Developing a Mathematical Model for Bobbin Lace

Bobbin lace is a fibre art form in which intricate and delicate patterns are created by braiding together many threads. An overview of how bobbin lace is made is presented and illustrated with a simple, traditional bookmark design. Research on the topology of textiles and braid theory form a base for the current work and is briefly summarized. We define a new mathematical model that supports the enumeration and generation of bobbin lace patterns using an intelligent combinatorial search. Results of this new approach are presented and, by comparison to existing bobbin lace patterns, it is demonstrated that this model reveals new patterns that have never been seen before. Finally, we apply our new patterns to an original bookmark design and propose future areas for exploration.Comment: 20 pages, 18 figures, intended audience includes Artists as well as Computer Scientists and Mathematician

Generating toric noncommutative crepant resolutions

We present an algorithm that finds all toric noncommutative crepant resolutions of a given toric 3-dimensional Gorenstein singularity. The algorithm embeds the quivers of these algebras inside a real 3-dimensional torus such that the relations are homotopy relations. One can project these embedded quivers down to a 2-dimensional torus to obtain the corresponding dimer models. We discuss some examples and use the algorithm to show that all toric noncommutative crepant resolutions of a finite quotient of the conifold singularity can be obtained by mutating one basic dimer model. We also discuss how this algorithm might be extended to higher dimensional singularities

Predictability of evolutionary trajectories in fitness landscapes

Experimental studies on enzyme evolution show that only a small fraction of all possible mutation trajectories are accessible to evolution. However, these experiments deal with individual enzymes and explore a tiny part of the fitness landscape. We report an exhaustive analysis of fitness landscapes constructed with an off-lattice model of protein folding where fitness is equated with robustness to misfolding. This model mimics the essential features of the interactions between amino acids, is consistent with the key paradigms of protein folding and reproduces the universal distribution of evolutionary rates among orthologous proteins. We introduce mean path divergence as a quantitative measure of the degree to which the starting and ending points determine the path of evolution in fitness landscapes. Global measures of landscape roughness are good predictors of path divergence in all studied landscapes: the mean path divergence is greater in smooth landscapes than in rough ones. The model-derived and experimental landscapes are significantly smoother than random landscapes and resemble additive landscapes perturbed with moderate amounts of noise; thus, these landscapes are substantially robust to mutation. The model landscapes show a deficit of suboptimal peaks even compared with noisy additive landscapes with similar overall roughness. We suggest that smoothness and the substantial deficit of peaks in the fitness landscapes of protein evolution are fundamental consequences of the physics of protein folding.Comment: 14 pages, 7 figure

Constructs and evaluation strategies for intelligent speculative parallelism - armageddon revisited

This report addresses speculative parallelism (the assignment of spare processing resources to tasks which are not known to be strictly required for the successful completion of a computation) at the user and application level. At this level, the execution of a program is seen as a (dynamic) tree —a graph, in general. A solution for a problem is a traversal of this graph from the initial state to a node known to be the answer. Speculative parallelism then represents the assignment of resources to múltiple branches of this graph even if they are not positively known to be on the path to a solution. In highly non-deterministic programs the branching factor can be very high and a naive assignment will very soon use up all the resources. This report presents work assignment strategies other than the usual depth-first and breadth-first. Instead, best-first strategies are used. Since their definition is application-dependent, the application language contains primitives that allow the user (or application programmer) to a) indícate when intelligent OR-parallelism should be used; b) provide the functions that define "best," and c) indícate when to use them. An abstract architecture enables those primitives to perform the search in a "speculative" way, using several processors, synchronizing them, killing the siblings of the path leading to the answer, etc. The user is freed from worrying about these interactions. Several search strategies are proposed and their implementation issues are addressed. "Armageddon," a global pruning method, is introduced, together with both a software and a hardware implementation for it. The concepts exposed are applicable to áreas of Artificial Intelligence such as extensive expert systems, planning, game playing, and in general to large search problems. The proposed strategies, although showing promise, have not been evaluated by simulation or experimentation

Rapid Mixing for Lattice Colorings with Fewer Colors

We provide an optimally mixing Markov chain for 6-colorings of the square lattice on rectangular regions with free, fixed, or toroidal boundary conditions. This implies that the uniform distribution on the set of such colorings has strong spatial mixing, so that the 6-state Potts antiferromagnet has a finite correlation length and a unique Gibbs measure at zero temperature. Four and five are now the only remaining values of q for which it is not known whether there exists a rapidly mixing Markov chain for q-colorings of the square lattice.Comment: Appeared in Proc. LATIN 2004, to appear in JSTA