11,033 research outputs found
The Lin-Ni's problem for mean convex domains
We prove some refined asymptotic estimates for postive blowing up solutions
to on , on ; being a smooth bounded domain of \rn,
. In particular, we show that concentration can occur only on boundary
points with nonpositive mean curvature when or . As a direct
consequence, we prove the validity of the Lin-Ni's conjecture in dimension
and for mean convex domains and with bounded energy. Recent
examples by Wang-Wei-Yan show that the bound on the energy is a necessary
condition.Comment: To appear in "Memoirs of the AMS
A Rational Deconstruction of Landin's SECD Machine with the J Operator
Landin's SECD machine was the first abstract machine for applicative
expressions, i.e., functional programs. Landin's J operator was the first
control operator for functional languages, and was specified by an extension of
the SECD machine. We present a family of evaluation functions corresponding to
this extension of the SECD machine, using a series of elementary
transformations (transformation into continu-ation-passing style (CPS) and
defunctionalization, chiefly) and their left inverses (transformation into
direct style and refunctionalization). To this end, we modernize the SECD
machine into a bisimilar one that operates in lockstep with the original one
but that (1) does not use a data stack and (2) uses the caller-save rather than
the callee-save convention for environments. We also identify that the dump
component of the SECD machine is managed in a callee-save way. The caller-save
counterpart of the modernized SECD machine precisely corresponds to Thielecke's
double-barrelled continuations and to Felleisen's encoding of J in terms of
call/cc. We then variously characterize the J operator in terms of CPS and in
terms of delimited-control operators in the CPS hierarchy. As a byproduct, we
also present several reduction semantics for applicative expressions with the J
operator, based on Curien's original calculus of explicit substitutions. These
reduction semantics mechanically correspond to the modernized versions of the
SECD machine and to the best of our knowledge, they provide the first syntactic
theories of applicative expressions with the J operator
A minimal-variable symplectic integrator on spheres
We construct a symplectic, globally defined, minimal-coordinate, equivariant
integrator on products of 2-spheres. Examples of corresponding Hamiltonian
systems, called spin systems, include the reduced free rigid body, the motion
of point vortices on a sphere, and the classical Heisenberg spin chain, a
spatial discretisation of the Landau-Lifschitz equation. The existence of such
an integrator is remarkable, as the sphere is neither a vector space, nor a
cotangent bundle, has no global coordinate chart, and its symplectic form is
not even exact. Moreover, the formulation of the integrator is very simple, and
resembles the geodesic midpoint method, although the latter is not symplectic
Collective symplectic integrators
We construct symplectic integrators for Lie-Poisson systems. The integrators
are standard symplectic (partitioned) Runge--Kutta methods. Their phase space
is a symplectic vector space with a Hamiltonian action with momentum map
whose range is the target Lie--Poisson manifold, and their Hamiltonian is
collective, that is, it is the target Hamiltonian pulled back by . The
method yields, for example, a symplectic midpoint rule expressed in 4 variables
for arbitrary Hamiltonians on . The method specializes in
the case that a sufficiently large symmetry group acts on the fibres of ,
and generalizes to the case that the vector space carries a bifoliation.
Examples involving many classical groups are presented
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