49,251 research outputs found
Asymptotically polynomial solutions of difference equations of neutral type
Asymptotic properties of solutions of difference equation of the form are studied. We give
sufficient conditions under which all solutions, or all solutions with
polynomial growth, or all nonoscillatory solutions are asymptotically
polynomial. We use a new technique which allows us to control the degree of
approximation
Oscillatory Properties of Solutions of the Fourth Order Difference Equations with Quasidifferences
A class of fourth--order neutral type difference equations with
quasidifferences and deviating arguments is considered. Our approach is based
on studying the considered equation as a system of a four--dimensional
difference system. The sufficient conditions under which the considered
equation has no quickly oscillatory solutions are given
Control of functional differential equations with function space boundary conditions
Problems involving functional differential equations with terminal conditions in function space are considered. Their application to mechanical and electrical systems is discussed. Investigations of controllability, existence of optimal controls, and necessary and sufficient conditions for optimality are reported
Invariant Measures for Dissipative Dynamical Systems: Abstract Results and Applications
In this work we study certain invariant measures that can be associated to
the time averaged observation of a broad class of dissipative semigroups via
the notion of a generalized Banach limit. Consider an arbitrary complete
separable metric space which is acted on by any continuous semigroup
. Suppose that possesses a global
attractor . We show that, for any generalized Banach limit
and any distribution of initial
conditions , that there exists an invariant probability measure
, whose support is contained in , such that for all
observables living in a suitable function space of continuous mappings
on .
This work is based on a functional analytic framework simplifying and
generalizing previous works in this direction. In particular our results rely
on the novel use of a general but elementary topological observation, valid in
any metric space, which concerns the growth of continuous functions in the
neighborhood of compact sets. In the case when does not
possess a compact absorbing set, this lemma allows us to sidestep the use of
weak compactness arguments which require the imposition of cumbersome weak
continuity conditions and limits the phase space to the case of a reflexive
Banach space. Two examples of concrete dynamical systems where the semigroup is
known to be non-compact are examined in detail.Comment: To appear in Communications in Mathematical Physic
On non-uniform smeared black branes
We investigate charged dilatonic black -branes smeared on a transverse
circle. The system can be reduced to neutral vacuum black branes, and we
perform static perturbations for the reduced system to construct non-uniform
solutions. At each order a single master equation is derived, and the
Gregory-Laflamme critical wavelength is determined. Based on the non-uniform
solutions, we discuss thermodynamic properties of this system and argue that in
a microcanonical ensemble the non-uniform smeared branes are entropically
disfavored even near the extremality, if the spacetime dimension is , which is the critical dimension for the vacuum case. However, the critical
dimension is not universal. In a canonical ensemble the vacuum non-uniform
black branes are thermodynamically favorable at , whereas the
non-uniform smeared branes are favorable at near the extremality.Comment: 24 pages, 2 figures; v2: typos corrected, submitted to
Class.Quant.Gra
Neutral and charged matter in equilibrium with black holes
We study the conditions of a possible static equilibrium between spherically
symmetric, electrically charged or neutral black holes and ambient matter. The
following kinds of matter are considered: (1) neutral and charged matter with a
linear equation of state p_r = w\rho (for neutral matter the results of our
previous work are reproduced), (2) neutral and charged matter with p_r \sim
\rho^m, m > 1, and (3) the possible presence of a "vacuum fluid" (the
cosmological constant or, more generally, anything that satisfies the equality
T^0_0 = T^1_1 at least at the horizon). We find a number of new cases of such
an equilibrium, including those generalizing the well-known Majumdar-Papapetrou
conditions for charged dust. It turns out, in particular, that ultraextremal
black holes cannot be in equilibrium with any matter in the absence of a vacuum
fluid; meanwhile, matter with w > 0, if it is properly charged, can surround an
extremal charged black hole.Comment: 12 pages, no figures, final version published in PR
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