For each d-1 there is a constant Ca > 0 such that any finite set X c R a contains a subset YcX, IYI<-[~d(d+3)J +1 having the following property: if E = Y is an ellipsoid, then IE nXt-> c~JXl.

If G is any finite group and k is a field, there is a natural construction of a Hopf algebra over k associated to G, the Drinfel’d double D(G). We prove that if G is any finite real reflection group, with Drinfel’d double D(G) over an algebraically closed field k of characteristic not 2, then every simple D(G)-module has Frobenius-Schur indicator +1. This generalizes the classical results for m...

It is, in general, very hard to predict the features of the continued fraction expansion of a given positive real number. If the number in question is of the form √ d, where d is a positive integer which is not a square, then its continued fraction expansion is of the form [a0, {a1, . . . , ar−1, 2a0}], where we use {. . . } to emphasize the period of the expansion. It is known that a1, . . . ,...

In this paper, we describe a new infinite family of q −1 2 -tight sets in the hyperbolic quadrics Q(5, q), for q ≡ 5 or 9 mod 12. Under the Klein correspondence, these correspond to Cameron–Liebler line classes of PG(3, q) having parameter q −1 2 . This is the second known infinite family of nontrivial Cameron–Liebler line classes, the first family having been described by Bruen and Drudge with...

We point out an interplay between Fq-Frobenius non-classical plane curves and complete (k, d)-arcs in P2(Fq). A typical example that shows how this works is the one concerning an Hermitian curve. We present some other examples here which give rise to the existence of new complete (k, d)-arcs with parameters k = d(q − d + 2) and d = (q − 1)/(q − 1), q being a power of the characteristic. In addi...

Let A denote the ring of differential operators on the affine line with its two usual generators t and d dt given degrees +1 and −1 respectively. Let X be the stack having coarse moduli space the affine line Spec k[z] and isotropy groups Z/2 at each integer point. Then the category of graded A-modules is equivalent to the category of quasicoherent sheaves on X .

We suppose that p = 2α3β +1, where α ≥ 1, β ≥ 0, and p ≥ 7 is a prime number. Then we prove that the simple groups An, where n = p,p+1, or p+2, and finite groups Sn, where n = p,p+1, are also uniquely determined by their order components. As corollaries of these results, the validity of a conjecture of J. G. Thompson and a conjecture of Shi and Bi (1990) both on An, where n= p,p+1, or p+2, is o...

Given a family F and a host graph H, a graph G ⊆ H is F-saturated relative to H if no subgraph of G lies in F but adding any edge from E(H)−E(G) to G creates such a subgraph. In the F-saturation game on H, players Max and Min alternately add edges of H to G, avoiding subgraphs in F , until G becomes F-saturated relative to H. They aim to maximize or minimize the length of the game, respectively...