Unitary expansion of the time evolution operator

We propose an expansion of the unitary evolution operator, associated to a given Schr\"odinger equation, in terms of a finite product of explicit unitary operators. In this manner, this unitary expansion can be truncated at the desired level of approximation, as shown in the given examples.Comment: 6 pages, 7 figures. Updated version, minor final change

Structure and forcing of the overflow at the Storfjorden sill and its connection to the Arctic coastal polynya in Storfjorden

Storfjorden (Svalbard) is a sill-fjord with an active polynya and exemplifies the dense water formation process over the Arctic shelves. Here we report on our simulations of Storfjorden covering the freezing season of 1999–2000 using an eddy-permitting 3-D ocean circulation model with a fully coupled dynamic and thermodynamic sea-ice model. The model results in the polynya region and of the dense water plume flowing over the sill crest are compared to observations. The connections of the overflow at the sill to the dense water production at the polynya and to the local wind forcing are investigated. Both the overflow and the polynya dynamics are found to be sensitive to wind forcing. In response to freezing and brine rejection over the polynya, the buoyancy forcing initiates an abrupt positive density anomaly. While the ocean integrates the buoyancy forcing over several polynya events (about 25 days), the wind forcing dominates the overflow response at the sill at weather scale. In the model, the density excess is diluted in the basin and leads to a gradual build-up of dense water behind the sill. The overflow transport is typically inferred from observations using a single current profiler at the sill crest. Despite the significant variability of the plume width, we show that a constant overflow width of 15 km produces realistic estimates of the overflow volume transport. Another difficulty in monitoring the overflow is measuring the plume thickness in the absence of hydrographic profiles. Volume flux estimates assuming a constant plume width and the thickness inferred from velocity profiles explain 58% of the modelled overflow volume flux variance and agrees to within 10% when averaged over the overflow season

Transfinite reductions in orthogonal term rewriting systems

Strongly convergent reduction is the fundamental notion of reduction in infinitary orthogonal term rewriting systems (OTRSs). For these we prove the Transfinite Parallel Moves Lemma and the Compressing Lemma. Strongness is necessary as shown by counterexamples. Normal forms, which we allow to be infinite, are unique, in contrast to ω-normal forms. Strongly converging fair reductions result in normal forms. In general OTRSs the infinite Church-Rosser Property fails for strongly converging reductions. However for Böhm reduction (as in Lambda Calculus, subterms without head normal forms may be replaced by ⊥) the infinite Church-Rosser property does hold. The infinite Church-Rosser Property for non-unifiable OTRSs follows. The top-terminating OTRSs of Dershowitz c.s. are examples of non-unifiable OTRSs

Infinitary lambda calculus

In a previous paper we have established the theory of transfinite reduction for orthogonal term rewriting systems. In this paper we perform the same task for the lambda calculus. From the viewpoint of infinitary rewriting, the Böhm model of the lambda calculus can be seen as an infinitary term model. In contrast to term rewriting, there are several different possible notions of infinite term, which give rise to different Böhm-like models, which embody different notions of lazy or eager computation