No abstract uploaded!

We provide a refinement of the Poincaré inequality on the torus $\mathbb{T}^{d}$ : there exists a set $\mathcal{B} \subset \mathbb{T} ^{d}$ of directions such that for every $\alpha \in \mathcal{B}$ there is a $c_{\alpha } > 0$ with $$\begin{aligned} \|\nabla f\|_{L^{2}(\mathbb{T}^{d})}^{d-1} \| \langle \nabla f, \alpha \rangle \|_{L^{2}(\mathbb{T}^{d})} \geq c_{\alpha }\|f\| _{L^{2}(\mathbb{T}^{d})}^{d} \quad \mbox{for all}~f\in H^{1}\bigl( \mathbb{T}^{d}\bigr)~ \mbox{with mean 0.} \end{aligned}$$ The derivative $\langle \nabla f, \alpha \rangle $ does not detect any oscillation in directions orthogonal to $\alpha $ , however, for certain $\alpha $ the geodesic flow in direction $\alpha $ is sufficiently mixing to compensate for that defect. On the two-dimensional torus $\mathbb{T}^{2}$ the inequality holds for $\alpha = (1, \sqrt{2})$ but is not true for $\alpha = (1,e)$ . Similar results should hold at a great level of generality on very general domains.

The existence of the well-known Jacquet–Langlands correspondence was established by Jacquet and Langlands via the trace formula method in 1970. An explicit construction of such a correspondence was obtained by Shimizu via theta series in 1972. In this paper, we extend the automorphic descent method of Ginzburg–Rallis–Soudry to a new setting. As a consequence, we recover the classical Jacquet–Langlands correspondence for PGL(2) via a new explicit construction.

The estimation of large covariance and precision matrices is fundamental in modern multivariate analysis. However, problems arise from the statistical analysis of large panel economic and financial data. The covariance matrix reveals marginal correlations between variables, while the precision matrix encodes conditional correlations between pairs of variables given the remaining variables. In this paper, we provide a selective review of several recent developments on the estimation of large covariance and precision matrices. We focus on two general approaches: a rankbased method and a factormodelbased method. Theories and applications of both approaches are presented. These methods are expected to be widely applicable to the analysis of economic and financial data.

Japan’s lunar probe satellite “Kaguya”, China’s lunar probe satellite “Chang’ e-1 lunar probe” left the earth on September 14th, 2007 and October 24th, 2007, began the exploration to the moon. The launching of the China’s lunar satellite is the proud to the global Chinese. However, at the same time as we are proud of the event, we must admit that the pictures published which were sent back by Japanese are clearer and more beautiful than Chinese. While we are saying that the Japan has a better technology, maybe we also need to say that the Japanese can hold better chance than Chinese when taking the pictures. The following two pictures were sent back by the satellite. Picture one is sent back by Japan’s satellite, Picture two is sent back by China’s. It can be seen that there is a long way for China if we want to become stronger at science technology. We also hope that we can have some contribution in the prosperity of China now and in the future.