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Modulation of Kekul\'e adatom ordering due to strain in graphene
Intervalley scattering of carriers in graphene at `top' adatoms may give rise
to a hidden Kekul\'e ordering pattern in the adatom positions. This ordering is
the result of a rapid modulation in the electron-mediated interaction between
adatoms at the wavevector , which has been shown experimentally and
theoretically to dominate their spatial distribution. Here we show that the
adatom interaction is extremely sensitive to strain in the supporting graphene,
which leads to a characteristic spatial modulation of the Kekul\'e order as a
function of adatom distance. Our results suggest that the spatial distributions
of adatoms could provide a way to measure the type and magnitude of strain in
graphene and the associated pseudogauge field with high accuracy.Comment: 9 pages, 7 figure
Lower and upper bounds for the first eigenvalue of nonlocal diffusion problems in the whole space
We find lower and upper bounds for the first eigenvalue of a nonlocal
diffusion operator of the form T(u) = - \int_{\rr^d} K(x,y) (u(y)-u(x)) \,
dy. Here we consider a kernel where
is a bounded, nonnegative function supported in the unit ball and means a
diffeomorphism on \rr^d. A simple example being a linear function .
The upper and lower bounds that we obtain are given in terms of the Jacobian of
and the integral of . Indeed, in the linear case we
obtain an explicit expression for the first eigenvalue in the whole \rr^d and
it is positive when the the determinant of the matrix is different from
one. As an application of our results, we observe that, when the first
eigenvalue is positive, there is an exponential decay for the solutions to the
associated evolution problem. As a tool to obtain the result, we also study the
behaviour of the principal eigenvalue of the nonlocal Dirichlet problem in the
ball and prove that it converges to the first eigenvalue in the whole
space as
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