Throughout, we work in the Euclidean vector space V = R, the space of column vectors with n real entries. As inner product, we will only use the dot product v ·w = vw and corresponding Euclidean norm ‖v ‖ = √v · v . Two vectors v,w ∈ V are called orthogonal if their inner product vanishes: v ·w = 0. In the case of vectors in Euclidean space, orthogonality under the dot product means that they m...

We propose an optimal computational complexity algorithm for the solution of quadratic programming problems with equality constraints arising from partial differential equations. The algorithm combines a variant of the semi–monotonic augmented Lagrangian (SMALE) method with adaptive precision control and a multigrid preconditioning for the Hessian of the cost function and for the inner product ...

Let A be a Banach algebra and X a Banach A-bimodule, the derivation D : A → X is semi-inner if there are ξ, μ ∈ X such that D(a) = a.ξ − μ.a, (a ∈ A). A is called semi-amenable if every derivation D : A → X∗ is semi-inner. The dual Banach algebra A is Connes semi-amenable (resp. approximately semi-amenable) if, every D ∈ Z1w _ (A,X), for each normal, dual Banach A-bimodule X, is semi -inner (re...