Singular Poisson equations on Finsler-Hadamard manifolds

In the first part of the paper we study the reflexivity of Sobolev spaces on non-compact and not necessarily reversible Finsler manifolds. Then, by using direct methods in the calculus of variations, we establish uniqueness, location and rigidity results for singular Poisson equations involving the Finsler-Laplace operator on Finsler-Hadamard manifolds having finite reversibility constant.Comment: Published in: Calc. Var. Partial Differential Equations 54 (2015), no. 2, 1219-124

A computer-based approach to the classification of nilpotent Lie algebras

We adopt the $p$-group generation algorithm to classify small-dimensional nilpotent Lie algebras over small fields. Using an implementation of this algorithm, we list the nilpotent Lie algebras of dimension at most~9 over \F_2 and those of dimension at most~7 over \F_3 and \F_5.Comment: submitte

A Multipartite Hajnal-Szemer\'edi Theorem

The celebrated Hajnal-Szemer\'edi theorem gives the precise minimum degree threshold that forces a graph to contain a perfect K_k-packing. Fischer's conjecture states that the analogous result holds for all multipartite graphs except for those formed by a single construction. Recently, we deduced an approximate version of this conjecture from new results on perfect matchings in hypergraphs. In this paper, we apply a stability analysis to the extremal cases of this argument, thus showing that the exact conjecture holds for any sufficiently large graph.Comment: Final version, accepted to appear in JCTB. 43 pages, 2 figure

Evidence for fresh frost layer on the bare nucleus of comet Hale--Bopp at 32 AU distance

Here we report that the activity of comet Hale--Bopp ceased between late 2007 and March, 2009, at about 28 AU distance from the Sun. At that time the comet resided at a distance from the Sun that exceeded the freeze-out distance of regular comets by an order of magnitude. A Herschel Space Observatory PACS scan was taken in mid-2010, in the already inactive state of the nucleus. The albedo has been found to be surprisingly large (8.1$\pm$0.9%{}), which exceeds the value known for any other comets. With re-reduction of archive HST images from 1995 and 1996, we confirm that the pre-perihelion albedo resembled that of an ordinary comet, and was smaller by a factor of two than the post-activity albedo. Our further observations with the Very Large Telescope (VLT) also confirmed that the albedo increased significantly by the end of the activity. We explain these observations by proposing gravitational redeposition of icy grains towards the end of the activity. This is plausible for such a massive body in a cold environment, where gas velocity is lowered to the range of the escape velocity. These observations also show that giant comets are not just the upscaled versions of the comets we know but can be affected by processes that are yet to be fully identified.Comment: 7 pages, 6 figures, accepted for publication in Ap