Hankel determinants, Pad\'e approximations, and irrationality exponents

The irrationality exponent of an irrational number $\xi$, which measures the approximation rate of $\xi$ by rationals, is in general extremely difficult to compute explicitly, unless we know the continued fraction expansion of $\xi$. Results obtained so far are rather fragmentary, and often treated case by case. In this work, we shall unify all the known results on the subject by showing that the irrationality exponents of large classes of automatic numbers and Mahler numbers (which are transcendental) are exactly equal to $2$. Our classes contain the Thue--Morse--Mahler numbers, the sum of the reciprocals of the Fermat numbers, the regular paperfolding numbers, which have been previously considered respectively by Bugeaud, Coons, and Guo, Wu and Wen, but also new classes such as the Stern numbers and so on. Among other ingredients, our proofs use results on Hankel determinants obtained recently by Han.Comment: International Mathematics Research Notices 201

Multi-Modal Multi-Scale Deep Learning for Large-Scale Image Annotation

Image annotation aims to annotate a given image with a variable number of class labels corresponding to diverse visual concepts. In this paper, we address two main issues in large-scale image annotation: 1) how to learn a rich feature representation suitable for predicting a diverse set of visual concepts ranging from object, scene to abstract concept; 2) how to annotate an image with the optimal number of class labels. To address the first issue, we propose a novel multi-scale deep model for extracting rich and discriminative features capable of representing a wide range of visual concepts. Specifically, a novel two-branch deep neural network architecture is proposed which comprises a very deep main network branch and a companion feature fusion network branch designed for fusing the multi-scale features computed from the main branch. The deep model is also made multi-modal by taking noisy user-provided tags as model input to complement the image input. For tackling the second issue, we introduce a label quantity prediction auxiliary task to the main label prediction task to explicitly estimate the optimal label number for a given image. Extensive experiments are carried out on two large-scale image annotation benchmark datasets and the results show that our method significantly outperforms the state-of-the-art.Comment: Submited to IEEE TI

Massive Dirac fermions and spin physics in an ultrathin film of topological insulator

We study transport and optical properties of the surface states which lie in the bulk energy gap of a thin-film topological insulator. When the film thickness is comparable with the surface state decay length into the bulk, the tunneling between the top and bottom surfaces opens an energy gap and form two degenerate massive Dirac hyperbolas. Spin dependent physics emerges in the surface bands which are vastly different from the bulk behavior. These include the surface spin Hall effects, spin dependent orbital magnetic moment, and spin dependent optical transition selection rule which allows optical spin injection. We show a topological quantum phase transition where the Chern number of the surface bands changes when varying the thickness of the thin film.Comment: 7 pages, 5 figure

Asymmetric superradiant scattering and abnormal mode amplification induced by atomic density distortion

The superradiant Rayleigh scattering using a pump laser incident along the short axis of a Bose-Einstein condensate with a density distortion is studied, where the distortion is formed by shocking the condensate utilizing the residual magnetic force after the switching-off of the trapping potential. We find that very small variation of the atomic density distribution would induce remarkable asymmetrically populated scattering modes by the matter-wave superradiance with long time pulse. The optical field in the diluter region of the atomic cloud is more greatly amplified, which is not an ordinary mode amplification with the previous cognition. Our numerical simulations with the density envelop distortion are consistent with the experimental results. This supplies a useful method to reflect the geometric symmetries of the atomic density profile by the superradiance scattering.Comment: 7pages,4 figures, Optical Express 21,(2013)1437

Quantum Hall Conductivity in a Landau Type Model with a Realistic Geometry

In this paper, we revisit some quantum mechanical aspects related to the Quantum Hall Effect. We consider a Landau type model, paying a special attention to the experimental and geometrical features of Quantum Hall experiments. The resulting formalism is then used to compute explicitely the Hall conductivity from a Kubo formula.Comment: LaTeX, 1 eps figur