Optimal eigenvalue estimate for the Dirac-Witten operator on bounded domains with smooth boundary

Eigenvalue estimate for the Dirac-Witten operator is given on bounded domains (with smooth boundary) of spacelike hypersurfaces satisfying the dominant energy condition, under four natural boundary conditions (MIT, APS, modified APS, and chiral conditions). This result is a generalisation of Friedrich's inequality for the usual Dirac operator. The limiting cases are also investigated.Comment: 2007, 18 pages, submitted 02 June 200

The Dirac operator on untrapped surfaces

We establish a sharp extrinsic lower bound for the first eigenvalue of the Dirac operator of an untrapped surface in initial data sets without apparent horizon in terms of the norm of its mean curvature vector. The equality case leads to rigidity results for the constraint equations with spherical boundary as well as uniqueness results for constant mean curvature surfaces in Minkowski space.Comment: 16 page

Optimal eigenvalues estimate for the Dirac operator on domains with boundary

We give a lower bound for the eigenvalues of the Dirac operator on a compact domain of a Riemannian spin manifold under the \MIT bag boundary condition. The limiting case is characterized by the existence of an imaginary Killing spinor.Comment: 10 page

Generic metrics and the mass endomorphism on spin three-manifolds

Let $(M,g)$ be a closed Riemannian spin manifold. The constant term in the expansion of the Green function for the Dirac operator at a fixed point $p\in M$ is called the mass endomorphism in $p$ associated to the metric $g$ due to an analogy to the mass in the Yamabe problem. We show that the mass endomorphism of a generic metric on a three-dimensional spin manifold is nonzero. This implies a strict inequality which can be used to avoid bubbling-off phenomena in conformal spin geometry.Comment: 8 page

Enhanced Canny edge detection for Covid-19 and pneumonia X-Ray images

In image processing, one of the most fundamental technique is edge detection. It is a process to detect edges from images by identifying discontinuities in brightness. In this research, we present an enhanced Canny edge detection technique. This method integrates local morphological contrast enhancement and Canny edge detection. Furthermore, the proposed edge detection technique was also applied for pneumonia and COVID-19 detection in digital x-ray images by utilising convolutional neural networks. Results show that this enhanced Canny edge detection technique is better than the traditional Canny technique. Also, we were able to produce classifiers that can classify edge x-ray images into COVID-19, normal, and pneumonia classes with high accuracy, sensitivity, and specificity

Extended Formulations in Mixed-integer Convex Programming

We present a unifying framework for generating extended formulations for the polyhedral outer approximations used in algorithms for mixed-integer convex programming (MICP). Extended formulations lead to fewer iterations of outer approximation algorithms and generally faster solution times. First, we observe that all MICP instances from the MINLPLIB2 benchmark library are conic representable with standard symmetric and nonsymmetric cones. Conic reformulations are shown to be effective extended formulations themselves because they encode separability structure. For mixed-integer conic-representable problems, we provide the first outer approximation algorithm with finite-time convergence guarantees, opening a path for the use of conic solvers for continuous relaxations. We then connect the popular modeling framework of disciplined convex programming (DCP) to the existence of extended formulations independent of conic representability. We present evidence that our approach can yield significant gains in practice, with the solution of a number of open instances from the MINLPLIB2 benchmark library.Comment: To be presented at IPCO 201

Spectral Bounds for Dirac Operators on Open Manifolds

We extend several classical eigenvalue estimates for Dirac operators on compact manifolds to noncompact, even incomplete manifolds. This includes Friedrich's estimate for manifolds with positive scalar curvature as well as the author's estimate on surfaces.Comment: pdflatex, 14 pages, 3 figure

Instantons and Killing spinors

We investigate instantons on manifolds with Killing spinors and their cones. Examples of manifolds with Killing spinors include nearly Kaehler 6-manifolds, nearly parallel G_2-manifolds in dimension 7, Sasaki-Einstein manifolds, and 3-Sasakian manifolds. We construct a connection on the tangent bundle over these manifolds which solves the instanton equation, and also show that the instanton equation implies the Yang-Mills equation, despite the presence of torsion. We then construct instantons on the cones over these manifolds, and lift them to solutions of heterotic supergravity. Amongst our solutions are new instantons on even-dimensional Euclidean spaces, as well as the well-known BPST, quaternionic and octonionic instantons.Comment: 40 pages, 2 figures v2: author email addresses and affiliations adde

The Ly6 Protein Coiled Is Required for Septate Junction and Blood Brain Barrier Organisation in Drosophila

Background: Genetic analysis of the Drosophila septate junctions has greatly contributed to our understanding of the mechanisms controlling the assembly of these adhesion structures, which bear strong similarities with the vertebrate tight junctions and the paranodal septate junctions. These adhesion complexes share conserved molecular components and have a common function: the formation of paracellular barriers restraining the diffusion of solutes through epithelial and glial envelopes. Methodology/Principal Findings: In this work we characterise the function of the Drosophila cold gene, that codes for a protein belonging to the Ly6 superfamily of extracellular ligands. Analysis of cold mutants shows that this gene is specifically required for the organisation of the septate junctions in epithelial tissues and in the nervous system, where its contribution is essential for the maintenance of the blood-brain barrier. We show that cold acts in a cell autonomous way, and we present evidence indicating that this protein could act as a septate junction component. Conclusion/Significance: We discuss the specific roles of cold and three other Drosophila members of the Ly6 superfamily that have been shown to participate in a non-redundant way in the process of septate junction assembly. We propose tha