Limit-(quasi)periodic point sets as quasicrystals with p-adic internal spaces

Model sets (or cut and project sets) provide a familiar and commonly used method of constructing and studying nonperiodic point sets. Here we extend this method to situations where the internal spaces are no longer Euclidean, but instead spaces with p-adic topologies or even with mixed Euclidean/p-adic topologies. We show that a number of well known tilings precisely fit this form, including the chair tiling and the Robinson square tilings. Thus the scope of the cut and project formalism is considerably larger than is usually supposed. Applying the powerful consequences of model sets we derive the diffractive nature of these tilings.Comment: 11 pages, 2 figures; dedicated to Peter Kramer on the occasion of his 65th birthda

Substitution Delone Sets

This paper addresses the problem of describing aperiodic discrete structures that have a self-similar or self-affine structure. Substitution Delone set families are families of Delone sets (X_1, ..., X_n) in R^d that satisfy an inflation functional equation under the action of an expanding integer matrix in R^d. This paper studies such functional equation in which each X_i is a discrete multiset (a set whose elements are counted with a finite multiplicity). It gives necessary conditions on the coefficients of the functional equation for discrete solutions to exist. It treats the case where the equation has Delone set solutions. Finally, it gives a large set of examples showing limits to the results obtained.Comment: 34 pages, latex file; some results in Sect 5 rearranged and theorems reformulate

On the noncommutative geometry of tilings

This is a chapter in an incoming book on aperiodic order. We review results about the topology, the dynamics, and the combinatorics of aperiodically ordered tilings obtained with the tools of noncommutative geometry

Expression cartography of human tissues using self organizing maps

Background: The availability of parallel, high-throughput microarray and sequencing experiments poses a challenge how to best arrange and to analyze the obtained heap of multidimensional data in a concerted way. Self organizing maps (SOM), a machine learning method, enables the parallel sample- and gene-centered view on the data combined with strong visualization and second-level analysis capabilities. The paper addresses aspects of the method with practical impact in the context of expression analysis of complex data sets.
Results: The method was applied to generate a SOM characterizing the whole genome expression profiles of 67 healthy human tissues selected from ten tissue categories (adipose, endocrine, homeostasis, digestion, exocrine, epithelium, sexual reproduction, muscle, immune system and nervous tissues). SOM mapping reduces the dimension of expression data from ten thousands of genes to a few thousands of metagenes where each metagene acts as representative of a minicluster of co-regulated single genes. Tissue-specific and common properties shared between groups of tissues emerge as a handful of localized spots in the tissue maps collecting groups of co-regulated and co-expressed metagenes. The functional context of the spots was discovered using overrepresentation analysis with respect to pre-defined gene sets of known functional impact. We found that tissue related spots typically contain enriched populations of gene sets well corresponding to molecular processes in the respective tissues. Analysis techniques normally used at the gene-level such as two-way hierarchical clustering provide a better signal-to-noise ratio and a better representativeness of the method if applied to the metagenes. Metagene-based clustering analyses aggregate the tissues into essentially three clusters containing nervous, immune system and the remaining tissues. 
Conclusions: The global view on the behavior of a few well-defined modules of correlated and differentially expressed genes is more intuitive and more informative than the separate discovery of the expression levels of hundreds or thousands of individual genes. The metagene approach is less sensitive to a priori selection of genes. It can detect a coordinated expression pattern whose components would not pass single-gene significance thresholds and it is able to extract context-dependent patterns of gene expression in complex data sets.
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Expression cartography of human tissues using self organizing maps

Background: The availability of parallel, high-throughput microarray and sequencing experiments poses a challenge how to best arrange and to analyze the obtained heap of multidimensional data in a concerted way. Self organizing maps (SOM), a machine learning method, enables the parallel sample- and gene-centered view on the data combined with strong visualization and second-level analysis capabilities. The paper addresses aspects of the method with practical impact in the context of expression analysis of complex data sets.
Results: The method was applied to generate a SOM characterizing the whole genome expression profiles of 67 healthy human tissues selected from ten tissue categories (adipose, endocrine, homeostasis, digestion, exocrine, epithelium, sexual reproduction, muscle, immune system and nervous tissues). SOM mapping reduces the dimension of expression data from ten thousands of genes to a few thousands of metagenes where each metagene acts as representative of a minicluster of co-regulated single genes. Tissue-specific and common properties shared between groups of tissues emerge as a handful of localized spots in the tissue maps collecting groups of co-regulated and co-expressed metagenes. The functional context of the spots was discovered using overrepresentation analysis with respect to pre-defined gene sets of known functional impact. We found that tissue related spots typically contain enriched populations of gene sets well corresponding to molecular processes in the respective tissues. Analysis techniques normally used at the gene-level such as two-way hierarchical clustering provide a better signal-to-noise ratio and a better representativeness of the method if applied to the metagenes. Metagene-based clustering analyses aggregate the tissues into essentially three clusters containing nervous, immune system and the remaining tissues. 
Conclusions: The global view on the behavior of a few well-defined modules of correlated and differentially expressed genes is more intuitive and more informative than the separate discovery of the expression levels of hundreds or thousands of individual genes. The metagene approach is less sensitive to a priori selection of genes. It can detect a coordinated expression pattern whose components would not pass single-gene significance thresholds and it is able to extract context-dependent patterns of gene expression in complex data sets.
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Cellular mixing with bounded palenstrophy

We study the problem of optimal mixing of a passive scalar $\rho$ advected by an incompressible flow on the two dimensional unit square. The scalar $\rho$ solves the continuity equation with a divergence-free velocity field $u$ with uniform-in-time bounds on the homogeneous Sobolev semi-norm $\dot{W}^{s,p}$, where $s>1$ and $1< p \leq \infty$. We measure the degree of mixedness of the tracer $\rho$ via the two different notions of mixing scale commonly used in this setting, namely the functional and the geometric mixing scale. For velocity fields with the above constraint, it is known that the decay of both mixing scales cannot be faster than exponential. Numerical simulations suggest that this exponential lower bound is in fact sharp, but so far there is no explicit analytical example which matches this result. We analyze velocity fields of cellular type, which is a special localized structure often used in constructions of explicit analytical examples of mixing flows and can be viewed as a generalization of the self-similar construction by Alberti, Crippa and Mazzucato. We show that for any velocity field of cellular type both mixing scales cannot decay faster than polynomially.Comment: 20 pages, 5 figure

Automata, Groups, Limit Spaces, and Tilings

We explore the connections between automata, groups, limit spaces of self-similar actions, and tilings. In particular, we show how a group acting ``nicely'' on a tree gives rise to a self-covering of a topological groupoid, and how the group can be reconstructed from the groupoid and its covering. The connection is via finite-state automata. These define decomposition rules, or self-similar tilings, on leaves of the solenoid associated with the covering.Comment: to appear in J. Algebr

Re-using Digital Narrative Content in Interactive Games

This paper presents a model, called Scene-Driver, for the reuse of film and television material. We begin by exploring general issues surrounding the ways in which content can be sub-divided into meaningful units for re-use and how criteria might then be applied to the selection and ordering of these units. We also identify and discuss the different means by which a user might interact with the content to create novel and engaging experiences. The Scene-Driver model has been instantiated using content from an animated children’s television series called Tiny Planets, which is aimed at children of 5-7 years old. This type of material, being story-based itself, lends itself particularly well to the application of narrative constraints to scene reordering, to provide coherence to the experience of interacting with the content. We propose an interactive narrative-driven game architecture, in which a user generates novel narratives from existing content by placing “domino” like tiles. These tiles act as “glue” between scenes and each tile-choice dictates certain properties of the next scene to be shown within a game. There are three different game-types, based on three different ways in which tiles can be matched to scenes. We introduce algorithms for generating legal tile-sets for each of these three game-types, which can be extended to include narrative constraints. This ensures that all novel orderings adhere to a minimum narrative plan, which has been identified based on analysis of the Tiny Planets series and on narrative theories. We also suggest ways in which basic narratives can be enhanced by the inclusion of directorial techniques and by the use of more complex plot structures. In our evaluation studies with children in the target age-range, our game compared favourably with other games that the children enjoyed playing