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We study the moments of equilibrium measures for iterated function systems (IFSs) and draw connections to operator theory. Our main object of study is the infinite matrix which encodes all the moment data of a Borel measure on We first examine the classical existence problem for moments, culminating in a new proof of the existence of a Borel measure on

We correlate an overlapping period of earthshine measurements of Earth's reflectance (from 1999 through mid-2001) with satellite observations of global cloud properties to construct from the latter a proxy measure of Earth's global shortwave reflectance. This proxy shows a steady decrease in Earth's reflectance from 1984 to 2000, with a strong climatologically significant drop after 1995. From 2001 to 2003, only earthshine data are available, and they indicate a complete reversal of the decline. Understanding how the causes of these decadal changes are apportioned between natural variability, direct forcing, and feedbacks is fundamental to confidently assessing and predicting climate change.

With the rapid developments in the exoplanet field, more and more terrestrial exoplanets are being detected. Characterizing their atmospheres using transit observations will become a key datum in the quest for detecting an Earth-like exoplanet. The atmospheric transmission spectrum of our Earth will be an ideal template for comparison with future exo-Earth candidates. By observing a lunar eclipse, which offers a similar configuration to that of an exoplanet transit, we have obtained a high-resolution and high signal-to-noise ratio (SNR) transmission spectrum of the Earth's atmosphere. This observation was performed with the High Resolution Spectrograph at Xinglong Station, China during the total lunar eclipse in December 2011. We compare the observed transmission spectrum with our atmospheric model, and determine the characteristics of the various atmospheric species in detail. In the transmission spectrum, O2, O3, O2 · O2, NO2 and H2O are detected, and their column densities are measured and compared with the satellites data. The visible Chappuis band of ozone produces the most prominent absorption feature, which suggests that ozone is a promising molecule for the future exo-Earth characterization. Due to the high resolution and high SNR of our spectrum, several novel details of the Earth atmosphere's transmission spectrum are presented. The individual O2 lines are resolved and O2 isotopes are clearly detected. Our new observations do not confirm the absorption features of Ca II or Na I which have been reported in previous lunar eclipse observations. However, features in these and some other strong Fraunhofer line positions do occur in the observed spectrum. We propose that these are due to a Raman-scattered component in the forward-scattered sunlight appearing in the lunar umbral spectrum. Water vapour absorption is found to be rather weak in our spectrum because the atmosphere we probed is relatively dry, which prompts us to discuss the detectability of water vapour in Earth-like exoplanet atmospheres.

We present a photometric follow-up of transiting exoplanets HAT-P-3b and TrES-3b, observed by using several optical and near-infrared filters, with four small-class telescopes (D = 36-152 cm) in the Northern Hemisphere. Two of the facilities present their first scientific results. New 10 HAT-P-3b light curves and new 26 TrES-3b light curves are reduced and combined by filter to improve the quality of the photometry. Combined light curves fitting is carried out independently by using two different analysis packages, allowing the corroboration of the orbital and physical parameters in the literature. Results find no differences in the relative radius with the observing filter. In particular, we report for HAT-P-3b a first estimation of the planet-to-star radius R p R * = 0.1112 − 0.0026 + 0.0025 in the B band which is coherent with values found in the VRIz′JH filters. Concerning TrES-3b, we derive a value for the orbital period of P = 1.3061862 0.0000001 days which shows no linear variations over nine years of photometric observations.

In this paper, we summarize fractal calculus on fractal curves and nonstandard analysis. Using nonstandard analysis which includes hyperreal and hyperinteger numbers, we define left and right limits and derivatives on fractal curves. Fractal integral and differential forms are defined using nonstandard analysis. Some examples are solved to show details.

Here, we report the NIR transmission spectrum of Jupiter, with high signal-to-noise ratio, as if it were a transiting planet. Our technique was to observe Ganymede, when crossing Jupiter’s shadow. During the eclipse, the spectral features of the Jovian atmosphere are imprinted in the sunlight that, after passing through Jupiter’s planetary limb, is reflected from Ganymede towards the Earth. The ratio spectrum of Ganymede before and during the eclipse removes the spectral features of the Sun, the local telluric atmosphere on top of the telescopes, and the spectral albedo of Ganymede. Ganymede is a practically atmosphere-less body and do not suffer any significant weather variability.

In this paper, we analyze interacting multi functional extreme learning machines by applying graph theory, groupoid theory, representation theory, operator theory, operator algebra theory and free probability.

This book introduces the study of algebra induced by combinatorial objects called directed graphs. These graphs are used as tools in the analysis of graph-theoretic problems and in the characterization and solution of analytic problems. The book presents recent research in operator algebra theory connected with discrete and combinatorial mathematic

The Moon's changeable aspect during a lunar eclipse is largely attributable to variations in the refracted unscattered sunlight absorbed by the terrestrial atmosphere that occur as the satellite crosses the Earth's shadow. The contribution to the Moon's aspect from sunlight scattered at the Earth's terminator is generally deemed minor. However, our analysis of a published spectrum of the 16 August 2008 lunar eclipse shows that diffuse sunlight is a major component of the measured spectrum at wavelengths shorter than 600 nm. The conclusion is supported by two distinct features, namely the spectrum's tail at short wavelengths and the unequal absorption by an oxygen collisional complex at two nearby bands. Our findings are consistent with the presence of the volcanic cloud reported at high northern latitudes following the 7–8 August 2008 eruption in Alaska of the Kasatochi volcano. The cloud both attenuates the unscattered sunlight and enhances moderately the scattered component, thus modifying the contrast between the two contributions. Key Points Scattered sunlight contributes to the illumination of the eclipsed Moon The Kasatochi volcanic cloud leaves a signature on the eclipse spectrum The effect of meteoritic dust is likely masked by the volcanic cloud

Let $N\\in\\mathbb{N}$, $N\\geq2$, be given. Motivated by wavelet analysis, we consider a class of normal representations of the $C^{\\ast}$-algebra $\\mathfrak {A}_{N}$ on two unitary generators $U$, $V$ subject to the relation $UVU^{-1}=V^{N}$. The representations are in one-to-one correspondence with solutions $h\\in L^{1}\\left(\\mathbb{T}\\right)$, $h\\geq0$, to $R\\left(h\\right)=h$ where $R$ is a certain transfer operator (positivity-preserving) which was studied previously by D. Ruelle. The representations of $\\mathfrak {A}_{N}$ may also be viewed as representations of a certain (discrete) $N$-adic $ax+b$ group which was considered recently by J.-B. Bost and A. Connes.