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Matrix inequalities from a two variables functional
Several matrix/operator inequalies are given. Most of them are unexpected
extensions of the Araki Log-majorization theorem, obtained thanks to a new
log-majorization for positive linear maps and normal operators (Theorem 2.9).
The main idea and technical tool is a two variables log-convex norm functional
(Theorem 1.2).Comment: Final version, to appear in International J. Math: Two corollaries on
Schur products have been added at the end of Section
Pinchings and Positive linear maps
We employ the pinching theorem, ensuring that some operators A admit any
sequence of contractions as an operator diagonal of A, to deduce/improve two
recent theorems of Kennedy-Skoufranis and Loreaux-Weiss for conditional
expectations onto a masa in the algebra of operators on a Hilbert space. We
also get a few results for sums in a unitary orbit
Rattling and freezing in a 1-D transport model
We consider a heat conduction model introduced in \cite{Collet-Eckmann 2009}.
This is an open system in which particles exchange momentum with a row of
(fixed) scatterers. We assume simplified bath conditions throughout, and give a
qualitative description of the dynamics extrapolating from the case of a single
particle for which we have a fairly clear understanding. The main phenomenon
discussed is {\it freezing}, or the slowing down of particles with time. As
particle number is conserved, this means fewer collisions per unit time, and
less contact with the baths; in other words, the conductor becomes less
effective. Careful numerical documentation of freezing is provided, and a
theoretical explanation is proposed. Freezing being an extremely slow process,
however, the system behaves as though it is in a steady state for long
durations. Quantities such as energy and fluxes are studied, and are found to
have curious relationships with particle density
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