Search results for "Polytope"
showing 10 items of 25 documents
A closer look at mirrors and quotients of Calabi-Yau threefolds
2016
Let X be the toric variety (P1)4 associated with its four-dimensional polytope 1. Denote by X˜ the resolution of the singular Fano variety Xo associated with the dual polytope 1o. Generically, anticanonical sections Y of X and anticanonical sections Y˜ of X˜ are mirror partners in the sense of Batyrev. Our main result is the following: the Hodge-theoretic mirror of the quotient Z associated to a maximal admissible pair (Y, G) in X is not a quotient Z˜ associated to an admissible pair in X˜ . Nevertheless, it is possible to construct a mirror orbifold for Z by means of a quotient of a suitable Y˜. Its crepant resolution is a Calabi-Yau threefold with Hodge numbers (8, 4). Instead, if we star…
Robust control of uncertain multi-inventory systems via linear matrix inequality
2008
We consider a continuous time linear multi inventory system with unknown demands bounded within ellipsoids and controls bounded within ellipsoids or polytopes. We address the problem of "-stabilizing the inventory since this implies some reduction of the inventory costs. The main results are certain conditions under which "-stabilizability is possible through a saturated linear state feedback control. All the results are based on a Linear Matrix Inequalities (LMIs) approach and on some recent techniques for the modeling and analysis of polytopic systems with saturations.
The polyhedral Hodge number $h^{2,1}$ and vanishing of obstructions
2000
We prove a vanishing theorem for the Hodge number $h^{2,1}$ of projective toric varieties provided by a certain class of polytopes. We explain how this Hodge number also gives information about the deformation theory of the toric Gorenstein singularity derived from the same polytope. In particular, the vanishing theorem for $h^{2,1}$ implies that these deformations are unobstructed.
The Bernstein Basis and its applications in solving geometric constraint systems
2012
International audience; This article reviews the properties of Tensorial Bernstein Basis (TBB) and its usage, with interval analysis, for solving systems of nonlinear, univariate or multivariate equations resulting from geometric constraints. TBB are routinely used in computerized geometry for geometric modelling in CAD-CAM, or in computer graphics. They provide sharp enclosures of polynomials and their derivatives. They are used to reduce domains while preserving roots of polynomial systems, to prove that domains do not contain roots, and to make existence and uniqueness tests. They are compatible with standard preconditioning methods and fit linear program- ming techniques. However, curre…
Skeleta of affine hypersurfaces
2014
A smooth affine hypersurface Z of complex dimension n is homotopy equivalent to an n-dimensional cell complex. Given a defining polynomial f for Z as well as a regular triangulation of its Newton polytope, we provide a purely combinatorial construction of a compact topological space S as a union of components of real dimension n, and prove that S embeds into Z as a deformation retract. In particular, Z is homotopy equivalent to S.
Approximation of the Feasible Parameter Set in worst-case identification of Hammerstein models
2005
The estimation of the Feasible Parameter Set (FPS) for Hammerstein models in a worst-case setting is considered. A bounding procedure is determined both for polytopic and ellipsoidic uncertainties. It consists in the projection of the FPS of the extended parameter vector onto suitable subspaces and in the solution of convex optimization problems which provide Uncertainties Intervals of the model parameters. The bounds obtained are tighter than in the previous approaches. hes.
An upper bound for nonlinear eigenvalues on convex domains by means of the isoperimetric deficit
2010
We prove an upper bound for the first Dirichlet eigenvalue of the p-Laplacian operator on convex domains. The result implies a sharp inequality where, for any convex set, the Faber-Krahn deficit is dominated by the isoperimetric deficit.
Pure Functions in C: A Small Keyword for Automatic Parallelization
2017
AbstractThe need for parallel task execution has been steadily growing in recent years since manufacturers mainly improve processor performance by increasing the number of installed cores instead of scaling the processor’s frequency. To make use of this potential, an essential technique to increase the parallelism of a program is to parallelize loops. Several automatic loop nest parallelizers have been developed in the past such as PluTo. The main restriction of these tools is that the loops must be statically analyzable which, among other things, disallows function calls within the loops. In this article, we present a seemingly simple extension to the C programming language which marks fun…
Polyhedral results for a vehicle routing problem
1991
Abstract The Vehicle Routing Problem is a well known, and hard, combinatorial problem, whose polyhedral structure has deserved little attention. In this paper we consider the particular case in which all the demands are equal (since in the general case the associated polytope may be empty). From a known formulation of the problem we obtain the dimension of the corresponding polytope and we study the facetial properties of every inequality in it.
OPTIMIZATIONS FOR TENSORIAL BERNSTEIN–BASED SOLVERS BY USING POLYHEDRAL BOUNDS
2010
The tensorial Bernstein basis for multivariate polynomials in n variables has a number 3n of functions for degree 2. Consequently, computing the representation of a multivariate polynomial in the tensorial Bernstein basis is an exponential time algorithm, which makes tensorial Bernstein-based solvers impractical for systems with more than n = 6 or 7 variables. This article describes a polytope (Bernstein polytope) with a number of faces, which allows to bound a sparse, multivariate polynomial expressed in the canonical basis by solving several linear programming problems. We compare the performance of a subdivision solver using domain reductions by linear programming with a solver using a c…