On the approximation by Lüroth Series

Let x 2 (0; 1] and pn=qn; n 1 be its sequence of Luroth Series convergents. Dene the approximation coecients n = n(x) by n = qnx. In [BBDK] the limiting distribution of the sequence (n)n1 was obtained for a.e. x using the natural extension of the ergodic system underlying the L? uroth Series expansion. Here we show that this can be done without the natural extension. We also will get a bound on the speed of convergence. Using the natural extension we will study the distribution for a.e. x of the sequence (n; n+1)n1 and related sequences like (n + n+1)n1. It turns out that for a.e. x the sequence (n; n+1)n1 is distributed according to a continuous singular distribution function G. Furthermore we will see that two consecutive 's are positively correlated

A Gauss-Kusmin theorem for optimal continued fractions

One of the first and still one of the most important results in the metrical theory of continued fractions is the so-called Gauss-Kusmin theorem. Let and let be the regular continued fraction (RCF) expansion of then it was observed by Gauss in 1800 that

'The mother of all continued fractions'

In this paper we give the relationship between the regular continued fraction and the Lehner fractions using a procedure known as insertion Starting from the regular continued fraction expansion of any real irrational x when the maximal number of insertions is applied one obtains the Lehner fraction of x Insertions and singularizations show how these and other continued fractions expansions are related We will also investigate the relation between the Lehner fractions and the Farey expansion and obtain the ergodic system underlying the Farey expansio

The natural extension of the β-transformation

For each real number β>1 the β-transformation is dened by Tβx = βx(mod1). In this paper the natural extension Tβ of the ergodic system underlying Tβ is explicitly given. Furthermore, it is shown that a certain induced system of this natural extension is Bernoulli. Since Tβ is weakly mixing, due to W. Parry, it follows from a deep result of A. Saleski that the natural extension is also Bernoulli, a result previously obtained by M. Smorodinsky

Natural extensions and entropy of α\alpha-continued fractions

We construct a natural extension for each of Nakada's $\alpha$-continued fractions and show the continuity as a function of $\alpha$ of both the entropy and the measure of the natural extension domain with respect to the density function $(1+xy)^{-2}$. In particular, we show that, for all $0 < \alpha \le 1$, the product of the entropy with the measure of the domain equals $\pi^2/6$. As a key step, we give the explicit relationship between the $\alpha$-expansion of $\alpha-1$ and of $\alpha$

Scale space consistency of piecewise constant least squares estimators -- another look at the regressogram

We study the asymptotic behavior of piecewise constant least squares regression estimates, when the number of partitions of the estimate is penalized. We show that the estimator is consistent in the relevant metric if the signal is in $L^2([0,1])$, the space of c\`{a}dl\`{a}g functions equipped with the Skorokhod metric or $C([0,1])$ equipped with the supremum metric. Moreover, we consider the family of estimates under a varying smoothing parameter, also called scale space. We prove convergence of the empirical scale space towards its deterministic target.Comment: Published at http://dx.doi.org/10.1214/074921707000000274 in the IMS Lecture Notes Monograph Series (http://www.imstat.org/publications/lecnotes.htm) by the Institute of Mathematical Statistics (http://www.imstat.org

Cross sections for geodesic flows and \alpha-continued fractions

We adjust Arnoux's coding, in terms of regular continued fractions, of the geodesic flow on the modular surface to give a cross section on which the return map is a double cover of the natural extension for the \alpha-continued fractions, for each $\alpha$ in (0,1]. The argument is sufficiently robust to apply to the Rosen continued fractions and their recently introduced \alpha-variants.Comment: 20 pages, 2 figure

Beyond small-scale transients: a closer look at the diffuse quiet solar corona

Within the quiet Sun corona imaged at 1 MK, much of the field of view consists of diffuse emission that appears to lack the spatial structuring that is so evident in coronal loops or bright points. We seek to determine if these diffuse regions are categorically different in terms of their intensity fluctuations and spatial configuration from the more well-studied dynamic coronal features. We analyze a time series of observations from Solar Orbiter's High Resolution Imager in the Extreme Ultraviolet to quantify the characterization of the diffuse corona at high spatial and temporal resolutions. We then compare this to the dynamic features within the field of view, mainly a coronal bright point. We find that the diffuse corona lacks visible structuring, such as small embedded loops, and that this is persistent over the 25 min duration of the observation. The intensity fluctuations of the diffuse corona, which are within +/-5%, are significantly smaller in comparison to the coronal bright point. Yet, the total intensity observed in the diffuse corona is of the same order as the bright point. It seems inconsistent with our data that the diffuse corona is a composition of small loops or jets or that it is driven by discrete small heating events that follow a power-law-like distribution. We speculate that small-scale processes like MHD turbulence might be energizing the diffuse regions, but at this point we cannot offer a conclusive explanation for the nature of this feature.Comment: Accepted for publication in A&A. 10 pages, 8 figure

Small-scale EUV features as the drivers of coronal upflows in the quiet Sun

Context. Coronal upflows in the quiet Sun are seen in a wide range of features, including jets and filament eruptions. The in situ measurements from Parker Solar Probe within ≈0.2 au have demonstrated that the solar wind is highly structured, showing abrupt and near-ubiquitous magnetic field reversals (i.e., switchbacks) on different timescales. The source of these structures has been associated with supergranular structures on the solar disc. This raises the question of whether there are additional small coronal features that contribute energy to the corona and produce plasma that potentially feeds into the solar wind. / Aims. During the Solar Orbiter first science perihelion, high-resolution images of the solar corona were recorded using the Extreme Ultraviolet High Resolution Imager (HRIEUV) from the Extreme Ultraviolet Imager (EUI). The Hinode spacecraft was also observing at the same location providing coronal spectroscopic measurements. Combining the two datasets allows us to determine the cause of the weak upflows observed in the quiet Sun and the associated activity. / Methods. We used a multi-spacecraft approach to characterise regions of upflows. The upflows were identified in the Fe XII emission line by the Hinode EUV Imaging Spectrometer (EIS). We then used imaging data from the Atmospheric Imaging Assembly on board the Solar Dynamics Observatory (SDO/AIA) and the High Resolution Imagers (HRI) from EUI on board the Solar Orbiter to identify coronal features and magnetic field data from the SDO Helioseismic and Magnetic Imager (HMI). Interface Region Imaging Spectrograph (IRIS) observations were also used to understand the photospheric and chromospheric driving mechanisms. / Results. We have identified two regions of coronal upflows in the quiet Sun, with respective sizes and lifetimes of (20 Mm2, 20 min) and (180 Mm2, several hours), which are contrasting dynamic events. Both examples show weak flux cancellation, indicating that the source of the upflows and enhancements is related to the magnetic field changes. The first event, a larger upflow region, shows velocities of up to −8.6 km s−1 at the footpoint of a complex loop structure. We observe several distinct extreme ultraviolet (EUV) features including frequent loop brightenings and plasma blobs travelling along closed coronal loops. The second upflow region has velocities of up to −7.2 km s−1. Within it, a complex EUV feature that lasts for about 20 min can be seen. This main feature has several substructures. During its appearance, a clear mini-filament eruption takes place at its location, before the EUV feature disappears. / Conclusions. Two features, with contrasting properties, show upflows with comparable magnitudes. The first event, a complex loop structure, shares several similarities with active region upflows. The second one, a complex small-scale feature that could not have been well resolved with previous instruments, triggered a cascade of events, including a mini-filament that lead to a measurable upflow. This is remarkable for an EUV feature that many instruments can barely resolve. The complexity of the two events, including small loop brightenings and travelling plasma blobs for the first and EUV small-scale loops and mini-filament for the second one would not have been identifiable as the sources of upflow without an instrument with the spatial resolution of HRIEUV at this distance to the Sun. These results reinforce the importance of the smallest-scale features in the Sun and their potential relevance for and impact on the solar corona and the solar wind