49,251 research outputs found

    Asymptotically polynomial solutions of difference equations of neutral type

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    Asymptotic properties of solutions of difference equation of the form Δm(xn+unxn+k)=anf(n,xσ(n))+bn \Delta^m(x_n+u_nx_{n+k})=a_nf(n,x_{\sigma(n)})+b_n 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

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    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

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    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

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    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 XX which is acted on by any continuous semigroup {S(t)}t0\{S(t)\}_{t \geq 0}. Suppose that §(t)}t0\S(t)\}_{t \geq 0} possesses a global attractor A\mathcal{A}. We show that, for any generalized Banach limit LIMT\underset{T \rightarrow \infty}{\rm{LIM}} and any distribution of initial conditions m0\mathfrak{m}_0, that there exists an invariant probability measure m\mathfrak{m}, whose support is contained in A\mathcal{A}, such that Xϕ(x)dm(x)=LIMT1T0TXϕ(S(t)x)dm0(x)dt, \int_{X} \phi(x) d\mathfrak{m} (x) = \underset{T\to \infty}{\rm{LIM}} \frac{1}{T}\int_0^T \int_X \phi(S(t) x) d \mathfrak{m}_0(x) d t, for all observables ϕ\phi living in a suitable function space of continuous mappings on XX. 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 {S(t)}t0\{S(t)\}_{t \geq 0} 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 XX 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

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    We investigate charged dilatonic black pp-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 D13+pD \le 13 +p, 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 D>12+pD > 12+p, whereas the non-uniform smeared branes are favorable at D>14+pD > 14+p 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

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    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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