897,855 research outputs found
Inverse obstacle problem for the non-stationary wave equation with an unknown background
We consider boundary measurements for the wave equation on a bounded domain
or on a compact Riemannian surface, and introduce a method to
locate a discontinuity in the wave speed. Assuming that the wave speed consist
of an inclusion in a known smooth background, the method can determine the
distance from any boundary point to the inclusion. In the case of a known
constant background wave speed, the method reconstructs a set contained in the
convex hull of the inclusion and containing the inclusion. Even if the
background wave speed is unknown, the method can reconstruct the distance from
each boundary point to the inclusion assuming that the Riemannian metric tensor
determined by the wave speed gives simple geometry in . The method is based
on reconstruction of volumes of domains of influence by solving a sequence of
linear equations. For \tau \in C(\p M) the domain of influence is
the set of those points on the manifold from which the distance to some
boundary point is less than .Comment: 4 figure
Envelopes of holomorphy and holomorphic discs
The envelope of holomorphy of an arbitrary domain in a two-dimensional Stein
manifold is identified with a connected component of the set of equivalence
classes of analytic discs immersed into the Stein manifold with boundary in the
domain. This implies, in particular, that for each of its points the envelope
of holomorphy contains an embedded (non-singular) Riemann surface (and also an
immersed analytic disc) passing through this point with boundary contained in
the natural embedding of the original domain into its envelope of holomorphy.
Moreover, it says, that analytic continuation to a neighbourhood of an
arbitrary point of the envelope of holomorphy can be performed by applying the
continuity principle once. Another corollary concerns representation of certain
elements of the fundamental group of the domain by boundaries of analytic
discs. A particular case is the following. Given a contact three-manifold with
Stein filling, any element of the fundamental group of the contact manifold
whose representatives are contractible in the filling can be represented by the
boundary of an immersed analytic disc.Comment: 39 pages, 9 figure
A Fundamental Domain for V_3
We describe a fundamental domain for the punctured Riemann surface
which parametrises (up to M\"obius conjugacy) the set of quadratic rational
maps with numbered critical points, such that the first critical point has
period three, and such that the second critical point is not mapped in
iterates or less to the periodic orbit of the first. This gives, in turn, a
description, up to topological conjugacy, of all dynamics in all type III
hyperbolic components in , and gives indications of a topological model
for , together with the hyperbolic components contained in it.Comment: 120 page
Robust stability at the Swallowtail singularity
Consider the set of monic fourth-order real polynomials transformed so that the constant term is one. In the three-dimensional space of the coefficients describing this set, the domain of asymptotic stability is bounded by a surface with the Whitney umbrella singularity. The maximum of the real parts of the roots of these polynomials is globally minimized at the Swallowtail singular point of the discriminant surface of the set corresponding to a negative real root of multiplicity four. Motivated by this example, we review recent works on robust stability, abscissa optimization, heavily damped systems, dissipation-induced instabilities, and eigenvalue dynamics in order to point out some connections that appear to be not widely known
The Mean-Field Limit for Solid Particles in a Navier-Stokes Flow
We propose a mathematical derivation of Brinkman's force for a cloud of
particles immersed in an incompressible fluid. Our starting point is the Stokes
or steady Navier-Stokes equations set in a bounded domain with the disjoint
union of N balls of radius 1/N removed, and with a no-slip boundary condition
for the fluid at the surface of each ball. The large N limit of the fluid
velocity field is governed by the same (Navier-)Stokes equations in the whole
domain, with an additional term (Brinkman's force) that is (minus) the total
drag force exerted by the fluid on the particle system. This can be seen as a
generalization of Allaire's result in [Arch. Rational Mech. Analysis 113
(1991), 209-259] who treated the case of motionless, periodically distributed
balls. Our proof is based on slightly simpler, though similar homogenization
techniques, except that we avoid the periodicity assumption and use instead the
phase-space empirical measure for the particle system. Similar equations are
used for describing the fluid phase in various models for sprays
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