14,012 research outputs found
Around supersymmetry for semiclassical second order differential operators
Let be a semiclassical scalar differential operator of order
. The existence of a supersymmetric structure given by a matrix was
exhibited in \cite{HeHiSj13} under rather general assumptions. In this note we
give a sufficient condition on its coefficient so that the matrix
enjoys some nice estimates with respect to the semiclassical parameter
On Kronecker's density theorem, primitive points and orbits of matrices
We discuss recent quantitative results in connexion with Kronecker's theorem
on the density of subgroups in R^n and with Dani and Raghavan's theorem on the
density of orbits in the spaces of frames. We also propose several related
problems. The case of the natural linear action of the unimodular group SL_2(Z)
on the real plane is investigated more closely. We then establish an intriguing
link between the configuration of (discrete) orbits of primitive points and the
rate of density of dense orbits
On inhomogeneous Diophantine approximation and Hausdorff dimension
Let be a dense subgroup with rank in and let
denote the exponent of uniform simultaneous rational approximation
to the point . We show that for any real number , the
Hausdorff dimension of the set of points in which are
-approximable with respect to , is equal to
Spectral analysis of random walk operators on euclidian space
We study the operator associated to a random walk on endowed with a
probability measure. We give a precise description of the spectrum of the
operator near and use it to estimate the total variation distance between
the iterated kernel and its stationary measure. Our study contains the case of
Gaussian densities on .Comment: 19 page
Semi-classical analysis of a random walk on a manifold
We prove a sharp rate of convergence to stationarity for a natural random
walk on a compact Riemannian manifold . The proof includes a detailed
study of the spectral theory of the associated operator.Comment: Published in at http://dx.doi.org/10.1214/09-AOP483 the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Approximation to points in the plane by SL(2,Z)-orbits
Let x be a point in R^2 with irrational slope and let \Gamma denote the
lattice SL(2,Z) acting linearly on R^2. Then, the orbit \Gamma x is dense in
R^2. We give efective results on the approximation of a point y in R^2 by
points of the form \gamma x, where \gamma belongs to \Gamma, in terms of the
size of \gamma
Exponents of Diophantine Approximation and Sturmian Continued Fractions
Let x be a real number and let n be a positive integer. We define four
exponents of Diophantine approximation, which complement the exponents w_n(x)
and w_n^*(x) defined by Mahler and Koksma. We calculate their six values when
n=2 and x is a real number whose continued fraction expansion coincides with
some Sturmian sequence of positive integers, up to the initial terms. In
particular, we obtain the exact exponent of approximation to such a continued
fraction x by quadratic surds.Comment: 25 page
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