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Two Finiteness Theorems for Periodic Tilings of dDimensional Euclidean Space
, 1996
"... Consider the ddimensional euclidean space E d . Two main results are presented: First, for any N 2 N, the number of types of periodic equivariant tilings (T ; \Gamma) that have precisely N orbits of (2; 4; 6 : : :)flags with respect to the symmetry group \Gamma, is finite. Second, for any N 2 N, ..."
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Cited by 2 (2 self)
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Consider the ddimensional euclidean space E d . Two main results are presented: First, for any N 2 N, the number of types of periodic equivariant tilings (T ; \Gamma) that have precisely N orbits of (2; 4; 6 : : :)flags with respect to the symmetry group \Gamma, is finite. Second, for any N 2 N
Documenta Math. 275 Large Parallel Volumes of Finite and Compact Sets in dDimensional Euclidean Space
, 2012
"... Abstract. The rparallel volume V(Cr) of a compact subset C in ddimensional Euclidean space is the volume of the set Cr of all points of Euclidean distance at most r> 0 from C. According to Steiner’s formula, V(Cr) is a polynomial in r when C is convex. For finite sets C satisfying a certain geo ..."
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Abstract. The rparallel volume V(Cr) of a compact subset C in ddimensional Euclidean space is the volume of the set Cr of all points of Euclidean distance at most r> 0 from C. According to Steiner’s formula, V(Cr) is a polynomial in r when C is convex. For finite sets C satisfying a certain
3 Spectral Properties of the Dirichlet Operator∑d i=1(−∂2i)s on Domains in dDimensional Euclidean Space
"... ar ..."
Some problems on random walk in space
 Pvoc. Second Berkeley Symposium 0% Mathematical Statistics and Probability
, 1951
"... Consider the lattice formed by all points whose coordinates are integers in ddimensional Euclidean space, and let a point S j(n) perform a move randomly ..."
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Cited by 69 (1 self)
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Consider the lattice formed by all points whose coordinates are integers in ddimensional Euclidean space, and let a point S j(n) perform a move randomly
Approximate Nearest Neighbors: Towards Removing the Curse of Dimensionality
, 1998
"... The nearest neighbor problem is the following: Given a set of n points P = fp 1 ; : : : ; png in some metric space X, preprocess P so as to efficiently answer queries which require finding the point in P closest to a query point q 2 X. We focus on the particularly interesting case of the ddimens ..."
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Cited by 1017 (40 self)
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The nearest neighbor problem is the following: Given a set of n points P = fp 1 ; : : : ; png in some metric space X, preprocess P so as to efficiently answer queries which require finding the point in P closest to a query point q 2 X. We focus on the particularly interesting case of the ddimensional
Databasefriendly Random Projections
, 2001
"... A classic result of Johnson and Lindenstrauss asserts that any set of n points in ddimensional Euclidean space can be embedded into kdimensional Euclidean space  where k is logarithmic in n and independent of d  so that all pairwise distances are maintained within an arbitrarily small factor. Al ..."
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Cited by 240 (3 self)
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A classic result of Johnson and Lindenstrauss asserts that any set of n points in ddimensional Euclidean space can be embedded into kdimensional Euclidean space  where k is logarithmic in n and independent of d  so that all pairwise distances are maintained within an arbitrarily small factor
Unit distances and diameters in euclidean spaces, Discrete and
, 2008
"... Abstract. We show that the maximum number of unit distances or of diameters in a set of n points in ddimensional Euclidean space is attained only by specific types of Lenz constructions, for all d≥4 and n sufficiently large, depending on d. As a corollary we determine the exact maximum number of un ..."
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Cited by 6 (3 self)
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Abstract. We show that the maximum number of unit distances or of diameters in a set of n points in ddimensional Euclidean space is attained only by specific types of Lenz constructions, for all d≥4 and n sufficiently large, depending on d. As a corollary we determine the exact maximum number
Fast kNN Classification Rule Using Metrics on SpaceFilling Curves
"... A fast nearest neighbor algorithm for pattern classification is proposed and tested on real data. The patterns (points in ddimensional Euclidean space) are sorted along a spacefilling curve. This way the multidimensional problem is compressed to the simplest case of the nearest neighbor search in ..."
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A fast nearest neighbor algorithm for pattern classification is proposed and tested on real data. The patterns (points in ddimensional Euclidean space) are sorted along a spacefilling curve. This way the multidimensional problem is compressed to the simplest case of the nearest neighbor search
Convex bodies, economic cap coverings, random polytopes
 Mathematika
, 1988
"... in the ddimensional Euclidean space Rd and let x, ,..., xn be random points in K, independently and uniformly distributed. Define Kn = conv {xx,..., xn}. Our main concern in this paper will be the behaviour of the deviation of vol Kn from vol K as a function of n, more precisely, the expectation of ..."
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Cited by 49 (15 self)
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in the ddimensional Euclidean space Rd and let x, ,..., xn be random points in K, independently and uniformly distributed. Define Kn = conv {xx,..., xn}. Our main concern in this paper will be the behaviour of the deviation of vol Kn from vol K as a function of n, more precisely, the expectation
Results 1  10
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1,255,462