Good data locality is an important aspect of obtaining scalable performance for multigrid methods. However, locality can be difficult to achieve, especially when working with unstructured grids and sparse matrices whose structure is not known until runtime. Our previous work developed full sparse tiling, a runtime reordering and rescheduling technique for improving locality. We applied full spa...

This work focuses on the complexity of computing exact loop bounds in Multilevel Tiling. Conventional tiling techniques implement tiling using first strip-mining and afterwards loop interchange. Multilevel tiling has typically been implemented applying tiling level by level. We present a new way to implement Multilevel Tiling that deals with all levels simultaneously, performing a loop intercha...

This paper examines a recomposition of the rhombic Penrose aperiodic protoset due to Robert Ammann. We show that the three prototiles that result from the recomposition form an aperiodic protoset in their own right without adjacency rules and that every tiling admitted by this protoset (here called an Ammann tiling) is mutually locally derivable with a Penrose tiling. Although these Ammann tili...

In this account recent progress in enhancing the complexity of liquid crystal self-assembly is highlighted. The discussed superstructures are formed mainly by polyphilic T-shaped and X-shaped molecules composed of a rod-like core, tethered with glycerol units at both ends and flexible non-polar chain(s) in lateral position, but also related inverted molecular structures are considered. A series...

We introduce thèsquare tiling group' and use it to give necessary conditions on a planar polygon to be tileable with squares. We deene square tilings on Riemann surfaces, and compute the Euclidean structures on a torus which are square-tileable. We give necessary conditions on surfaces of higher genus to be tileable. A higher-dimensional version of the square-tiling group yields necessary condi...

Abdus-Salam School of Mathematical Sciences, GCU, 35-C-2 Gulberg -III Lahore, Pakistan [email protected] [email protected] key words: rectangle, tiling, trapped block. Abstract. In this paper, we prove that if a finite number of rectangles, every of which has at least one integer side, perfectly tile a big rectangle then there exists a strategy which reduces the number of these tiles...

Let τ1 be the Penrose tiling on R 2 [G77] (using kites and darts) with edges of lengths 1 and φ, where φ is the golden ratio 1+ √ 5 2 . Let U be a square of the form [j1, j1 + l] × [j2, j2 + l] where l = 2, i, j1, j2 ∈ N. Notice that when we have a tiling of the plane (so that the diameter of its tiles is bounded) we can produce a separated net from it. We do it by placing one point in each til...