by elementary arguments that they generate a subgroup V ..."
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Cited by 7 (5 self)
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ABSTRACT. We study the elliptic curve E given by y 2 = x(x + 1)(x + t) over the rational function field k(t) and its extensions Kd = k(µd, t 1/d). When k is finite of characteristic p and d = p f + 1, we write down explicit points on E and show by elementary arguments that they generate a subgroup
calculus with explicit points and approximations
 Previously appeared in Fixed Points in Computer Science, FICS '02
"... Abstract. We present a Gentzenstyle sequent calculus for program verication which accomodates both model checkinglike verication based on global state space exploration, and compositional reasoning. To handle the complexities arrising from the presence of xedpoint formulas, programs with dynamic ..."
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Cited by 9 (6 self)
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with dynamically evolving architecture, and cut rules we use transition assertions, and introduce xedpoint approximants explicitly into the assertion language. We address, in a gamebased manner, the semantical basis of this approach, as it applies to the entailment subproblem. Soundness and completeness results
EXPLICIT POINTS ON THE LEGENDRE CURVE II
"... ABSTRACT. Let E be the elliptic curve y2 = x(x + 1)(x + t) over the field Fp(t) where p is an odd prime. We study the arithmetic of E over extensions Fq(t1/d) where q is a power of p and d is an integer prime to p. The rank of E is given in terms of an elementary property of the subgroup of (Z/dZ) × ..."
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Cited by 1 (1 self)
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/dZ) × generated by p. We show that for many values of d the rank is large. For example, if d divides 2(pf − 1) and 2(pf − 1)/d is odd, then the rank is at least d/2. When d = 2(pf − 1), we exhibit explicit points generating a subgroup of E(Fq(t1/d)) of finite index, and we bound the index as well as the order
µCalculus with Explicit Points and Approximations
, 2000
"... We present a Gentzenstyle sequent calculus for program verification which accomodates both model checkinglike verification based on global state space exploration, and compositional reasoning. To handle the complexities arrising from the presence of fixedpoint formulas, programs with dynamically ..."
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Cited by 6 (2 self)
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evolving architecture, and cut rules we use transition assertions, and introduce fixedpoint approximants explicitly into the assertion language. We address, in a gamebased manner, the semantical basis of this approach, as it applies to the entailment subproblem. Soundness and completeness results
OPTICS: Ordering Points To Identify the Clustering Structure
, 1999
"... Cluster analysis is a primary method for database mining. It is either used as a standalone tool to get insight into the distribution of a data set, e.g. to focus further analysis and data processing, or as a preprocessing step for other algorithms operating on the detected clusters. Almost all of ..."
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Cited by 527 (51 self)
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the intrinsic clustering structure accurately. We introduce a new algorithm for the purpose of cluster analysis which does not produce a clustering of a data set explicitly; but instead creates an augmented ordering of the database representing its densitybased clustering structure. This cluster
Plenoptic Modeling: An ImageBased Rendering System
, 1995
"... Imagebased rendering is a powerful new approach for generating realtime photorealistic computer graphics. It can provide convincing animations without an explicit geometric representation. We use the “plenoptic function” of Adelson and Bergen to provide a concise problem statement for imagebased ..."
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Cited by 760 (20 self)
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Imagebased rendering is a powerful new approach for generating realtime photorealistic computer graphics. It can provide convincing animations without an explicit geometric representation. We use the “plenoptic function” of Adelson and Bergen to provide a concise problem statement for image
Symbolic Model Checking: 10^20 States and Beyond
, 1992
"... Many different methods have been devised for automatically verifying finite state systems by examining stategraph models of system behavior. These methods all depend on decision procedures that explicitly represent the state space using a list or a table that grows in proportion to the number of st ..."
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Cited by 758 (41 self)
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Many different methods have been devised for automatically verifying finite state systems by examining stategraph models of system behavior. These methods all depend on decision procedures that explicitly represent the state space using a list or a table that grows in proportion to the number
Laplacian eigenmaps and spectral techniques for embedding and clustering.
 Proceeding of Neural Information Processing Systems,
, 2001
"... Abstract Drawing on the correspondence between the graph Laplacian, the LaplaceBeltrami op erator on a manifold , and the connections to the heat equation , we propose a geometrically motivated algorithm for constructing a representation for data sampled from a low dimensional manifold embedded in ..."
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Cited by 668 (7 self)
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retrieval and data mining, one is often confronted with intrinsically low dimensional data lying in a very high dimensional space. For example, gray scale n x n images of a fixed object taken with a moving camera yield data points in rn: n2 . However , the intrinsic dimensionality of the space of all images
Results 1  10
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11,993