No abstract uploaded!

No abstract uploaded!

For spacetimes that are not asymptotic to anti-de Sitter Space (non AAdS), we adapt the Lewkowycz-Maldacena procedure to find the holographic entanglement entropy. The key observation, which to our knowledge is not very well appreciated, is that asymptotic boundary conditions play an essential role on extending the replica trick to the bulk. For non AAdS, we expect the following three main modifications: (1) the expansion near the special surface has to be compatible with the asymptotic expansion; (2) periodic conditions are imposed to coordinates on the phase space with diagonalized symplectic structure, not to all fields appearing in the action; (3) evaluating the entanglement functional using the boundary term method amounts to evaluating the presymplectic structure at the special surface, where some additional exact form may contribute. An explicit calculation is carried out for three-dimensional warped anti-de Sitter spacetime (WAdS3) in a consistent truncation of string theory, the so-called S-dual dipole theory. It turns out that the generalized gravitational entropy in WAdS3 is captured by the least action of a charged particle in WAdS3 space, or equivalently, by the geodesic length in an auxiliary AdS3. Consequently, the bulk calculation agrees with the CFT results, providing another piece of evidence for the WAdS3/CFT2 correspondence.

We study the boundedness problem for maximal operators $ \mathcal{M} $ associated with averages along smooth hypersurfaces$S$of finite type in 3-dimensional Euclidean space. For$p$> 2, we prove that if no affine tangent plane to$S$passes through the origin and$S$is analytic, then the associated maximal operator is bounded on $ {L^p}\left( {{\mathbb{R}^3}} \right) $ if and only if$p$>$h$($S$), where$h$($S$) denotes the so-called height of the surface$S$(defined in terms of certain Newton diagrams). For non-analytic$S$we obtain the same statement with the exception of the exponent$p$=$h$($S$). Our notion of height$h$($S$) is closely related to A. N. Varchenko’s notion of height$h$($ϕ$) for functions$ϕ$such that$S$can be locally represented as the graph of$ϕ$after a rotation of coordinates.

We introduce a non-linear injective transformation τ from the set of non-vanishing normalized Hausdorff moment sequences to the set of normalized Stieltjes moment sequences by the formula$T$[($a$_{$n$})_{$n$=1}^{∞}]_{$n$}= 1/$a$_{1}...$a$_{$n$}. Special cases of this transformation have appeared in various papers on exponential functionals of Lévy processes, partly motivated by mathematical finance. We give several examples of moment sequences arising from the transformation and provide the corresponding measures, some of which are related to$q$-series.