On periodic homeomorphisms of spheres

The purpose of this paper is to study how small orbits of periodic homemorphisms of spheres can be.Comment: Published by Algebraic and Geometric Topology at http://www.maths.warwick.ac.uk/agt/AGTVol1/agt-1-22.abs.htm

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

Photochemistry of the PAH pyrene in water ice: the case for ion-mediated solid-state astrochemistry

Context. Icy dust grains play an important role in the formation of complex inter- and circumstellar molecules. Observational studies show that polycyclic aromatic hydrocarbons (PAHs) are abundantly present in the ISM in the gas phase. It is likely that these non-volatile species freeze out onto dust grains as well and participate in the astrochemical solid-state network, but experimental PAH ice studies are largely lacking. Methods. Near UV/VIS spectroscopy is used to track the in situ VUV driven photochemistry of pyrene containing ices at temperatures ranging from 10 to 125 K. Results. The main photoproducts of VUV photolyzed pyrene ices are spectroscopically identified and their band positions are listed for two host ices, \water and CO. Pyrene ionisation is found to be most efficient in \water ices at low temperatures. The reaction products, triplet pyrene and the 1-hydro-1-pyrenyl radical are most efficiently formed in higher temperature water ices and in low temperature CO ice. Formation routes and band strength information of the identified species are discussed. Additionally, the oscillator strengths of Py, Py^+ and PyH are derived and a quantitative kinetic analysis is performed by fitting a chemical reaction network to the experimental data. Conclusions. Pyrene is efficiently ionised in water ice at temperatures below 50 K. Hydrogenation reactions dominate the chemistry in low temperature CO ice with trace amounts of water. The results are put in an astrophysical context by determining the importance of PAH ionisation in a molecular cloud. The photoprocessing of a sample PAH in ice described in this manuscript indicates that PAH photoprocessing in the solid state should also be taken into account in astrochemical models.Comment: 11 pages, 8 figures, accepted for publication in A&

An orbital-free molecular dynamics study of melting in K_20, K_55, K_92, K_142, Rb_55 and Cs_55 clusters

The melting-like transition in potasium clusters K_N, with N=20, 55, 92 and 142, is studied by using an orbital-free density-functional constant-energy molecular dynamics simulation method, and compared to previous theoretical results on the melting-like transition in sodium clusters of the same sizes. Melting in potasium and sodium clusters proceeds in a similar way: a surface melting stage develops upon heating before the homogeneous melting temperature is reached. Premelting effects are nevertheless more important and more easily established in potasium clusters, and the transition regions spread over temperature intervals which are wider than in the case of sodium. For all the sizes considered, the percentage melting temperature reduction when passing from Na to K clusters is substantially larger than in the bulk. Once those two materials have been compared for a number of different cluster sizes, we study the melting-like transition in Rb_55 and Cs_55 clusters and make a comparison with the melting behavior of Na_55 and K_55. As the atomic number increases, the height of the specific heat peaks decreases, their width increases, and the melting temperature decreases as in bulk melting, but in a more pronounced way.Comment: LaTeX file. 6 pages with 17 pictures. Final version with minor change

Bounding Helly numbers via Betti numbers

We show that very weak topological assumptions are enough to ensure the existence of a Helly-type theorem. More precisely, we show that for any non-negative integers $b$ and $d$ there exists an integer $h(b,d)$ such that the following holds. If $\mathcal F$ is a finite family of subsets of $\mathbb R^d$ such that $\tilde\beta_i\left(\bigcap\mathcal G\right) \le b$ for any $\mathcal G \subsetneq \mathcal F$ and every $0 \le i \le \lceil d/2 \rceil-1$ then $\mathcal F$ has Helly number at most $h(b,d)$. Here $\tilde\beta_i$ denotes the reduced $\mathbb Z_2$-Betti numbers (with singular homology). These topological conditions are sharp: not controlling any of these $\lceil d/2 \rceil$ first Betti numbers allow for families with unbounded Helly number. Our proofs combine homological non-embeddability results with a Ramsey-based approach to build, given an arbitrary simplicial complex $K$, some well-behaved chain map $C_*(K) \to C_*(\mathbb R^d)$.Comment: 29 pages, 8 figure

Self-dual Maps III: projective links

In this paper, we present necessary and sufficient combinatorial conditions for a link to be projective, that is, a link in $RP^3$. This characterization is closely related to the notions of antipodally self-dual and antipodally symmetric maps. We also discuss the notion of symmetric cycle, an interesting issue arising in projective links leading us to an easy condition to prevent a projective link to be alternating.Comment: arXiv admin note: substantial text overlap with arXiv:2106.0537