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

Existence and uniqueness of the solution to the Lp Minkowski problem for the electrostatic p-capacity are proved when p > 1 and 1 < p < n. These results are nonlinear extensions of the very recent solution to the Lp Minkowski problem for p-capacity when p = 1 and 1 < p < n by Colesanti et al. and Akman et al., and the classical solution to the Minkowski problem for electrostatic capacity when p = 1 and p = 2 by Jerison.

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The GKZ system for the Hesse pencil of elliptic curves has more solutions than the period integrals. In this work we give different realizations and interpretations of the extra solution, in terms of oscillating integral, Eichler integral, chain integral on the elliptic curve, limit of a period of a certain compact Calabi-Yau threefold geometry, etc. We also highlight the role played by the orbifold singularity on the moduli space and its relation to the GKZ system.

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