In particular, this means that smaller entries of A lead to random variables with smaller variance. On the other hand, the bound on ∥∥∥A− Â∥∥∥ 2 depends on the maximum variance. Thus, to improve the results, one idea is to keep entries of Aij with probability pij ≤ p, so that all entries in  have roughly the same variance. This will help us to get sparser matrices, while keeping similar qualit...

In this paper, we present a framework for fitting multivariate Hawkes processes for large-scale problems, both in the number of events in the observed history n and the number of event types d (i.e. dimensions). The proposed Scalable LowRank Hawkes Process (SLRHP) framework introduces a lowrank approximation of the kernel matrix that allows to perform the nonparametric learning of the d trigger...

We use this theorem to derive the same corollaries for the theories covered by the theorem as were derived for the strongly minimal case in [2]. We also note that the theorem is in some senses optimal. Specifically we can easily construct trivial Morley Rank 1 theories which are not categorical and for which the conclusion of the theorem fails. Also Marker in [3] constructs trivial totally cate...

Let A be an abelian variety defined over a number field k and F a finite Galois extension of k. Let p be a prime number. Then under certain not-too-stringent conditions on A and F we investigate the explicit Galois structure of the p-primary Selmer group of A over F . We also use the results so obtained to derive new bounds on the growth of the Selmer rank of A over extensions of k.

The article is devoted to different aspects of the question: ”What can be done with a matrix by a low rank perturbation?” It is proved that one can change a geometrically simple spectrum drastically by a rank 1 perturbation, but the situation is quite different if one restricts oneself to normal matrices. Also the Jordan normal form of a perturbed matrix is discussed. It is proved that with res...

We prove pointwise bounds for L eigenfunctions of the Laplace-Beltrami operator on locally symmetric spaces with Q-rank one if the corresponding eigenvalues lie below the continuous part of the L spectrum. Furthermore, we use these bounds in order to obtain some results concerning the L spectrum.

We introduce various notions of rank for a high order symmetric tensor, namely: rank, border rank, catalecticant rank, generalized rank, scheme length, border scheme length, extension rank and smoothable rank. We analyze the stratification induced by these ranks. The mutual relations between these stratifications, allow us to describe the hierarchy among all the ranks. We show that strict inequ...

We determine the p-rank of the incidence matrix of hyperplanes of PG(n, p) and points of a nondegenerate Hermitian variety. As a corollary, we obtain new bounds for the size of caps and the existence of ovoids in finite unitary spaces. This paper is a companion to [2], in which Blokhuis and this author derive the analogous p-ranks for quadrics.

LetG be a Polish group, τ a Polish topology on a spaceX, G acting continuously on (X, τ), with B ⊂ X G-invariant and in the Borel algebra generated by τ . Then there is a larger Polish topology τ ⊃ τ on X so that B is open with respect to τ, G still acts continuously on (X, τ), and τ has a basis consisting of sets that are of the same Borel rank as B relative to τ . 1

We prove that the smallest degree of an apolar 0-dimensional scheme to a general cubic form in n+ 1 variables is at most 2n+ 2, when n ≥ 8, and therefore smaller than the rank of the form. When n = 8 we show that the bound is sharp, i.e. the smallest degree of an apolar subscheme is 18.