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Tilings associated with non-Pisot matrices

Maki Furukado, Shunji Ito, E. Arthur Robinson (2006)

Annales de l’institut Fourier

Suppose A G l d ( ) has a 2-dimensional expanding subspace E u , satisfies a regularity condition, called “good star”, and has A * 0 , where A * is an oriented compound of A . A morphism θ of the free group on { 1 , 2 , , d } is called a non-abelianization of A if it has structure matrix A . We show that there is a tiling substitution Θ whose “boundary substitution” θ = Θ is a non-abelianization of A . Such a tiling substitution Θ leads to a self-affine tiling of E u 2 with A u : = A | E u G L 2 ( ) as its expansion. In the last section we find conditions on A so...

Tilings of convex polygons

Richard Kenyon (1997)

Annales de l'institut Fourier

Call a polygon rational if every pair of side lengths has rational ratio. We show that a convex polygon can be tiled with rational polygons if and only if it is itself rational. Furthermore we give a necessary condition for an arbitrary polygon to be tileable with rational polygons: we associate to any polygon P a quadratic form q ( P ) , which must be positive semidefinite if P is tileable with rational polygons.The above results also hold replacing the rationality condition with the following: a polygon...

Topologies faciales dans les convexes compacts. Calcul fonctionnel et décomposition spectrale dans le centre d’un espace A ( X )

Marc Rogalski (1972)

Annales de l'institut Fourier

Cet article étudie, sur l’ensemble 𝒮 ( X ) des points extrémaux d’un convexe compact X , des topologies faciales dont les fermés sont les traces de faces F “parallélisables” (il existe une plus grande face F ' disjointe de F , et tout x de X s’écrit x = λ y + ( 1 - λ ) y ' , y F , y ' F ' , avec λ unique). Les topologies faciales uniformisables sont en bijection avec les sous-espaces réticulés fermés et contenant 1 de l’espace A ( X ) des fonctions affines continues sur X . Ceci redonne des résultats classiques sur les simplexes, et permet une étude...

Topology and geometry of nontrivial rank-one convex hulls for two-by-two matrices

Carl-Friedrich Kreiner, Johannes Zimmer (2006)

ESAIM: Control, Optimisation and Calculus of Variations

Continuing earlier work by Székelyhidi, we describe the topological and geometric structure of so-called T4-configurations which are the most prominent examples of nontrivial rank-one convex hulls. It turns out that the structure of T4-configurations in 2 × 2 is very rich; in particular, their collection is open as a subset of ( 2 × 2 ) 4 . Moreover a previously purely algebraic criterion is given a geometric interpretation. As a consequence, we sketch an improved algorithm to detect T4-configurations. ...

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