General Galois geometries
01 Jan 2016-
TL;DR: In this paper, the authors define Hermitian varieties, Grassmann varieties, Veronese and Segre varieties, and embedded geometries for finite projective spaces of three dimensions.
Abstract: Terminology Quadrics Hermitian varieties Grassmann varieties Veronese and Segre varieties Embedded geometries Arcs and caps Appendix VI. Ovoids and spreads of finite classical polar spaces Appendix VII. Errata for Finite projective spaces of three dimensions and Projective geometries over finite fields Bibliography Index of notation Author index General index.
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TL;DR: In this article, the functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation, which is equivalent to the completely discretized classical 2D Toda lattice with open boundaries.
Abstract: The standard objects of quantum integrable systems are identified with elements of classical nonlinear integrable difference equations. The functional relation for commuting quantum transfer matrices of quantum integrable models is shown to coincide with classical Hirota's bilinear difference equation. This equation is equivalent to the completely discretized classical 2D Toda lattice with open boundaries. Elliptic solutions of Hirota's equation give a complete set of eigenvalues of the quantum transfer matrices. Eigenvalues of Baxter's Q-operator are solutions to the auxiliary linear problems for classical Hirota's equation. The elliptic solutions relevant to the Bethe ansatz are studied. The nested Bethe ansatz equations for A
k-1
-type models appear as discrete time equations of motions for zeros of classical τ-functions and Baker-Akhiezer functions. Determinant representations of the general solution
to bilinear discrete Hirota's equation are analysed and a new determinant formula for eigenvalues of the quantum transfer matrices is obtained. Difference equations for eigenvalues of the Q-operators which generalize Baxter's three-term T−Q-relation are derived.
323 citations
01 Jan 2001
TL;DR: In this paper, the authors updated the 1998 survey on the packing problem, up to 1995, and showed that considerable progress has been made on different kinds of subconfigurations over the last few decades.
Abstract: This article updates the authors’ 1998 survey [134] on the same theme that was written for the Bose Memorial Conference (Colorado, June 7–11, 1995). That article contained the principal results on the packing problem, up to 1995. Since then, considerable progress has been made on different kinds of subconfigurations.
236 citations
TL;DR: In this paper, the authors considered the problem of finding the largest size of a point set in a set of points in a projective space and showed that it is equivalent to the packing problem in geometry.
143 citations
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31 Dec 1996
TL;DR: A chronology of key events and figures from the year in the history of the United States, as well as some of the individuals and institutions that were involved in the manufacture and distribution of goods and services, are recalled.
Abstract: Pooling (or “group testing”) designs for screening clone libraries for rare “positives” are described and compared. We focus on non-adaptive designs in which, in order both to facilitate automation and to minimize the total number of pools required in multiple screenings, all the pools are specified in advance of the experiments. The designs considered include deterministic designs, such as set-packing designs, the widely-used “row and column” designs and the more general “transversal” designs, as well as random designs such as “random incidence” and “random k- set” designs. A range of possible performance measures is considered, including the expected numbers of unresolved positive and negative clones, and the probability of a one-pass solution. We describe a flexible strategy in which the experimenter chooses a compromise between the random k-set and the set-packing designs. In general, the latter have superior performance while the former are nearly as efficient and are easier to construct.
114 citations
01 Jan 1995
TL;DR: In this article, the authors focus on projective geometry over a finite field and define a k-arc in projective plane, PG (n, q) is a set K of k points with k ≥ n + 1 such that no n+ 1 points of K lie in a hyperplane.
Abstract: Publisher Summary
This chapter focuses on projective geometry over a finite field. A k-arc in projective plane, PG (n, q) is a set K of k points with k ≥ n + 1 such that no n + 1 points of K lie in a hyperplane. An arc K is complete if it is not properly contained in a larger arc. A normal rational curve of PG(2, q) is an irreducible conic; a normal rational curve of PG(3, q) is a twisted cubic. It is well known that any (n + 3)-arc of PG(n, q) is contained in a unique normal rational curve of this space. For q > n + 1, the osculating hyperplane of the normal rational curve C at the point x ϵ C is the unique hyperplane through x intersecting C at x with multiplicity n. In PG(n, q), n ≥ 3, a set K of k points no three of which are collinear is a k-cap. A k-cap is complete if it is not contained in a (k + 1)-cap. A line of PG(n, q) is a secant, a tangent, or an external line of a k-cap as it meets K in 2, 1, or 0 points.
111 citations