We apply our model of quantum gravity to a Kerr-AdS spacetime of dimension $2 m+1$, $m\ge2$, where all rotational parameters are equal, resulting in a wave equation in a quantum spacetime which has a sequence of solutions that can be expressed as a product of stationary and temporal eigenfunctions. The stationary eigenfunctions can be interpreted as radiation and the temporal as gravitational waves. The event horizon corresponds in the quantum model to a Cauchy hypersurface that can be crossed by causal curves in both directions such that the information paradox does not occur. We also prove that the Kerr-AdS spacetime can be maximally extended by replacing in a generalized Boyer-Lindquist coordinate system the $r$ variable by $\rho=r^2$ such that the extended spacetime has a timelike curvature singularity in $\rho=-a^2$.
Surface parameterizations have been widely applied to computer graphics and digital geometry processing. In this paper, we propose a novel stretch energy minimization (SEM) algorithm for the computation of equiareal parameterizations of simply connected open surfaces with a very small area distortion and a highly improved computational efficiency. In addition, the existence of nontrivial limit points of the SEM algorithm is guaranteed under some mild assumptions of the mesh quality. Numerical experiments indicate that the efficiency, accuracy, and robustness of the proposed SEM algorithm outperform other state-of-the-art algorithms. Applications of the SEM on surface remeshing and surface registration for simply connected open surfaces are demonstrated thereafter. Thanks to the SEM algorithm, the computations for these applications can be carried out efficiently and robustly.
We give a classification of birational transformations on smooth projective surfaces
which have a Zariski-dense set of noncritical periodic points. In particular, we show
that if the first dynamical degree is greater than one, the union of all noncritical periodic
orbits is Zariski-dense.
The EM algorithm is a widely used tool in maximum-likelihood estimation
in incomplete data problems. Existing theoretical work has focused on
conditions under which the iterates or likelihood values converge, and the
associated rates of convergence. Such guarantees do not distinguish whether
the ultimate fixed point is a near global optimum or a bad local optimum of
the sample likelihood, nor do they relate the obtained fixed point to the global
optima of the idealized population likelihood (obtained in the limit of infinite
data). This paper develops a theoretical framework for quantifying when and
how quickly EM-type iterates converge to a small neighborhood of a given
global optimum of the population likelihood. For correctly specified models,
such a characterization yields rigorous guarantees on the performance of certain
two-stage estimators in which a suitable initial pilot estimator is refined
with iterations of the EM algorithm. Our analysis is divided into two parts:
a treatment of the EM and first-order EM algorithms at the population level,
followed by results that apply to these algorithms on a finite set of samples.
Our conditions allow for a characterization of the region of convergence of
EM-type iterates to a given population fixed point, that is, the region of the
parameter space over which convergence is guaranteed to a point within a
small neighborhood of the specified population fixed point. We verify our
conditions and give tight characterizations of the region of convergence for
three canonical problems of interest: symmetric mixture of two Gaussians,
symmetric mixture of two regressions and linear regression with covariates
missing completely at random.
In this paper we study the p-adic analytic geometry of the basic unitary
group Rapoport–Zink spaces MK with signature (1, n − 1). Using the theory of
Harder–Narasimhan filtration of finite flat groups developed in Fargues (Journal für die
reine und angewandteMathematik 645:1–39, 2010), Fargues (Théorie de la réduction
pour les groupes p-divisibles, prépublications. http://www.math.jussieu.fr/~fargues/
Prepublications.html, 2010), and the Bruhat–Tits stratification of the reduced special
fiber Mred defined in Vollaard and Wedhorn (Invent. Math. 184:591–627, 2011),
we find some relatively compact fundamental domain DK in MK for the action of
G(Qp)×Jb(Qp), the product of the associated p-adic reductive groups, and prove that
MK admits a locally finite cell decomposition. By considering the action of regular
elliptic elements on these cells, we establish a Lefschetz trace formula for these spaces
by applying Mieda’s main theorem in Mieda (Lefschetz trace formula for open adic
spaces (Preprint). arXiv:1011.1720, 2013).