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We study the generalized trace regression with a near low-rank regression coefficient matrix, which extends notion of sparsity for regression coefficient vectors. Specifically, given a matrix covariate X, the probability density function of the response Y is f (Y| X)= c (Y) exp ( 1 Y + b ()), where = tr ( T X). This model accommodates various types of responses and embraces many important problem setups such as reduced-rank regression, matrix regression that accommodates a panel of regressors, matrix completion, among others. We estimate through minimizing empirical negative log-likelihood plus nuclear norm penalty. We first establish a general theory and then for each specific problem, we derive explicitly the statistical rate of the proposed estimator. They all match the minimax rates in the linear trace regression up to logarithmic factors. Numerical studies confirm the rates we established and

In view of such problems as lacking interaction between figures and text and difficulty in simultaneous positioning of explanation with figures in chinese geometry exercise solution, a Win32 C++ voice lecturing tool integrated with HTML5 dynamic geometry software is designed to realize free control of lecturing progress, random skipping through double-click, simultaneous interaction and highlighting of figure and text, and automatic synchronized reading. On the basis of JavaScript regular expression, the semantic comprehension of geometry exercise text is realized and the production rule set of automatic plotting is designed. During voice synthesising line by line, the mathematic language undergoes normalization processing and figure highlighting information extraction. The study can shed some light on the GeoGebra integrated development and semantic comprehension-based automatic plotting application, and it can be applied to innovative geometry teaching and knowledge base structuring fields.

We show that each limiting semiclassical measure obtained from a sequence of eigenfunctions of the Laplacian on a compact hyperbolic surface is supported on the entire cosphere bundle. The key new ingredient for the proof is the fractal uncertainty principle, first formulated by Dyatlov-Zahl and proved for porous sets in Bourgain-Dyatlov.

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