The standard eigenfunctions φλ = ei〈λ,x〉 on flat tori Rn/L have L-norms bounded independently of the eigenvalue. In the case of irrational flat tori, it follows that L2normalized eigenfunctions have uniformly bounded L-norms. Similar bases exist on other flat manifolds. Does this property characterize flat manifolds? We give an affirmative answer for compact Riemannian manifolds with quantum co...

Let Xt be any additive process in R d. There are finite indices δi, βi, i = 1, 2 and a function u, all of which are defined in terms of the characteristics of Xt, such that lim inf t→0 u(t) −1/η X * t = 0, if η > δ1, ∞, if η < δ2, lim sup t→0 u(t) −1/η X * t = 0, if η > β2, ∞, if η < β1, a.s., where X * t = sup 0≤s≤t |Xs|. When Xt is a Lévy process with X0 = 0, δ1 = δ2, β1 = β2 and u(t) = t. Th...

In Section 2 replace the definition of ∗◦Q in Definition 1 by x∗◦Q y = inf{u ∗◦ v | u > x, v > y}. It is defined only if the infimum exists. Proposition 1 remains unchanged. Theorem 0. Let (X, ∗◦,→∗◦,≤) be a commutative residuated semigroup on a complete chain equipped with the order topology. Let a, b, c ∈ X be such that a = b→∗◦ c. Let (x, y) ∈ X × X be such that 1. neither x nor y equals the...

We show that the displacement and strain formulations of the displacement-traction problem of three-dimensional linearized elasticity can be viewed as Legendre-Fenchel dual problems to the stress formulation of the same problem. We also show that each corresponding Lagrangian has a saddle-point, thus fully justifying this new approach to elasticity by means of Legendre-Fenchel duality. Résumé D...