Stationary black holes: Large DD analysis

We consider the effective theory of the large D stationary black hole. By solving Einstein equation with a cosmological constant using the 1/D expansion in near zone of a black hole we obtain the effective equation for the stationary black hole. The effective equation describes the Myers-Perry black hole, bumpy black holes and, possibly, the black ring solution as its solutions. In this effective theory the black hole is represented as the embedded membrane in the background, i.e., Minkowski or Anti-de Sitter spacetime and its mean curvature is given by the redshifted surface gravity by the background geometry and the local Lorentz boost. The local Lorentz boost property of the effective equation is observed also in the metric. In fact we show that the leading order metric of the Einstein equation in the 1/D expansion is generically regarded as the Lorentz boosted Schwarzschild black hole. We apply this Lorentz boost property of the stationary black hole solution to solve the perturbation equation. As a result we obtain the analytic formula for the quasinormal mode of the singly rotating Myers-Perry black hole in the 1/D expansion.Comment: 45 pages, 6 figures, published version in JHE

Non-uniform black strings and the critical dimension in the 1/D1/D expansion

Non-uniform black strings (NUBS) are studied by the large $D$ effective theory approach. By solving the near-horizon geometry in the $1/D$ expansion, we obtain the effective equation for the deformed horizon up to the next-to-next-to-leading order (NNLO) in $1/D$. We also solve the far-zone geometry by the Newtonian approximation. Matching the near and far zones, the thermodynamic variables are computed in the $1/D$ expansion. As the result, the large $D$ analysis gives a critical dimension $D_*\simeq13.5$ at which the translation-symmetry-breaking phase transition changes between first and second order. This value of $D_*$ agrees perfectly, within the precision of the $1/D$ expansion, with the result previously obtained by E. Sorkin through the numerical resolution. We also compare our NNLO results for the thermodynamics of NUBS to earlier numerical calculations, and find good agreement within the expected precision.Comment: 33 pages, 8 figures, Ancillary Mathematica notebook contains details of NNLO results; v2: Published versio

An estimate for derivative of the de la Vallee Poussin mean

In this paper, we discuss derivative of the de la Vallee Poussin mean for exponential weights on real line. When we lead an inequality, an estimate for the Christoffel function plays an important role.Comment: 18 page

Decoupling and non-decoupling dynamics of large D black holes

The limit of large number of dimensions localizes the gravitational field of a black hole in a well-defined region near the horizon. The perturbative dynamics of the black hole can then be characterized in terms of states in the near-horizon geometry. We investigate this by computing the spectrum of quasinormal modes of the Schwarzschild black hole in the 1/D expansion, which we find splits into two classes. Most modes are non-decoupled modes: non-normalizable states of the near-horizon geometry that straddle between the near-horizon zone and the asymptotic zone. They have frequency of order D/r_0 (with r_0 the horizon radius), and are also present in a large class of other black holes. There also exist a much smaller number of decoupled modes: normalizable states of the near-horizon geometry that are strongly suppressed in the asymptotic region. They have frequency of order 1/r_0, and are specific of each black hole. Our results for their frequencies are in excellent agreement with numerical calculations, in some cases even in D=4.Comment: 30 pages, 13 figures; v2: minor correction

Mixture of Expert/Imitator Networks: Scalable Semi-supervised Learning Framework

The current success of deep neural networks (DNNs) in an increasingly broad range of tasks involving artificial intelligence strongly depends on the quality and quantity of labeled training data. In general, the scarcity of labeled data, which is often observed in many natural language processing tasks, is one of the most important issues to be addressed. Semi-supervised learning (SSL) is a promising approach to overcoming this issue by incorporating a large amount of unlabeled data. In this paper, we propose a novel scalable method of SSL for text classification tasks. The unique property of our method, Mixture of Expert/Imitator Networks, is that imitator networks learn to "imitate" the estimated label distribution of the expert network over the unlabeled data, which potentially contributes a set of features for the classification. Our experiments demonstrate that the proposed method consistently improves the performance of several types of baseline DNNs. We also demonstrate that our method has the more data, better performance property with promising scalability to the amount of unlabeled data.Comment: Accepted by AAAI 201

Instability of rotating black holes: large D analysis

We study the stability of odd-dimensional rotating black holes with equal angular momenta by performing an expansion in the inverse of the number of dimensions D. Universality at large $D$ allows us to calculate analytically the complex frequency of quasinormal modes to next-to-leading order in the expansion. We identify the onset of non-axisymmetric, bar-mode instabilities at a specific finite rotation, and axisymmetric instabilities at larger rotation. The former occur at the threshold where the modes become superradiant, and before the ultraspinning regime is reached. Our results fully confirm the picture found in numerical studies, with very good quantitative agreement. We extend the analysis to the same class of black holes in Anti-deSitter space, and find the same qualitative features. We also discuss the appearance at high frequencies of the universal set of (stable) quasinormal modes.Comment: 38 pages, 14 figures. v3: NLO results included so the instability is shown to occur before the ultraspinning regime of rotation. Significant improvements in accuracy. Ancillary Mathematica notebook contains details of NLO result

Quasinormal modes of (Anti-)de Sitter black holes in the 1/D expansion

We use the inverse-dimensional expansion to compute analytically the frequencies of a set of quasinormal modes of static black holes of Einstein-(Anti-)de Sitter gravity, including the cases of spherical, planar or hyperbolic horizons. The modes we study are decoupled modes localized in the near-horizon region, which are the ones that capture physics peculiar to each black hole (such as their instabilities), and which in large black holes contain hydrodynamic behavior. Our results also give the unstable Gregory-Laflamme frequencies of Ricci-flat black branes to two orders higher in 1/D than previous calculations. We discuss the limits on the accuracy of these results due to the asymptotic but not convergent character of the 1/D expansion, which is due to the violation of the decoupling condition at finite D. Finally, we compare the frequencies for AdS black branes to calculations in the hydrodynamic expansion in powers of the momentum k. Our results extend up to k^9 for the sound mode and to k^8 for the shear mode.Comment: 20 pages, 3 figure