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Polyhedral Divisors and Algebraic Torus Actions
 Math. Ann
, 2006
"... Abstract. We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our approach extends classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, ..."
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Abstract. We provide a complete description of normal affine varieties with effective algebraic torus action in terms of what we call proper polyhedral divisors on semiprojective varieties. Our approach extends classical cone constructions of Dolgachev, Demazure and Pinkham to the multigraded case, and it comprises the theory of affine toric varieties.
An Invitation to Toric Topology: Vertex Four of a Remarkable Tetrahedron
, 2008
"... Motivation. Sometime around the turn of the recent millennium, those of us in Manchester and Moscow who had been collaborating since the mid1990s began using the term toric topology to describe our widening interests in certain wellbehaved actions of the torus. Little did we realise that, within s ..."
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Motivation. Sometime around the turn of the recent millennium, those of us in Manchester and Moscow who had been collaborating since the mid1990s began using the term toric topology to describe our widening interests in certain wellbehaved actions of the torus. Little did we realise that, within seven years, a significant international conference would be planned with the subject as its theme, and delightful Japanese hospitality at its heart. When first asked to prepare this article, we fantasised about an authoritative and comprehensive survey; one that would lead readers carefully through the foothills above which the subject rises, and provide techniques for gaining sufficient height to glimpse its extensive mathematical vistas. All this, and more, would be illuminated by references to the wonderful Osaka lectures! Soon afterwards, however, reality took hold, and we began to appreciate that such a task could not be completed to our satisfaction within the timescale available. Simultaneously, we understood that at least as valuable a service could be rendered to conference participants by an invitation to a wider mathematical audience an invitation to savour the atmosphere and texture of the subject, to
Construction of planar triangulations with minimum degree 5
, 1969
"... In this article we describe a method of constructing all simple triangulations of the sphere with minimum degree 5; equivalently, 3connected planar cubic graphs with girth 5. We also present the results of a computer program based on this algorithm, including counts of convex polytopes of minimum d ..."
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In this article we describe a method of constructing all simple triangulations of the sphere with minimum degree 5; equivalently, 3connected planar cubic graphs with girth 5. We also present the results of a computer program based on this algorithm, including counts of convex polytopes of minimum degree 5. Key words: planar triangulation, cubic graph, generation, fullerene
Complete enumeration of small realizable oriented matroids
"... Point configurations and convex polytopes play central roles in computational geometry and discrete geometry. For many problems, their combinatorial structures, i.e., the underlying oriented matroids up to isomorphism, ..."
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Point configurations and convex polytopes play central roles in computational geometry and discrete geometry. For many problems, their combinatorial structures, i.e., the underlying oriented matroids up to isomorphism,
Finding Nash Equilibria of Bimatrix Games
"... This thesis concerns the computational problem of finding one Nash equilibrium of a bimatrix game, a twoplayer game in strategic form. Bimatrix games are among the most basic models in noncooperative game theory, and finding a Nash equilibrium is important for their analysis. The Lemke–Howson algo ..."
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This thesis concerns the computational problem of finding one Nash equilibrium of a bimatrix game, a twoplayer game in strategic form. Bimatrix games are among the most basic models in noncooperative game theory, and finding a Nash equilibrium is important for their analysis. The Lemke–Howson algorithm is the classical method for finding one Nash equilibrium of a bimatrix game. In this thesis, we present a class of square bimatrix games for which this algorithm takes, even in the best case, an exponential number of steps in the dimension d of the game. Using polytope theory, the games are constructed using pairs of dual cyclic polytopes with 2d suitably labelled facets in dspace. The construction is extended to two classes of nonsquare games where, in addition to exponentially long Lemke–Howson computations, finding an equilibrium by support enumeration takes exponential time on average. The Lemke–Howson algorithm, which is a complementary pivoting algorithm, finds at least one solution to the linear complementarity problem (LCP) derived from a bimatrix
Cooperative provision of indivisible public goods
, 2012
"... A community faces the obligation of providing an indivisible public good that each of its members is able to provide at a certain cost. The solution is to rely on the member who can provide the public good at the lowest cost, with a due compensation from the other members. This problem has been stud ..."
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A community faces the obligation of providing an indivisible public good that each of its members is able to provide at a certain cost. The solution is to rely on the member who can provide the public good at the lowest cost, with a due compensation from the other members. This problem has been studied in a noncooperative setting by Kleindorfer and Sertel (1994). They propose an auction mechanism that results in an interval of possible individual contributions whose lower bound is the equal division. Here, instead we take a cooperative stand point by modelling this problem as a cost sharing game that turns out to be a "reverse" airport game whose core is shown to have a regular structure. This enables an easy calculation of the nucleolus that happens to define the upper bound of the KleindorferSertel interval. The Shapley value instead is not an appropriate solution in this context because it may imply compensations to nonproviders.
Affine and toric arrangements
"... Abstract. We extend the Billera–Ehrenborg–Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For toric arrangements, we also generalize Zaslavsky’s fundamental results on the number of regions. Résumé. Nous ..."
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Abstract. We extend the Billera–Ehrenborg–Readdy map between the intersection lattice and face lattice of a central hyperplane arrangement to affine and toric hyperplane arrangements. For toric arrangements, we also generalize Zaslavsky’s fundamental results on the number of regions. Résumé. Nous étendons l’opérateur de Billera–Ehrenborg–Readdy entre la trellis d’intersection et la trellis de faces d’un arrangement hyperplans centrals aux arrangements affines et toriques. Pour les arrangements toriques, nous généralisons aussi les résultats fondamentaux de Zaslavsky sur le nombre de régions.
Defuzzification using Steiner points
, 2006
"... A defuzzification function assigns to each fuzzy set a crisp value in a way that this value may intuitively be understood as the “centre” of the fuzzy set. In the present paper, this vague concept is put into a mathematically rigorous form. To this end, we proceed analogously to the case of sharply ..."
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A defuzzification function assigns to each fuzzy set a crisp value in a way that this value may intuitively be understood as the “centre” of the fuzzy set. In the present paper, this vague concept is put into a mathematically rigorous form. To this end, we proceed analogously to the case of sharply bordered subsets, for which the Steiner point is frequently used. The function assigning to each convex subset its Steiner point is characterised by three properties; here, we study functions whose domains consist of fuzzy sets and which fulfil analogous properties. Although uniqueness can no longer be achieved, we give a complete characterisation of what we call Steiner points of fuzzy sets. 1