Number-resolved master equation approach to quantum measurement and quantum transport

In addition to the well-known Landauer-Buttiker scattering theory and the nonequilibrium Green's function technique for mesoscopic transports, an alternative (and very useful) scheme is quantum master equation approach. In this article, we review the particle-number (n)-resolved master equation (n-ME) approach and its systematic applications in quantum measurement and quantum transport problems. The n-ME contains rich dynamical information, allowing efficient study of topics such as shot noise and full counting statistics analysis. Moreover, we also review a newly developed master equation approach (and its n-resolved version) under self-consistent Born approximation. The application potential of this new approach is critically examined via its ability to recover the exact results for noninteracting systems under arbitrary voltage and in presence of strong quantum interference, and the challenging non-equilibrium Kondo effect.Comment: 24 pages, 16 figures; review article to appear in Frontiers of Physic

Masses of doubly heavy baryons in the Bethe-Salpeter equation approach

A doubly heavy baryon can be regarded as composed of a heavy diquark and a light quark. In this picture, we study the masses of the doubly heavy diquarkes in the Bethe-Salpeter (BS) formalism first, which are then used as one of the inputs in studying the masses of the doubly heavy baryons in the quark-diquark model. We establish the BS equations for both the heavy diquarks and the heavy baryons with and without taking the heavy quark limit, respectively. These equations are solved numerically with the kernel containing the scalar confinement and one-gluon-exchange terms. The mass of the doubly charmed baryon $\Xi_{cc}^{(\ast)}$ is obtained in both approaches, $3.60\sim3.65\,\rm GeV$ ($\Xi_{cc}^{(\ast)}$) under the heavy quark limit, $3.53\sim3.56\,\rm GeV$ for $\Xi_{cc}$ and $3.61\sim3.63\,\rm GeV$ for $\Xi_{cc}^\ast$ without taking the heavy quark limit. The masses of $\Xi_{bc}^\prime$, $\Xi_{bc}^{(\ast)}$, $\Xi_{bb}^{(\ast)}$, $\Omega_{cc}^{(\ast)}$, $\Omega_{bc}^\prime$, $\Omega_{bc}^{(\ast)}$ and $\Omega_{bb}^{(\ast)}$ are also predicted in the same way. We find that the corrections to the results in the heavy quark limit are about $-0.02\,\rm GeV\sim-0.11\,\rm GeV$ for the masses of the doubly heavy baryons

An Orthogonal Discrete Auditory Transform

An orthogonal discrete auditory transform (ODAT) from sound signal to spectrum is constructed by combining the auditory spreading matrix of Schroeder et al and the time one map of a discrete nonlocal Schr\"odinger equation. Thanks to the dispersive smoothing property of the Schr\"odinger evolution, ODAT spectrum is smoother than that of the discrete Fourier transform (DFT) consistent with human audition. ODAT and DFT are compared in signal denoising tests with spectral thresholding method. The signals are noisy speech segments. ODAT outperforms DFT in signal to noise ratio (SNR) when the noise level is relatively high.Comment: 11 pages, 4 figure

Global well-posedness and multi-tone solutions of a class of nonlinear nonlocal cochlear models in hearing

We study a class of nonlinear nonlocal cochlear models of the transmission line type, describing the motion of basilar membrane (BM) in the cochlea. They are damped dispersive partial differential equations (PDEs) driven by time dependent boundary forcing due to the input sounds. The global well-posedness in time follows from energy estimates. Uniform bounds of solutions hold in case of bounded nonlinear damping. When the input sounds are multi-frequency tones, and the nonlinearity in the PDEs is cubic, we construct smooth quasi-periodic solutions (multi-tone solutions) in the weakly nonlinear regime, where new frequencies are generated due to nonlinear interaction. When the input is two tones at frequencies $f_1$, $f_2$ ($f_1 < f_2$), and high enough intensities, numerical results illustrate the formation of combination tones at $2 f_1 -f_2$ and $2f_2 -f_1$, in agreement with hearing experiments. We visualize the frequency content of solutions through the FFT power spectral density of displacement at selected spatial locations on BM.Comment: 23 pages,4 figure

Signal extraction approach for sparse multivariate response regression

In this paper, we consider multivariate response regression models with high dimensional predictor variables. One way to model the correlation among the response variables is through the low rank decomposition of the coefficient matrix, which has been considered by several papers for the high dimensional predictors. However, all these papers focus on the singular value decomposition of the coefficient matrix. Our target is the decomposition of the coefficient matrix which leads to the best lower rank approximation to the regression function, the signal part in the response. Given any rank, this decomposition has nearly the smallest expected prediction error among all approximations to the the coefficient matrix with the same rank. To estimate the decomposition, we formulate a penalized generalized eigenvalue problem to obtain the first matrix in the decomposition and then obtain the second one by a least squares method. In the high-dimensional setting, we establish the oracle inequalities for the estimates. Compared to the existing theoretical results, we have less restrictions on the distribution of the noise vector in each observation and allow correlations among its coordinates. Our theoretical results do not depend on the dimension of the multivariate response. Therefore, the dimension is arbitrary and can be larger than the sample size and the dimension of the predictor. Simulation studies and application to real data show that the proposed method has good prediction performance and is efficient in dimension reduction for various reduced rank models.Comment: 28 pages, 4 figure

Sparse Fisher's discriminant analysis with thresholded linear constraints

Various regularized linear discriminant analysis (LDA) methods have been proposed to address the problems of the classic methods in high-dimensional settings. Asymptotic optimality has been established for some of these methods in high dimension when there are only two classes. A major difficulty in proving asymptotic optimality for multiclass classification is that the classification boundary is typically complicated and no explicit formula for classification error generally exists when the number of classes is greater than two. For the Fisher's LDA, one additional difficulty is that the covariance matrix is also involved in the linear constraints. The main purpose of this paper is to establish asymptotic consistency and asymptotic optimality for our sparse Fisher's LDA with thresholded linear constraints in the high-dimensional settings for arbitrary number of classes. To address the first difficulty above, we provide asymptotic optimality and the corresponding convergence rates in high-dimensional settings for a large family of linear classification rules with arbitrary number of classes, and apply them to our method. To overcome the second difficulty, we propose a thresholding approach to avoid the estimate of the covariance matrix. We apply the method to the classification problems for multivariate functional data through the wavelet transformations

An Invertible Discrete Auditory Transform

A discrete auditory transform (DAT) from sound signal to spectrum is presented and shown to be invertible in closed form. The transform preserves energy, and its spectrum is smoother than that of the discrete Fourier transform (DFT) consistent with human audition. DAT and DFT are compared in signal denoising tests with spectral thresholding method. The signals are noisy speech segments. It is found that DAT can gain 5 to 7 decibel (dB) in signal to noise ratio (SNR) over DFT except when the noise level is relatively low.Comment: 13 pages, 5 figure

A Many to One Discrete Auditory Transform

A many to one discrete auditory transform is presented to map a sound signal to a perceptually meaningful spectrum on the scale of human auditory filter band widths (critical bands). A generalized inverse is constructed in closed analytical form, preserving the band energy and band signal to noise ratio of the input sound signal. The forward and inverse transforms can be implemented in real time. Experiments on speech and music segments show that the inversion gives a perceptually equivalent though mathematically different sound from the input.Comment: 23 pages, 7 figures, 2 table

Asymptotic optimality of sparse linear discriminant analysis with arbitrary number of classes

Many sparse linear discriminant analysis (LDA) methods have been proposed to overcome the major problems of the classic LDA in high-dimensional settings. However, the asymptotic optimality results are limited to the case that there are only two classes, which is due to the fact that the classification boundary of LDA is a hyperplane and explicit formulas exist for the classification error in this case. In the situation where there are more than two classes, the classification boundary is usually complicated and no explicit formulas for the classification errors exist. In this paper, we consider the asymptotic optimality in the high-dimensional settings for a large family of linear classification rules with arbitrary number of classes under the situation of multivariate normal distribution. Our main theorem provides easy-to-check criteria for the asymptotic optimality of a general classification rule in this family as dimensionality and sample size both go to infinity and the number of classes is arbitrary. We establish the corresponding convergence rates. The general theory is applied to the classic LDA and the extensions of two recently proposed sparse LDA methods to obtain the asymptotic optimality. We conduct simulation studies on the extended methods in various settings

Volume growth, eigenvalue and compactness for self-shrinkers

In this paper, we show an optimal volume growth for self-shrinkers, and estimate a lower bound of the first eigenvalue of $\mathcal{L}$ operator on self-shrinkers, inspired by the first eigenvalue conjecture on minimal hypersurfaces in the unit sphere by Yau \cite{SY}. By the eigenvalue estimates, we can prove a compactness theorem on a class of compact self-shrinkers in \ir{3} obtained by Colding-Minicozzi under weaker conditions.Comment: 17 page