A Hardy inequality in twisted waveguides

We show that twisting of an infinite straight three-dimensional tube with non-circular cross-section gives rise to a Hardy-type inequality for the associated Dirichlet Laplacian. As an application we prove certain stability of the spectrum of the Dirichlet Laplacian in locally and mildly bent tubes. Namely, it is known that any local bending, no matter how small, generates eigenvalues below the essential spectrum of the Laplacian in the tubes with arbitrary cross-sections rotated along a reference curve in an appropriate way. In the present paper we show that for any other rotation some critical strength of the bending is needed in order to induce a non-empty discrete spectrum.Comment: LaTeX, 20 page

Heat Kernel on Homogeneous Bundles over Symmetric Spaces

We consider Laplacians acting on sections of homogeneous vector bundles over symmetric spaces. By using an integral representation of the heat semi-group we find a formal solution for the heat kernel diagonal that gives a generating function for the whole sequence of heat invariants. We show explicitly that the obtained result correctly reproduces the first non-trivial heat kernel coefficient as well as the exact heat kernel diagonals on two-dimensional sphere $S^2$ and the hyperbolic plane $H^2$. We argue that the obtained formal solution correctly reproduces the exact heat kernel diagonal after a suitable regularization and analytical continuation.Comment: 55 page

GET: The Connection between Monogenic Scale-Space and Gaussian derivatives

In this paper we propose a new operator which combines advantages of monogenic scale-space and Gaussian scale-space, of the monogenic signal and the structure tensor. The gradient energy tensor (GET) defined in this paper is based on Gaussian derivatives up to third order using different scales. These filters are commonly available, separable, and have an optimal uncertainty. The response of this new operator can be used like the monogenic signal to estimate the local amplitude, the local phase, and the local orientation of an image, but it also allows to measure the coherence of image regions as in the case of the structure tensor. Both theoretically and in experiments the new approach compares favourably with existing methods

Targeting of Rough Endoplasmic Reticulum Membrane Proteins and Ribosomes in Invertebrate Neurons

The endoplasmic reticulum (ER) is divided into rough and smooth domains (RER and SER). The two domains share most proteins, but RER is enriched in some membrane proteins by an unknown mechanism. We studied RER protein targeting by expressing fluorescent protein fusions to ER membrane proteins in Caenorhabditis elegans. In several cell types RER and general ER proteins colocalized, but in neurons RER proteins were concentrated in the cell body, whereas general ER proteins were also found in neurites. Surprisingly RER membrane proteins diffused rapidly within the cell body, indicating they are not localized by immobilization. Ribosomes were also concentrated in the cell body, suggesting they may be in part responsible for targeting RER membrane proteins