Counting Singular Matrices with Primitive Row Vectors

We solve an asymptotic problem in the geometry of numbers, where we count the number of singular n×n matrices where row vectors are primitive and of length at most T. Without the constraint of primitivity, the problem was solved by Y. Katznelson. We show that as T, the number is asymptotic to for n3. The 3-dimensional case is the most problematic and we need to invoke an equidistribution theorem due to W. M. Schmidt.     

Lie point symmetries and some group invariant solutions of the quasilinear equation involving the infinity Laplacian

Using the classical Lie method we obtain the full Lie point symmetry group of the Aronsson equation in two independent variables. Some group invariant solutions of this equation are found and a conjecture on the Lie point symmetry group of the Aronsson equation in Rⁿ is presented.     

On the local asymptotics of Faber polynomials

We study a local asymptotics of the (generalized) Faber polynomials at the boundary of the associated domain, under certain mild smoothness conditions on the weight function and geometric conditions on the boundary. The main result exhibits how this asymptotics depends on the corners at the boundary. Its proof is based on the continuity properties of the Visser-Ostrowski quotient at the corners.

     

A class of dimension-skipping graphs

Forn≧6 there exists a graphG with dimG=n, dimG*≧n+2, whereG* isG with a certain edge added.     

High-frequency 263 GHz PELDOR

Pulsed electron–electron double resonance (PELDOR/DEER) at high frequencies can provide information on the relative orientation of paramagnetic centres or spin labels, if those are rigidly oriented in a host biomolecule and experiments are performed with sufficient orientation selectivity. We present the first comparative PELDOR study at 263 and 94 GHz on a model RNA system containing rigid nitroxides. We show that at 263 GHz still considerable modulation depth is observed and orientation selectivity is significant, particularly in g _x–g _y plane of the nitroxides.     

Removability of isolated singularity for anisotropic parabolic equations with absorption

We study the problem of removability of isolated singularity for general quasilinear anisotropic parabolic equations with absorption. The precise conditions on the behaviour of the absorption term that ensure the non-existence of solutions with point singularities are established.     

A canonical frame for nonholonomic rank two distributions of maximal class

In 1910 E. Cartan constructed the canonical frame and found the most symmetric case for maximally nonholonomic rank 2 distributions in ${\mathbb{R}}^5$. We solve the analogous problems for rank 2 distributions in ${\mathbb{R}}^n$ for arbitrary $n>5$. Our method is a kind of symplectification of the problem and it is completely different from the Cartan method of equivalence. To cite this article: B. Doubrov, I. Zelenko, C. R. Acad. Sci. Paris, Ser. I 342 (2006).