Proper morphism | EPFL Graph Search

In algebraic geometry, a proper morphism between schemes is an analog of a proper map between complex analytic spaces. Some authors call a proper variety over a field k a complete variety. For example, every projective variety over a field k is proper over k. A scheme X of finite type over the complex numbers (for example, a variety) is proper over C if and only if the space X(C) of complex points with the classical (Euclidean) topology is compact and Hausdorff. A closed immersion is proper. A morphism is finite if and only if it is proper and quasi-finite. A morphism f: X → Y of schemes is called universally closed if for every scheme Z with a morphism Z → Y, the projection from the fiber product is a closed map of the underlying topological spaces. A morphism of schemes is called proper if it is separated, of finite type, and universally closed ([EGA] II, 5.4.1 ). One also says that X is proper over Y. In particular, a variety X over a field k is said to be proper over k if the morphism X → Spec(k) is proper. For any natural number n, projective space Pn over a commutative ring R is proper over R. Projective morphisms are proper, but not all proper morphisms are projective. For example, there is a smooth proper complex variety of dimension 3 which is not projective over C. Affine varieties of positive dimension over a field k are never proper over k. More generally, a proper affine morphism of schemes must be finite. For example, it is not hard to see that the affine line A1 over a field k is not proper over k, because the morphism A1 → Spec(k) is not universally closed. Indeed, the pulled-back morphism (given by (x,y) ↦ y) is not closed, because the image of the closed subset xy = 1 in A1 × A1 = A2 is A1 − 0, which is not closed in A1. In the following, let f: X → Y be a morphism of schemes. The composition of two proper morphisms is proper. Any base change of a proper morphism f: X → Y is proper. That is, if g: Z → Y is any morphism of schemes, then the resulting morphism X ×Y Z → Z is proper.

Real-Time Optimization Based on Adaptation of Surrogate Models

Dominique Bonvin, Timm Faulwasser, Alejandro Gabriel Marchetti, Martand Singhal

Recently, different real-time optimization (RTO) schemes that guarantee feasibility of all RTO iterates and monotonic convergence to the optimal plant operating point have been proposed. However, simulations reveal that these schemes converge very slowly t ...

Elsevier Science Bv2016