12 research outputs found
The BPHZ Theorem for Regularity Structures via the Spectral Gap Inequality
We provide a relatively compact proof of the BPHZ theorem for regularity
structures of decorated trees in the case where the driving noise satisfies a
suitable spectral gap property, as in the Gaussian case. This is inspired by
the recent work [LOTT21] in the multi-index setting, but our proof relies
crucially on a novel version of the reconstruction theorem for a space of
"pointed Besov modelled distributions". As a consequence, the analytical core
of the proof is quite short and self-contained, which should make it easier to
adapt the proof to different contexts (such as the setting of discrete models)
Non-optimality of the Greedy Algorithm for subspace orderings in the method of alternating projections
The method of alternating projections involves projecting an element of a
Hilbert space cyclically onto a collection of closed subspaces. It is known
that the resulting sequence always converges in norm and that one can obtain
estimates for the rate of convergence in terms of quantities describing the
geometric relationship between the subspaces in question, namely their pairwise
Friedrichs numbers. We consider the question of how best to order a given
collection of subspaces so as to obtain the best estimate on the rate of
convergence. We prove, by relating the ordering problem to a variant of the
famous Travelling Salesman Problem, that correctness of a natural form of the
Greedy Algorithm would imply that , before presenting a
simple example which shows that, contrary to a claim made in the influential
paper [Kayalar-Weinert, Math. Control Signals Systems, vol. 1(1), 1988], the
result of the Greedy Algorithm is not in general optimal. We go on to establish
sharp estimates on the degree to which the result of the Greedy Algorithm can
differ from the optimal result. Underlying all of these results is a
construction which shows that for any matrix whose entries satisfy certain
natural assumptions it is possible to construct a Hilbert space and a
collection of closed subspaces such that the pairwise Friedrichs numbers
between the subspaces are given precisely by the entries of that matrix.Comment: To appear in Results in Mathematic
The measure has sub-Gaussian tails
We provide a very simple argument showing that the measure does
have quartic exponential tails, as expected from its formal expression. This
completes the programme of recovering the verification of the Osterwalder and
Schrader axioms for that measure based purely on SPDE techniques
Managing racial equality and diversity in the UK construction industry
Managing racial equality and diversity in the UK construction industr