The second order spectrum and optimal convergence

The Second Order Spectrum and Optimal Convergence

The method of second order relative spectra has been shown to reliably approximate the discrete spectrum for a self-adjoint operator. We extend the method to normal operators and find optimal convergence rates for eigenvalues and eigenspaces. The convergence to eigenspaces is new, while the convergence rate for eigenvalues improves on the previous estimate by an order of magnitude.

The second order spectrum and optimal convergence

Second Order Sliding Mode Control With Finite Time Convergence

In this paper, a new smooth second order sliding mode control is proposed. This algorithm is a modified form of Super Twisting algorithm. The Super Twisting guarantees the asymptotic stability, but the finite time stability of proposed method is proved with introducing a new particular Lyapunov function. The Proposed algorithm which is able to control nonlinear systems with matched structured u...

Second Order Stability for the Monge-ampère Equation and Strong Sobolev Convergence of Optimal Transport Maps

The aim of this note is to show that Alexandrov solutions of the Monge-Ampère equation, with right hand side bounded away from zero and infinity, converge strongly in W 2,1 loc if their right hand side converge strongly in Lloc. As a corollary we deduce strong W 1,1 loc stability of optimal transport maps.

Second Order General Slow-roll Power Spectrum

Recent combined results from theWilkinson Microwave Anisotropy Probe (WMAP) and Sloan Digital Sky Survey (SDSS) provide a remarkable set of data which requires more accurate and general investigation. Here we derive formulae for the power spectrum P(k) of the density perturbations produced during inflation in the general slowroll approximation with second order corrections. Also, using the resu...