Context Aware Machine Learning

We propose a principle for exploring context in machine learning models. Starting with a simple assumption that each observation may or may not depend on its context, a conditional probability distribution is decomposed into two parts: context-free and context-sensitive. Then by employing the log-linear word production model for relating random variables to their embedding space representation and making use of the convexity of natural exponential function, we show that the embedding of an observation can also be decomposed into a weighted sum of two vectors, representing its context-free and context-sensitive parts, respectively. This simple treatment of context provides a unified view of many existing deep learning models, leading to revisions of these models able to achieve significant performance boost. Specifically, our upgraded version of a recent sentence embedding model not only outperforms the original one by a large margin, but also leads to a new, principled approach for compositing the embeddings of bag-of-words features, as well as a new architecture for modeling attention in deep neural networks. More surprisingly, our new principle provides a novel understanding of the gates and equations defined by the long short term memory model, which also leads to a new model that is able to converge significantly faster and achieve much lower prediction errors. Furthermore, our principle also inspires a new type of generic neural network layer that better resembles real biological neurons than the traditional linear mapping plus nonlinear activation based architecture. Its multi-layer extension provides a new principle for deep neural networks which subsumes residual network (ResNet) as its special case, and its extension to convolutional neutral network model accounts for irrelevant input (e.g., background in an image) in addition to filtering

Hermitian representations of the extended affine Lie algebra \widetilde{\frak{gl}_{2}(\bc_q)}

We use the idea of free fields to obtain highest weight representations for the extended affine Lie algebra \widetilde{\frak{gl}_{2}(\bc_q)} coordinatized by the quantum torus \bc_q and go on to construct a contravariant hermitian form. We further give a necessary and sufficient condition such that the contravariant hermitian form is positive definite.Comment: 24 page

On the connection problem for the second Painlev\'e equation with large initial data

We consider two special cases of the connection problem for the second Painlev\'e equation (PII) using the method of uniform asymptotics proposed by Bassom et al.. We give a classification of the real solutions of PII on the negative (positive) real axis with respect to their initial data. By product, a rigorous proof of a property associate with the nonlinear eigenvalue problem of PII on the real axis, recently revealed by Bender and Komijani, is given by deriving the asymptotic behavior of the Stokes multipliers.Comment: 25 pages,4 figures. arXiv admin note: text overlap with arXiv:1612.0135

Fine Residual Carrier Frequency and Sampling Frequency Estimation in Wireless OFDM Systems

This paper presents a novel algorithm for residual phase estimation in wireless OFDM systems, including the carrier frequency offset (CFO) and the sampling frequency offset (SFO). The subcarriers are partitioned into several regions which exhibit pairwise correlations. The phase increment between successive OFDM blocks is exploited which can be estimated by two estimators with different computational loads. Numerical results of estimation variance are presented. Simulations indicate performance improvement of the proposed technique over several conventional schemes in a multipath channel.Comment: Submitted to ICC 2013, Budapes

Application of uniform asymptotics to the connection formulas of the fifth Painlev\'{e} equation

We apply the uniform asymptotics method proposed by Bassom, Clarkson, Law and McLeod to a special Painlev\'{e} V equation, and we provide a simpler and more rigorous proof of the connection formulas for a special solution of the equation, which have been established earlier by McCoy and Tang via the isomonodromy and WKB methods.Comment: 18 page

B-spline one-center method for molecular Hartree-Fock calculations

We introduce one-center method in spherical coordinates to carry out Hartree-Fock calculations. Both the radial wave function and the angular wave function are expanded by B-splines, and the radial knots and angular knots are adjusted to deal with cusps properly, resulting in the significant improvement of convergence for several typical closed-shell diatomic molecules. B-splines could represent both the bound state and continuum state wave function properly, and the present approach has been applied to investigating ionization dynamics for H$_2$ in the intense laser field adopting single-active-electron model

Effect of phantom dark energy on the holographic thermalization

Gravitational collapse of a shell of charged dust surrounded by the phantom dark energy is probed by the minimal area surface, which is dual to probe the thermalization in the boundary quantum field by expectation values of Wilson loop in the framework of the AdS/CFT correspondence. We investigated mainly the effect of the phantom dark energy parameter and chemical potential on the thermalization. The result shows that the smaller the phantom dark energy parameter is, the easier the plasma thermalizes as the chemical potential is fixed, and the larger the chemical potential is, the harder the plasma thermalizes as the dark energy parameter is fixed. We get the fitting function of the thermalization curve and with it, the thermalization velocity and thermalization acceleration are discussed.Comment: arXiv admin note: text overlap with arXiv:1412.3878, arXiv:1405.5745 by other author

A note on the connection problem of some special Painlev\'e V functions

As a new application of the method of "uniform asymptotics" proposed by Bassom, Clarkson, Law and McLeod, we provide a simpler and more rigorous proof of the connection formulas of some special solutions of the fifth Painlev\'e equation, which have been established earlier by Andreev and Kitaev

Weak cosmic censorship conjecture with pressure and volume in the Gauss-Bonnet AdS black hole

With the Hamilton-Jacobi equation, we obtain the energy-momentum relation of a charged particle as it is absorbed by the Gauss-Bonnet AdS black hole. On the basis of the energy-momentum relation at the event horizon, we investigate the first law, second law, and weak cosmic censorship conjecture in both the normal phase space and extended phase space. Our results show that the first law, second law as well as the weak cosmic censorship conjecture are valid in the normal phase space for all the initial states are black holes. However, in the extended phase space, the second law is violated for the extremal and near-extremal black holes, and the weak cosmic censorship conjecture is violable for the near-extremal black hole, though the first law is still valid. In addition, in both the the normal and extended phase spaces, we find the absorbed particle changes the configuration of the near-extremal black hole, while don't change that of the extremal black hole.Comment: typos have been fixe

Fitting the trajectories of particles in the equatorial plane of a magnetic dipole with epicycloids

In this paper we discuss epicycloid approximation of the trajectories of charged particles in axisymmetric magnetic fields. Epicycloid trajectories are natural in the Guiding Center approximation and we study in detail the errors arising in this approach. We also discuss how using exact results for particle motion in the equatorial plane of a magnetic dipole the accuracy of this approximation can be significantly extended. We also show that the epicycloids approximate the trajectory of a charged particle more accurately than the position of the particle along the trajectory.Comment: 8 pages, 10 figure