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Packing Lines, Planes, etc.: Packings in Grassmannian Spaces
, 1996
"... We address the question: How should N ndimensional subspaces of mdimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of N; n; m are described, as well as a reformulation of the problem that was suggested by th ..."
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Cited by 86 (10 self)
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We address the question: How should N ndimensional subspaces of mdimensional Euclidean space be arranged so that they are as far apart as possible? The results of extensive computations for modest values of N; n; m are described, as well as a reformulation of the problem that was suggested by these computations. The reformulation gives a way to describe n dimensional subspaces of mspace as points on a sphere in dimension (m \Gamma 1)(m+2), which provides a (usually) lowerdimensional representation than the Pl ucker embedding, and leads to a proof that many of the new packings are optimal. The results have applications to the graphical display of multidimensional data via Asimov's grand tour method.
A Concept Exploration Method for Product Family Design
 in Mechanical Engineering. Atlanta, GA: Georgia Institute of Technology
, 1998
"... ii ..."
Adaptive Tuning of Numerical Weather Prediction Models: Simultaneous Estimation of Weighting, Smoothing and Physical Parameters
, 1996
"... In Wahba et al (1995) it was shown how the randomized trace method could be used to adaptively tune numerical weather prediction models via generalized cross validation (GCV ) and related methods. In this paper a `toy' four dimensional data assimilation model is developed (actually one space and one ..."
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Cited by 23 (7 self)
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In Wahba et al (1995) it was shown how the randomized trace method could be used to adaptively tune numerical weather prediction models via generalized cross validation (GCV ) and related methods. In this paper a `toy' four dimensional data assimilation model is developed (actually one space and one time variable), consisting of an equivalent barotropic vorticity equation on a latitude circle, and used to demonstrate how this technique may be used to simultaneously tune weighting, smoothing and physical parameters. Analyses both with the model as a strong constraint (corresponding to the usual 4DVar approach) and as a weak constraint (corresponding theoretically to a fixedinterval Kalman smoother) are carried out. The conclusions are limited to the particular toy problem considered but it can be seen how more elaborate experiments could be carried out, as well as how the method might be applied in practice. We have considered five adjustable parameters, two related to a distributed c...
McLaren’s improved snub cube and other new spherical designs in three dimensions
 Discrete and Computational Geometry
, 1996
"... Evidence is presented to suggest that, in three dimensions, spherical 6designs with N ..."
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Cited by 15 (1 self)
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Evidence is presented to suggest that, in three dimensions, spherical 6designs with N
Packing planes in four dimensions and other mysteries
 in Proceedings of the Conference on Algebraic Combinatorics and Related Topics
, 1997
"... How should you choose a good set of (say) 48 planes in four dimensions? More generally, how do you find packings in Grassmannian spaces? In this article I give a brief introduction to the work that I have been doing on this problem in collaboration with A. R. Calderbank, J. H. Conway, R. H. Hardin, ..."
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Cited by 9 (0 self)
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How should you choose a good set of (say) 48 planes in four dimensions? More generally, how do you find packings in Grassmannian spaces? In this article I give a brief introduction to the work that I have been doing on this problem in collaboration with A. R. Calderbank, J. H. Conway, R. H. Hardin, E. M. Rains and P. W. Shor. We have found many nice examples of specific packings (70 4spaces in 8space, for instance), several general constructions, and an embedding theorem which shows that a packing in Grassmannian space G(m,n) is a subset of a sphere in R D, D = (m + 2)(m − 1)/2, and leads to a proof that many of our packings are optimal. There are a number of interesting unsolved problems. 1.
Operating manual for Gosset: A general purpose program for constructing experimental designs
 AT&T Bell Laboratories
, 1994
"... This is the second edition of the operating manual for gosset, a flexible and powerful computer program for constructing experimental designs. Variables may be discrete or continuous (or both), discrete variables may be numeric or symbolic (or both), and continuous variables may range over a cube or ..."
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Cited by 3 (0 self)
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This is the second edition of the operating manual for gosset, a flexible and powerful computer program for constructing experimental designs. Variables may be discrete or continuous (or both), discrete variables may be numeric or symbolic (or both), and continuous variables may range over a cube or a ball (or both). The variables may be required to satisfy linear equalities or inequalities, and the model to be fitted may be any low degree polynomial (e.g. a quadratic). The number of observations is specified by the user. The design may be required to include a specified set of points (so a sequence of designs can be found, each of which is optimal given that the earlier measurements have been made). The region where the model is to be fitted need not be the same as the region where measurements are to be made (so the designs can be used for interpolation or extrapolation). The following types of designs can be requested: I, A, D or Eoptimal; the same but with protection against the loss of one run; or packings (when no model is available). Block designs, and designs with correlated errors can also be constructed. The algorithm is powerful enough to routinely minimize functions of 1000 variables (e.g. can find optimal or nearly optimal designs for a quadratic model involving 12 variables). An extensive library of precomputed optimal designs is included for linear and quadratic designs in the cube, ball and simplex. The user does not have to specify starting points for the search. The user also has control over how much effort is expended by the algorithm, and can monitor the progress of the search. Applications so far include VLSI production, conductivity of diamond films, growth of protein crystals, flow through a catalytic converter, laser welding, etc.
New Spherical 4Designs
 Discrete Math
, 1992
"... This paper gives a number of new spherical 4designs, and presents numerical evidence that spherical 4designs containing n points in kdimensional space with k G 8 exist precisely for the following values of n and k: n even and 22 for k = 1; n 2 5 for k = 2; n = 12, 14,>I6 for k=3;n~2Ofork=4;n>29fo ..."
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Cited by 3 (1 self)
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This paper gives a number of new spherical 4designs, and presents numerical evidence that spherical 4designs containing n points in kdimensional space with k G 8 exist precisely for the following values of n and k: n even and 22 for k = 1; n 2 5 for k = 2; n = 12, 14,>I6 for k=3;n~2Ofork=4;n>29fork=5;n=27,36, ~39fork=6;n~53fork=7;andn~69 for k = 8. 1.
How Much Internalization of Nuclear Risk Through Liability Insurance? Working Paper No 0211
, 2002
"... An important source of conflict surrounding nuclear energy is that with a very small probability, a largescale nuclear accident may occur. One way to internalize the associated financial risks is through mandating nuclear operators to have liability insurance. This paper presents estimates of consu ..."
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Cited by 3 (0 self)
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An important source of conflict surrounding nuclear energy is that with a very small probability, a largescale nuclear accident may occur. One way to internalize the associated financial risks is through mandating nuclear operators to have liability insurance. This paper presents estimates of consumers ’ willingness to pay for increased financial security provided by an extension of coverage, based on the ‘stated choice ’ approach. A Swiss citizen with median characteristics may be willing to pay 0.14 US cents per kwh to increase coverage beyond the current CHF 0.7 billion (bn.) (US $ 0.47 bn.). Marginal willingness to pay declines with higher coverage but exceeds marginal cost at least up to CHF 4 bn. (US $ 2.7 bn.). An extension of nuclear liability insurance coverage therefore may be efficiencyenhancing.
Heuristics in Monte Carlo Go
 In Proceedings of the 2007 International Conference on Artificial Intelligence, CSREA
, 2007
"... Writing programs to play the classical Asian game of Go is considered one of the grand challenges of artificial intelligence. Traditional game tree search methods have failed to conquer Go because the search space is so vast and because static evaluation of board positions is extremely difficult. Th ..."
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Cited by 2 (1 self)
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Writing programs to play the classical Asian game of Go is considered one of the grand challenges of artificial intelligence. Traditional game tree search methods have failed to conquer Go because the search space is so vast and because static evaluation of board positions is extremely difficult. There has been considerable progress recently in using Monte Carlo sampling to select moves. This paper presents four heuristics used to bias the selection of moves during Monte Carlo sampling: the proximity heuristic (play near the last move), the avoidthefirsttwolines heuristic (don't play at the edge of the board), the firstorder history heuristic (play the move that has fared best elsewhere in the tree), and the secondorder history heuristic (play the move that has fared best elsewhere in the tree in response to a particular move from the opponent). Experimental results show that the use of these heuristics significantly improves the ability of our program to defeat GNU Go, a widelyused Go program based on traditional, knowledgeintensive techniques.
Combinatorial Designs to Explore Large Experimental Search Spaces
 Systems Biology
, 2003
"... Genomic and proteomic studies take advantage of omic level techniques such as microarrays to achieve specieswide scale. Obtaining an operational model (e.g., a virtual animal) however requires a search in a space consisting of many factors and many values of each factor, all of which may interac ..."
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Cited by 2 (1 self)
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Genomic and proteomic studies take advantage of omic level techniques such as microarrays to achieve specieswide scale. Obtaining an operational model (e.g., a virtual animal) however requires a search in a space consisting of many factors and many values of each factor, all of which may interact.