We show that the space Lip0(R) is the dual space of L(R;R)/N where N is the subspace of L(R;R) consisting of vector fields whose divergence vanishes identically. We prove that although the quotient space L(R;R)/N is weakly sequentially complete, the subspace N is not nicely placed in other words, its unit ball is not closed for the topology τm of local convergence in measure. We prove that if Ω...

Motivated by a question of Vincent Lafforgue, we study the Banach spaces X satisfying the following property: there is a function ε → ∆ X (ε) tending to zero with ε > 0 such that every operator T : L 2 → L 2 with T ≤ ε that is simultaneously contractive (i.e. of norm ≤ 1) on L 1 and on L ∞ must be of norm ≤ ∆ X (ε) on L 2 (X). We show that ∆ X (ε) ∈ O(ε α) for some α > 0 iff X is isomorphic to ...

We want to compute a worst case ε-approximation to the solution of the Helmholtz equation −1u+qu = f over the unit d-cube Id , subject to Neumann boundary conditions ∂νu = g on ∂Id . Let card(ε, d) denote the minimal number of evaluations of f , g, and q needed to compute an absolute or normalized ε-approximation, assuming that f , g, and q vary over balls of weighted reproducing kernel Hilbert...

Using a recent result of Batson, Spielman and Srivastava, We obtain a tight estimate on the dimension of lp , p an even integer, needed to almost isometrically contain all k-dimensional subspaces of Lp. In a recent paper [BSS] Batson, Spielman and Srivastava introduced a new method for sparsification of graphs which already proved to have functional analytic applications. Here we bring one more...

We show that if X is a tight subspace of C(K) then X has the Pe lczyński property and X is weakly sequentially complete. We apply this result to the space U of uniformly convergent Taylor series on the unit circle and using a minimal amount of Fourier theory prove that U has the Pe lczyński property and U is weakly sequentially complete. Using separate methods, we prove U and U have the Dunford...

Covariance inflation is one of the necessary tools enabling success ensemble Kalman filters (EnKFs) in high dimensional spaces and presence model error. Inflation maintains variance to a sufficiently large value, counteracting damping at analysis times its underestimate arising from sampling errors. In this work we investigate effect on dynamics EnKF ensemble. When focus recursive full cycle fo...

A topological space X is called discretely generated if for every subset A⊂X we have A=∪{D:D⊂A and D is a discrete subspace of X}. We say that X is weakly discretely generated if A⊂X and A6=A implies D\A6=∅ for some discrete D⊂A. It is established that sequential spaces, monotonically normal spaces and compact countably tight spaces are discretely generated. We also prove that every compact spa...

Let Φ ∈ Rm×n be a sparse Johnson-Lindenstrauss transform [KN] with s non-zeroes per column. For T a subset of the unit sphere, ε ∈ (0, 1/2) given, we study settings for m, s required to ensure E Φ sup x∈T ∣∣‖Φx‖22 − 1∣∣ < ε, i.e. so that Φ preserves the norm of every x ∈ T simultaneously and multiplicatively up to 1 + ε. In particular, our most general theorem shows that it suffices to set m = ...

Over the past few years, attempts to scale up infinite-horizon DECPOMDPs with discounted rewards are mainly due to approximate algorithms, but without the theoretical guarantees of their exact counterparts. In contrast, ε-optimal methods have only theoretical significance but are not efficient in practice. In this paper, we introduce an algorithmic framework (β-PI) that exploits the scalability...