Let X be a d×d symmetric random matrix with independent but non-identically distributed Gaussian entries. It has been conjectured by Lata la that the spectral norm of X is always of the same order as the largest Euclidean norm of its rows. A positive resolution of this conjecture would provide a sharp understanding of the probabilistic mechanisms that control the spectral norm of inhomogeneous ...

We present a new scaleable algorithm for approximating the H∞ norm, an important robust stability measure for linear dynamical systems with input and output. Our spectral value set based method uses a novel hybrid expansion-contraction scheme that, under reasonable assumptions, is guaranteed to converge to a stationary point of the optimization problem defining the H∞ norm, and, in practice, ty...

The main result of this paper is that the weak membership problem in the unit ball of a given norm is NP-hard if and only if the weak membership problem in the unit ball of the dual norm is NP-hard. Equivalently, the approximation of a given norm is polynomial time if and only if the approximation of the dual norm is polynomial time. Using the NP-hardness of the approximation of spectral norm o...

In this article, we design a novel linearized and momentum-preserving Fourier pseudo-spectral scheme to solve the Rosenau-Korteweg de Vries equation. With aid of new semi-norm equivalence between method finite difference method, prior bound numerical solution in discrete L ∞ $$ {L}^{\infty } -norm is obtained from momentum conservation law. Subsequently, based on energy solution, show that, wit...

We study the spectra of p×p random matrices K with off-diagonal (i, j) entry equal to n−1/2k(XT i Xj/n ), where Xi’s are the rows of a p× n matrix with i.i.d. entries and k is a scalar function. It is known that under mild conditions, as n and p increase proportionally, the empirical spectral measure of K converges to a deterministic limit μ. We prove that if k is a polynomial and the distribut...

The H∞ norm of a transfer matrix of a control system is the reciprocal of the largest value of ε such that the associated ε-spectral value set is contained in the stability region for the dynamical system (the left half-plane in the continuous-time case and the unit disk in the discrete-time case). We extend an algorithm recently introduced by Guglielmi and Overton [GO11] for approximating the ...

We derive a refinement of the spectral expansion of Arthur’s trace formula. The expression is absolutely convergent with respect to the trace norm.

We prove a H\"older-type inequality for Hamiltonian diffeomorphisms relating the $C^0$ norm, norm of derivative, and Hofer/spectral norm. obtain as consequence that sufficiently fast convergence in metric forces convergence. The second theme our paper is study pseudo-rotations arise from Anosov-Katok method. As an application inequality, we rigidity result such pseudo-rotations.

We derive a refinement of the spectral expansion of Arthur’s trace formula. The expression is absolutely convergent with respect to the trace norm.