90 research outputs found
On approximate solutions of semilinear evolution equations
A general framework is presented to discuss the approximate solutions of an
evolution equation in a Banach space, with a linear part generating a semigroup
and a sufficiently smooth nonlinear part. A theorem is presented, allowing to
infer from an approximate solution the existence of an exact solution.
According to this theorem, the interval of existence of the exact solution and
the distance of the latter from the approximate solution can be evaluated
solving a one-dimensional "control" integral equation, where the unknown gives
a bound on the previous distance as a function of time. For example, the
control equation can be applied to the approximation methods based on the
reduction of the evolution equation to finite-dimensional manifolds: among
them, the Galerkin method is discussed in detail. To illustrate this framework,
the nonlinear heat equation is considered. In this case the control equation is
used to evaluate the error of the Galerkin approximation; depending on the
initial datum, this approach either grants global existence of the solution or
gives fairly accurate bounds on the blow up time.Comment: 33 pages, 10 figures. To appear in Rev. Math. Phys. (Shortened
version; the proof of Prop. 3.4. has been simplified
Rotational symmetries of crystals with defects
I use the theory of Lie groups/algebras to discuss the symmetries of crystals with uniform distributions of defects
Phase transitions in geometrothermodynamics
Using the formalism of geometrothermodynamics, we investigate the geometric
properties of the equilibrium manifold for diverse thermodynamic systems.
Starting from Legendre invariant metrics of the phase manifold, we derive
thermodynamic metrics for the equilibrium manifold whose curvature becomes
singular at those points where phase transitions of first and second order
occur. We conclude that the thermodynamic curvature of the equilibrium
manifold, as defined in geometrothermodynamics, can be used as a measure of
thermodynamic interaction in diverse systems with two and three thermodynamic
degrees of freedom
On observability of Renyi's entropy
Despite recent claims we argue that Renyi's entropy is an observable
quantity. It is shown that, contrary to popular belief, the reported domain of
instability for Renyi entropies has zero measure (Bhattacharyya measure). In
addition, we show the instabilities can be easily emended by introducing a
coarse graining into an actual measurement. We also clear up doubts regarding
the observability of Renyi's entropy in (multi--)fractal systems and in systems
with absolutely continuous PDF's.Comment: 18 pages, 1 EPS figure, REVTeX, minor changes, accepted to Phys. Rev.
Polyhedral Cosmic Strings
Quantum field theory is discussed in M\"obius corner kaleidoscopes using the
method of images. The vacuum average of the stress-energy tensor of a free
field is derived and is shown to be a simple sum of straight cosmic string
expressions, the strings running along the edges of the corners. It does not
seem possible to set up a spin-half theory easily.Comment: 15 pages, 4 text figures not include
Non-perturbative calculations for the effective potential of the symmetric and non-Hermitian field theoretic model
We investigate the effective potential of the symmetric
field theory, perturbatively as well as non-perturbatively. For the
perturbative calculations, we first use normal ordering to obtain the first
order effective potential from which the predicted vacuum condensate vanishes
exponentially as in agreement with previous calculations. For the
higher orders, we employed the invariance of the bare parameters under the
change of the mass scale to fix the transformed form totally equivalent to
the original theory. The form so obtained up to is new and shows that all
the 1PI amplitudes are perurbative for both and regions. For
the intermediate region, we modified the fractal self-similar resummation
method to have a unique resummation formula for all values. This unique
formula is necessary because the effective potential is the generating
functional for all the 1PI amplitudes which can be obtained via and thus we can obtain an analytic calculation for the 1PI
amplitudes. Again, the resummed from of the effective potential is new and
interpolates the effective potential between the perturbative regions.
Moreover, the resummed effective potential agrees in spirit of previous
calculation concerning bound states.Comment: 20 page
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