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 ...

Braunstein and Caves showed how quantum information can be used to define a metric for quantum states, relying on attainability of a quantum information bound. We show that the bound is not generally attainable, but that a two-stage procedure of repeated measurements achieves the bound in the limit. We connect to the question of 'non-locality without entanglement': can a joint measurement on n ...

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 ...

Claim A.2. Fix any σ ∈ Ω(R̂). Also, assume that, for each i, outcome ã ∈ A used in the construction of S above satisfies condition (7) in LS. Then, for any t and θ(t), if g = ĝ then m θ(t),θt i = (·, 0) and m θ(t),θt j = (·, 0) for any θ. Proof. Suppose not; then, for some t, θ(t) and θ, g = ĝ and the continuation regime next period at h(θ(t), θ) is either D or S for some i. By similar reasoning...

Let {X(t), 1 ≤ t ≤ T} be a sequence of Rp-valued independent random variables. Let Θ ⊆ Rd be an open, non-empty convex parameter space equipped with the Euclidean inner product 〈·, ·〉, and norm‖ · ‖2. We will also use the `1-norm ‖θ‖1 def = ∑d j=1 |θj |, and the `∞-norm ‖θ‖∞ def = max1≤j≤d |θj |. We assume that there exists a change point τ? ∈ {1, . . . , T − 1}, parameters θ ? , θ (2) ? ∈ Θ, s...

Some considerable damage to steel structures during the Hyogo-ken Nanbu Earthquake occurred. Among them, many exposed-type column bases failed in several consistent patterns, such as brittle base plate fracture, excessive bolt elongation, unexpected early bolt failure, and inferior construction work, etc. The lessons from these phenomena led to the need for improved understanding of column base...

Let Q be an infinite subset of Z, let Ψ : Z→ [0,∞), and let θ ∈ R. Define E(Q,Ψ, θ) = {x ∈ R : ‖qx− θ‖ ≤ Ψ(q) for infinitely many q ∈ Q}. We prove a lower bound on the Fourier dimension of E(Q,Ψ, θ). This generalizes theorems of Kaufman and Bluhm and yields new explicit examples of Salem sets. We give applications to metrical Diophantine approximation, including determining the Hausdorff dimens...

Let κ be a regular cardinal. Consider the Baire numbers of the spaces (2)κ for various θ ≥ κ. Let l be the number of such different Baire numbers. Models of set theory with l = 1 or l = 2 are known and it is also known that l is finite. We show here that if κ > ω, then l could be any given finite number. The Baire number of a topological space with no isolated points is the minimal cardinality ...

In this paper, we give an open book decomposition for the contact structures on some Brieskorn manifolds, in particular for the contact structures of Ustilovsky. The decomposition uses right-handed Dehn twists as conjectured by Giroux. 0. Introduction At the ICM of 2002 Giroux announced some of his results concerning a correspondence between contact structures on manifolds and open book structu...