94 research outputs found
Wind reversals in turbulent Rayleigh-Benard convection
The phenomenon of irregular cessation and subsequent reversal of the
large-scale circulation in turbulent Rayleigh-B\'enard convection is
theoretically analysed. The force and thermal balance on a single plume
detached from the thermal boundary layer yields a set of coupled nonlinear
equations, whose dynamics is related to the Lorenz equations. For Prandtl and
Rayleigh numbers in the range and 10^{7} \leq
\Ra \leq 10^{12}, the model has the following features: (i) chaotic reversals
may be exhibited at Ra ; (ii) the Reynolds number based on the
root mean square velocity scales as \Re_{rms} \sim \Ra^{[0.41 ...
0.47]} (depending on Pr), and as
(depending on Ra); and (iii) the mean reversal frequency follows an effective
scaling law \omega / (\nu L^{-2}) \sim \Pr^{-(0.64 \pm 0.01)} \Ra^{0.44 \pm
0.01}. The phase diagram of the model is sketched, and the observed
transitions are discussed.Comment: 4 pages, 5 figure
A System F accounting for scalars
The Algebraic lambda-calculus and the Linear-Algebraic lambda-calculus extend
the lambda-calculus with the possibility of making arbitrary linear
combinations of terms. In this paper we provide a fine-grained, System F-like
type system for the linear-algebraic lambda-calculus. We show that this
"scalar" type system enjoys both the subject-reduction property and the
strong-normalisation property, our main technical results. The latter yields a
significant simplification of the linear-algebraic lambda-calculus itself, by
removing the need for some restrictions in its reduction rules. But the more
important, original feature of this scalar type system is that it keeps track
of 'the amount of a type' that is present in each term. As an example of its
use, we shown that it can serve as a guarantee that the normal form of a term
is barycentric, i.e that its scalars are summing to one
Towards a unified theory of Sobolev inequalities
We discuss our work on pointwise inequalities for the gradient which are
connected with the isoperimetric profile associated to a given geometry. We
show how they can be used to unify certain aspects of the theory of Sobolev
inequalities. In particular, we discuss our recent papers on fractional order
inequalities, Coulhon type inequalities, transference and dimensionless
inequalities and our forthcoming work on sharp higher order Sobolev
inequalities that can be obtained by iteration.Comment: 39 pages, made some changes to section 1
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