In this paper, we investigate traveling wave behavior in symmetric stratospheric zonal flows governed by the two-dimensional quasi-geostrophic equations under the β-plane approximation. We show that if the stream function is spatially symmetric with respect to a moving axis, then the flow necessarily defines a traveling wave. The proof relies on maximum-principle arguments, the construction of a translated flow field, and the structural properties of the governing equations.
We study sparse leading eigenvalue maximization for general symmetric matrices, with sparse principal component analysis as an important special case. The main purpose of this paper is to provide global optimality characterizations of this class of nonconvex optimization problems under a rank-one spiked model. In the saturated and undersaturated cases, global optimality is characterized through distinct comparisons of the leading eigenvalues of principal submatrices. Both characterizations follow from the same condition based on the restricted spectral norm of the perturbation. The required bound on this restricted spectral norm holds with exponentially high probability when the perturbation is Gaussian Wigner, and analogous bounds are established for sub-Gaussian and sample-covariance perturbations.
