The spectral order on$R$^{$n$}induces a natural partial ordering on the manifold $\mathcal{H}_{n}$ of monic hyperbolic polynomials of degree$n$. We show that all differential operators of Laguerre–Pólya type preserve the spectral order. We also establish a global monotony property for infinite families of deformations of these operators parametrized by the space ℓ^{∞}of real bounded sequences. As a consequence, we deduce that the monoid $\mathcal{A}^{\prime}$ of linear operators that preserve averages of zero sets and hyperbolicity consists only of differential operators of Laguerre–Pólya type which are both extensive and isotonic. In particular, these results imply that any hyperbolic polynomial is the global minimum of its $\mathcal{A}^{\prime}$ -orbit and that Appell polynomials are characterized by a global minimum property with respect to the spectral order.

We study a non-convex low-rank promoting penalty function, the transformed Schatten- 1 (TS1), and its applications in matrix completion. The TS1 penalty, as a matrix quasi-norm defined on its singular values, interpolates the rank and the nuclear norm through a nonnegative parameter a∈ (0,+∞). We consider the unconstrained TS1 regularized low-rank matrix recovery problem and develop a fixed point representation for its global minimizer. The TS1 thresholding functions are in closed analytical form for all parameter values. The TS1 threshold values differ in subcritical (supercritical) parameter regime where the TS1 threshold functions are continuous (discontinuous). We propose TS1 iterative thresholding algorithms and compare them with some state-of-the-art algorithms on matrix completion test problems. For problems with known rank, a fully adaptive TS1 iterative thresholding algorithm consistently performs the best under different conditions, where ground truth matrices are generated by multivariate Gaussian, (0,1) uniform and Chi-square distributions. For problems with unknown rank, TS1 algorithms with an additional rank estimation procedure approach the level of IRucL-q which is an iterative reweighted algorithm, non-convex in nature and best in performance.

For a one parameter family of Calabi-Yau threefolds, Green et al. (2009) expressed the total singularities in terms of the degrees of Hodge bundles and Euler number of the general fiber. In this paper, we show that the total singularities can be expressed by the sum of asymptotic values of BCOV (Bershadsky-Cecotti-Ooguri-Vafa) invariants, studied by Fang et al. (2008). On the other hand, by using Yau's Schwarz lemma, we prove Arakelov type inequalities and Euler number bound for Calabi-Yau family over a compact Riemann surface.

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In this paper, we adapt a simple weighted essentially non-oscillatory (WENO) limiter, originally designed for discontinuous Galerkin (DG) schemes, to the correction procedure via reconstruction (CPR) framework for solving conservation laws. The objective of this simple WENO limiter is to simultaneously maintain uniform high order accuracy of the CPR framework in smooth regions and control spurious numerical oscillations near discontinuities. The WENO limiter we adopt in this paper is particularly simple to implement and will not harm the conservativeness of the CPR framework. Also, it uses information only from the target cell and its immediate neighbors, thus can maintain the compactness of the CPR framework. Since the CPR framework with the WENO limiter does not in general preserve positivity of the solution, we also extend the positivity preserving limiters to the CPR framework. Numerical results in one and two dimensions are provided to illustrate the good behavior of this procedure.