Koszul Complexes and Symmetric Forms over the Punctured Affine Space

Let X be a regular separated scheme of finite Krull dimension and let $U^{n}_{X} \subset A^{n}_{X}$ be the punctured affine n-space over X. We show that the total graded Witt ring of $U^{n}_{X}$ is a free graded module over the total graded Witt ring of X with two generators 1 and ε. The second generator satisfies the equation ε2 = 1 when n = 1 and ε2 = 0 when n ≥ 2. 2000 Mathematics Subject Classification 11E81, 19G1

Pairings in triangular Witt theory

AbstractGiven a product ⊠:K×L→M between triangulated categories with duality, we show that under some conditions there exist naturally two different pairings Wi(K)×Wj(L)→Wi+j(M), where W∗ denotes the triangulated Witt functor of Balmer [P. Balmer, K-theory 19 (2000) 311–363]. Our main example of such a situation is the case that K=L=M is the bounded derived category of vector bundles over a scheme X and ⊠ is the (derived) tensor product. The derived Witt groups of this scheme W∗(X):=⊕i∈ZWi(X) become a graded skew-commutative ring with two different product structures which are both equally natural. In the last section we prove then a projection formula for our product and show as an application that a Springer-type theorem is true for the derived Witt groups, too

On the splitting principle for cohomoligical invariants of reflection groups

Let k0 be a field of characteristic not two, (V, b) a finite-dimensional regular bilinear space over k0, and W a subgroup of the orthogonal group of (V, b) with the property that the subring of W-invariants of the symmetric algebra of V is a polynomial algebra over k0. We prove that Serre’s splitting principle holds for cohomological invariants of W with values in Rost’s cycle modules

Automatisches Differenzieren und minimal erweiterte Systeme zur Berechnung singulärer Punkte

Zur Bestimmung singulärer Punkte eines bestimmten Typs muss eine zugehörige reduzierte Funktion und deren Ableitungen bestimmte Bedingungen erfüllen. Dabei ist diese reduzierte Funktion implizit durch ein nichtlineares Gleichungssystem definiert. Man erhält letztendlich ein minimal erweitertes System, das auch Ableitungen der reduzierten Funktion enthält,und den singulären Punkt als reguläre Lösung besitzt. In der vorliegenden Arbeit wird die Technik des automatischen Differenzierens für die Vorwärtsmethode dargestellt, insbesondere wird die Differentiation iterativer Verfahren untersucht. Es wird ein Überblick über die Theorie von singulären Punkten gegeben und das Erkennungsproblem definiert. Ein zweistufiges Verfahren zur Bestimmung singulärer Punkte wird auf Basis der Vorwärtsmethode und des Newton-Verfahrens beschrieben und wurde an verschiedenen Typen von singulären Punkten getestet

Rost nilpotence and free theories

We introduce coherent cohomology theories h_* and prove that if such a theory is moreover generically constant then the Rost nilpotence principle holds for projective homogeneous varieties in the category of h_*-motives. Examples of such theories are algebraic cobordism and its descendants the free theories

Characteristics of colliding sea breeze gravity current fronts : a laboratory study

Author Posting. © The Author(s), 2017. This is the author's version of the work. It is posted here under a nonexclusive, irrevocable, paid-up, worldwide license granted to WHOI. It is made available for personal use, not for redistribution. The definitive version was published in Quarterly Journal of the Royal Meteorological Society 143 (2017): 1434–1441, doi:10.1002/qj.3015.Sea and land breeze circulations driven by surface temperature differences between land and sea often evolve into gravity currents with sharp fronts. Along narrow peninsulas, islands and enclosed seas, sea/land breeze fronts from opposing shorelines may converge and collide and may initiate deep convection and heavy precipitation. Here we investigate the collision of two sea breeze gravity current fronts in an analogue laboratory setting. We examine these collisions by means of ‘lock-exchange’ experiments in a rectangular channel. The effects of differences in gravity current density and height are studied. Upon collision, a sharp front separating the two currents develops. For symmetric collisions (the same current densities and heights) this front is vertical and stationary. For asymmetric collisions (density differences, similar heights) the front is tilted, changes shape in time and propagates in the same direction as the heavier current before the collision. Both symmetric and asymmetric collisions lead to upward displacement of fluid from the gravity currents and mixing along the plane of contact. The amount of mixing along the collision front decreases with asymmetry. Height differences impact post-collision horizontal propagation: there is significant propagation in the same direction as the higher current before collision, independent of density differences. Collisions of two gravity current fronts force sustained ascending motions which increase the potential for deep convection. From our experiments we conclude that this potential is larger in stationary collision fronts from symmetric sea breeze collisions than in propagating collision fronts from asymmetric sea breeze collisions.National Science Foundation Grant Number: OCE-0824636; Office of Naval Research Grant Number: N00014-09-1-0844; National Aeronautics and Space Administration Grant Number: NASA NNX14A078