Turan's method and compressive sampling

Turan's method, as expressed in his books, is a careful study of trigonometric polynomials from different points of view. The present article starts from a problem asked by Turan: how to construct a sequence of real numbers x(j) (j= 1,2,...n) such that the almost periodic polynomial whose frequencies are the x(j) and the coefficients are 1 are small (say, their absolute values are less than n d, d< given) for all integral values of the variable m between 1 and M= M(n,d) ? The best known answer is a random choice of the x(j) modulo 1. Using the random choice as Turan (and before him Erd\"os and Renyi), we improve the estimate of M (n, d) and we discuss an explicit construction derived from another chapter of Turan's book. The main part of the paper deals with the corresponding problem when R / Z is replaced by Z / NZ, N prime, and m takes all integral values modulo 1 except 0. Then it has an interpretation in signal theory, when a signal is representad by a function on the cyclic goup G = Z / NZ and the frequencies by the dual cyclic group G^ : knowing that the signal is carried by T points, to evaluate the probability that a random choice of a set W of frequencies allows to recover the signal x from the restriction of its Fourier tranform to W by the process of minimal extrapolation in the Wiener algebra of G^(process of Cand\`es, Romberg and Tao). Some random choices were considered in the original article of CRT and the corresponding probabilities were estimated in two preceding papers of mine. Here we have another random choice, associated with occupancy problems

On uniform convergence of Fourier series

We consider the space $U(\mathbb T)$ of all continuous functions on the circle $\mathbb T$ with uniformly convergent Fourier series. We show that if $\varphi: \mathbb T\rightarrow\mathbb T$ is a continuous piecewise linear but not linear map, then $\|e^{in\varphi}\|_{U(\mathbb T)}\simeq\log n$

Packing-Dimension Profiles and Fractional Brownian Motion

In order to compute the packing dimension of orthogonal projections Falconer and Howroyd (1997) introduced a family of packing dimension profiles ${\rm Dim}_s$ that are parametrized by real numbers $s>0$. Subsequently, Howroyd (2001) introduced alternate $s$-dimensional packing dimension profiles \hbox{{\rm P}$-$\dim}_s and proved, among many other things, that \hbox{{\rm P}$-$\dim}_s E={\rm Dim}_s E for all integers $s>0$ and all analytic sets $E\subseteq\R^N$. The goal of this article is to prove that \hbox{{\rm P}$-$\dim}_s E={\rm Dim}_s E for all real numbers $s>0$ and analytic sets $E\subseteq\R^N$. This answers a question of Howroyd (2001, p. 159). Our proof hinges on a new property of fractional Brownian motion

The census of complex organic molecules in the solar type protostar IRAS16293-2422

Complex Organic Molecules (COMs) are considered crucial molecules, since they are connected with organic chemistry, at the basis of the terrestrial life. More pragmatically, they are molecules in principle difficult to synthetize in the harsh interstellar environments and, therefore, a crucial test for astrochemical models. Current models assume that several COMs are synthesised on the lukewarm grain surfaces ($\gtrsim$30-40 K), and released in the gas phase at dust temperatures $\gtrsim$100 K. However, recent detections of COMs in $\lesssim$20 K gas demonstrate that we still need important pieces to complete the puzzle of the COMs formation. We present here a complete census of the oxygen and nitrogen bearing COMs, previously detected in different ISM regions, towards the solar type protostar IRAS16293-2422. The census was obtained from the millimeter-submillimeter unbiased spectral survey TIMASSS. Six COMs, out of the 29 searched for, were detected: methyl cyanide, ketene, acetaldehyde, formamide, dimethyl ether, and methyl formate. The multifrequency analysis of the last five COMs provides clear evidence that they are present in the cold ($\lesssim$30 K) envelope of IRAS16293-2422, with abundances 0.03-2 $\times 10^{-10}$. Our data do not allow to support the hypothesis that the COMs abundance increases with increasing dust temperature in the cold envelope, as expected if COMs were predominately formed on the lukewarm grain surfaces. Finally, when considering also other ISM sources, we find a strong correlation over five orders of magnitude, between the methyl formate and dimethyl ether and methyl formate and formamide abundances, which may point to a link between these two couples of species, in cold and warm gas

The abortion-crime link: evidence from England and Wales

We use panel data from 1983 to 1997 for the 42 police force areas in England and Wales to test the hypothesis that legalizing abortion contributes to lower crime rates. We provide an advance on previous work by focusing on the impact of possible endogeneity of effective abortion rates with respect to crime. Our use of U.K. data allows us to exploit regional differences in the provision of free abortions to identify abortion rates. When we use a similar model and estimation methodology, we are able to replicate the negative association between abortion rates and reported crime found by Donohue and Levitt for the U.S. However, when we allow for the potential endogeneity of effective abortion rates with respect to crime, we find no clear connection between the two.