The problem of finding the unique low dimensional decomposition a given matrix has been fundamental and recurrent in many areas. In this paper, we study seeking rank $Y\in \mathbb{R}^{p\times n}$ that admits sparse representation. Specifically, consider $Y = A X\in where $A\in r}$ full column rank, with $r < \min\{n,p\}$, $X\in \mathbb{R}^{r\times is element-wise sparse. We prove $Y$ can be uni...

Abstract We prove a general local law for Wigner matrices that optimally handles observables of arbitrary rank and thus unifies the well-known averaged isotropic laws. As an application, we central limit theorem in quantum unique ergodicity (QUE): is, show quadratic forms deterministic matrix A on bulk eigenvectors have approximately Gaussian fluctuation. For spectrum, generalise our previous r...

Asymptotic formula are derived for the number of n x m matrices of fixed rank k with rational integral coefficients that are contained in a Euclidean ball of radius T in R" *m . It is assumed that n > m > k > 1 are fixed, and the asymptotics are valid as T tends to infinity. The methods used are elementary.

In this paper we present a generalization of the Kalman rank condition for linear ordinary differential systems to the case of systems of n coupled parabolic equations (posed in the time interval (0,T ) with T > 0) where the coupling matrices A and B depend on the time variable t . To be precise, we will prove that the Kalman rank condition rank [A|B](t0) = n , with t0 ∈ [0,T ] , is a sufficien...