We study the space of L 2 harmonic forms on complete manifolds with metrics of fibred boundary or fibred cusp type. These metrics generalize the geometric structures at infinity of several different well-known classes of metrics, including asymptotically locally Euclidean manifolds, the (known types of) gravitational instantons, and also Poincaré metrics on Qrank 1 ends of locally symmetric spaces and on the complements of smooth divisors in Kähler manifolds. The answer in all cases is given in terms of intersection cohomology of a stratified compactification of the manifold. The L 2 signature formula implied by our result is closely related to the one proved by Dai [25] and more generally by Vaillant [67], and identifies Dai's τ invariant directly in terms of intersection cohomology of differing perversities. This work is also closely related to a recent paper of Carron [12] and the forthcoming paper of Cheeger and Dai [17]. We apply our results to a number of examples, gravitational instantons among them, arising in predictions about L 2 harmonic forms in duality theories in string theory.

Technologies that can detect and characterise particulates in liquids have applications in health, food and environmental monitoring. Simply counting the numbers of cells or particles however is not sufficient for most applications, and other physical properties must also be measured. Typically, it is necessary to compromise between chemical and biological specificity of the sensor and its speed. Here we present a low-cost and high-throughput multiuse counter that classifies a particle's size, concentration, porosity and shape. Using an additive manufacturing process, we have assembled a reusable flow resistive pulse sensor capable of being be tuned in real time to measure particles from 2-30 m, across a range of salt concentrations i.e. 2.5 × 10-4 to 0.1M. The device remains stable for several days with repeat measurements. We demonstrate its use for characterising algae with spherical and rod structures as well as microplastics shed from teabags. We present a methodology that results in a specific signal for microplastics, namely a conductive pulse, in contrast to particles with smooth surfaces such as calibration particles or algae, allowing the presence of microplastics to be easily confirmed and quantified. In addition, the shape of the signal and particle are correlated, giving an extra physical property to characterise suspended particulates. The technology can rapidly screen volumes of liquid, 1 mL/ min, for the presence of microplastics and algae.

Currently, researchers spend significant time manually searching through large volumes of data produced during scanning probe imaging to identify specific patterns and motifs formed via self-assembly and self-organization. Here, we use a combination of Monte Carlo simulations, general statistics, and machine learning to automatically distinguish several spatially correlated patterns in a mixed, highly varied data set of real AFM images of self-organized nanoparticles. We do this regardless of feature-scale and without the need for manually labeled training data. Provided that the structures of interest can be simulated, the strategy and protocols we describe can be easily adapted to other self-organized systems and data sets.

The discovery and characterisation of nanomaterials represents a multidisciplinary problem, here we apply predictive logistic regression models with resistive pulse sensing to create an rapid analysis technology.

Over the past fifty years, Hodge and signature theorems have been proved for various classes of noncompact and incomplete Riemannian manifolds. Two of these classes are manifolds with incomplete cylindrical ends and manifolds with cone bundle ends, that is, whose ends have the structure of a fibre bundle over a compact oriented manifold, where the fibres are cones on a second fixed compact oriented manifold. In this paper, we prove Hodge and signature theorems for a family of metrics on a manifold M with fibre bundle boundary that interpolates between the incomplete cylindrical metric and the cone bundle metric on M . We show that the Hodge and signature theorems for this family of metrics interpolate naturally between the known Hodge and signature theorems for the extremal metrics. The Hodge theorem involves intersection cohomology groups of varying perversities on the conical pseudomanifold X that completes the cone bundle metric on M . The signature theorem involves the summands i of Dai's invariant [10] that are defined as signatures on the pages of the Leray-Serre spectral sequence of the boundary fibre bundle of M . The two theorems together allow us to interpret the i in terms of perverse signatures, which are signatures defined on the intersection cohomology groups of varying perversities on X .