40 research outputs found

    Uniform coloured hypergraphs and blocking sets

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    On organizing principles of Discrete Differential Geometry. Geometry of spheres

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    Discrete differential geometry aims to develop discrete equivalents of the geometric notions and methods of classical differential geometry. In this survey we discuss the following two fundamental Discretization Principles: the transformation group principle (smooth geometric objects and their discretizations are invariant with respect to the same transformation group) and the consistency principle (discretizations of smooth parametrized geometries can be extended to multidimensional consistent nets). The main concrete geometric problem discussed in this survey is a discretization of curvature line parametrized surfaces in Lie geometry. We find a discretization of curvature line parametrization which unifies the circular and conical nets by systematically applying the Discretization Principles.Comment: 57 pages, 18 figures; In the second version the terminology is slightly changed and umbilic points are discusse

    Multi-scale Geometric Modeling of Ambiguous Shapes with Toleranced Balls and Compoundly Weighted alpha-shapes

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    Dealing with ambiguous data is a challenge in Science in general and geometry processing in particular. One route of choice to extract information from such data consists of replacing the ambiguous input by a continuum, typically a one-parameter family, so as to mine stable geometric and topological features within this family. This work follows this spirit and introduces a novel framework to handle 3D ambiguous geometric data which are naturally modeled by balls. First, we introduce {\em toleranced balls} to model ambiguous geometric objects. A toleranced ball consists of two concentric balls, and interpolating between their radii provides a way to explore a range of possible geometries. We propose to model an ambiguous shape by a collection of toleranced balls, and show that the aforementioned radius interpolation is tantamount to the growth process associated with an additively-multiplicatively weighted Voronoi diagram (also called compoundly weighted or CW). Second and third, we investigate properties of the CW diagram and the associated CW α\alpha-complex, which provides a filtration called the λ\lambda-complex. Fourth, we propose a naive algorithm to compute the CW VD. Finally, we use the λ\lambda-complex to assess the quality of models of large protein assemblies, as these models inherently feature ambiguities

    Studying Evolutionary Change: Transdisciplinary Advances in Understanding and Measuring Evolution

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    Evolutionary processes can be found in almost any historical, i.e. evolving, system that erroneously copies from the past. Well studied examples do not only originate in evolutionary biology but also in historical linguistics. Yet an approach that would bind together studies of such evolving systems is still elusive. This thesis is an attempt to narrowing down this gap to some extend. An evolving system can be described using characters that identify their changing features. While the problem of a proper choice of characters is beyond the scope of this thesis and remains in the hands of experts we concern ourselves with some theoretical as well data driven approaches. Having a well chosen set of characters describing a system of different entities such as homologous genes, i.e. genes of same origin in different species, we can build a phylogenetic tree. Consider the special case of gene clusters containing paralogous genes, i.e. genes of same origin within a species usually located closely, such as the well known HOX cluster. These are formed by step- wise duplication of its members, often involving unequal crossing over forming hybrid genes. Gene conversion and possibly other mechanisms of concerted evolution further obfuscate phylogenetic relationships. Hence, it is very difficult or even impossible to disentangle the detailed history of gene duplications in gene clusters. Expanding gene clusters that use unequal crossing over as proposed by Walter Gehring leads to distinctive patterns of genetic distances. We show that this special class of distances helps in extracting phylogenetic information from the data still. Disregarding genome rearrangements, we find that the shortest Hamiltonian path then coincides with the ordering of paralogous genes in a cluster. This observation can be used to detect ancient genomic rearrangements of gene clus- ters and to distinguish gene clusters whose evolution was dominated by unequal crossing over within genes from those that expanded through other mechanisms. While the evolution of DNA or protein sequences is well studied and can be formally described, we find that this does not hold for other systems such as language evolution. This is due to a lack of detectable mechanisms that drive the evolutionary processes in other fields. Hence, it is hard to quantify distances between entities, e.g. languages, and therefore the characters describing them. Starting out with distortions of distances, we first see that poor choices of the distance measure can lead to incorrect phylogenies. Given that phylogenetic inference requires additive metrics we can infer the correct phylogeny from a distance matrix D if there is a monotonic, subadditive function ζ such that ζ^−1(D) is additive. We compute the metric-preserving transformation ζ as the solution of an optimization problem. This result shows that the problem of phylogeny reconstruction is well defined even if a detailed mechanistic model of the evolutionary process is missing. Yet, this does not hinder studies of language evolution using automated tools. As the amount of available and large digital corpora increased so did the possibilities to study them automatically. The obvious parallels between historical linguistics and phylogenetics lead to many studies adapting bioinformatics tools to fit linguistics means. Here, we use jAlign to calculate bigram alignments, i.e. an alignment algorithm that operates with regard to adjacency of letters. Its performance is tested in different cognate recognition tasks. Using pairwise alignments one major obstacle is the systematic errors they make such as underestimation of gaps and their misplacement. Applying multiple sequence alignments instead of a pairwise algorithm implicitly includes more evolutionary information and thus can overcome the problem of correct gap placement. They can be seen as a generalization of the string-to-string edit problem to more than two strings. With the steady increase in computational power, exact, dynamic programming solutions have become feasible in practice also for 3- and 4-way alignments. For the pairwise (2-way) case, there is a clear distinction between local and global alignments. As more sequences are consid- ered, this distinction, which can in fact be made independently for both ends of each sequence, gives rise to a rich set of partially local alignment problems. So far these have remained largely unexplored. Thus, a general formal frame- work that gives raise to a classification of partially local alignment problems is introduced. It leads to a generic scheme that guides the principled design of exact dynamic programming solutions for particular partially local alignment problems

    Subject Index Volumes 1–200

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    Culture and Cognition : Essays in Honor of Peter Damerow

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