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...

Abstract We study the global topology and geometry of horofunction compactification classes symmetric spaces under Finsler distances in three settings: bounded domains form $$B=B_1\times \cdots \times B_r$$ B = 1 × ⋯ <mm...

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 ...

A problem for many kernel-based methods is that the amount of computation required to find the solution scales as O(n), where n is the number of training examples. We develop and analyze an algorithm to compute an easily-interpretable low-rank approximation to an n × n Gram matrix G such that computations of interest may be performed more rapidly. The approximation is of the form G̃k = CW + k C ...

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