Time-independence of gravitational Rényi entropies and unitarity in quantum gravity

The Hubeny-Rangamani-Takayanagi surface γHRT computing the entropy S(D) of a domain of dependence D on an asymptotically AdS boundary is known to be causally inaccessible from D. We generalize this gravitational result to higher replica numbers n > 1 by considering the replica-invariant surfaces (aka ‘splitting surfaces’) γ of real-time replica-wormhole saddle-points computing Rényi entropies Sn(D) and showing that there is a sense in which D must again be causally inaccessible from γ when the saddle preserves both replica and conjugation symmetry. This property turns out to imply the Sn(D) to be independent of any choice of any Cauchy surface ΣD for D, and also that the Sn(D) are independent of the choice of boundary sources within D. This is a key hallmark of unitary evolution in any dual field theory. Furthermore, from the bulk point of view it adds to the evidence that time evolution of asymptotic observables in quantum gravity is implemented by a unitary operator in each baby universe superselection sector. Though we focus here on pure Einstein-Hilbert gravity and its Kaluza-Klein reductions, we expect the argument to extend to any two-derivative theory who satisfies the null convergence condition. We consider both classical saddles and the effect of back-reaction from quantum corrections.

Integrability and bifurcation of a three-dimensional circuit differential system

<p style='text-indent:20px;'>We study integrability and bifurcations of a three-dimensional circuit differential system. The emerging of periodic solutions under Hopf bifurcation and zero-Hopf bifurcation is investigated using the center manifolds and the averaging theory. The zero-Hopf equilibrium is non-isolated and lies on a line filled in with equilibria. A Lyapunov function is found and the global stability of the origin is proven in the case when it is a simple and locally asymptotically stable equilibrium. We also study the integrability of the model and the foliations of the phase space by invariant surfaces. It is shown that in an invariant foliation at most two limit cycles can bifurcate from a weak focus.</p>

Description of aspheric surfaces

Aspheric surfaces, in particular rotationally invariant surfaces, can be described according to the ISO standard 10110 Part 12 as sagitta functions of the surface coordinates. Usually, such functions are standardized as a combination of conic terms and power series or orthogonal polynomials. Similar functions are applied for surface forms, which are not rotationally invariant as cylindric and toric surfaces. In the following, different forms of describing aspheric surfaces as given in the standard as well as other forms will be presented and compared in an overview, and their special features will be discussed.