The eigenvalues and eigenspaces of some discrete div- and curl-related operators are investigated. The discrete operators give some good discrete analogues of the continuous counterparts and play an important role in developing finite volume schemes for solving div-curl equations and electromagnetic systems. Knowledge of the eigenvalues and eigenspaces is very useful in the numerical analysis of finite volume methods for electromagetic systems in <i>nonhomogeneous</i> media.

In this paper, we continue our study of the canonical metrics on the moduli space of curves. We first prove the bounded geometry of the Ricci and perturbed Ricci metrics. By carefully choosing the pertubation constant and by studying the asymptotics, we show that the Ricci and holomorphic sectional curvatures of the perturbed Ricci metric are bounded from above and below by negative constants. Based on our understanding of the KhlerEinstein metric, we show that the logarithmic cotangent bundle over the DeligneMumford moduli space is stable with respect to the canonical polarization. Finally, in the last section, we prove the strongly bounded geometry of the KhlerEinstein metric by using the KhlerRicci flow and a priori estimates of the complex Monge-Ampere equation.

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 Schrodinger equation. Thanks to the dispersive smoothing property of the Schrodinger evolution, ODAT spectrum is a 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.

Existing domain adaptation methods aim at learning features that can be generalized among domains. These methods commonly require to update source classifier to adapt to the target domain and do not properly handle the trade-off between the source domain and the target domain. In this work, instead of training a classifier to adapt to the target domain, we use a separable component called data calibrator to help the fixed source classifier recover discrimination power in the target domain, while preserving the source domain’s performance. When the difference between two domains is small, the source classifier’s representation is sufficient to perform well in the target domain and outperforms GAN-based methods in digits. Otherwise, the proposed method can leverage synthetic images generated by GANs to boost performance and achieve state-of-the-art performance in digits datasets and driving scene semantic segmentation. Our method also empirically suggests the potential connection between domain adaptation and adversarial attacks. Code release is available at https://github.com/yeshaokai/Calibrator-Domain-Adaptation

In this paper, we present and prove the first closed formula bounding the degree of regularity of an HFE system over an arbitrary finite field. Though these bounds are not necessarily optimal, they can be used to deduce 1. if D, the degree of the corresponding HFE polynomial, and q, the size of the corresponding finite field, are fixed, inverting HFE system is polynomial for all fields; 2. if D is of the scale O(n^α) where n is the number of variables in an HFE system, and q is fixed, inverting HFE systems is quasi-polynomial for all fields. We generalize and prove rigorously similar results by Granboulan, Joux and Stern in the case when q = 2 that were communicated at Crypto 2006.