Navigation of Distinct Euclidean Particles via Hierarchical Clustering

We present a centralized online (completely reactive) hybrid navigation algorithm for bringing a swarm of n perfectly sensed and actuated point particles in Euclidean d space (for arbitrary n and d) to an arbitrary goal configuration with the guarantee of no collisions along the way. Our construction entails a discrete abstraction of configurations using cluster hierarchies, and relies upon two prior recent constructions: (i) a family of hierarchy-preserving control policies and (ii) an abstract discrete dynamical system for navigating through the space of cluster hierarchies. Here, we relate the (combinatorial) topology of hierarchical clusters to the (continuous) topology of configurations by constructing “portals” — open sets of configurations supporting two adjacent hierarchies. The resulting online sequential composition of hierarchy-invariant swarming followed by discrete selection of a hierarchy “closer” to that of the destination along with its continuous instantiation via an appropriate portal configuration yields a computationally effective construction for the desired navigation policy

Relative blocking in posets

Poset-theoretic generalizations of set-theoretic committee constructions are presented. The structure of the corresponding subposets is described. Sequences of irreducible fractions associated to the principal order ideals of finite bounded posets are considered and those related to the Boolean lattices are explored; it is shown that such sequences inherit all the familiar properties of the Farey sequences.Comment: 29 pages. Corrected version of original publication which is available at http://www.springerlink.com, see Corrigendu

Toric Construction of Global F-Theory GUTs

We systematically construct a large number of compact Calabi-Yau fourfolds which are suitable for F-theory model building. These elliptically fibered Calabi-Yaus are complete intersections of two hypersurfaces in a six dimensional ambient space. We first construct three-dimensional base manifolds that are hypersurfaces in a toric ambient space. We search for divisors which can support an F-theory GUT. The fourfolds are obtained as elliptic fibrations over these base manifolds. We find that elementary conditions which are motivated by F-theory GUTs lead to strong constraints on the geometry, which significantly reduce the number of suitable models. The complete database of models is available at http://hep.itp.tuwien.ac.at/f-theory/. We work out several examples in more detail.Comment: 35 pages, references adde

On the use of cartographic projections in visualizing phylo-genetic tree space

Phylogenetic analysis is becoming an increasingly important tool for biological research. Applications include epidemiological studies, drug development, and evolutionary analysis. Phylogenetic search is a known NP-Hard problem. The size of the data sets which can be analyzed is limited by the exponential growth in the number of trees that must be considered as the problem size increases. A better understanding of the problem space could lead to better methods, which in turn could lead to the feasible analysis of more data sets. We present a definition of phylogenetic tree space and a visualization of this space that shows significant exploitable structure. This structure can be used to develop search methods capable of handling much larger data sets

A Calabi-Yau Database: Threefolds Constructed from the Kreuzer-Skarke List

Kreuzer and Skarke famously produced the largest known database of Calabi-Yau threefolds by providing a complete construction of all 473,800,776 reflexive polyhedra that exist in four dimensions [1]. These polyhedra describe the singular limits of ambient toric varieties in which Calabi-Yau threefolds can exist as hypersurfaces. In this paper, we review how to extract topological and geometric information about Calabi-Yau threefolds using the toric construction, and we provide, in a companion online database (see http://​nuweb1.​neu.​edu/​cydatabase), a detailed inventory of these quantities which are of interest to physicists. Many of the singular ambient spaces described by the Kreuzer-Skarke list can be smoothed out into multiple distinct toric ambient spaces describing different Calabi-Yau threefolds. We provide a list of the different Calabi-Yau threefolds which can be obtained from each polytope, up to current computational limits. We then give the details of a variety of quantities associated to each of these Calabi-Yau such as Chern classes, intersection numbers, and the Kähler and Mori cones, in addition to the Hodge data. This data forms a useful starting point for a number of physical applications of the Kreuzer-Skarke list

Universal Artifacts Affect the Branching of Phylogenetic Trees, Not Universal Scaling Laws

The superficial resemblance of phylogenetic trees to other branching structures allows searching for macroevolutionary patterns. However, such trees are just statistical inferences of particular historical events. Recent meta-analyses report finding regularities in the branching pattern of phylogenetic trees. But is this supported by evidence, or are such regularities just methodological artifacts? If so, is there any signal in a phylogeny?In order to evaluate the impact of polytomies and imbalance on tree shape, the distribution of all binary and polytomic trees of up to 7 taxa was assessed in tree-shape space. The relationship between the proportion of outgroups and the amount of imbalance introduced with them was assessed applying four different tree-building methods to 100 combinations from a set of 10 ingroup and 9 outgroup species, and performing covariance analyses. The relevance of this analysis was explored taking 61 published phylogenies, based on nucleic acid sequences and involving various taxa, taxonomic levels, and tree-building methods.All methods of phylogenetic inference are quite sensitive to the artifacts introduced by outgroups. However, published phylogenies appear to be subject to a rather effective, albeit rather intuitive control against such artifacts. The data and methods used to build phylogenetic trees are varied, so any meta-analysis is subject to pitfalls due to their uneven intrinsic merits, which translate into artifacts in tree shape. The binary branching pattern is an imposition of methods, and seldom reflects true relationships in intraspecific analyses, yielding artifactual polytomies in short trees. Above the species level, the departure of real trees from simplistic random models is caused at least by two natural factors--uneven speciation and extinction rates; and artifacts such as choice of taxa included in the analysis, and imbalance introduced by outgroups and basal paraphyletic taxa. This artifactual imbalance accounts for tree shape convergence of large trees.There is no evidence for any universal scaling in the tree of life. Instead, there is a need for improved methods of tree analysis that can be used to discriminate the noise due to outgroups from the phylogenetic signal within the taxon of interest, and to evaluate realistic models of evolution, correcting the retrospective perspective and explicitly recognizing extinction as a driving force. Artifacts are pervasive, and can only be overcome through understanding the structure and biological meaning of phylogenetic trees. Catalan Abstract in Translation S1