Majorisation with applications to the calculus of variations

This paper explores some connections between rank one convexity, multiplicative quasiconvexity and Schur convexity. Theorem 5.1 gives simple necessary and sufficient conditions for an isotropic objective function to be rank one convex on the set of matrices with positive determinant. Theorem 6.2 describes a class of possible non-polyconvex but multiplicative quasiconvex isotropic functions. This class is not contained in a well known theorem of Ball (6.3 in this paper) which gives sufficient conditions for an isotropic and objective function to be polyconvex. We show here that there is a new way to prove directly the quasiconvexity (in the multiplicative form). Relevance of Schur convexity for the description of rank one convex hulls is explained.Comment: 13 page

The little Grothendieck theorem and Khintchine inequalities for symmetric spaces of measurable operators

We prove the little Grothendieck theorem for any 2-convex noncommutative symmetric space. Let \M be a von Neumann algebra equipped with a normal faithful semifinite trace \t, and let $E$ be an r.i. space on (0, \8). Let E(\M) be the associated symmetric space of measurable operators. Then to any bounded linear map $T$ from E(\M) into a Hilbert space $\mathcal H$ corresponds a positive norm one functional f\in E_{(2)}(\M)^* such that \forall x\in E(\M)\quad \|T(x)\|^2\le K^2 \|T\|^2 f(x^*x+xx^*), where $E_{(2)}$ denotes the 2-concavification of $E$ and $K$ is a universal constant. As a consequence we obtain the noncommutative Khintchine inequalities for E(\M) when $E$ is either 2-concave or 2-convex and $q$-concave for some q<\8. We apply these results to the study of Schur multipliers from a 2-convex unitary ideal into a 2-concave one.Comment: 14 pages. To appear in J. Funct. Ana

Transformed Primal-Dual Methods For Nonlinear Saddle Point Systems

A transformed primal-dual (TPD) flow is developed for a class of nonlinear smooth saddle point system. The flow for the dual variable contains a Schur complement which is strongly convex. Exponential stability of the saddle point is obtained by showing the strong Lyapunov property. Several TPD iterations are derived by implicit Euler, explicit Euler, and implicit-explicit methods of the TPD flow. Generalized to the symmetric TPD iterations, linear convergence rate is preserved for convex-concave saddle point systems under assumptions that the regularized functions are strongly convex. The effectiveness of augmented Lagrangian methods can be explained as a regularization of the non-strongly convexity and a preconditioning for the Schur complement. The algorithm and convergence analysis depends crucially on appropriate inner products of the spaces for the primal variable and dual variable. A clear convergence analysis with nonlinear inexact inner solvers is also developed

Portfolio Diversification and Value at Risk Under Thick-Tailedness

We present a unified approach to value at risk analysis under heavy-tailedness using new majorization theory for linear combinations of thick-tailed random variables that we develop. Among other results, we show that the stylized fact that portfolio diversification is always preferable is reversed for extremely heavy-tailed risks or returns. The stylized facts on diversification are nevertheless robust to thick-tailedness of risks or returns as long as their distributions are not extremely long-tailed. We further demonstrate that the value at risk is a coherent measure of risk if distributions of risks are not extremely heavy-tailed. However, coherency of the value at risk is always violated under extreme thick-tailedness. Extensions of the results to the case of dependence, including convolutions of a-symmetric distributions and models with common stochs are provided.

On inequalities for normalized Schur functions

We prove a conjecture of Cuttler et al.~[2011] [A. Cuttler, C. Greene, and M. Skandera; \emph{Inequalities for symmetric means}. European J. Combinatorics, 32(2011), 745--761] on the monotonicity of \emph{normalized Schur functions} under the usual (dominance) partial-order on partitions. We believe that our proof technique may be helpful in obtaining similar inequalities for other symmetric functions.Comment: This version fixes the error of the previous on