Sketching is a prominent algorithmic tool for processinglarge data. In this paper, we study the problem of sketchingmatrix norms. We consider two sketching models. The firstis bilinear sketching, in which there is a distribution overpairs of r×n matrices S and n× s matrices T such that forany fixed n×n matrix A, from S ·A ·T one can approximate‖A‖p up to an approxima...

We define ε-factors in the de Rham setting and calculate the determinant of the Gauß-Manin connection for a family of (affine) curves and a vector bundle equipped with a flat connection. “Ordentliche Leute pflegten ihren Schatten mit sich zu nehmen, wenn sie in die Sonne gingen.” A. v. Chamisso, from ”Peter Schlemihls wundersame Geschichte”

We generalize the L spectral cluster bounds of Sogge for the Laplace–Beltrami operator on compact Riemannian manifolds to systems of orthonormal functions. The optimality of these new bounds is also discussed. These spectral cluster bounds follow from Schatten-type bounds on oscillatory integral operators.

2 Linear Algebra 3 2.1 Basics. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3 2.2 Norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.3 Vector norms. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 2.4 Induced matrix norms. . . . . . . . . . . . . . . . . . . . . . . . ....

We discuss the connection between the expansion of small sets in graphs, and the Schatten norms of their adjacency matrices. In conjunction with a variant of the Azuma inequality for uniformly smooth normed spaces, we deduce improved bounds on the small-set isoperimetry of Abelian Alon–Roichman random Cayley graphs.