Set Of Tiles Research Articles

Combinatorial Proofs of Various q-Pell Identities via Tilings

Recently, Benjamin, Plott, and Sellers proved a variety of identities involving sums of Pell numbers combinatorially by interpreting both sides of a given identity as enumerators of certain sets of tilings using white squares, black squares, and gray dominoes. In this article, we state and prove q-analogues of several Pell identities via weighted tilings.

The effect of malformed tiles on tile assemblies within the kinetic tile assembly model

Many different constructions of proofreading tile sets have been proposed in the literature to reduce the effect of deviations from ideal behaviour of the dynamics of the molecular tile self-assembly process. In this paper, we consider the effect on the tile assembly process of a different kind of non-ideality, namely, imperfections in the tiles themselves. We assume a scenario in which some small proportion of the tiles in a tile set are "malformed". We study, through simulations, the effect of such malformed tiles on the self-assembly process within the kinetic Tile Assembly Model (kTAM). Our simulation results show that some tile set constructions show greater error-resilience in the presence of malformed tiles than others. For example, the 2- and 3-way overlay compact proofreading tile sets of Reif et al. (DNA Computing 10, Lecture Notes in Computer Science, vol 3384. Springer, 2005) are able to handle malformed tiles quite well. On the other hand, the snaked proofreading tile set of Chen and Goel (DNA Computing 10, Lecture Notes in Computer Science, vol 3384. Springer, 2005) fails to form even moderately sized tile assemblies when malformed tiles are present. We show how the Chen---Goel construction may be modified to yield new snaked proofreading tile sets that are resilient not only to errors intrinsic to the assembly process, but also to errors caused by malformed tiles.

Tilings: simulation and universality

Wang tiles are unit-size squares with coloured edges. In this paper, we approach one aspect of the study of tiling computability: the quest for a universal tile set. Using a complex construction, based on Robinson's classical construction and its different modifications, we build a tile set (pronounced ayin) that almost always simulates any tile set. By way of Banach–Mazur games on tilings topological spaces, we prove that the set of -tilings that do not satisfy the universality condition is meagre in the set of -tilings.

Wang Tile-based improved texture synthesis

Wang Tiles are constructed from four texture samples and get ready for matching other Tiles.Because they are precomputed Wang Tiles is a real-time way to generate textures.However matching problems occur within Tiles and at the corners of adjacent Tiles.Therefore a new technique of improving Wang Tile was proposed which generated Tile set and synthesized a large texture by replacing the center of edge-matching texture samples with a new sample and using the Image Quilting path-finding algorithm to find the instead path.The experimental results indicate that the rendered texture successfully overcomes these problems and enhances texture diversity to some extent.

2D quasi-periodic Rauzy tiling as a section of 3D periodic tiling

A method for constructing a 2D quasi-periodic Rauzy tiling Til as a section of some 3D periodic tiling Til3D is considered. The translation lattice of the tiling Til3D and its connectivity graph are constructed using the discrete modeling of packings. The calculation of the layer-by-layer growth polyhedron for the tiling Til3D made it possible to estimate from upper the shape of a growth polygon for the tiling Til. As a result, the growth shape in six out of eight growth sectors has been rigorously proven. A set of quasi-periodic tilings (locally indistinguishable from the Rauzy tiling Til), including seven centrosymmetric tilings, has been obtained.

Optimization of supply diversity for the self-assembly of simple objects in two and three dimensions

The field of algorithmic self-assembly is concerned with the design and analysis of self-assembly systems from a computational perspective, that is, from the perspective of mathematical problems whose study may give insight into the natural processes through which elementary objects self-assemble into more complex ones. One of the main problems of algorithmic self-assembly is the minimum tile set problem, which in the extended formulation we consider, here referred to as MTSP, asks for a collection of types of elementary objects (called tiles) to be found for the self-assembly of an object having a pre-established shape. Such a collection is to be as concise as possible, thus minimizing supply diversity, while satisfying a set of stringent constraints having to do with important properties of the self-assembly process from its tile types. We present a study of what, to the best of our knowledge, is the first practical approach to MTSP. Our study starts with the introduction of an evolutionary heuristic to tackle MTSP and includes selected results from extensive experimentation with the heuristic on the self-assembly of simple objects in two and three dimensions, including the possibility of tile rotation. The heuristic we introduce combines classic elements from the field of evolutionary computation with a problem-specific variant of Pareto dominance into a multi-objective approach to MTSP.

Immobilization and one-dimensional arrangement of virus capsids with nanoscale precision using DNA origami.

DNA origami was used as a scaffold to arrange spherical virus capsids into one-dimensional arrays with precise nanoscale positioning. To do this, we first modified the interior surface of bacteriophage MS2 capsids with fluorescent dyes as a model cargo. An unnatural amino acid on the external surface was then coupled to DNA strands that were complementary to those extending from origami tiles. Two different geometries of DNA tiles (rectangular and triangular) were used. The capsids associated with tiles of both geometries with virtually 100% efficiency under mild annealing conditions, and the location of capsid immobilization on the tile could be controlled by the position of the probe strands. The rectangular tiles and capsids could then be arranged into one-dimensional arrays by adding DNA strands linking the corners of the tiles. The resulting structures consisted of multiple capsids with even spacing (approximately 100 nm). We also used a second set of tiles that had probe strands at both ends, resulting in a one-dimensional array of alternating capsids and tiles. This hierarchical self-assembly allows us to position the virus particles with unprecedented control and allows the future construction of integrated multicomponent systems from biological scaffolds using the power of rationally engineered DNA nanostructures.

Comparison and calibration of transcriptome data from RNA-Seq and tiling arrays

BackgroundTiling arrays have been the tool of choice for probing an organism's transcriptome without prior assumptions about the transcribed regions, but RNA-Seq is becoming a viable alternative as the costs of sequencing continue to decrease. Understanding the relative merits of these technologies will help researchers select the appropriate technology for their needs.ResultsHere, we compare these two platforms using a matched sample of poly(A)-enriched RNA isolated from the second larval stage of C. elegans. We find that the raw signals from these two technologies are reasonably well correlated but that RNA-Seq outperforms tiling arrays in several respects, notably in exon boundary detection and dynamic range of expression. By exploring the accuracy of sequencing as a function of depth of coverage, we found that about 4 million reads are required to match the sensitivity of two tiling array replicates. The effects of cross-hybridization were analyzed using a "nearest neighbor" classifier applied to array probes; we describe a method for determining potential "black list" regions whose signals are unreliable. Finally, we propose a strategy for using RNA-Seq data as a gold standard set to calibrate tiling array data. All tiling array and RNA-Seq data sets have been submitted to the modENCODE Data Coordinating Center.ConclusionsTiling arrays effectively detect transcript expression levels at a low cost for many species while RNA-Seq provides greater accuracy in several regards. Researchers will need to carefully select the technology appropriate to the biological investigations they are undertaking. It will also be important to reconsider a comparison such as ours as sequencing technologies continue to evolve.

Semi‐Stochastic Tilings for Example‐Based Texture Synthesis

AbstractWe investigate semi‐stochastic tilings based on Wang or corner tiles for the real‐time synthesis of example‐based textures. In particular, we propose two new tiling approaches: (1) to replace stochastic tilings with pseudo‐random tilings based on the Halton low‐discrepancy sequence, and (2) to allow the controllable generation of tilings based on a user‐provided probability distribution. Our first method prevents local repetition of texture content as common with stochastic approaches and yields better results with smaller sets of utilized tiles. Our second method allows to directly influence the synthesis result which—in combination with an enhanced tile construction method that merges multiple source textures—extends synthesis tasks to globally‐varying textures. We show that both methods can be implemented very efficiently in connection with tile‐based texture mapping and also present a general rule that allows to significantly reduce resulting tile sets.

Migration and deposition of 13C in the full-tungsten ASDEX Upgrade tokamak

The migration of carbon in low-density, low-confinement plasmas of ASDEX Upgrade was studied by injecting 13C into the main chamber of the torus at the end of the 2007 experimental campaign. A selection of standard tungsten-coated lower-divertor and main-chamber tiles as well as a complete set of lower-divertor tiles with an uncoated poloidal marker stripe were removed from one poloidal cross section and analysed using secondary ion mass spectrometry. The poloidal deposition profiles of 13C on both the tungsten-coated tiles and on the uncoated graphite areas of the marker tiles were measured and compared. For the W-coated lower-divertor tiles, 13C was deposited mainly on the high-field side tiles, while barely detectable amounts of 13C were observed on low-field side samples. In contrast, on the uncoated marker stripes the deposition was equally pronounced in the high-field and low-field side divertor. The marker-tile results are in agreement with those obtained from graphite tiles after the 2003 and 2005 13C experiments in ASDEX Upgrade. In the case of W-coated tiles, the 13C measurements were complemented by determining the total amount of deposited carbon (12C) on the tiles, which also shows strong deposition at the inner parts of the lower divertor. The estimated deposition of 13C on W at the divertor areas was less than 1.5% of the injected amount of 13C atoms. The 13C analyses of the main-chamber tiles and small silicon samples mounted in remote areas revealed significant deposition in the upper divertor, in upper parts of the heat shield, in the limiter region close to the injection valve, and below the roof baffle. Approximately 8% of the injected 13C is estimated to have accumulated in these regions. Possible reasons for the different deposition patterns on W and on graphite in different regions of the torus are discussed.

Hexagonal Lattice Model of the Patterns Formed by Hydrogen-Bonded Molecules on the Surface

We model the two-dimensional self-assembly of planar molecules capable of complementary interactions (like hydrogen bonding) as a set of hexagonal tiles on a hexagonal lattice. We use Monte Carlo simulations to study the phase diagrams of three model systems. The phases are characterized using a variety of order parameters, and they are studied as a function of the strength of the complementary interaction energy. This simplified model is proven to be capable of reproducing the phases encountered in real systems, unifying within the same framework most of the structures encountered experimentally.

Programmable Control of Nucleation for Algorithmic Self-Assembly

Algorithmic self-assembly, a generalization of crystal growth processes, has been proposed as a mechanism for autonomous DNA computation and for bottom-up fabrication of complex nanostructures. A “program” for growing a desired structure consists of a set of molecular “tiles” designed to have specific binding interactions. A key challenge to making algorithmic self-assembly practical is designing tile set programs that make assembly robust to errors that occur during initiation and growth. One method for the controlled initiation of assembly, often seen in biology, is the use of a seed or catalyst molecule that reduces an otherwise large kinetic barrier to nucleation. Here we show how to program algorithmic self-assembly similarly, such that seeded assembly proceeds quickly but there is an arbitrarily large kinetic barrier to unseeded growth. We demonstrate this technique by introducing a family of tile sets for which we rigorously prove that, under the right physical conditions, linearly increasing the size of the tile set exponentially reduces the rate of spurious nucleation. Simulations of these “zig-zag” tile sets suggest that under plausible experimental conditions, it is possible to grow large seeded crystals in just a few hours such that less than 1 percent of crystals are spuriously nucleated. Simulation results also suggest that zig-zag tile sets could be used for detection of single DNA strands. Together with prior work showing that tile sets can be made robust to errors during properly initiated growth, this work demonstrates that growth of objects via algorithmic self-assembly can proceed both efficiently and with an arbitrarily low error rate, even in a model where local growth rules are probabilistic.

Parallel growth and healing of DNA self-assembly for interconnects

Self-assembly has been employed in nano-technology to build crystals using individual components (commonly referred to as tiles) with limited control. Templates of regular lattice structures for two-dimensional scaffolds and interconnects have been recently implemented by self-assembly. This study proposes a diagonally based growth scheme that is applicable to these templates of interconnects (as an example). Differently from previous techniques (mostly sequential in execution), growth is allowed along two different directions in the aggregate, thus permitting a parallel mode of operation. This is made possible by utilising a tile set and binding scheme to allow multiple seed tiles to grow along the main diagonal of the pattern. The conditions by which this type of new growth is possible at a reduced error occurrence in mismatched tiles, are presented; error tolerance is achieved by employing healing and so-called robust generation of the seed tiles, thus ensuring that pattern growth is controlled along both directions. Simulation results are presented under different scenarios of growth direction (inclusive of backward growth for healing).

Wang Tile Based Geometric Texture Synthesis

This paper presents geometric texture Wang Tiles to generate geometric textures on different object surfaces at run-time. A set of Wang Tiles is pre-computed from a specific geometric texture and used to generate geometric textures on different object surfaces. Although tile-based approaches are used for image textures, constructing geometric texture Wang Tiles presents additional challenges due to the differences between geometric textures and image textures. The geometric texture Wang Tiles is built automatically by using constrained synthesis to ensure geometric continuity across all possible combinations of tile arrangements. Once a tile set is built, it can be reused for texturing different output objects. This texturing algorithm consumes much less storage and computation than the existing methods.

A hierarchical strongly aperiodic set of tiles in the hyperbolic plane

We give a new construction of strongly aperiodic set of tiles in H 2 , exhibiting a kind of hierarchical structure, simplifying the central framework of Margenstern’s proof that the Domino Problem is undecidable in the hyperbolic plane (Margenstern (2008) [16]).

GIzMOs: Genuine Image Mosaics with Adaptive Tiling

Abstract We present a method that splits an input image into a set of tiles. Each tile is then replaced by another image from a large database such that, when viewed from a distance, the original image is reproduced as well as possible. While the general concept of image mosaics is not new, we consider our results as ‘genuine image mosaics’ (or short GIzMOs) in the sense that the images from the database are not modified in any way. This is different from previous work, where the image tiles are usually colour shifted or overlaid with the high‐frequency content of the input image. Besides the regular alignment of the tiles we propose a greedy approach for adaptive tiling where larger tiles are placed in homogenous image regions. By this we avoid the visual periodicity, which is induced by the equal spacing of the image tiles in the completely regular setting. Our overall system addresses also the cleaning of the image database by removing all unwanted images with no meaningful content. We apply differently sophisticated image descriptors to find the best matching image for each tile. For aesthetic and artistic reasons we classify each tile as ‘feature’ or ‘non‐feature’ and then apply a suitable image descriptor. In a user study we have verified that our descriptors lead to mosaics that are significantly better recognizable than just taking, e.g. average colour values.

DIVIMP-B2-EIRENE modelling of 13C migration and deposition in ASDEX Upgrade L-mode plasmas

Carbon transport and migration were studied experimentally and numerically in a high-density, low-confinement mode plasma in the ASDEX Upgrade tokamak. On the last day of plasma operation of the 2004–2005 experimental campaign, 13CH4 was injected into the vacuum vessel from the low field side midplane. A poloidal set of tiles was subsequently removed and analysed for 13C deposition. In this work the measured deposition profile is interpreted using the impurity transport code DIVIMP. The simulated poloidal distribution of 13C deviates significantly from the measured profile. The simulations indicate that 13C is promptly deposited at the wall in the vicinity of the injection port, and is transported to the low field side divertor plate predominately via the scrape-off layer. The B2-EIRENE plasma solution produce stagnant plasma flow in the main scrape-off layer, in contrast to measurements in ASDEX Upgrade and other tokamaks. This is the likely cause of the discrepancy between the measured and the calculated poloidal distribution of the 13C deposition. Key model parameters of DIVIMP were varied to determine their effect on the simulated deposition profile.

Counting Fixed-Height Tatami Tilings

A tatami tiling is an arrangement of $1 \times 2$ dominoes (or mats) in a rectangle with $m$ rows and $n$ columns, subject to the constraint that no four corners meet at a point. For fixed $m$ we present and use Dean Hickerson's combinatorial decomposition of the set of tatami tilings — a decomposition that allows them to be viewed as certain classes of restricted compositions when $n \ge m$. Using this decomposition we find the ordinary generating functions of both unrestricted and inequivalent tatami tilings that fit in a rectangle with $m$ rows and $n$ columns, for fixed $m$ and $n \ge m$. This allows us to verify a modified version of a conjecture of Knuth. Finally, we give explicit solutions for the count of tatami tilings, in the form of sums of binomial coefficients.

Procedural Generation of Rock Piles using Aperiodic Tiling

AbstractIn this paper, we present a tiling method for generating piles of rocks without any computationally demanding physically‐based simulation. Previous techniques rely on a periodic tiling of rocks and generate unrealistic repetitive patterns. In contrast, our approach relies on a modified corner cube algorithm to generate a set of aperiodic tiles. We generalize the construction method so that the geometry of rocks should straddle corner cubes with a view to avoiding unrealistic gaps in the arrangement of rocks. Moreover, we propose an original technique to control the shape of rocks into contact by computing the Voronoï cells using a parameterized anisotropic distance. Our method has been successfully used to generate landscapes and stone huts and walls with thousands of rocks piled together.

An information-bearing seed for nucleating algorithmic self-assembly

Self-assembly creates natural mineral, chemical, and biological structures of great complexity. Often, the same starting materials have the potential to form an infinite variety of distinct structures; information in a seed molecule can determine which form is grown as well as where and when. These phenomena can be exploited to program the growth of complex supramolecular structures, as demonstrated by the algorithmic self-assembly of DNA tiles. However, the lack of effective seeds has limited the reliability and yield of algorithmic crystals. Here, we present a programmable DNA origami seed that can display up to 32 distinct binding sites and demonstrate the use of seeds to nucleate three types of algorithmic crystals. In the simplest case, the starting materials are a set of tiles that can form crystalline ribbons of any width; the seed directs assembly of a chosen width with >90% yield. Increased structural diversity is obtained by using tiles that copy a binary string from layer to layer; the seed specifies the initial string and triggers growth under near-optimal conditions where the bit copying error rate is <0.2%. Increased structural complexity is achieved by using tiles that generate a binary counting pattern; the seed specifies the initial value for the counter. Self-assembly proceeds in a one-pot annealing reaction involving up to 300 DNA strands containing >17 kb of sequence information. In sum, this work demonstrates how DNA origami seeds enable the easy, high-yield, low-error-rate growth of algorithmic crystals as a route toward programmable bottom-up fabrication.