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Neighborliness of RandomlyProjected Simplices in High Dimensions
 Proc. Natl. Acad. Sci. USA
, 2005
"... Let A be a d by n matrix, d < n. Let T = T n−1 be the standard regular simplex in R n. We count the faces of the projected simplex AT in the case where the projection is random, the dimension d is large and n and d are comparable: d ∼ δn, δ ∈ (0, 1). The projector A is chosen uniformly at random ..."
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Cited by 92 (18 self)
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Let A be a d by n matrix, d < n. Let T = T n−1 be the standard regular simplex in R n. We count the faces of the projected simplex AT in the case where the projection is random, the dimension d is large and n and d are comparable: d ∼ δn, δ ∈ (0, 1). The projector A is chosen uniformly at random
Dimension expanders
 Journal of Algebra
"... Abstract. We show that there exists k ∈ N and 0 < ɛ ∈ R such that for every field F of characteristic zero and for every n ∈ N, there exists explicitly given linear transformations T1,...,Tk: F n → F n satisfying the following: For every subspace W of F n of dimension less or equal n 2, dim(W + k ..."
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Cited by 29 (1 self)
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Abstract. We show that there exists k ∈ N and 0 < ɛ ∈ R such that for every field F of characteristic zero and for every n ∈ N, there exists explicitly given linear transformations T1,...,Tk: F n → F n satisfying the following: For every subspace W of F n of dimension less or equal n 2, dim
Valuation of Mortgage Backed Securities Using Brownian Bridges to Reduce Effective Dimension
, 1997
"... The quasiMonte Carlo method for financial valuation and other integration problems has error bounds of size O((log N) k N \Gamma1 ), or even O((log N) k N \Gamma3=2 ), which suggests significantly better performance than the error size O(N \Gamma1=2 ) for standard Monte Carlo. But in hig ..."
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Cited by 100 (15 self)
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The quasiMonte Carlo method for financial valuation and other integration problems has error bounds of size O((log N) k N \Gamma1 ), or even O((log N) k N \Gamma3=2 ), which suggests significantly better performance than the error size O(N \Gamma1=2 ) for standard Monte Carlo
On the essential dimension of a finite group
 COMPOSITIO MATH
, 1997
"... Let f(x) = aix i be a monic polynomial of degree n whose coefficients are algebraically independent variables over a base field k of characteristic 0. We say that a polynomial g(x) is generating (for the symmetric group) if it can be obtained from f(x) by a nondegenerate Tschirnhaus transformation. ..."
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Cited by 72 (21 self)
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generalizations) is based on the idea of the “essential dimension ” of a finite group G: the smallest possible dimension of an algebraic Gvariety over k to which one can “compress ” a faithful linear representation of G. We show that dk(n) is just the essential dimension of the symmetric group Sn. We give
Approximate clustering via coresets
 In Proc. 34th Annu. ACM Sympos. Theory Comput
, 2002
"... In this paper, we show that for several clustering problems one can extract a small set of points, so that using those coresets enable us to perform approximate clustering efficiently. The surprising property of those coresets is that their size is independent of the dimension. Using those, we pre ..."
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Cited by 141 (16 self)
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present a ¡ 1 ¢ ε £approximation algorithms for the kcenter clustering and kmedian clustering problems in Euclidean space. The running time of the new algorithms has linear or near linear dependency on the number of points and the dimension, and exponential dependency on 1 ¤ ε and k. As such, our
Optimal inequalities in probability theory: A convex optimization approach
 SIAM Journal of Optimization
"... Abstract. We propose a semidefinite optimization approach to the problem of deriving tight moment inequalities for P (X ∈ S), for a set S defined by polynomial inequalities and a random vector X defined on Ω ⊆Rn that has a given collection of up to kthorder moments. In the univariate case, we provi ..."
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Cited by 110 (11 self)
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provide optimal bounds on P (X ∈ S), when the first k moments of X are given, as the solution of a semidefinite optimization problem in k + 1 dimensions. In the multivariate case, if the sets S and Ω are given by polynomial inequalities, we obtain an improving sequence of bounds by solving semidefinite
The number of faces of simplicial convex polytopes
 Advances in Math. 35
, 1980
"... Let P be a simplicial convex dpolytope with fi = fi(P) faces of dimension i. The vector f(P) = (f., fi,..., fdel) is called the fvector of P. In 1971 McMullen [6; 7, p. 1791 conjectured that a certain condition on a vector f = (f., fi,..., fd...J of integers was necessary and sufficient for f to ..."
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Cited by 149 (2 self)
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(P) = (f., fi,..., f&, where we set fel = 1. The vector h(P) = (h, , h,,..., hd) is called the hvector of P 181. The DehnSommerville equations, which hold for any simplicial convex polytope, are equivalent to the statement that hi = hdpi, 0 < i,< d [7, Sect. 5.11. If k and i are positive
DivideandConquer in Multidimensional Space
, 1976
"... We investigate a divideandconquer technique in multidimensional space which decomposes a geometric problem on N points in k dimensions into two problems on N/2 points in k dimensions plus a single problem on N points in k1 dimension. Special structure of the subproblems is exploited to obtain an ..."
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Cited by 38 (0 self)
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We investigate a divideandconquer technique in multidimensional space which decomposes a geometric problem on N points in k dimensions into two problems on N/2 points in k dimensions plus a single problem on N points in k1 dimension. Special structure of the subproblems is exploited to obtain
Anomaly cancellation in sixdimensions
 J. Math. Phys
, 1994
"... I show that anomaly cancellation conditions are sufficient to determine the two most important topological numbers relevant for CalabiYau compactification to six dimensions. This reflects the fact that K3 is the only nontrivial CY manifold in two complex dimensions. I explicitly construct the Gree ..."
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Cited by 46 (0 self)
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I show that anomaly cancellation conditions are sufficient to determine the two most important topological numbers relevant for CalabiYau compactification to six dimensions. This reflects the fact that K3 is the only nontrivial CY manifold in two complex dimensions. I explicitly construct
On Dimension in the Cube
 Discr. Math. (submitted
"... For integers 0 k < r n, let K and R denote the families of all subsets of [n] = f1; 2; : : : ; ng of size k and r, respectively. Denote by P (k; r; n) the containment order on K [ R (A < B whenever A B) and let d(k; r; n) be its order dimension. We characterize those linear extensions of ..."
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Cited by 2 (1 self)
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For integers 0 k < r n, let K and R denote the families of all subsets of [n] = f1; 2; : : : ; ng of size k and r, respectively. Denote by P (k; r; n) the containment order on K [ R (A < B whenever A B) and let d(k; r; n) be its order dimension. We characterize those linear extensions
Results 11  20
of
5,857