arXiv subjects

Eugenie Hunsicker

Publications and source records attributed to Eugenie Hunsicker.

13 recordsLinked to original sources

arXiv2020

Improving the Segmentation of Scanning Probe Microscope Images using Convolutional Neural Networks

A wide range of techniques can be considered for segmentation of images of nanostructured surfaces. Manually segmenting these images is time-consuming and results in a user-dependent segmentation bias, while there is currently no consensus on the best automated segmentation methods for particular techniques, image classes, and samples. Any image segmentation approach must minimise the noise in the images to ensure accurate and meaningful statistical analysis can be carried out. Here we develop protocols for the segmentation of images of 2D assemblies of gold nanoparticles formed on silicon surfaces via deposition from an organic solvent. The evaporation of the solvent drives far-from-equilibrium self-organisation of the particles, producing a wide variety of nano- and micro-structured patterns. We show that a segmentation strategy using the U-Net convolutional neural network outperforms traditional automated approaches and has particular potential in the processing of images of nanostructured systems.

eess.IV↗

arXiv2020

Fast and automated biomarker detection in breath samples with machine learning

Volatile organic compounds (VOCs) in human breath can reveal a large spectrum of health conditions and can be used for fast, accurate and non-invasive diagnostics. Gas chromatography-mass spectrometry (GC-MS) is used to measure VOCs, but its application is limited by expert-driven data analysis that is time-consuming, subjective and may introduce errors. We propose a system to perform GC-MS data analysis that exploits deep learning pattern recognition ability to learn and automatically detect VOCs directly from raw data, thus bypassing expert-led processing. The new proposed approach showed to outperform the expert-led analysis by detecting a significantly higher number of VOCs in just a fraction of time while maintaining high specificity. These results suggest that the proposed method can help the large-scale deployment of breath-based diagnosis by reducing time and cost, and increasing accuracy and consistency.

cs.CV↗

arXiv2020

Architecture Definition in Complex System Design Using Model Theory

Architecture Definition, which is central to system design, is one of the two most used technical processes in the practice of model-based systems engineering. In this paper a fundamental approach to architecture definition is presented and demonstrated. The success of its application to engineering problems depends on a precise but practical definition of the term architecture. In the standard for Architecture Description, ISO/IEC/IEEE 42010:2011, a definition was adopted that has been subsumed into later standards. In 2018 the working group JTC1/SC7/WG42 on System Architecture began a review of the adopted definition, holding sessions late in the year. This paper extends and complements a position paper submitted during the meetings; in which Tarski model theory and ISO/IEC 24707:2018 (logic-based languages) were used to better understand relationships between system models and concepts related to architecture. Independent from the working group, it now contributes intuitive fundamental definitions of the terms architecture and system that are used to specify a mathematically based technical process for architecture definition. The engineering utility and benefits to complex system design are demonstrated in a diesel engine emissions reduction case study.

cs.SE↗

arXiv2017

Isospectrality For Orbifold Lens Spaces

We answer Mark Kac's famous question, "can one hear the shape of a drum?" in the positive for orbifolds that are 3-dimensional and 4-dimensional lens spaces; we thus complete the answer to this question for orbifold lens spaces in all dimensions. We also show that the coefficients of the asymptotic expansion of the trace of the heat kernel are not sufficient to determine the above results.

math.DG↗

arXiv2015

Hodge Theory for Intersection Space Cohomology

Given a perversity function in the sense of intersection homology theory, the method of intersection spaces assigns to certain oriented stratified spaces cell complexes whose ordinary reduced homology with real coefficients satisfies Poincaré duality across complementary perversities. The resulting homology theory is well-known not to be isomorphic to intersection homology. For a two-strata pseudomanifold with product link bundle, we give a description of the cohomology of intersection spaces as a space of weighted $L^2$ harmonic forms on the regular part, equipped with a fibred scattering metric. Some consequences of our methods for the signature are discussed as well.

math.AT↗

arXiv2013

A Parametrix Construction for the Laplacian on Q-rank 1 Locally Symmetric Space

This paper presents the construction of parametrices for the Gauss-Bonnet and Hodge Laplace operators on noncompact manifolds modelled on Q-rank 1 locally symmetric spaces. These operators are, up to a scalar factor, $ϕ$-differential operators, that is, they live in the generalised $ϕ$-calculus studied by the authors in a previous paper, which extends work of Melrose and Mazzeo. However, because they are not totally elliptic elements in this calculus, it is not possible to construct parametrices for these operators within the $ϕ$-calculus. We construct parametrices for them in this paper using a combination of the $b$-pseudodifferential operator calculus of R. Melrose and the $ϕ$-pseudodifferential operator calculus. The construction simplifies and generalizes the construction done by Vaillant in his thesis for the Dirac operator. In addition, we study the mapping properties of these operators and determine the appropriate Hlibert spaces between which the Gauss-Bonnet and Hodge Laplace operators are Fredholm. Finally, we establish regularity results for elements of the kernels of these operators.

math.AP↗

arXiv2012

Analysis of Schrödinger operators with inverse square potentials {II}: FEM and approximation of eigenfunctions in the periodic case

Let $V$ be a {\em periodic} potential on $\RR^3$ that is smooth everywhere except at a discrete set $\maS$ of points, where it has singularities of the form $Z/ρ^2$, with $ρ(x) = |x - p|$ for $x$ close to $p$ and $Z$ is continuous, $Z(p) > -1/4$ for $p \in \maS$. We also assume that $ρ$ and $Z$ are smooth outside $\maS$ and $Z$ is smooth in polar coordinates around each singular point. Let us denote by $Λ$ the periodicity lattice and set $\TT := \RR^3/ Λ$. In the first paper of this series \cite{HLNU1}, we obtained regularity results in weighted Sobolev space for the eigenfunctions of the Schrödinger-type operator $H = -Δ+ V$ acting on $L^2(\TT)$, as well as for the induced $\vt k$--Hamiltonians $\Hk$ obtained by resticting the action of $H$ to Bloch waves. In this paper we present two related applications: one to the Finite Element approximation of the solution of $(L+\Hk) v = f$ and one to the numerical approximation of the eigenvalues, $λ$, and eigenfunctions, $u$, of $\Hk$. We give optimal, higher order convergence results for approximation spaces defined piecewise polynomials. Our numerical tests are in good agreement with the theoretical results.

math-ph↗

arXiv2012

Analysis of Schrödinger operators with inverse square potentials I: regularity results in 3D

Let $V$ be a potential on $\RR^3$ that is smooth everywhere except at a discrete set $\maS$ of points, where it has singularities of the form $Z/ρ^2$, with $ρ(x) = |x - p|$ for $x$ close to $p$ and $Z$ continuous on $\RR^3$ with $Z(p) > -1/4$ for $p \in \maS$. Also assume that $ρ$ and $Z$ are smooth outside $\maS$ and $Z$ is smooth in polar coordinates around each singular point. We either assume that $V$ is periodic or that the set $\maS$ is finite and $V$ extends to a smooth function on the radial compactification of $\RR^3$ that is bounded outside a compact set containing $\maS$. In the periodic case, we let $Λ$ be the periodicity lattice and define $\TT := \RR^3/ Λ$. We obtain regularity results in weighted Sobolev space for the eigenfunctions of the Schrödinger-type operator $H = -Δ+ V$ acting on $L^2(\TT)$, as well as for the induced $\vt k$--Hamiltonians $\Hk$ obtained by restricting the action of $H$ to Bloch waves. Under some additional assumptions, we extend these regularity and solvability results to the non-periodic case. We sketch some applications to approximation of eigenfunctions and eigenvalues that will be studied in more detail in a second paper.

math.NA↗

arXiv2011

Additivity and non-additivity for perverse signatures

A well-known property of the signature of closed oriented 4n-dimensional manifolds is Novikov additivity, which states that if a manifold is split into two manifolds with boundary along an oriented smooth hypersurface, then the signature of the original manifold equals the sum of the signatures of the resulting manifolds with boundary. Wall showed that this property is not true of signatures on manifolds with boundary and that the difference from additivity could be described as a certain Maslov triple index. Perverse signatures are signatures defined for any stratified pseudomanifold, using the intersection homology groups of Goresky and MacPherson. In the case of Witt spaces, the middle perverse signature is the same as the Witt signature. This paper proves a generalization to perverse signatures of Wall's non-additivity theorem for signatures of manifolds with boundary. Under certain topological conditions on the dividing hypersurface,Novikov additivity for perverse signatures may be deduced as a corollary. In particular, Siegel's version of Novikov additivity for Witt signatures is a special case of this corollary.

math.GT↗

arXiv2009

Pseudodifferential operator calculus for generalized Q-rank 1 locally symmetric spaces, I

This paper is the first of two papers constructing a calculus of pseudodifferential operators suitable for doing analysis on Q-rank 1 locally symmetric spaces and Riemannian manifolds generalizing these. This generalization is the interior of a manifold with boundary, where the boundary has the structure of a tower of fibre bundles. The class of operators we consider on such a space includes those arising naturally from metrics which degenerate to various orders at the boundary, in directions given by the tower of fibrations. As well as Q-rank 1 locally symmetric spaces, examples include Ricci-flat metrics on the complement of a divisor in a smooth variety constructed by Tian and Yau. In this first part of the calculus construction, parametrices are found for "fully elliptic differential \bfa-operators", which are uniformly elliptic operators on these manifolds that satisfy an additional invertibility condition at infinity. In the second part we will consider operators that do not satisfy this condition.

math.AP↗

arXiv2005

Harmonic forms on manifolds with edges

Let $(X,g)$ be a compact Riemannian stratified space with simple edge singularity. Thus a neighbourhood of the singular stratum is a bundle of truncated cones over a lower dimensional compact smooth manifold. We calculate the various polynomially weighted de Rham cohomology spaces of $X$, as well as the associated spaces of harmonic forms. In the unweighted case, this is closely related to recent work of Cheeger and Dai \cite{CD}. Because the metric $g$ is incomplete, this requires a consideration of the various choices of ideal boundary conditions at the singular set. We also calculate the space of $L^2$ harmonic forms for any complete edge metric on the regular part of $X$.

math.DG↗

arXiv2005

Hodge and signature theorems for a family of manifolds with fibration boundary

Let $\bar{M}$ be a manifold with boundary $Y$ which is the total space of a fibre bundle, and is defined by the vanishing of a boundary defining function, $x$. We prove $L^2$ Hodge and signature theorems for $M$ endowed with a metric of the form $dx^2 + x^{2c} h + k$, where $k$ is the lift to $Y$ of the metric on the base of the fibre bundle, $h$ is a two form on $Y$ which restricts to a metric on each fibre, and $0 \leq c \leq 1$. These metrics interpolate between the case when $c=0$, in which case the metric near the boundary is a cylinder, and the case where $c=1$, in which case the metric near the boundary is that of a cone bundle over the base of the boundary fibration. We show that the $L^2$ Hodge theorems for the cohomologies given by the maximal and minimal extensions of $d$ with respect to these metrics and the $L^2$ signature theorem for the image of the minimal cohomology in the maximal cohomology interpolate between known results for $L^2$ Hodge and signature theorems for cylindrical and cone bundle type metrics. In particular, the Hodge theorems all relate the related spaces of $L^2$ harmonic forms to intersection cohomology of varying perversities for $X$, the space formed from $\bar{M}$ by collapsing the fibres of $Y$ at the boundary. The signature theorem involves variations on the $τ$ invariant described by Dai.

math.GT↗

arXiv2003

Hodge cohomology of gravitational instantons

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 Q-rank 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 [dai] and more generally by Vaillant [Va], and identifies Dai's tau invariant directly in terms of intersection cohomology of differing perversities. This work is also closely related to a recent paper of Carron [Car] and the forthcoming paper of Cheeger and Dai [CD]. 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.

math.DG↗