124,959 research outputs found
A Uniqueness Theorem for Constraint Quantization
This work addresses certain ambiguities in the Dirac approach to constrained
systems. Specifically, we investigate the space of so-called ``rigging maps''
associated with Refined Algebraic Quantization, a particular realization of the
Dirac scheme. Our main result is to provide a condition under which the rigging
map is unique, in which case we also show that it is given by group averaging
techniques. Our results comprise all cases where the gauge group is a
finite-dimensional Lie group.Comment: 23 pages, RevTeX, further comments and references added (May 26. '99
Uniqueness of canonical tensor model with local time
Canonical formalism of the rank-three tensor model has recently been
proposed, in which "local" time is consistently incorporated by a set of first
class constraints. By brute-force analysis, this paper shows that there exist
only two forms of a Hamiltonian constraint which satisfies the following
assumptions: (i) A Hamiltonian constraint has one index. (ii) The kinematical
symmetry is given by an orthogonal group. (iii) A consistent first class
constraint algebra is formed by a Hamiltonian constraint and the generators of
the kinematical symmetry. (iv) A Hamiltonian constraint is invariant under time
reversal transformation. (v) A Hamiltonian constraint is an at most cubic
polynomial function of canonical variables. (vi) There are no disconnected
terms in a constraint algebra. The two forms are the same except for a slight
difference in index contractions. The Hamiltonian constraint which was obtained
in the previous paper and behaved oddly under time reversal symmetry can
actually be transformed to one of them by a canonical change of variables. The
two-fold uniqueness is shown up to the potential ambiguity of adding terms
which vanish in the limit of pure gravitational physics.Comment: 21 pages, 12 figures. The final result unchanged. Section 5 rewritten
for clearer discussions. The range of uniqueness commented in the final
section. Some other minor correction
Impermeability effects in three-dimensional vesicles
We analyse the effects that the impermeability constraint induces on the
equilibrium shapes of a three-dimensional vesicle hosting a rigid inclusion. A
given alteration of the inclusion and/or vesicle parameters leads to shape
modifications of different orders of magnitude, when applied to permeable or
impermeable vesicles. Moreover, the enclosed-volume constraint wrecks the
uniqueness of stationary equilibrium shapes, and gives rise to pear-shaped or
stomatocyte-like vesicles.Comment: 16 pages, 7 figure
Quasivariational solutions for first order quasilinear equations with gradient constraint
We prove the existence of solutions for an evolution quasi-variational
inequality with a first order quasilinear operator and a variable convex set,
which is characterized by a constraint on the absolute value of the gradient
that depends on the solution itself. The only required assumption on the
nonlinearity of this constraint is its continuity and positivity. The method
relies on an appropriate parabolic regularization and suitable {\em a priori}
estimates. We obtain also the existence of stationary solutions, by studying
the asymptotic behaviour in time. In the variational case, corresponding to a
constraint independent of the solution, we also give uniqueness results
On determinism and well-posedness in multiple time dimensions
We study the initial value problem for the wave equation and the
ultrahyperbolic equation for data posed on initial surface of mixed signature
(both spacelike and timelike). Under a nonlocal constraint, we show that the
Cauchy problem on codimension-one hypersurfaces has global unique solutions in
the Sobolev spaces , thus it is well-posed. In contrast, we show that
the initial value problem on higher codimension hypersurfaces is ill-posed, at
least when specifying a finite number of derivatives of the data, due to the
failure of uniqueness. This is in contrast to a uniqueness result which Courant
and Hilbert deduce from Asgeirsson's mean value theorem, for which we give an
independent derivation. The proofs use Fourier synthesis and the Holmgren-John
uniqueness theorem
Well-Posed Initial-Boundary Value Problem for a Constrained Evolution System and Radiation-Controlling Constraint-Preserving Boundary Conditions
A well-posed initial-boundary value problem is formulated for the model
problem of the vector wave equation subject to the divergence-free constraint.
Existence, uniqueness and stability of the solution is proved by reduction to a
system evolving the constraint quantity statically, i.e., the second time
derivative of the constraint quantity is zero. A new set of
radiation-controlling constraint-preserving boundary conditions is constructed
for the free evolution problem. Comparison between the new conditions and the
standard constraint-preserving boundary conditions is made using the
Fourier-Laplace analysis and the power series decomposition in time. The new
boundary conditions satisfy the Kreiss condition and are free from the
ill-posed modes growing polynomially in time.Comment: To appear in the Journal of Hyperbolic Differential Equations. In
response to the reviewers request, a theorem on well-posedness of the free
evolution problem has been added, definitions clarified in Sections 4 and 5,
as well as a typo was removed from Section
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