Volume 15, Issue 3

Recently Andrews introduced the concept of signed partition: a signed partition is a finite sequence of integers a k , . . . , a 1 , a −1 , . . . , a −l such that a k ≥ ... ≥ a 1 > 0 > a −1 ≥ ... ≥ a −l . So far the signed partitions have been studied from an arithmetical point of view. In this paper we first generalize the concept of signed partition and we next use such a generalization to introduce a partial order on the set of all the signed partitions. Furthermore, we show that this order has many remarkable properties and that it generalizes the classical order on the Young lattice. Keywords Graded lattices, integer partitions, Young diagrams, sand piles models

We use vector-bundle techniques in order to compute dimW 1 d (C) where C is general and smooth in a linear system on an unnodal Enriques surface. We furthermore find new examples of smooth curves on Enriques surfaces with an infinite number of g 1 gon(C) ’s.

We establish versions of Michael’s Selection Theorem and Tietze’s Extension Theorem in the category of semilinear maps.

We prove that among the quasi-simple Lie groups only the group SL 4 (ℝ) occurs as the multiplication group of 3-dimensional connected topological loops L. These loops L are homeomorphic to the sphere S 3 . Moreover, there does not exist any connected topological loop having an at most 8-dimensional quasi-simple Lie groups as its multiplication group.

The finite field Kakeya problem asks both the minimum size of a point set inAG(2, q)which contains a line in every direction, as well as a characterization of the examples. Blokhuis and Mazzocca [2] solved this problem, and a subsequent paper [1] addresses the stability of this solution for even order planes, i.e. the spectrum of sizes near the minimum size of a Kakeya set for which non-minimum Kakeya sets exist. In this paper we provide some computational results in small order planes to determine the full spectrum of sizes of Kakeya sets. We then address some spectrum issues on the upper end of possible sizes, providing some bounds and new constructions.We also address the question of minimality, i.e.whether a given Kakeya set contains any smaller Kakeya set.

We give necessary and sufficient criteria for a smooth Enriques surface S ⊂ ℙ r to be scheme-theoretically an intersection of quadrics. Moreover we prove in many cases that, when S contains plane cubic curves, the intersection of the quadrics containing S is the union of S and the 2-planes spanned by the plane cubic curves. We also give a new (very quick) proof of the projective normality of S if deg S ≥ 12.

We introduce topological contact dynamics of a smooth manifold carrying a cooriented contact structure, generalizing previous work in the case of a symplectic structure [27] or a contact form [5]. A topological contact isotopy is not generated by a vector field; nevertheless, the group identities, the transformation law, and classical uniqueness results in the smooth case extend to topological contact isotopies and homeomorphisms, giving rise to an extension of smooth contact dynamics to topological dynamics. Our approach is via symplectization of a contact manifold, and our main tools are an energy-capacity inequality we prove for contact diffeomorphisms, combined with techniques from measure theory on oriented manifolds. We establish non-degeneracy of a Hofer-like bi-invariant pseudo-metric on the group of strictly contact diffeomorphisms constructed in [4]. The topological automorphism group of the contact structure exhibits rigidity properties analogous to those of symplectic diffeomorphisms, including C 0 -rigidity of contact and strictly contact diffeomorphisms.

We study the existence of invariant metrics with holonomy G* 2(2) ⊂ SO(4, 3) on compact nilmanifolds, i.e. on compact quotients of nilpotent Lie groups by discrete subgroups. We prove that, up to isomorphism, there exists only one indecomposable nilpotent Lie algebra admitting a torsion-free G* 2(2) -structure such that the center is definite with respect to the induced inner product. In particular, we show that the associated compact nilmanifold admits a 3-parameter family of invariant metrics with full holonomy G* 2(2) .