Torsion and the second fundamental form for distributions

The second fundamental form of Riemannian geometry is generalised to the case of a manifold with a linear connection and an integrable distribution. This bilinear form is generally not symmetric and its skew part is the torsion. The form itself is closely related to the shape map of the connection. The codimension one case generalises the traditional shape operator of Riemannian geometry

Shape maps for second order partial differential equations

We analyse the singularity formation of congruences of solutions of systems of second order PDEs via the construction of \emph{shape maps}. The trace of such maps represents a congruence volume whose collapse we study through an appropriate evolution equation, akin to Raychaudhuri's equation. We develop the necessary geometric framework on a suitable jet space in which the shape maps appear naturally associated with certain linear connections. Explicit computations are given, along with a nontrivial example

Self Super-Resolution for Magnetic Resonance Images using Deep Networks

High resolution magnetic resonance~(MR) imaging~(MRI) is desirable in many clinical applications, however, there is a trade-off between resolution, speed of acquisition, and noise. It is common for MR images to have worse through-plane resolution~(slice thickness) than in-plane resolution. In these MRI images, high frequency information in the through-plane direction is not acquired, and cannot be resolved through interpolation. To address this issue, super-resolution methods have been developed to enhance spatial resolution. As an ill-posed problem, state-of-the-art super-resolution methods rely on the presence of external/training atlases to learn the transform from low resolution~(LR) images to high resolution~(HR) images. For several reasons, such HR atlas images are often not available for MRI sequences. This paper presents a self super-resolution~(SSR) algorithm, which does not use any external atlas images, yet can still resolve HR images only reliant on the acquired LR image. We use a blurred version of the input image to create training data for a state-of-the-art super-resolution deep network. The trained network is applied to the original input image to estimate the HR image. Our SSR result shows a significant improvement on through-plane resolution compared to competing SSR methods.Comment: Accepted by IEEE International Symposium on Biomedical Imaging (ISBI) 201

The Hox Paradox: More Complex(es) Than Imagined

AbstractAn understanding of the origin of different body plans requires knowledge of how the genes and genetic pathways that control embryonic development have evolved. The Hox genes provide an appealing starting point for such studies because they play a well-understood causal role in the regionalization of the body plan of all bilaterally symmetric animals. Vertebrate evolution has been characterized by gene, and possibly genome, duplication events, which are believed to have provided raw genetic material for selection to act upon. It has recently been established that the Hox gene organization of ray-finned fishes, such as the zebrafish, differs dramatically from that of their lobe-finned relatives, a group that includes humans and all the other widely used vertebrate model systems. This unusual Hox gene organization of zebrafish is the result of a duplication event within the ray-finned fish lineage. Thus, teleosts, such as zebrafish, have more Hox genes arrayed over more clusters (or “complexes”) than do tetrapod vertebrates. Here, I review our understanding of Hox cluster architecture in different vertebrates and consider the implications of gene duplication for Hox gene regulation and function and the evolution of different body plans

Civil War Canon: Sites of Confederate Memory in South Carolina

Confederate Commemoration in the Palmetto State First, this book is not about artillery. Nor is it a visitors’ guide to Civil War sites in South Carolina. Rather, Civil War Canon examines the evolution of Confederate memory and commemoration in the Palmetto State from the 1850s to t...

Catalogue of solar activity during 1957

Catalog of major solar activity during 195