Topological transversals to a family of convex sets

Let $\mathcal F$ be a family of compact convex sets in $\mathbb R^d$. We say that $\mathcal F$ has a \emph{topological $\rho$-transversal of index $(m,k)$} ($\rho<m$, $0<k\leq d-m$) if there are, homologically, as many transversal $m$-planes to $\mathcal F$ as $m$-planes containing a fixed $\rho$-plane in $\mathbb R^{m+k}$. Clearly, if $\mathcal F$ has a $\rho$-transversal plane, then $\mathcal F$ has a topological $\rho$-transversal of index $(m,k),$ for $\rho<m$ and $k\leq d-m$. The converse is not true in general. We prove that for a family $\mathcal F$ of $\rho+k+1$ compact convex sets in $\mathbb R^d$ a topological $\rho$-transversal of index $(m,k)$ implies an ordinary $\rho$-transversal. We use this result, together with the multiplication formulas for Schubert cocycles, the Lusternik-Schnirelmann category of the Grassmannian, and different versions of the colorful Helly theorem by B\'ar\'any and Lov\'asz, to obtain some geometric consequences

On the heterochromatic number of hypergraphs associated to geometric graphs and to matroids

The heterochromatic number hc(H) of a non-empty hypergraph H is the smallest integer k such that for every colouring of the vertices of H with exactly k colours, there is a hyperedge of H all of whose vertices have different colours. We denote by nu(H) the number of vertices of H and by tau(H) the size of the smallest set containing at least two vertices of each hyperedge of H. For a complete geometric graph G with n > 2 vertices let H = H(G) be the hypergraph whose vertices are the edges of G and whose hyperedges are the edge sets of plane spanning trees of G. We prove that if G has at most one interior vertex, then hc(H) = nu(H) - tau(H) + 2. We also show that hc(H) = nu(H) - tau(H) + 2 whenever H is a hypergraph with vertex set and hyperedge set given by the ground set and the bases of a matroid, respectively

Affine configurations and pure braids

We show that the fundamental group of the space of ordered affine-equivalent configurations of at least five points in the real plane is isomorphic to the pure braid group modulo its centre. In the case of four points this fundamental group is free with eleven generators.Comment: 5 pages, 1 figure, final version; to appear in Discrete & Computational Geometry, available from the publishers at http://www.springerlink.com/content/384516n7q24811ph

Lines pinning lines

A line g is a transversal to a family F of convex polytopes in 3-dimensional space if it intersects every member of F. If, in addition, g is an isolated point of the space of line transversals to F, we say that F is a pinning of g. We show that any minimal pinning of a line by convex polytopes such that no face of a polytope is coplanar with the line has size at most eight. If, in addition, the polytopes are disjoint, then it has size at most six. We completely characterize configurations of disjoint polytopes that form minimal pinnings of a line.Comment: 27 pages, 10 figure

Acremonium phylogenetic overview and revision of Gliomastix, Sarocladium, and Trichothecium

AbstractOver 200 new sequences are generated for members of the genus Acremonium and related taxa including ribosomal small subunit sequences (SSU) for phylogenetic analysis and large subunit (LSU) sequences for phylogeny and DNA-based identification. Phylogenetic analysis reveals that within the Hypocreales, there are two major clusters containing multiple Acremonium species. One clade contains Acremonium sclerotigenum, the genus Emericellopsis, and the genus Geosmithia as prominent elements. The second clade contains the genera Gliomastix sensu stricto and Bionectria. In addition, there are numerous smaller clades plus two multi-species clades, one containing Acremonium strictum and the type species of the genus Sarocladium, and, as seen in the combined SSU/LSU analysis, one associated subclade containing Acremonium breve and related species plus Acremonium curvulum and related species. This sequence information allows the revision of three genera. Gliomastix is revived for five species, G. murorum, G. polychroma, G. tumulicola, G. roseogrisea, and G. masseei. Sarocladium is extended to include all members of the phylogenetically distinct A. strictum clade including the medically important A. kiliense and the protective maize endophyte A. zeae. Also included in Sarocladium are members of the phylogenetically delimited Acremonium bacillisporum clade, closely linked to the A. strictum clade. The genus Trichothecium is revised following the principles of unitary nomenclature based on the oldest valid anamorph or teleomorph name, and new combinations are made in Trichothecium for the tightly interrelated Acremonium crotocinigenum, Spicellum roseum, and teleomorph Leucosphaerina indica. Outside the Hypocreales, numerous Acremonium-like species fall into the Plectosphaerellaceae, and A. atrogriseum falls into the Cephalothecaceae

Notes about the Caratheodory number

In this paper we give sufficient conditions for a compactum in $\mathbb R^n$ to have Carath\'{e}odory number less than $n+1$, generalizing an old result of Fenchel. Then we prove the corresponding versions of the colorful Carath\'{e}odory theorem and give a Tverberg type theorem for families of convex compacta

The repulsive lattice gas, the independent-set polynomial, and the Lov\'asz local lemma

We elucidate the close connection between the repulsive lattice gas in equilibrium statistical mechanics and the Lovasz local lemma in probabilistic combinatorics. We show that the conclusion of the Lovasz local lemma holds for dependency graph G and probabilities {p_x} if and only if the independent-set polynomial for G is nonvanishing in the polydisc of radii {p_x}. Furthermore, we show that the usual proof of the Lovasz local lemma -- which provides a sufficient condition for this to occur -- corresponds to a simple inductive argument for the nonvanishing of the independent-set polynomial in a polydisc, which was discovered implicitly by Shearer and explicitly by Dobrushin. We also present some refinements and extensions of both arguments, including a generalization of the Lovasz local lemma that allows for "soft" dependencies. In addition, we prove some general properties of the partition function of a repulsive lattice gas, most of which are consequences of the alternating-sign property for the Mayer coefficients. We conclude with a brief discussion of the repulsive lattice gas on countably infinite graphs.Comment: LaTex2e, 97 pages. Version 2 makes slight changes to improve clarity. To be published in J. Stat. Phy

Seascape genomics and phylogeography of the sailfish (Istiophorus platypterus)

Permeable phylogeographic barriers characterize the vast open ocean, boosting gene flow and counteracting population differentiation and speciation of widely distributed and migratory species. However, many widely distributed species consists of distinct populations throughout their distribution, evidencing that our understanding of how the marine environment triggers population and species divergence are insufficient. The sailfish is a circumtropical and highly migratory billfish that inhabits warm and productive areas. Despite its ecological and socioeconomic importance as a predator and fishery resource, the species is threatened by overfishing, requiring innovative approaches to improve their management and conservation status. Thus, we presented a novel high-quality reference genome for the species and applied a seascape genomics approach to understand how marine environmental features may promote local adaptation and how it affects gene flow between populations. We delimit two populations between the Atlantic and Indo-Western Pacific oceans and detect outlier loci correlated with sea surface temperature, salinity, oxygen, and chlorophyll concentrations. However, the most significant explanatory factor that explains the differences between populations was isolation by distance. Despite recent population drops, the sailfish populations are not inbred. For billfishes in general, genome-wide heterozygosity was found to be relatively low compared to other marine fishes, evidencing the need to counteract overfishing effects. In addition, in a climate change scenario, management agencies must implement state-of-the-art sequencing methods, consider our findings in their management plans, and monitor genome-wide heterozygosity over time to improve sustainable fisheries and the long-term viability of its populations.LA/P/0101/2020info:eu-repo/semantics/publishedVersio