On a Problem of Littlewood

AbstractWe prove: If 0≤a1≤a2≤···≤aNandAn=∑ni=1ai, then ∑Nn=1anA2n[∑Nm=na3/2m]2≤2∑Nn=1a2nA4n. G. Bennett had proved this result, but with the factor “4” instead of “2.

Infinite-dimensional stochastic differential equations arising from Airy random point fields

We identify infinite-dimensional stochastic differential equations (ISDEs) describing the stochastic dynamics related to Airy$_{\beta }$ random point fields with $\beta =1,2,4$. We prove the existence of unique strong solutions of these ISDEs. When $\beta = 2$, this solution is equal to the stochastic dynamics defined by the space-time correlation functions obtained by Spohn and Johansson among others. We develop a new method to construct a unique, strong solution of ISDEs. We expect that our approach is valid for other soft-edge scaling limits of stochastic dynamics arising from the random matrix theory.Comment: 55 page

Generalized Forward-Backward Splitting with Penalization for Monotone Inclusion Problems

We introduce a generalized forward-backward splitting method with penalty term for solving monotone inclusion problems involving the sum of a finite number of maximally monotone operators and the normal cone to the nonempty set of zeros of another maximal monotone operator. We show weak ergodic convergence of the generated sequence of iterates to a solution of the considered monotone inclusion problem, provided the condition corresponded to the Fitzpatrick function of the operator describing the set of the normal cone is fulfilled. Under strong monotonicity of an operator, we show strong convergence of the iterates. Furthermore, we utilize the proposed method for minimizing a large-scale hierarchical minimization problem concerning the sum of differentiable and nondifferentiable convex functions subject to the set of minima of another differentiable convex function. We illustrate the functionality of the method through numerical experiments addressing constrained elastic net and generalized Heron location problems