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A proof of the Kepler conjecture
 Math. Intelligencer
, 1994
"... This section describes the structure of the proof of ..."
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Cited by 112 (11 self)
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This section describes the structure of the proof of
What color is your Jacobian? Graph coloring for computing derivatives
 SIAM REV
, 2005
"... Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specific ..."
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Cited by 41 (7 self)
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Graph coloring has been employed since the 1980s to efficiently compute sparse Jacobian and Hessian matrices using either finite differences or automatic differentiation. Several coloring problems occur in this context, depending on whether the matrix is a Jacobian or a Hessian, and on the specifics of the computational techniques employed. We consider eight variant vertexcoloring problems here. This article begins with a gentle introduction to the problem of computing a sparse Jacobian, followed by an overview of the historical development of the research area. Then we present a unifying framework for the graph models of the variant matrixestimation problems. The framework is based upon the viewpoint that a partition of a matrixinto structurally orthogonal groups of columns corresponds to distance2 coloring an appropriate graph representation. The unified framework helps integrate earlier work and leads to fresh insights; enables the design of more efficient algorithms for many problems; leads to new algorithms for others; and eases the task of building graph models for new problems. We report computational results on two of the coloring problems to support our claims. Most of the methods for these problems treat a column or a row of a matrixas an atomic entity, and partition the columns or rows (or both). A brief review of methods that do not fit these criteria is provided. We also discuss results in discrete mathematics and theoretical computer science that intersect with the topics considered here.
Computers, Reasoning and Mathematical Practice
"... ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of ..."
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Cited by 6 (2 self)
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ion in itself is not the goal: for Whitehead [117]"it is the large generalisation, limited by a happy particularity, which is the fruitful conception." As an example consider the theorem in ring theory, which states that if R is a ring, f(x) is a polynomial over R and f(r) = 0 for every element of r of R then R is commutative. Special cases of this, for example f(x) is x 2 \Gamma x or x 3 \Gamma x, can be given a first order proof in a few lines of symbol manipulation. The usual proof of the general result [20] (which takes a semester's postgraduate course to develop from scratch) is a corollary of other results: we prove that rings satisfying the condition are semisimple artinian, apply a theorem which shows that all such rings are matrix rings over division rings, and eventually obtain the result by showing that all finite division rings are fields, and hence commutative. This displays von Neumann's architectural qualities: it is "deep" in a way in which the symbol manipulati...
Joe Culberson's Graph Colouring Bibliography
, 1978
"... dence numbers of graphs. Journal of Combinatorial Theory Series B, 40(1):113 117, 1986. [406] Colin P. Williams and Tad Hogg. Using deep structure to locate hard problems. In Proc. of AAAI92, pages 472477, Menlo Park, CA, 1992. AAAI Press. [407] Colin P. Williams and Tad Hogg. Extending deep str ..."
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dence numbers of graphs. Journal of Combinatorial Theory Series B, 40(1):113 117, 1986. [406] Colin P. Williams and Tad Hogg. Using deep structure to locate hard problems. In Proc. of AAAI92, pages 472477, Menlo Park, CA, 1992. AAAI Press. [407] Colin P. Williams and Tad Hogg. Extending deep structure. In Proc. of AAAI93, pages 152157, Menlo Park, CA, 1993. AAAI Press. [408] M. R. Williams. The colouring of very large graphs. In Richard Guy, Haim Hanani, Norbert Sauer, and Johanen Schonheim, editors, Combinatorial Structures and Their Applications: Proceedings of the Calgary International Conference on Combinatorial Structures and Their Applications, pages 477478. Gordon and Breach, New York, 1970. [409] Brian Wilson. Linedistinguishing and harmonious colourings. In Roy Nelson and Robin J. Wilson, editors, Graph Colourings, Pitman Research Notes in Mathematics Series, pages 115133. Longman Scient
A categorical setting for the 4Colour Theorem
"... The 4Colour Theorem has been proved in the late seventies [2, 3], after more than a century of fruitless efforts. But the proof has provided very little new information about the map colouring itself. While trying to understand this phenomenon, we analyze colouring in terms of universal properties ..."
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The 4Colour Theorem has been proved in the late seventies [2, 3], after more than a century of fruitless efforts. But the proof has provided very little new information about the map colouring itself. While trying to understand this phenomenon, we analyze colouring in terms of universal properties and adjoint functors. It is well known that the 4colouring of maps is equivalent to the 3colouring of the edges of some graphs. We show that every slice of the category of 3coloured graphs is a topos. The forgetful functor to the category of graphs is cotripleable; every loopfree graph is covered by a 3coloured one in a universal way. In this context, the 4Color Theorem becomes a statement about the existence of coalgebra structure on graphs. In a sense, this approach seems complementary to the known combinatorial colouring procedures. Current address: Department of Computing, Imperial College, London SW7 2BZ, England; email: D.Pavlovic@doc.ic.ac.uk 1 Introduction: the meaning of t...
A Guide to the Discharging Method
, 2013
"... We provide a “howto” guide to the use and application of the Discharging Method. Our aim is not to exhaustively survey results that have been proved by this technique, but rather to demystify the technique and facilitate its wider use. Along the way, we present some new proofs and new problems. ..."
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We provide a “howto” guide to the use and application of the Discharging Method. Our aim is not to exhaustively survey results that have been proved by this technique, but rather to demystify the technique and facilitate its wider use. Along the way, we present some new proofs and new problems.