Interpolation by Linear Functions on an nn-Dimensional Ball

By $B=B(x^{(0)};R)$ we denote the Euclidean ball in ${\mathbb R}^n$ given by the inequality $\|x-x^{(0)}\|\leq R$. Here $x^{(0)}\in{\mathbb R}^n, R>0$, $\|x\|:=\left(\sum_{i=1}^n x_i^2\right)^{1/2}$. We mean by $C(B)$ the space of continuous functions $f:B\to{\mathbb R}$ with the norm $\|f\|_{C(B)}:=\max_{x\in B}|f(x)|$ and by $\Pi_1\left({\mathbb R}^n\right)$ the set of polynomials in $n$ variables of degree $\leq 1$, i.e., linear functions on ${\mathbb R}^n$. Let $x^{(1)}, \ldots, x^{(n+1)}$ be the vertices of $n$-dimensional nondegenerate simplex $S\subset B$. The interpolation projector $P:C(B)\to \Pi_1({\mathbb R}^n)$ corresponding to $S$ is defined by the equalities $Pf\left(x^{(j)}\right)=f\left(x^{(j)}\right).$ We obtain the formula to compute the norm of $P$ as an operator from $C(B)$ into $C(B)$ via $x^{(0)}$, $R$ and coefficients of basic Lagrange polynomials of $S$. In more details we study the case when $S$ is a regular simplex inscribed into $B_n=B(0,1)$.Comment: 17 pages, 6 figures, 1 tabl

FitSKIRT: genetic algorithms to automatically fit dusty galaxies with a Monte Carlo radiative transfer code

We present FitSKIRT, a method to efficiently fit radiative transfer models to UV/optical images of dusty galaxies. These images have the advantage that they have better spatial resolution compared to FIR/submm data. FitSKIRT uses the GAlib genetic algorithm library to optimize the output of the SKIRT Monte Carlo radiative transfer code. Genetic algorithms prove to be a valuable tool in handling the multi- dimensional search space as well as the noise induced by the random nature of the Monte Carlo radiative transfer code. FitSKIRT is tested on artificial images of a simulated edge-on spiral galaxy, where we gradually increase the number of fitted parameters. We find that we can recover all model parameters, even if all 11 model parameters are left unconstrained. Finally, we apply the FitSKIRT code to a V-band image of the edge-on spiral galaxy NGC4013. This galaxy has been modeled previously by other authors using different combinations of radiative transfer codes and optimization methods. Given the different models and techniques and the complexity and degeneracies in the parameter space, we find reasonable agreement between the different models. We conclude that the FitSKIRT method allows comparison between different models and geometries in a quantitative manner and minimizes the need of human intervention and biasing. The high level of automation makes it an ideal tool to use on larger sets of observed data.Comment: 14 pages, 10 figures; accepted for publication in Astronomy and Astrophysic

Rokhlin dimension for actions of residually finite groups

We introduce the concept of Rokhlin dimension for actions of residually finite groups on C*-algebras, extending previous notions of Rokhlin dimension for actions of finite groups and the integers, as introduced by Hirshberg, Winter and the third author. If the group has a box space of finite asymptotic dimension, then actions with finite Rokhlin dimension preserve the property of having finite nuclear dimension, when passing to the crossed product C*-algebra. A detailed study of the asymptotic dimension of box spaces shows that finitely generated, virtually nilpotent groups have box spaces with finite asymptotic dimension, providing a reasonably large class of examples. We then establish a relation between Rokhlin dimension of residually finite groups acting on compact metric spaces and amenability dimension of the action in the sense of Guentner, Willett and Yu. We show that for free actions of infinite, finitely generated, nilpotent groups on finite dimensional spaces, both these dimensional values are finite. In particular, the associated transformation group C*-algebras have finite nuclear dimension. This extends an analogous result about $\mathbb{Z}^m$-actions by the first author. We also provide some results concerning the genericity of finite Rokhlin dimension, and permanence properties with respect to the absorption of a strongly self-absorbing C*-algebra.Comment: 64 pages; some small corrections and added remark

Reconstruction methods for single-shot diffractive imaging of free nanostructures with ultrashort x-ray and XUV laser pulses

With x-ray and XUV single-shot diffractive imaging on free nanoparticles it is possible to investigate structure and shape of the particles. The scattering image of the nanoparticle only contains the intensity distribution but not the phase of the scattered light. Thus, numerical methods are required to infer information from experimental data. In the thesis, different reconstruction methods are implemented, advanced and applied to different scattering scenarios to characterize diffraction patterns for different laser parameters, reconstruct optical properties and 3d shapes of nanotargets.Mit Einzelschussmessungen an freien Nanoteilchen mit Hilfe von Röntgenstrahlung ist es möglich, die Teilchenstruktur zu untersuchen. Damit die Information aus Experimenten gewonnen werden kann, sind numerische Methoden notwendig, da das Streubild der Nanoteilchen nur die Inensitätsverteilung des gestreuten Lichtes, aber nicht die Phase bereitstellt. Es werden verschiedene Rekonstruktionsmethoden implementiert, weiterentwickelt und auf verschiedene Streuszenarien angewandt, um Streubilder für verschiedene Laserparameter zu charakterisieren, optische Eigenschaften und 3D-Formen zu rekonstruieren

GMOSS: All-sky model of spectral radio brightness based on physical components and associated radiative processes

We present Global MOdel for the radio Sky Spectrum (GMOSS) -- a novel, physically motivated model of the low-frequency radio sky from 22 MHz to 23 GHz. GMOSS invokes different physical components and associated radiative processes to describe the sky spectrum over 3072 pixels of $5^{\circ}$ resolution. The spectra are allowed to be convex, concave or of more complex form with contributions from synchrotron emission, thermal emission and free-free absorption included. Physical parameters that describe the model are optimized to best fit four all-sky maps at 150 MHz, 408 MHz, 1420 MHz and 23 GHz and two maps at 22 MHz and 45 MHz generated using the Global Sky Model of de Oliveira-Costa et al. (2008). The fractional deviation of model to data has a median value of $6\%$ and is less than $17\%$ for $99\%$ of the pixels. Though aimed at modeling of foregrounds for the global signal arising from the redshifted 21-cm line of Hydrogen during Cosmic Dawn and Epoch of Reionization (EoR) - over redshifts $150\lesssim z \lesssim 6$, GMOSS is well suited for any application that requires simulating spectra of the low-frequency radio sky as would be observed by the beam of any instrument. The complexity in spectral structure that naturally arises from the underlying physics of the model provides a useful expectation for departures from smoothness in EoR foreground spectra and hence may guide the development of algorithms for EoR signal detection. This aspect is further explored in a subsequent paper.Comment: 19 pages, 7 figure