95,975 research outputs found
Extending Whitney's extension theorem: nonlinear function spaces
We consider a global, nonlinear version of the Whitney extension problem for
manifold-valued smooth functions on closed domains , with non-smooth
boundary, in possibly non-compact manifolds. Assuming is a submanifold with
corners, or is compact and locally convex with rough boundary, we prove that
the restriction map from everywhere-defined functions is a submersion of
locally convex manifolds and so admits local linear splittings on charts. This
is achieved by considering the corresponding restriction map for locally convex
spaces of compactly-supported sections of vector bundles, allowing the even
more general case where only has mild restrictions on inward and outward
cusps, and proving the existence of an extension operator.Comment: 37 pages, 1 colour figure. v2 small edits, correction to Definition
A.3, which makes no impact on proofs or results. Version submitted for
publication. v3 small changes in response to referee comments, title
extended. v4 crucial gap filled, results not affected. v5 final version to
appear in Annales de l'Institut Fourie
The `Friction' of Vacuum, and other Fluctuation-Induced Forces
The static Casimir effect describes an attractive force between two
conducting plates, due to quantum fluctuations of the electromagnetic (EM)
field in the intervening space. {\it Thermal fluctuations} of correlated fluids
(such as critical mixtures, super-fluids, liquid crystals, or electrolytes) are
also modified by the boundaries, resulting in finite-size corrections at
criticality, and additional forces that effect wetting and layering phenomena.
Modified fluctuations of the EM field can also account for the `van der Waals'
interaction between conducting spheres, and have analogs in the
fluctuation--induced interactions between inclusions on a membrane. We employ a
path integral formalism to study these phenomena for boundaries of arbitrary
shape. This allows us to examine the many unexpected phenomena of the dynamic
Casimir effect due to moving boundaries. With the inclusion of quantum
fluctuations, the EM vacuum behaves essentially as a complex fluid, and
modifies the motion of objects through it. In particular, from the mechanical
response function of the EM vacuum, we extract a plethora of interesting
results, the most notable being: (i) The effective mass of a plate depends on
its shape, and becomes anisotropic. (ii) There is dissipation and damping of
the motion, again dependent upon shape and direction of motion, due to emission
of photons. (iii) There is a continuous spectrum of resonant cavity modes that
can be excited by the motion of the (neutral) boundaries.Comment: RevTex, 2 ps figures included. The presentation is completely
revised, and new sections are adde
- …