Smooth times of a flow in dimension 1

Let $\alpha$ be an irrational number and $I$ an interval of $\mathbb{R}$. If $\alpha$ is diophantine, we show that any one-parameter group of homeomorphisms of $I$ whose time-$1$ and $\alpha$ maps are $C^\infty$ is in fact the flow of a $C^\infty$ vector field. If $\alpha$ is Liouville on the other hand, we construct a one-parameter group of homeomorphisms of $I$ whose time-$1$ and $\alpha$ maps are $C^\infty$ but which is not the flow of a $C^2$ vector field (though, if $I$ has boundary, we explain that the hypotheses force it to be the flow of a $C^1$ vector field). We extend both results to families of irrational numbers, the critical arithmetic condition in this case being simultaneous "diophantinity". For one-parameter groups defining a free action of $(\mathbb{R},+)$ on $I$, these results follow from famous linearization theorems for circle diffeomorphisms. The novelty of this work concerns non-free actions.Comment: 35 pages, 8 figure

Bernstein von Mises Theorems for Gaussian Regression with increasing number of regressors

This paper brings a contribution to the Bayesian theory of nonparametric and semiparametric estimation. We are interested in the asymptotic normality of the posterior distribution in Gaussian linear regression models when the number of regressors increases with the sample size. Two kinds of Bernstein-von Mises Theorems are obtained in this framework: nonparametric theorems for the parameter itself, and semiparametric theorems for functionals of the parameter. We apply them to the Gaussian sequence model and to the regression of functions in Sobolev and $C^{\alpha}$ classes, in which we get the minimax convergence rates. Adaptivity is reached for the Bayesian estimators of functionals in our applications

Universal Coding on Infinite Alphabets: Exponentially Decreasing Envelopes

This paper deals with the problem of universal lossless coding on a countable infinite alphabet. It focuses on some classes of sources defined by an envelope condition on the marginal distribution, namely exponentially decreasing envelope classes with exponent $\alpha$. The minimax redundancy of exponentially decreasing envelope classes is proved to be equivalent to $\frac{1}{4 \alpha \log e} \log^2 n$. Then a coding strategy is proposed, with a Bayes redundancy equivalent to the maximin redundancy. At last, an adaptive algorithm is provided, whose redundancy is equivalent to the minimax redundanc

About adaptive coding on countable alphabets

This paper sheds light on universal coding with respect to classes of memoryless sources over a countable alphabet defined by an envelope function with finite and non-decreasing hazard rate. We prove that the auto-censuring AC code introduced by Bontemps (2011) is adaptive with respect to the collection of such classes. The analysis builds on the tight characterization of universal redundancy rate in terms of metric entropy % of small source classes by Opper and Haussler (1997) and on a careful analysis of the performance of the AC-coding algorithm. The latter relies on non-asymptotic bounds for maxima of samples from discrete distributions with finite and non-decreasing hazard rate

Optical Spectroscopy as a Probe of Gaps and Kinetic Electronic Energy in p- and n-type cuprates

The real part of the optical in-plane conductivity of p-- and n--type cuprates thin films at various doping levels was deduced from highly accurate reflectivity measurements. We present here a comprehensive set of optical spectral weight data as a function of the temperature $T (> T_c$), for underdoped and overdoped samples. The temperature dependence of the spectral weight is not universal. Using various cut-off frequencies for the spectral weight, we show that n--type Pr$_{2-x}$Ce$_x$CuO$_4$ and p--type Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$ exhibit both similarities and striking differences. The Fermi surface is closed in overdoped metallic samples. In underdoped Pr$_{2-x}$Ce$_x$CuO$_4$ samples, it clearly breaks into arcs, giving rise to a "pseudogap" signature. It is argued that such a signature is subtle in underdoped Bi$_2$Sr$_2$CaCu$_2$O$_{8+\delta}$.Comment: Proceedings M2SHTSCVIII, to appear in Physica

On the connectedness of the space of codimension one foliations on a closed 3-manifold

We study the topology of the space of smooth codimension one foliations on a closed 3-manifold. We regard this space as the space of integrable plane fields included in the space of all smooth plane fields. It has been known since the late 60's that every plane field can be deformed continuously to an integrable one, so the above inclusion induces a surjective map between connected components. We prove that this map is actually a bijection.Comment: 47 pages, 22 figure