Parallel Software to Offset the Cost of Higher Precision

Hardware double precision is often insufficient to solve large scientific problems accurately. Computing in higher precision defined by software causes significant computational overhead. The application of parallel algorithms compensates for this overhead. Newton's method to develop power series expansions of algebraic space curves is the use case for this application.

A Newmark space-time formulation in structural dynamics

In this contribution, we present a space-time formulation of the Newmark integration scheme for linear damped structures under both harmonic and transient excitations. The incremental set of equations of motion and the Newmark approximations are transformed into their corresponding space-time equivalents. The dynamic system is then represented by one algebraic space-time equation only. This equation is projected into a coupled pair of space-time equations, which is solved via the fixed point algorithm. The solution is iteratively assembled by enrichments, each of which is decomposed by a dyadic product of spatial and temporal enrichment vectors. The evolution of the spatial enrichment vectors is investigated during convergence and interpreted by comparing them to the set of linear modes of vibration. The new method is demonstrated by means of four numerical examples, presenting not only the excellent convergence behavior and the numerical efficiency but also the limits of the proposed approach.

The resolution property via Azumaya algebras

Using formal-local methods, we prove that a separated and normal tame Artin surface has the resolution property. By proving that normal tame Artin stacks can be rigidified, we ultimately reduce our analysis to establishing the existence of Azumaya algebras. Our construction passes through the case of tame Artin gerbes, tame Artin curves, and algebraic space surfaces, each of which we establish independently.

Quasi-Affineness and the 1-Resolution Property

We prove that, under mild hypothesis, every normal algebraic space that satisfies the $1$-resolution property is quasi-affine. More generally, we show that for algebraic stacks satisfying similar hypotheses, the 1-resolution property guarantees the existence of a finite flat cover by a quasi-affine scheme.