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 $\|e^{in\varphi}\|_{U(\mathbb T)}\simeq\log n$

Séries de Taylor et séries trigonométriques universelles au sens de Menchoff

RésuméLe résultat principal de cet article est qu'il existe une série trigonométrique universelle au sens de Menchoff qui est la restriction au cercle unité d'une série de Taylor dont les coefficients tendent vers zéro.Il nous a paru bon, avant de présenter ce résultat et ses variantes, de récapituler des éléments connus de la théorie des séries trigonométrique universelles et de celle des séries de Taylor universelles.AbstractHere is the principal result: there exists a trigonometric series of the Taylor type, with coefficients tending to zero, and universal in the sense of Mens̆ov. The article recapitulates known results on universal trigonometric series and universal Taylor series before presenting the main result and its developments

Approximated maximum likelihood estimation in multifractal random walks

We present an approximated maximum likelihood method for the multifractal random walk processes of [E. Bacry et al., Phys. Rev. E 64, 026103 (2001)]. The likelihood is computed using a Laplace approximation and a truncation in the dependency structure for the latent volatility. The procedure is implemented as a package in the R computer language. Its performance is tested on synthetic data and compared to an inference approach based on the generalized method of moments. The method is applied to estimate parameters for various financial stock indices.Comment: 8 pages, 3 figures, 2 table

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 $A_p(\mathbb R^m)$

KPZ in one dimensional random geometry of multiplicative cascades

We prove a formula relating the Hausdorff dimension of a subset of the unit interval and the Hausdorff dimension of the same set with respect to a random path matric on the interval, which is generated using a multiplicative cascade. When the random variables generating the cascade are exponentials of Gaussians, the well known KPZ formula of Knizhnik, Polyakov and Zamolodchikov from quantum gravity appears. This note was inspired by the recent work of Duplantier and Sheffield proving a somewhat different version of the KPZ formula for Liouville gravity. In contrast with the Liouville gravity setting, the one dimensional multiplicative cascade framework facilitates the determination of the Hausdorff dimension, rather than some expected box count dimension.Comment: 14 page

Macroscopic objects in quantum mechanics: A combinatorial approach

Why we do not see large macroscopic objects in entangled states? There are two ways to approach this question. The first is dynamic: the coupling of a large object to its environment cause any entanglement to decrease considerably. The second approach, which is discussed in this paper, puts the stress on the difficulty to observe a large scale entanglement. As the number of particles n grows we need an ever more precise knowledge of the state, and an ever more carefully designed experiment, in order to recognize entanglement. To develop this point we consider a family of observables, called witnesses, which are designed to detect entanglement. A witness W distinguishes all the separable (unentangled) states from some entangled states. If we normalize the witness W to satisfy |tr(W\rho)| \leq 1 for all separable states \rho, then the efficiency of W depends on the size of its maximal eigenvalue in absolute value; that is, its operator norm ||W||. It is known that there are witnesses on the space of n qbits for which ||W|| is exponential in n. However, we conjecture that for a large majority of n-qbit witnesses ||W|| \leq O(\sqrt{n logn}). Thus, in a non ideal measurement, which includes errors, the largest eigenvalue of a typical witness lies below the threshold of detection. We prove this conjecture for the family of extremal witnesses introduced by Werner and Wolf (Phys. Rev. A 64, 032112 (2001)).Comment: RevTeX, 14 pages, some additions to the published version: A second conjecture added, discussion expanded, and references adde

The geometry of fractal percolation

A well studied family of random fractals called fractal percolation is discussed. We focus on the projections of fractal percolation on the plane. Our goal is to present stronger versions of the classical Marstrand theorem, valid for almost every realization of fractal percolation. The extensions go in three directions: {itemize} the statements work for all directions, not almost all, the statements are true for more general projections, for example radial projections onto a circle, in the case $\dim_H >1$, each projection has not only positive Lebesgue measure but also has nonempty interior. {itemize}Comment: Survey submitted for AFRT2012 conferenc

BIMA CO Observation of EP Aqr the Semiregular Pulsating Star with a Double Component Line Profile

This paper reports the results of a Berkeley-Illinois-Maryland array interferometric observation of EP Aqr, a semiregular pulsating star with a double component line profile in the CO J=1-0 line. The broad component shows a flat-top profile, and the narrow component shows a spiky strong peak. Though the previous single dish observations suggested that the CO J=2-1 line exhibits a Gaussian-like profile, the CO J=1-0 line does not. The spatial distributions of both the narrow and broad components appears to be roughly round with the same peak positions. No significant velocity gradient is seen. The spatial-kinetic properties of the molecular envelope of EP Aqr are reminiscent of a multiple shell structure model rather than a bipolar flow and disk model. A problem of this interpretation is that no evidence of interaction between the narrow and broad component regions is seen. A Gaussian-like feature seen in the CO J=2-1 line might play a key role to understand the spatio-kinetic properties of the molecular envelope of EP Aqr.Comment: 10 pages, 3 figures; accepted for publication in Ap

Exact and explicit probability densities for one-sided Levy stable distributions

We study functions g_{\alpha}(x) which are one-sided, heavy-tailed Levy stable probability distributions of index \alpha, 0< \alpha <1, of fundamental importance in random systems, for anomalous diffusion and fractional kinetics. We furnish exact and explicit expression for g_{\alpha}(x), 0 \leq x < \infty, satisfying \int_{0}^{\infty} exp(-p x) g_{\alpha}(x) dx = exp(-p^{\alpha}), p>0, for all \alpha = l/k < 1, with k and l positive integers. We reproduce all the known results given by k\leq 4 and present many new exact solutions for k > 4, all expressed in terms of known functions. This will allow a 'fine-tuning' of \alpha in order to adapt g_{\alpha}(x) to a given experimental situation.Comment: 4 pages, 3 figures and 1 tabl

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