Holomorphic curves in exploded manifolds: regularity

The category of exploded manifolds is an extension of the category of smooth manifolds related to tropical geometry in which some adiabatic limits appear as smooth families. This paper studies the dbar equation on variations of a given family of curves in an exploded manifold. Roughly, we prove that the dbar equation on variations of an exploded family of curves behaves as nicely as the dbar equation on variations of a smooth family of smooth curves, even though exploded families of curves allow the development of normal crossing or log smooth singularities. The resulting regularity results are used in a series of separate papers to construct Gromov Witten invariants for exploded manifolds.Comment: 52 pages. v2: The construction of Gromov Witten invariants has been removed to another paper. v3: rewritten introduction, improved exposition. v4, v5: improved exposition v6, v7: Minor improvements and some expanded explanations, (including weakened hypothesis for Proposition 3.11), as suggested by an anonymous referee of a different paper. Final version to appear in Geometry and Topolog

Tip effects in scanning tunnelling microscopy of atomic-scale magnetic structures

The spin-polarized scanning tunnelling microscope (STM) can in principle resolve not only the electronic, but also the magnetic surface structure. We model recent STM measurements achieving magnetic resolution on the atomic scale by a first-principles method. It is shown that the signature of a specific magnetic or non-magnetic STM tip can unambiguously be identified. It is also established that the model of Tersoff and Hamann would yield an electronic as well as magnetic contrast of the surface which is well below STM resolution.Comment: Four pages (RevTeX) and three figures (EPS). For related papers see http://cmmp.phys.ucl.ac.uk/~wah/index.htm

Determining surface magnetization and local magnetic moments with atomic scale resolution

We propose a method to determine the direction of surface magnetization and local magnetic moments on the atomic scale. The method comprises high resolution scanning tunneling microscope experiments in conjunction with first principles simulations of the tunneling current. The potential of the method is demonstrated on a model system, antiferromagnetic Mn overlayers on W(110). We expect that it will ultimately allow to study the detailed changes of magnetic surface structures in the vicinity of dopants or impurities.Comment: Four pages (RevTeX) and five figures (EPS). For related papers see http://cmmp.phys.ucl.ac.uk/~wah

Pseudoholomorphic Strips in Symplectisations I: Asymptotic Behavior

This paper is part of a larger program, the investigation of the Chord Problem in three dimensional contact geometry. The main tool will be pseudoholomorphic strips in the symplectisation of a three dimensional contact manifold with two totally real submanifolds $L_0,L_1$ as boundary conditions. The submanifolds $L_0$ and $L_1$ do not intersect transversally. The subject of this paper is to study the asymptotic behavior of such pseudoholomorphic strips.Comment: An updated version of the preprint math.SG/0111142 will appear as four separate papers. This is the first part (53 pages). In this version one estimate is significantly improve

The ‘Curiouser and curiouser’ legal nature of non-UN sanctions : the case of the US sanctions against Russia

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