100,874 research outputs found
Interpolation of nonlinear maps
Let and be complex Banach couples and assume that
with norms satisfying for
some . For any , denote by
and the complex interpolation spaces and by
, the open ball of radius in
, centered at zero. Then for any analytic map such that and
are continuous and bounded by constants and , respectively, the
restriction of to , is
shown to be a map with values in which is analytic and bounded by
Schroedinger's Interpolating Dynamics and Burgers' Flows
We discuss a connection (and a proper place in this framework) of the
unforced and deterministically forced Burgers equation for local velocity
fields of certain flows, with probabilistic solutions of the so-called
Schr\"{o}dinger interpolation problem. The latter allows to reconstruct the
microscopic dynamics of the system from the available probability density data,
or the input-output statistics in the phenomenological situations. An issue of
deducing the most likely dynamics (and matter transport) scenario from the
given initial and terminal probability density data, appropriate e.g. for
studying chaos in terms of densities, is here exemplified in conjunction with
Born's statistical interpretation postulate in quantum theory, that yields
stochastic processes which are compatible with the Schr\"{o}dinger picture free
quantum evolution.Comment: Latex file, to appear in "Chaos, Solitons and Fractals
- …