Standard Tiling Research Articles

Algorithms for Efficient Reproducible Floating Point Summation

We define “reproducibility” as getting bitwise identical results from multiple runs of the same program, perhaps with different hardware resources or other changes that should not affect the answer. Many users depend on reproducibility for debugging or correctness. However, dynamic scheduling of parallel computing resources, combined with nonassociative floating point addition, makes reproducibility challenging even for summation, or operations like the BLAS. We describe a “reproducible accumulator” data structure (the “binned number”) and associated algorithms to reproducibly sum binary floating point numbers, independent of summation order. We use a subset of the IEEE Floating Point Standard 754-2008 and bitwise operations on the standard representations in memory. Our approach requires only one read-only pass over the data, and one reduction in parallel, using a 6-word reproducible accumulator (more words can be used for higher accuracy), enabling standard tiling optimization techniques. Summing n words with a 6-word reproducible accumulator requires approximately 9 n floating point operations (arithmetic, comparison, and absolute value) and approximately 3 n bitwise operations. The final error bound with a 6-word reproducible accumulator and our default settings can be up to 2 29 times smaller than the error bound for conventional (recursive) summation on ill-conditioned double-precision inputs.

Notes on tiled incompressible tori

Abstract Let Θ denote the class of essential tori in a closed braid complement which admit a standard tiling in the sense of Birman and Menasco [Birman J.S., Menasco W.W., Special positions for essential tori in link complements, Topology, 1994, 33(3), 525–556]. Moreover, let R denote the class of thin tiled tori in the sense of Ng [Ng K.Y., Essential tori in link complements, J. Knot Theory Ramifications, 1998, 7(2), 205–216]. We define the subclass B ⊂ Θ of typical tiled tori and show that R ⊂ B. We also describe a method allowing to construct new examples of tiled essential tori T which are outside the class B in the strong sense. In [Kazantsev A., Essential tori in link complements: detecting the satellite structure by monotonic simplification, preprint available at http://arxiv.org/abs/1005.5263], Kazantsev showed that the inclusion R ⊂ Θ is proper by giving the corresponding example of a nonthin tiled torus T. It turns out this torus T is inside the class B. We show that the inclusion B ⊂ Θ is proper. It follows that the tori from the class B do not provide the complete geometric description of the class Θ. The main results of the paper are Theorems 2.1 and 2.2 which give a constructive procedure for obtaining examples of nontypical tiled essential tori.

Remarks on tiled tori

J. S. Birman and W. W. Menasco [Topology, 33, 525–556 (1994)] introduced and studied a class of imbedded tori in closed braid complements that admit a standard tiling. K. Y. Ng [J. Knot Theory Ramifications, 7, 205–216 (1998)] showed that each essential torus in a closed braid complement that admits standard tiling possesses a staircase pattern. The combinatorial and geometric description of the tori from this class was partially given in [Fund. Math., 189, 195–226 (2006)]. In the present paper, we continue the study of the decomposition of tiled tori into building blocks and show that they are similar in a certain sense. In particular, for the class of geometric tiled tori described by Ng in [J. Knot Theory Ramifications, 7, 205–216 (1998)], we show that each torus of this type consists only of elementary blocks.

Essential tori admitting a standard tiling

Birman and Menasco (1994) introduced and studied a class of embedded tori in closed braid complements which admit a standard tiling. The geometric description of the tori from this class was not complete. Ng showed (1988) that each essential torus in a cl

A Tile with Surround Number 2

(2002). A Tile with Surround Number 2. The American Mathematical Monthly: Vol. 109, No. 4, pp. 383-388.

Electronic properties of the octagonal tiling : a new renormalization-group calculation

We present a new renormalization-group calculation of the octagonal tiling excitation spectrum. Two new self-similar octagonal tilings are introduced, a triangle-kite and a triangle-dart tiling, which are entangled with the standard rhombus-square tiling. The decimation procedure involved in the renormalization group calculation transforms each tiling onto a different one and goes back to the (rescaled) original tiling after three steps. The local and average density of states are calculated. In the case of a Laplacian-like tight-binding Hamiltonian, on the standard rhombussquare octagonal tiling, our results compare reasonably well to numerically derived spectra and improve some previous renormalization group calculations. In the pure hopping case, the renormalization is shown to be inaccurate by comparing to numerical density of states obtained by a continued fraction method.