The horizon problem for prevalent surfaces

We investigate the box dimensions of the horizon of a fractal surface defined by a function $f \in C[0,1]^2$. In particular we show that a prevalent surface satisfies the `horizon property', namely that the box dimension of the horizon is one less than that of the surface. Since a prevalent surface has box dimension 3, this does not give us any information about the horizon of surfaces of dimension strictly less than 3. To examine this situation we introduce spaces of functions with surfaces of upper box dimension at most \alpha, for \alpha $\in$ [2,3). In this setting the behaviour of the horizon is more subtle. We construct a prevalent subset of these spaces where the lower box dimension of the horizon lies between the dimension of the surface minus one and 2. We show that in the sense of prevalence these bounds are as tight as possible if the spaces are defined purely in terms of dimension. However, if we work in Lipschitz spaces, the horizon property does indeed hold for prevalent functions. Along the way, we obtain a range of properties of box dimensions of sums of functions

Construction and testing of self-drilled soil nails

Current standards and best practice guidance recognise that testing of self-drilled hollow bar soil nails can be problematic as conventional packers and debonded lengths cannot be constructed. As a result, this causes difficulty in testing and confirming the ultimate bond resistance within the passive zone of a soil-nailed slope, and thus the design soil nail lengths. This paper provides a summary and review of the various testing procedures adopted for a soil nail construction project in Scotland. The practical design considerations, and their validation through the installation and testing of 49 sacrificial test nails, are detailed. The construction issues associated with the nail installation and testing are also outlined and discussed in light of the results obtained using different testing approaches. The aim of this case study is to report on the experiences with installation and testing of hollow bar soil nails. The objectives are to develop an initial data base of available soil–grout bond strength of hollow bar soil nails based on the several practical installation procedures used in this project and to establish areas for improvement of installation, testing and quality control in order to perform comparable pullout tests on self-drilled hollow bar soil nails. </jats:p

Label-free, single molecule detection of cytokines using optical microcavities

Interleukin-2 (IL2) is a cytokine that regulates T-cell growth and is used in cancer therapies. By sensitizing a microcavity sensor surface with anti-IL2 and monitoring the resonant frequency, single molecules of IL2 can be detected

On the Assouad dimension of self-similar sets with overlaps

It is known that, unlike the Hausdorff dimension, the Assouad dimension of a self-similar set can exceed the similarity dimension if there are overlaps in the construction. Our main result is the following precise dichotomy for self-similar sets in the line: either the \emph{weak separation property} is satisfied, in which case the Hausdorff and Assouad dimensions coincide; or the \emph{weak separation property} is not satisfied, in which case the Assouad dimension is maximal (equal to one). In the first case we prove that the self-similar set is Ahlfors regular, and in the second case we use the fact that if the \emph{weak separation property} is not satisfied, one can approximate the identity arbitrarily well in the group generated by the similarity mappings, and this allows us to build a \emph{weak tangent} that contains an interval. We also obtain results in higher dimensions and provide illustrative examples showing that the `equality/maximal' dichotomy does not extend to this setting.Comment: 24 pages, 2 figure

Endogenous protease(s) in extracts of Neurospora mycella activate the exonuclease associated with a putative Rec-nuclease

Activation of exonuclease by endogenous protease