This article concerns the expressive power of depth in deep feed-forward neural nets with ReLU activations. Specifically, we answer the following question: for a fixed d ≥ 1, what is the minimal width w so that neural nets with ReLU activations, input dimension d, hidden layer widths at most w, and arbitrary depth can approximate any continuous function of d variables arbitrarily well. It turns...

In a recent paper, the authors have shown that the secondary reduction of W-algebras provides a natural framework for the linearization of W-algebras. In particular, it allows in a very simple way the calculation of the linear algebra W(G, H) ≥0 associated to a wide class of W(G,H) algebras, as well as the expression of the W generators of W(G,H) in terms of the generators of W(G, H) ≥0. In thi...

Evaluation of bias-correction methods for ensemble streamflow volume forecasts T. Hashino, A. A. Bradley, and S. S. Schwartz University of Wisconsin, Department of Atmospheric and Ocean Sciences, Madison, Wisconsin, USA The University of Iowa, IIHR – Hydroscience & Engineering, Iowa City, Iowa, USA Center for Urban Environmental Research and Education, UMBC, Baltimore, Maryland, USA Received: 1...

Recent measurements of the properties of W + W − events produced in e + e − collisions at √ s ∼ 183 GeV at LEP are reviewed. The data are used to investigate the predicted effects of final state interactions, specifically " colour reconnection ". 1 Motivation Hadronic data in e + e − collisions can be characterised by event shape distributions and inclusive observables such as charged particle ...

Let {W (t), t ≥ 0} be a standard Brownian motion. If I is a bounded interval on which W has no zero, an almost sure lower bound to inf{|W (t)|, t ∈ I} can be provided, when I is taken from a given countable family of intervals covering the positive half-line. 1 Main Result Let {W (t), t ≥ 0} be a standard Brownian motion. Let I be some bounded interval of R. Suppose W (t) 6= 0, for all t ∈ I. W...

In this note we announce some results extending results of S. P. Novikov ([6] and [7]), the author [2], and C. T. C. Wall [8]. In the above papers it is shown how to characterize the homotopy type of 1-connected smooth closed manifolds of dimension w ^ S , « ^ 2 mod 4, and how to reduce the diffeomorphy classification of such manifolds to homotopy theory, with similar results for bounded manifo...

The Shumagin batholith is part of the Sanak-Baranof plutonic belt in Southwest Alaska that extends approximately 2200km from Sanak Island in the west to Baranof Island in the east. (Hudson, 1979; Hudson et al., 1979; Hill et al., 1980; Bradley et al., 2000; Haeussler et al., 2003). Previous work has indicated that the crystallization ages of the Sanak-Baranof plutons young from west to east. Th...

The tricyclic core of lycoposerramine S has been synthesised in 10 steps from a symmetrical cyclohexadiene precursor by way of a desymmetrising free-radical cyclisation and iodocyclisation.