11,696 research outputs found

### Around supersymmetry for semiclassical second order differential operators

Let $P(h),h\in]0,1]$ be a semiclassical scalar differential operator of order
$2$. The existence of a supersymmetric structure given by a matrix $G(x;h)$ was
exhibited in \cite{HeHiSj13} under rather general assumptions. In this note we
give a sufficient condition on its coefficient so that the matrix $G(x;h)$
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 $\Gamma = Z A +Z^n$ be a dense subgroup with rank $n+1$ in $R^n$ and let
$\omega(A)$ denote the exponent of uniform simultaneous rational approximation
to the point $A$. We show that for any real number $v\ge \omega(A)$, the
Hausdorff dimension of the set $B_v$ of points in $R^n$ which are
$v$-approximable with respect to $\Gamma$, is equal to $1/v$

### Spectral analysis of random walk operators on euclidian space

We study the operator associated to a random walk on $\R^d$ endowed with a
probability measure. We give a precise description of the spectrum of the
operator near $1$ 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 $\R^d$.Comment: 19 page

### 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

### Graph classes and forbidden patterns on three vertices

This paper deals with graph classes characterization and recognition. A
popular way to characterize a graph class is to list a minimal set of forbidden
induced subgraphs. Unfortunately this strategy usually does not lead to an
efficient recognition algorithm. On the other hand, many graph classes can be
efficiently recognized by techniques based on some interesting orderings of the
nodes, such as the ones given by traversals.
We study specifically graph classes that have an ordering avoiding some
ordered structures. More precisely, we consider what we call patterns on three
nodes, and the recognition complexity of the associated classes. In this
domain, there are two key previous works. Damashke started the study of the
classes defined by forbidden patterns, a set that contains interval, chordal
and bipartite graphs among others. On the algorithmic side, Hell, Mohar and
Rafiey proved that any class defined by a set of forbidden patterns can be
recognized in polynomial time. We improve on these two works, by characterizing
systematically all the classes defined sets of forbidden patterns (on three
nodes), and proving that among the 23 different classes (up to complementation)
that we find, 21 can actually be recognized in linear time.
Beyond this result, we consider that this type of characterization is very
useful, leads to a rich structure of classes, and generates a lot of open
questions worth investigating.Comment: Third version version. 38 page

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