Estimates in Beurling--Helson type theorems. Multidimensional case

We consider the spaces $A_p(\mathbb T^m)$ of functions $f$ on the $m$ -dimensional torus $\mathbb T^m$ such that the sequence of the Fourier coefficients $\hat{f}=\{\hat{f}(k), ~k \in \mathbb Z^m\}$ belongs to $l^p(\mathbb Z^m), ~1\leq p<2$. The norm on $A_p(\mathbb T^m)$ is defined by $\|f\|_{A_p(\mathbb T^m)}=\|\hat{f}\|_{l^p(\mathbb Z^m)}$. We study the rate of growth of the norms $\|e^{i\lambda\varphi}\|_{A_p(\mathbb T^m)}$ as $|\lambda|\rightarrow \infty, ~\lambda\in\mathbb R,$ for $C^1$ -smooth real functions $\varphi$ on $\mathbb T^m$ (the one-dimensional case was investigated by the author earlier). The lower estimates that we obtain have direct analogues for the spaces $A_p(\mathbb R^m)$

A spectral line survey in the 2 mm and 1.3 mm windows toward the carbon rich envelope of IRC +10216

We present the results of our spectral line surveys in the 2 mm and 1.3 mm windows toward the carbon rich envelope of IRC +10216. Totally 377 lines are detected, among which 360 lines are assigned to 57 known molecules (including 29 rare isotopomers and 2 cyclic isomers). Only 17 weak lines remain unidentified. Rotational lines of isotopomers 13CCH and HN13C are detected for the first time in IRC +10216. The detection of the formaldehyde lines in this star is also confirmed. Possible abundance difference among the three 13C substituted isotopic isomers of HC3N is reported. Isotopic ratios of C and O are confirmed to be non-solar while those of S and Si to be nearly solar. Column densities have been estimated for 15 molecular species. Modified spectroscopic parameters have been calculated for NaCN, Na13CN, KCN and SiC2. Transition frequencies from the present observations were used to improve the spectroscopic parameters of Si13CC, 29SiC2 and 30SiC2.Comment: 17 pages of text, 18 pages of 14 tables, 35 pages of 4 figures, a typo corrected in Abstrac

Shedding light on the formation of the pre-biotic molecule formamide with ASAI

Formamide (NH2CHO) has been proposed as a pre-biotic precursor with a key role in the emergence of life on Earth. While this molecule has been observed in space, most of its detections correspond to high-mass star-forming regions. Motivated by this lack of investigation in the low-mass regime, we searched for formamide, as well as isocyanic acid (HNCO), in 10 low- and intermediate-mass pre-stellar and protostellar objects. The present work is part of the IRAM Large Programme ASAI (Astrochemical Surveys At IRAM), which makes use of unbiased broadband spectral surveys at millimetre wavelengths. We detected HNCO in all the sources and NH2CHO in five of them. We derived their abundances and analysed them together with those reported in the literature for high-mass sources. For those sources with formamide detection, we found a tight and almost linear correlation between HNCO and NH2CHO abundances, with their ratio being roughly constant -between 3 and 10- across 6 orders of magnitude in luminosity. This suggests the two species are chemically related. The sources without formamide detection, which are also the coldest and devoid of hot corinos, fall well off the correlation, displaying a much larger amount of HNCO relative to NH2CHO. Our results suggest that, while HNCO can be formed in the gas phase during the cold stages of star formation, NH2CHO forms most efficiently on the mantles of dust grains at these temperatures, where it remains frozen until the temperature rises enough to sublimate the icy grain mantles. We propose hydrogenation of HNCO as a likely formation route leading to NH2CHO.Comment: 26 pages, 9 figures. Accepted by Monthly Notices of the Royal Astronomical Societ

Lognormal scale invariant random measures

In this article, we consider the continuous analog of the celebrated Mandelbrot star equation with lognormal weights. Mandelbrot introduced this equation to characterize the law of multiplicative cascades. We show existence and uniqueness of measures satisfying the aforementioned continuous equation; these measures fall under the scope of the Gaussian multiplicative chaos theory developed by J.P. Kahane in 1985 (or possibly extensions of this theory). As a by product, we also obtain an explicit characterization of the covariance structure of these measures. We also prove that qualitative properties such as long-range independence or isotropy can be read off the equation.Comment: 31 pages; Probability Theory and Related Fields (2012) electronic versio

The census of interstellar complex organic molecules in the Class I hot corino of SVS13-A

We present the first census of the interstellar Complex Organic Molecules (iCOMs) in the low-mass Class I protostar SVS13-A, obtained by analysing data from the IRAM-30m Large Project ASAI (Astrochemical Surveys At IRAM). They consist of an high-sensitivity unbiased spectral survey at the 1mm, 2mm and 3mm IRAM bands. We detected five iCOMs: acetaldehyde (CH$_3$CHO), methyl formate (HCOOCH$_3$), dimethyl ether (CH$_3$OCH$_3$), ethanol (CH$_3$CH$_2$OH) and formamide (NH$_2$CHO). In addition we searched for other iCOMs and ketene (H$_2$CCO), formic acid (HCOOH) and methoxy (CH$_3$O), whose only ketene was detected. The numerous detected lines, from 5 to 37 depending on the species, cover a large upper level energy range, between 15 and 254 K. This allowed us to carry out a rotational diagram analysis and derive rotational temperatures between 35 and 110 K, and column densities between $3\times 10^{15}$ and $1\times 10^{17}$ cm$^{-2}$ on the 0."3 size previously determined by interferometric observations of glycolaldehyde. These new observations clearly demonstrate the presence of a rich chemistry in the hot corino towards SVS13-A. The measured iCOMs abundances were compared to other Class 0 and I hot corinos, as well as comets, previously published in the literature. We find evidence that (i) SVS13-A is as chemically rich as younger Class 0 protostars, and (ii) the iCOMs relative abundances do not substantially evolve during the protostellar phase.Comment: 24 pages, MNRAS in pres

A multifractal random walk

We introduce a class of multifractal processes, referred to as Multifractal Random Walks (MRWs). To our knowledge, it is the first multifractal processes with continuous dilation invariance properties and stationary increments. MRWs are very attractive alternative processes to classical cascade-like multifractal models since they do not involve any particular scale ratio. The MRWs are indexed by few parameters that are shown to control in a very direct way the multifractal spectrum and the correlation structure of the increments. We briefly explain how, in the same way, one can build stationary multifractal processes or positive random measures.Comment: 5 pages, 4 figures, uses RevTe

Exponential and moment inequalities for U-statistics

A Bernstein-type exponential inequality for (generalized) canonical U-statistics of order 2 is obtained and the Rosenthal and Hoffmann-J{\o}rgensen inequalities for sums of independent random variables are extended to (generalized) U-statistics of any order whose kernels are either nonnegative or canonicalComment: 22 page

The maximum modulus of a trigonometric trinomial

Let Lambda be a set of three integers and let C_Lambda be the space of 2pi-periodic functions with spectrum in Lambda endowed with the maximum modulus norm. We isolate the maximum modulus points x of trigonometric trinomials T in C_Lambda and prove that x is unique unless |T| has an axis of symmetry. This permits to compute the exposed and the extreme points of the unit ball of C_Lambda, to describe how the maximum modulus of T varies with respect to the arguments of its Fourier coefficients and to compute the norm of unimodular relative Fourier multipliers on C_Lambda. We obtain in particular the Sidon constant of Lambda

How large are the level sets of the Takagi function?

Let T be Takagi's continuous but nowhere-differentiable function. This paper considers the size of the level sets of T both from a probabilistic point of view and from the perspective of Baire category. We first give more elementary proofs of three recently published results. The first, due to Z. Buczolich, states that almost all level sets (with respect to Lebesgue measure on the range of T) are finite. The second, due to J. Lagarias and Z. Maddock, states that the average number of points in a level set is infinite. The third result, also due to Lagarias and Maddock, states that the average number of local level sets contained in a level set is 3/2. In the second part of the paper it is shown that, in contrast to the above results, the set of ordinates y with uncountably infinite level sets is residual, and a fairly explicit description of this set is given. The paper also gives a negative answer to a question of Lagarias and Maddock by showing that most level sets (in the sense of Baire category) contain infinitely many local level sets, and that a continuum of level sets even contain uncountably many local level sets. Finally, several of the main results are extended to a version of T with arbitrary signs in the summands.Comment: Added a new Section 5 with generalization of the main results; some new and corrected proofs of the old material; 29 pages, 3 figure

Hubble Space Telescope Survey of Interstellar ^12CO/^13CO in the Solar Neighborhood

We examine 20 diffuse and translucent Galactic sight lines and extract the column densities of the ^12CO and ^13CO isotopologues from their ultraviolet A--X absorption bands detected in archival Space Telescope Imaging Spectrograph data with lambda/Deltalambda geq 46,000. Five more targets with Goddard High-Resolution Spectrograph data are added to the sample that more than doubles the number of sight lines with published Hubble Space Telescope observations of ^13CO. Most sight lines have 12-to-13 isotopic ratios that are not significantly different from the local value of 70 for ^12C/^13C, which is based on mm-wave observations of rotational lines in emission from CO and H_2CO inside dense molecular clouds, as well as on results from optical measurements of CH^+. Five of the 25 sight lines are found to be fractionated toward lower 12-to-13 values, while three sight lines in the sample are fractionated toward higher ratios, signaling the predominance of either isotopic charge exchange or selective photodissociation, respectively. There are no obvious trends of the ^12CO-to-^13CO ratio with physical conditions such as gas temperature or density, yet ^12CO/^13CO does vary in a complicated manner with the column density of either CO isotopologue, owing to varying levels of competition between isotopic charge exchange and selective photodissociation in the fractionation of CO. Finally, rotational temperatures of H_2 show that all sight lines with detected amounts of ^13CO pass through gas that is on average colder by 20 K than the gas without ^13CO. This colder gas is also sampled by CN and C_2 molecules, the latter indicating gas kinetic temperatures of only 28 K, enough to facilitate an efficient charge exchange reaction that lowers the value of ^12CO/^13CO.Comment: 1-column emulateapj, 23 pages, 9 figure