Variational approach to the excitonic phase transition in graphene

We analyze the Coulomb interacting problem in undoped graphene layers by using an excitonic variational ansatz. By minimizing the energy, we derive a gap equation which reproduces and extends known results. We show that a full treatment of the exchange term, which includes the renormalization of the Fermi velocity, tends to suppress the phase transition by increasing the critical coupling at which the excitonic instability takes place.Comment: 4 page

Non-archimedean Yomdin-Gromov parametrizations and points of bounded height

We prove an analogue of the Yomdin-Gromov Lemma for $p$-adic definable sets and more broadly in a non-archimedean, definable context. This analogue keeps track of piecewise approximation by Taylor polynomials, a nontrivial aspect in the totally disconnected case. We apply this result to bound the number of rational points of bounded height on the transcendental part of $p$-adic subanalytic sets, and to bound the dimension of the set of complex polynomials of bounded degree lying on an algebraic variety defined over $\mathbb{C} ((t))$, in analogy to results by Pila and Wilkie, resp. by Bombieri and Pila. Along the way we prove, for definable functions in a general context of non-archimedean geometry, that local Lipschitz continuity implies piecewise global Lipschitz continuity.Comment: 54 pages; revised, section 5.6 adde

Adaptive estimation of linear functionals in the convolution model and applications

We consider the model $Z_i=X_i+\varepsilon_i$, for i.i.d. $X_i$'s and $\varepsilon_i$'s and independent sequences $(X_i)_{i\in{\mathbb{N}}}$ and $(\varepsilon_i)_{i\in{\mathbb{N}}}$. The density $f_{\varepsilon}$ of $\varepsilon_1$ is assumed to be known, whereas the one of $X_1$, denoted by $g$, is unknown. Our aim is to estimate linear functionals of $g$, for a known function $\psi$. We propose a general estimator of and study the rate of convergence of its quadratic risk as a function of the smoothness of $g$, $f_{\varepsilon}$ and $\psi$. Different contexts with dependent data, such as stochastic volatility and AutoRegressive Conditionally Heteroskedastic models, are also considered. An estimator which is adaptive to the smoothness of unknown $g$ is then proposed, following a method studied by Laurent et al. (Preprint (2006)) in the Gaussian white noise model. We give upper bounds and asymptotic lower bounds of the quadratic risk of this estimator. The results are applied to adaptive pointwise deconvolution, in which context losses in the adaptive rates are shown to be optimal in the minimax sense. They are also applied in the context of the stochastic volatility model.Comment: Published in at http://dx.doi.org/10.3150/08-BEJ146 the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm

Asymptotic Theory for Multivariate GARCH Processes

We provide in this paper asymptotic theory for the multivariate GARCH (p,q) process. Strong consistency of the quasi-maximum likelihood estimator (MLE) is established by appealing to conditions given in Jeantheau [19] in conjunction with a result given by Boussama [9] concerning the existence of a stationary and ergodic solution to the multivariate GARCH (p,q) process. We prove asymptotic normality of the quasi-MLE when the initial state is either stationary or fixed.Asymptotic normality, BEKK, consistency, GARCH, Martingale CLT

Anisotropic adaptive kernel deconvolution

International audienceIn this paper, we consider a multidimensional convolution model for which we provide adaptive anisotropic kernel estimators of a signal density $f$ measured with additive error. For this, we generalize Fan's~(1991) estimators to multidimensional setting and use a bandwidth selection device in the spirit of Goldenschluger and Lepski's~(2011) proposal fr density estimation without noise. We consider first the pointwise setting and then, we study the integrated risk. Our estimators depend on an automatically selected random bandwidth. We assume both ordinary and super smooth components for measurement errors, which have known density. We also consider both anisotropic H\"{o}lder and Sobolev classes for $f$. We provide non asymptotic risk bounds and asymptotic rates for the resulting data driven estimator, which is proved to be adaptive. We provide an illustrative simulation study, involving the use of Fast Fourier Transform algorithms. We conclude by a proposal of extension of the method to the case of unknown noise density, when a preliminary pure noise sample is available

On small time asymptotics for rough differential equations driven by fractional Brownian motions

We survey existing results concerning the study in small times of the density of the solution of a rough differential equation driven by fractional Brownian motions. We also slightly improve existing results and discuss some possible applications to mathematical finance.Comment: This is a survey paper, submitted to proceedings in the memory of Peter Laurenc

Long-term low-dose ketoconazole treatment in bilateral macronodular adrenal hyperplasia

Medical therapy for Cushing's syndrome due to bilateral macronodular adrenal hyperplasia (BMAH) is generally administered for a limited time before surgery. Aberrant receptors antagonists show inconsistent efficacy in the long run to prevent adrenalectomy. We present a patient with BMAH, treated for 10 years with low doses of ketoconazole to control cortisol secretion. A 48-year-old woman presented with headaches and hypertension. Investigations showed the following: no clinical signs of Cushing's syndrome; enlarged lobulated adrenals; normal creatinine, potassium, and aldosterone; normal urinary aldosterone and metanephrines; elevated urinary free cortisol and steroid metabolites; and suppressed plasma renin activity and ACTH. A screening protocol for aberrant adrenal receptors failed to show any illegitimate hormone dependence. Ketoconazole caused rapid normalisation of cortisol and ACTH that persists over 10 years on treatment, while adrenals show no change in shape or size. Ketoconazole decreases cortisol in patients with Cushing's syndrome, and may prevent adrenal overgrowth. Steroid secretion in BMAH is inefficient as compared with normal adrenals or secreting tumours and can be controlled with low, well-tolerated doses of ketoconazole, as an alternative to surgery. LEARNING POINTS: Enlarged, macronodular adrenals are often incidentally found during the investigation of hypertension in patients harboring BMAH. Although laboratory findings include low ACTH and elevated cortisol, the majority of patients do not display cushingoid features.Bilateral adrenalectomy, followed by life-long steroid replacement, is the usual treatment of this benign condition, and alternative medical therapy is sought. Therapy based on aberrant adrenal receptors gives disappointing results, and inhibitors of steroidogenesis are not always well tolerated.However, ketoconazole at low, well-tolerated doses appeared appropriate to control adrenal steroid secretion indefinitely, while preventing adrenal overgrowth. This treatment probably constitutes the most convenient long-term alternative to surgery