Let R = k[x1, . . . , xr] be the polynomial ring in r variables over an infinite field k, and let M be the maximal ideal of R. Here a level algebra will be a graded Artinian quotient A of R having socle Soc(A) = 0 : M in a single degree j. The Hilbert function H(A) = (h0, h1, . . . , hj) gives the dimension hi = dimk Ai of each degree-i graded piece of A for 0 ≤ i ≤ j. The embedding dimension o...

We study a family of transcendental entire functions of genus zero, for which all of the zeros lie within a closed sector strictly smaller than a half-plane. In general these functions lie outside the Eremenko-Lyubich class. We show that for functions in this family the fast escaping set has Hausdorff dimension equal to two.

We investigate the question which (separable metrizable) spaces have a ‘large’ almost disjoint family of connected (and locally connected) sets. Every compact space of dimension at least 2 as well as all compact spaces containing an ‘uncountable star’ have such a family. Our results show that the situation for 1-dimensional compacta is unclear.  2004 Elsevier B.V. All rights reserved.

We compute the Minkowski dimension for a family of self-affine sets on R. Our result holds for every (rather than generic) set in the class. Moreover, we exhibit explicit open subsets of this class where we allow overlapping, and do not impose any conditions on the norms of the linear maps. The family under consideration was inspired by the theory of Kakeya sets.

In this paper we find the asymptotic behavior of the spectral counting function for the Steklov problem in a family of self similar domains with fractal boundaries. Using renewal theory, we show that the main term in the asymptotics depends on the Minkowski dimension of the boundary. Also, we compute explicitly a three term expansion for a family of self similar sets, and a two term asymptotic ...

We introduce a family of copulas which are locally piecewise uniform in the interior of the unit cube of any given dimension. Within that family, the simultaneous control of tail dependencies of all projections to faces of the cube is possible and we give an efficient sampling algorithm. The combination of these two properties may be appealing to risk modellers.

In order to compute the packing dimension of orthogonal projections Falconer and Howroyd (1997) introduced a family of packing dimension profiles Dims that are parametrized by real numbers s > 0. Subsequently, Howroyd (2001) introduced alternate s-dimensional packing dimension profiles P-dims and proved, among many other things, that P-dimsE = DimsE for all integers s > 0 and all analytic sets ...

We study the VC-dimension of the set system on the vertex set of some graph which is induced by the family of its k-connected subgraphs. In particular, we give tight upper and lower bounds for the VC-dimension. Moreover, we show that computing the VC-dimension is NP-complete and that it remains NP-complete for split graphs and for some subclasses of planar bipartite graphs in the cases k = 1 an...