Extension of formal conjugations between diffeomorphisms

We study the formal conjugacy properties of germs of complex analytic diffeomorphisms defined in the neighborhood of the origin of ${\mathbb C}^{n}$. More precisely, we are interested on the nature of formal conjugations along the fixed points set. We prove that there are formally conjugated local diffeomorphisms $\phi, \eta$ such that every formal conjugation $\hat{\sigma}$ (i.e. $\eta \circ \hat{\sigma} = \hat{\sigma} \circ \phi$) does not extend to the fixed points set $Fix (\phi)$ of $\phi$, meaning that it is not transversally formal (or semi-convergent) along $Fix (\phi)$. We focus on unfoldings of 1-dimensional tangent to the identity diffeomorphisms. We identify the geometrical configurations preventing formal conjugations to extend to the fixed points set: roughly speaking, either the unperturbed fiber is singular or generic fibers contain multiple fixed points.Comment: 34 page

Notes of a Meeting Held in the Diplomatic Reception Room of the Department of State on Monday, March 20, 1911

The document is a carbon transcript of notes from a meeting held in the State Department respecting the necessity of more such meetings, the present organization of the Department, expenditures, and esprit de corps. Present at the meeting were: The Assistant Secretaries, the Solicitor, the Director of the Consular Service, the Chief Clerk and the Chiefs and Assistant Chiefs of the Divisions and Bureaus of the Department.https://digitalcommons.ursinus.edu/fmhw_speeches/1004/thumbnail.jp

A remark on the dimension of the Bergman space of some Hartogs domains

Let D be a Hartogs domain of the form D={(z,w) \in CxC^N : |w| < e^{-u(z)}} where u is a subharmonic function on C. We prove that the Bergman space of holomorphic and square integrable functions on D is either trivial or infinite dimensional.Comment: 12 page

Ergodic Jacobi matrices and conformal maps

We study structural properties of the Lyapunov exponent $\gamma$ and the density of states $k$ for ergodic (or just invariant) Jacobi matrices in a general framework. In this analysis, a central role is played by the function $w=-\gamma+i\pi k$ as a conformal map between certain domains. This idea goes back to Marchenko and Ostrovskii, who used this device in their analysis of the periodic problem

Polya's inequalities, global uniform integrability and the size of plurisubharmonic lemniscates

First we prove a new inequality comparing uniformly the relative volume of a Borel subset with respect to any given complex euclidean ball \B \sub \C^n with its relative logarithmic capacity in \C^n with respect to the same ball \B. An analoguous comparison inequality for Borel subsets of euclidean balls of any generic real subspace of \C^n is also proved. Then we give several interesting applications of these inequalities. First we obtain sharp uniform estimates on the relative size of \psh lemniscates associated to the Lelong class of \psh functions of logarithmic singularities at infinity on \C^n as well as the Cegrell class of \psh functions of bounded Monge-Amp\`ere mass on a hyperconvex domain \W \Sub \C^n. Then we also deduce new results on the global behaviour of both the Lelong class and the Cegrell class of \psh functions.Comment: 25 page

Spinal involvement in mucopolysaccharidosis IVA (Morquio-Brailsford or Morquio A syndrome): presentation, diagnosis and management.

Mucopolysaccharidosis IVA (MPS IVA), also known as Morquio-Brailsford or Morquio A syndrome, is a lysosomal storage disorder caused by a deficiency of the enzyme N-acetyl-galactosamine-6-sulphate sulphatase (GALNS). MPS IVA is multisystemic but manifests primarily as a progressive skeletal dysplasia. Spinal involvement is a major cause of morbidity and mortality in MPS IVA. Early diagnosis and timely treatment of problems involving the spine are critical in preventing or arresting neurological deterioration and loss of function. This review details the spinal manifestations of MPS IVA and describes the tools used to diagnose and monitor spinal involvement. The relative utility of radiography, computed tomography (CT) and magnetic resonance imaging (MRI) for the evaluation of cervical spine instability, stenosis, and cord compression is discussed. Surgical interventions, anaesthetic considerations, and the use of neurophysiological monitoring during procedures performed under general anaesthesia are reviewed. Recommendations for regular radiological imaging and neurologic assessments are presented, and the need for a more standardized approach for evaluating and managing spinal involvement in MPS IVA is addressed

On the structure of codimension 1 foliations with pseudoeffective conormal bundle.

International audienceLet $X$ a projective manifold equipped with a codimension $1$ (maybe singular) distribution whose conormal sheaf is assumed to be pseudoeffective. By a theorem of Jean-Pierre Demailly, this distribution is actually integrable and thus defines a codimension $1$ holomorphic foliation \F. We aim at describing the structure of such a foliation, especially in the non abundant case: It turns out that \F is the pull-back of one of the "canonical foliations" on a Hilbert modular variety. This result remains valid for ''logarithmic foliated pairs''