arXiv subjects

Daniel Lenz

Publications and source records attributed to Daniel Lenz.

At least 55 records · Page 3Linked to original sources

arXiv2017

A new, large-scale map of interstellar reddening derived from HI emission

We present a new map of interstellar reddening, covering the 39\% of the sky with low {\rm HI} column densities ($N_{\rm HI} < 4\times10^{20}\,\rm cm^{-2}$ or $E(B-V)\approx 45\rm\, mmag$) at $16\overset{'}{.}1$ resolution, based on all-sky observations of Galactic HI emission by the HI4PI Survey. In this low column density regime, we derive a characteristic value of $N_{\rm HI}/E(B-V) = 8.8\times10^{21}\, \rm\, cm^{2}\, mag^{-1}$ for gas with $|v_{\rm LSR}| < 90\,\rm km\, s^{-1}$ and find no significant reddening associated with gas at higher velocities. We compare our HI-based reddening map with the Schlegel, Finkbeiner, and Davis (1998, SFD) reddening map and find them consistent to within a scatter of $\simeq 5\,\rm mmag$. Further, the differences between our map and the SFD map are in excellent agreement with the low resolution ($4\overset{\circ}{.}5$) corrections to the SFD map derived by Peek and Graves (2010) based on observed reddening toward passive galaxies. We therefore argue that our HI-based map provides the most accurate interstellar reddening estimates in the low column density regime to date. Our reddening map is made publicly available (http://dx.doi.org/10.7910/DVN/AFJNWJ).

astro-ph.GA↗

arXiv2017

On the decomposition principle and a Persson type theorem for general regular Dirichlet forms

We present a decomposition principle for general regular Dirichlet forms satisfying a spatial local compactness condition. We use the decomposition principle to derive a Persson type theorem for the corresponding Dirichlet forms. In particular our setting covers Laplace-Beltrami operators on Riemannian manifolds, and Dirichlet forms associated to $α$-stable processes in Euclidean space.

math.SP↗

arXiv2017

Averaging almost periodic functions along exponential sequences

The goal of this expository article is a fairly self-contained account of some averaging processes of functions along sequences of the form $(α^n x)^{}_{n\in\mathbb{N}}$, where $α$ is a fixed real number with $| α| > 1$ and $x\in\mathbb{R}$ is arbitrary. Such sequences appear in a multitude of situations including the spectral theory of inflation systems in aperiodic order. Due to the connection with uniform distribution theory, the results will mostly be metric in nature, which means that they hold for Lebesgue-almost every $x\in\mathbb{R}$.

math.NT↗

arXiv2016

On the spectrum of operator families on discrete groups over minimal dynamical systems

It is well known that, given an equivariant and continuous (in a suitable sense) family of selfadjoint operators in a Hilbert space over a minimal dynamical system, the spectrum of all operators from that family coincides. As shown recently similar results also hold for suitable families of non-selfadjoint operators in $\ell^p (\ZM)$. Here, we generalize this to a large class of bounded linear operator families on Banach-space valued $\ell^p$-spaces over countable discrete groups. We also provide equality of the pseudospectra for operators in such a family. A main tool for our analysis are techniques from limit operator theory.

math.SP↗

arXiv2016

Diffusion on Delone sets

We consider graphs associated to Delone sets in Euclidean space. Such graphs arise in various ways from tilings. Here, we provide a unified framework. In this context, we study the associated Laplace operators and show Gaussian heat kernel bounds for their semigroups. These results apply to both metric and discrete graphs.

math.SP↗

arXiv2016

On Weakly Almost Periodic Measures

We study the diffraction and dynamical properties of translation bounded weakly almost periodic measures. We prove that the dynamical hull of a weakly almost periodic measure is a weakly almost periodic dynamical system with unique minimal component given by the hull of the strongly almost periodic component of the measure. In particular the hull is minimal if and only if the measure is strongly almost periodic and the hull is always measurably conjugate to a torus and has pure point spectrum with continuous eigenfunctions. As an application we show the stability of the class of weighted Dirac combs with Meyer set or FLC support and deduce that such measures have either trivial or large pure point respectively continuous spectrum. We complement these results by investigating the Eberlein convolution of two weakly almost periodic measures. Here, we show that it is unique and a strongly almost periodic measure. We conclude by studying the Fourier-Bohr coefficients of weakly almost periodic measures.

math.DS↗

arXiv2016

An autocorrelation and discrete spectrum for dynamical systems on metric spaces

We study dynamical systems $(X,G,m)$ with a compact metric space $X$ and a locally compact, $σ$-compact, abelian group $G$. We show that such a system has discrete spectrum if and only if a certain space average over the metric is a Bohr almost periodic function. In this way, this average over the metric plays for general dynamical systems a similar role as the autocorrelation measure plays in the study of aperiodic order for special dynamical systems based on point sets.

math.DS↗

arXiv2016

Schreier graphs of Grigorchuk's group and a subshift associated to a non-primitive substitution

There is a recently discovered connection between the spectral theory of Schrö-dinger operators whose potentials exhibit aperiodic order and that of Laplacians associated with actions of groups on regular rooted trees, as Grigorchuk's group of intermediate growth. We give an overview of corresponding results, such as different spectral types in the isotropic and anisotropic cases, including Cantor spectrum of Lebesgue measure zero and absence of eigenvalues. Moreover, we discuss the relevant background as well as the combinatorial and dynamical tools that allow one to establish the afore-mentioned connection. The main such tool is the subshift associated to a substitution over a finite alphabet that defines the group algebraically via a recursive presentation by generators and relators.

math.GR↗

arXiv2016

Toeplitz flows and model sets

We show that binary Toeplitz flows can be interpreted as Delone dynamical systems induced by model sets and analyse the quantitative relations between the respective system parameters. This has a number of immediate consequences for the theory of model sets. In particular, we use our results in combination with special examples of irregular Toeplitz flows from the literature to demonstrate that irregular proper model sets may be uniquely ergodic and do not need to have positive entropy. This answers questions by Schlottmann and Moody.

math.DS↗

arXiv2016

Spectral notions of aperiodic order

Various spectral notions have been employed to grasp the structure of point sets, in particular non-periodic ones. In this article, we present them in a unified setting and explain the relations between them. For the sake of readability, we use Delone sets in Euclidean space as our main object class, and give generalisations in the form of further examples and remarks.

math.DS↗

arXiv2015

Dust in a compact, cold, high-velocity cloud: A new approach to removing foreground emission

Because isolated high-velocity clouds (HVCs) are found at great distances from the Galactic radiation field and because they have subsolar metallicities, there have been no detections of dust in these structures. A key problem in this search is the removal of foreground dust emission. Using the Effelsberg-Bonn HI Survey and the Planck far-infrared data, we investigate a bright, cold, and clumpy HVC. This cloud apparently undergoes an interaction with the ambient medium and thus has great potential to form dust. To remove the local foreground dust emission we used a regularised, generalised linear model and we show the advantages of this approach with respect to other methods. To estimate the dust emissivity of the HVC, we set up a simple Bayesian model with mildly informative priors to perform the line fit instead of an ordinary linear least-squares approach. We find that the foreground can be modelled accurately and robustly with our approach and is limited mostly by the cosmic infrared background. Despite this improvement, we did not detect any significant dust emission from this promising HVC. The 3-sigma upper limit to the dust emissivity is an order of magnitude below the typical values for the Galactic interstellar medium.

astro-ph.GA↗

arXiv2015

Note on short time behavior of semigroups associated to selfadjoint operators

We present a simple observation showing that the heat kernel on a locally finite graph behaves for short times $t$ roughly like $t^d$, where $d$ is the combinatorial distance. This is very different from the classical Varadhan type behavior on manifolds. Moreover, this also gives that short time behavior and global behavior of the heat kernel are governed by two different metrics whenever the degree of the graph is not uniformly bounded.

math.FA↗

arXiv2014

Note on the set of Bragg peaks with high intensity

We consider diffraction of Delone sets in Euclidean space. We show that the set of Bragg peaks with high intensity is always Meyer (if it is relatively dense). We use this to provide a new characterization for Meyer sets in terms of positive and positive definite measures. Our results are based on a careful study of positive definite measures, which may be of interest in its own right.

math-ph↗

arXiv2014

Expansion in generalized eigenfunctions for Laplacians on graphs and metric measure spaces

We consider an arbitrary selfadjoint operator on a separable Hilbert space. To this operator we construct an expansion in generalized eigenfunctions in which the original Hilbert space is decomposed as a direct integral of Hilbert spaces consisting of general eigenfunctions. This automatically gives a Plancherel type formula. For suitable operators on metric measure spaces we discuss some growth restrictions on the generalized eigenfunctions. For Laplacians on locally finite graphs the generalized eigenfunctions are exactly the solutions of the corresponding difference equation.

math.SP↗

arXiv2014

Note on spectra of non-selfadjoint operators over dynamical systems

We consider equivariant continuous families of discrete one-dimensional operators over arbitrary dynamical systems. We introduce the concept of a pseudo-ergodic element of a dynamical system. We then show that all operators associated to pseudo-ergodic elements have the same spectrum and that this spectrum agrees with their essential spectrum. As a consequence we obtain that the spectrum is constant and agrees with the essential spectrum for all elements in the dynamical system if minimality holds.

math.SP↗

arXiv2014

Global properties of Dirichlet forms in terms of Green's formula

We study global properties of Dirichlet forms such as uniqueness of the Dirichlet extension, stochastic completeness and recurrence. We characterize these properties by means of vanishing of a boundary term in Green's formula for functions from suitable function spaces and suitable operators arising from extensions of the underlying form. We first present results in the framework of general Dirichlet forms on $σ$-finite measure spaces. For regular Dirichlet forms our results can be strengthened as all operators from the previous considerations turn out to be restrictions of a single operator. Finally, the results are applied to graphs, weighted manifolds, and metric graphs, where the operators under investigation can be determined rather explicitly.

math.FA↗

arXiv2014

Far-infrared excess emission as a tracer of disk-halo interaction

Given the current and past star-formation in the Milky Way in combination with the limited gas supply, the re-fuelling of the reservoir of cool gas is an important aspect of Galactic astrophysics. The infall of \ion{H}{i} halo clouds can, among other mechanisms, contribute to solving this problem. We study the intermediate-velocity cloud IVC135+54 and its spatially associated high-velocity counterpart to look for signs of a past or ongoing interaction. Using the Effelsberg-Bonn \ion{H}{i} Survey data, we investigated the interplay of gas at different velocities. In combination with far-infrared Planck and IRIS data, we extended this study to interstellar dust and used the correlation of the data sets to infer information on the dark gas. The velocity structure indicates a strong compression and deceleration of the infalling high-velocity cloud (HVC), associated with far-infrared excess emission in the intermediate-velocity cloud. This excess emission traces molecular hydrogen, confirming that IVC135+54 is one of the very few molecular halo clouds. The high dust emissivity of IVC135+54 with respect to the local gas implies that it consists of disk material and does not, unlike the HVC, have an extragalactic origin. Based on the velocity structure of the HVC and the dust content of the IVC, a physical connection between them appears to be the logical conclusion. Since this is not compatible with the distance difference between the two objects, we conclude that this particular HVC might be much closer to us than complex C. Alternatively, the indicators for an interaction are misleading and have another origin.

astro-ph.GA↗

arXiv2014

Note on uniformly transient graphs

We study a special class of graphs with a strong transience feature called uniform transience. We characterize uniform transience via a Feller-type property and via validity of an isoperimetric inequality. We then give a further characterization via equality of the Royden boundary and the harmonic boundary and show that the Dirichlet problem has a unique solution for such graphs. The Markov semigroups and resolvents (with Dirichlet boundary conditions) on these graphs are shown to be ultracontractive. Moreover, if the underlying measure is finite, the semigroups and resolvents are trace class and their generators have $\ell^p$ independent pure point spectra (for $1 \leq p \leq \infty$). Examples of uniformly transient graphs include Cayley graphs of hyperbolic groups as well as trees and Euclidean lattices of dimension at least three. As a surprising consequence, the Royden compactification of such lattices turns out to be the one-point compacitifcation and the Laplacians of such lattices have pure point spectrum if the underlying measure is chosen to be finite.

math.FA↗