2,307 research outputs found
Asymptotic Uniqueness of Best Rational Approximants to Complex Cauchy Transforms in of the Circle
For all n large enough, we show uniqueness of a critical point in best
rational approximation of degree n, in the L^2-sense on the unit circle, to
functions f, where f is a sum of a Cauchy transform of a complex measure \mu
supported on a real interval included in (-1,1), whose Radon-Nikodym derivative
with respect to the arcsine distribution on its support is Dini-continuous,
non-vanishing and with and argument of bounded variation, and of a rational
function with no poles on the support of \mu.Comment: 28 page
Representation of Markov chains by random maps: existence and regularity conditions
We systematically investigate the problem of representing Markov chains by
families of random maps, and which regularity of these maps can be achieved
depending on the properties of the probability measures. Our key idea is to use
techniques from optimal transport to select optimal such maps. Optimal
transport theory also tells us how convexity properties of the supports of the
measures translate into regularity properties of the maps via Legendre
transforms. Thus, from this scheme, we cannot only deduce the representation by
measurable random maps, but we can also obtain conditions for the
representation by continuous random maps. Finally, we present conditions for
the representation of Markov chain by random diffeomorphisms.Comment: 22 pages, several changes from the previous version including
extended discussion of many detail
On sharp lower bounds for Calabi type functionals and destabilizing properties of gradient flows
Let be a compact K\"ahler manifold with a given ample line bundle . In
\cite{Don05}, Donaldson proved that the Calabi energy of a K\"ahler metric in
is bounded from below by the supremum of a normalized version of the
minus Donaldson--Futaki invariants of test configurations of . He also
conjectured that the bound is sharp. In this paper, we prove a metric analogue
of Donaldson's conjecture, we show that if we enlarge the space of test
configurations to the space of geodesic rays in and replace the
Donaldson--Futaki invariant by the radial Mabuchi K-energy , then a
similar bound holds and the bound is indeed sharp. Moreover, we construct
explicitly a minimizer of . On a Fano manifold, a similar sharp
bound for the Ricci--Calabi energy is also derived.Comment: Final version. Statement of Theorem 4.1 corrected. To appear on
Analysis & PD
On admissibility criteria for weak solutions of the Euler equations
We consider solutions to the Cauchy problem for the incompressible Euler
equations satisfying several additional requirements, like the global and local
energy inequalities. Using some techniques introduced in an earlier paper we
show that, for some bounded compactly supported initial data, none of these
admissibility criteria singles out a unique weak solution.
As a byproduct we show bounded initial data for which admissible solutions to
the p-system of isentropic gas dynamics in Eulerian coordinates are not unique
in more than one space dimension.Comment: 33 pages, 1 figure; v2: 35 pages, corrected typos, clarified proof
Minimum Riesz energy problems for a condenser with "touching plates"
Minimum Riesz energy problems in the presence of an external field are
analyzed for a condenser with touching plates. We obtain sufficient and/or
necessary conditions for the solvability of these problems in both the
unconstrained and the constrained settings, investigate the properties of
minimizers, and prove their uniqueness. Furthermore, characterization theorems
in terms of variational inequalities for the weighted potentials are
established. The results obtained are illustrated by several examples.Comment: 32 pages, 1 figur
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