Linear convergence of accelerated conditional gradient algorithms in spaces of measures

A class of generalized conditional gradient algorithms for the solution of optimization problem in spaces of Radon measures is presented. The method iteratively inserts additional Dirac-delta functions and optimizes the corresponding coefficients. Under general assumptions, a sub-linear $\mathcal{O}(1/k)$ rate in the objective functional is obtained, which is sharp in most cases. To improve efficiency, one can fully resolve the finite-dimensional subproblems occurring in each iteration of the method. We provide an analysis for the resulting procedure: under a structural assumption on the optimal solution, a linear $\mathcal{O}(\zeta^k)$ convergence rate is obtained locally.Comment: 30 pages, 7 figure

Biogenesis of mitochondrial c-type cytochromes

Cytochromesc andc 1 are essential components of the mitochondrial respiratory chain. In both cytochromes the heme group is covalently linked to the polypeptide chain via thioether bridges. The location of the two cytochromes is in the intermembrane space; cytochromec is loosely attached to the surface of the inner mitochondrial membrane, whereas cytochromec 1 is firmly anchored to the inner membrane. Both cytochromec andc 1 are encoded by nuclear genes, translated on cytoplasmic ribosomes, and are transported into the mitochondria where they become covalently modified and assembled. Despite the many similarities, the import pathways of cytochromec andc 1 are drastically different. Cytochromec 1 is made as a precursor with a complex bipartite presequence. In a first step the precursor is directed across outer and inner membranes to the matrix compartment of the mitochondria where cleavage of the first part of the presequence takes place. In a following step the intermediate-size form is redirected across the inner membrane; heme addition then occurs on the surface of the inner membrane followed by the second processing reaction. The import pathway of cytochromec is exceptional in practically all aspects, in comparison with the general import pathway into mitochondria. Cytochromec is synthesized as apocytochromec without any additional sequence. It is translocated selectively across the outer membrane. Addition of the heme group, catalyzed by cytochromec heme lyase, is a requirement for transport. In summary, cytochromec 1 import appears to follow a conservative pathway reflecting features of cytochromec 1 sorting in prokaryotic cells. In contrast, cytochromec has invented a rather unique pathway which is essentially non-conservative

Foundations for an iteration theory of entire quasiregular maps

The Fatou-Julia iteration theory of rational functions has been extended to quasiregular mappings in higher dimension by various authors. The purpose of this paper is an analogous extension of the iteration theory of transcendental entire functions. Here the Julia set is defined as the set of all points such that complement of the forward orbit of any neighbourhood has capacity zero. It is shown that for maps which are not of polynomial type the Julia set is non-empty and has many properties of the classical Julia set of transcendental entire functions.Comment: 31 page

Maximal uniform convergence rates in parametric estimation problems

This paper considers parametric estimation problems with independent, identically,non-regularly distributed data. It focuses on rate-effciency, in the sense of maximal possible convergence rates of stochastically bounded estimators, as an optimality criterion,largely unexplored in parametric estimation.Under mild conditions, the Hellinger metric,defined on the space of parametric probability measures, is shown to be an essentially universally applicable tool to determine maximal possible convergence rates. These rates are shown to be attainable in general classes of parametric estimation problems.

Maximal uniform convergence rates in parametric estimation problems

This paper considers parametric estimation problems with i.i.d. data. It focusses on rate-effciency, in the sense of maximal possible convergence rates of stochastically bounded estimators, as an optimality criterion, largely unexplored in parametric estimation. Under mild conditions, the Hellinger metric, defined on the space of parametric probability measures, is shown to be an essentially universally applicable tool to determine maximal possible convergence rates.parametric estimators, uniform convergence, Hellinger distance, Locally Asymptotically Quadratic (LAQ) Families

Expected neutrino fluence from short Gamma-Ray Burst 170817A and off-axis angle constraints

We compute the expected neutrino fluence from SGRB 170817A, associated with the gravitational wave event GW 170817, directly based on Fermi observations in two scenarios: structured jet and off-axis (observed) top-hat jet. While the expected neutrino fluence for the structured jet case is very small, large off-axis angles imply high radiation densities in the jet, which can enhance the neutrino production efficiency. In the most optimistic allowed scenario, the neutrino fluence can reach only $10^{-4}$ of the sensitivity of the neutrino telescopes. We furthermore demonstrate that the fact that gamma-rays can escape limits the baryonic loading (energy in protons versus photons) and the off-axis angle for the internal shock scenario. In particular, for a baryonic loading of ten, the off-axis angle is more strongly constrained by the baryonic loading than by the time delay between the gravitational wave event and the onset of the gamma-ray emission.Comment: 9 pages, 6 figure

Numerical Analysis of Sparse Initial Data Identification for Parabolic Problems

In this paper we consider a problem of initial data identification from the final time observation for homogeneous parabolic problems. It is well-known that such problems are exponentially ill-posed due to the strong smoothing property of parabolic equations. We are interested in a situation when the initial data we intend to recover is known to be sparse, i.e. its support has Lebesgue measure zero. We formulate the problem as an optimal control problem and incorporate the information on the sparsity of the unknown initial data into the structure of the objective functional. In particular, we are looking for the control variable in the space of regular Borel measures and use the corresponding norm as a regularization term in the objective functional. This leads to a convex but non-smooth optimization problem. For the discretization we use continuous piecewise linear finite elements in space and discontinuous Galerkin finite elements of arbitrary degree in time. For the general case we establish error estimates for the state variable. Under a certain structural assumption, we show that the control variable consists of a finite linear combination of Dirac measures. For this case we obtain error estimates for the locations of Dirac measures as well as for the corresponding coefficients. The key to the numerical analysis are the sharp smoothing type pointwise finite element error estimates for homogeneous parabolic problems, which are of independent interest. Moreover, we discuss an efficient algorithmic approach to the problem and show several numerical experiments illustrating our theoretical results.Comment: 43 pages, 10 figure

Effects of Smooth Boundaries on Topological Edge Modes in Optical Lattices

Since the experimental realization of synthetic gauge fields for neutral atoms, the simulation of topologically non-trivial phases of matter with ultracold atoms has become a major focus of cold atom experiments. However, several obvious differences exist between cold atom and solid state systems, for instance the finite size of the atomic cloud and the smooth confining potential. In this article we show that sharp boundaries are not required to realize quantum Hall or quantum spin Hall physics in optical lattices and, on the contrary, that edge states which belong to a smooth confinement exhibit additional interesting properties, such as spatially resolved splitting and merging of bulk bands and the emergence of robust auxiliary states in bulk gaps to preserve the topological quantum numbers. In addition, we numerically validate that these states are robust against disorder. Finally, we analyze possible detection methods, with a focus on Bragg spectroscopy, to demonstrate that the edge states can be detected and that Bragg spectroscopy can reveal how topological edge states are connected to the different bulk bands.Comment: 12 pages, 10 figures, updated figures and minor text correction