劍橋大學彼得豪斯學院

Mishra et al. have recently synthesized a "pi-extended Clar's goblet". This is a "concealed" non-Kekulean condensed benzenoid hydrocarbon formed by (a) adding to Clar's goblet some extra layers of benzenoid rings and (b) simultaneously changing the size of the moiety at the center of the structure from one hexagon to two hexagons-thereby changing that moiety from "benzenoid" to "naphthalenic". This idea is generalized and the "topological" ring-currents in Clar's goblet, and several of its homologues and oxidation states, are compared with those arising from these new structures. Quantitative calculations of topological ring-currents in these open-shell systems necessarily incorporate the technique of "Configurational State Averaging". Their predictions are compared with those from pictorial current maps generated by more sophisticated ab initio and pseudo-pi calculations. It is found that (a) adding or removing electrons from frontier nonbonding orbital shells makes no difference to the ring current; (b) the number of unpaired electrons in a given goblet depends on the number of layers in it, but not on its girth; (c) as the goblet becomes "fatter", the current within the central moiety increases and the central rings become "less empty"; there is, therefore, a greater mixing of current between the upper and lower triangles within each goblet as their girths become larger; and (d) the Huckel-London "topological" calculations predict that the direction of the current around the periphery of any goblet will be counterclockwise (diamagnetic)-a conclusion that is invariably confirmed by calculations at the ipso-centric pseudo-pi level.

New and overlooked records from the Aldabra group of islands, Seychelles (Aldabra, Assumption, Cosmoledo, and Astove) extend the species list for the islands by four species, two of which are described as new: Tropidophora fryeri from Aldabra and Tropidophora menai from Cosmoledo. Records of Melampus caffra (Koster, 1843) are re-identified as M. parvulus L. Pfeiffer, 1856 and Truncatella cf valida var. minor G. Nevill, 1878 as Truncatella marginata Koster, 1855. Dupontia sp. is confirmed to be Microcystina rowsoni E. Gittenberger & van Bruggen, 2013. Thirty-five species are now recorded from the islands, 33 from Aldabra and smaller numbers on the other islands.

We investigate a random geometric graph model introduced by Bonato and Janssen. The vertices are the points of a countable dense set Sin a (necessarily separable) normed vector space X, and each pair of points are joined independently with some fixed probability p (with 0 < p < 1) if they are less than distance 1 apart. A countable dense set Sin a normed space is Rado, if the resulting graph is almost surely unique up to isomorphism: that is any two such graphs are, almost surely, isomorphic. Not surprisingly, understanding which sets are Rado is closely related to the geometry of the underlying normed space. It turns out that a key question is in which spaces must stepisometries (maps that preserve the integer parts of distances) on dense subsets necessarily be isometries. We answer this question for a large class of Banach spaces including all strictly convex reflexive spaces. In the process we prove results on the interplay between the norm topology , weak topology that may be of independent interest. As a consequence of these Banach space results we show that almost all countable dense sets in strictly convex reflexive spaces are strongly non-Rado (that is, any two graphs are almost surely non-isomorphic). However, we show that there do exist Rado sets even in l(2) . Finally we construct a Banach space in which all countable dense set are strongly non-Rado. (c) 2025 The Authors. Published by Elsevier Inc. This is an open access article under the CC BY license (http:// creativecommons.org/licenses/by/4.0/).