Smooth stable planes

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)This paper deals with smooth stable planes which generalize the notion of differentiable (affine or projective) planes. It is intended to be the first one of a series of papers on smooth incidence geometry based on the Habilitationsschrift of the author. It contains the basic definitions and results which are needed to build up a foundation for a systematic study of smooth planes. We define smooth stable planes, and we prove that point rows and line pencils are closed submanifolds of the point set and line set, respectively (Theorem (1.6)). Moreover, the flag space is a closed submanifold of the product manifold P×L (Theorem (1.14)), and the smooth structure on the set P of points and on the set L of lines is uniquely determined by the smooth structure of one single line pencil. In the second section it is shown that for any point p \te P the tangent space TpP carries the structure of a locally compact affine translation plane A p , see Theorem (2.5). Dually, we prove in Section 3 that for any line L∈L the tangent space T L L together with the set S L ={T L L p ∣p∈L} gives rise to some shear plane. It turned out that the translation planes A p are one of the most important tools in the investigation of smooth incidence geometries. The linearization theorems (3.9), (3.11), and (4.4) can be viewed as the main results of this paper. In the closing section we investigate some homogeneity properties of smooth projective planes

Semiaffine stable planes

A locally compact stable plane of positive topological dimension will be called semiaffine if for every line $L$ and every point $p$ not in $L$ there is at most one line passing through $p$ and disjoint from $L$. We show that then the plane is either an affine or projective plane or a punctured projective plane (i.e., a projective plane with one point deleted). We also compare this with the situation in general linear spaces (without topology), where P. Dembowski showed that the analogue of our main result is true for finite spaces but fails in general

Collineations of smooth stable planes

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch)Smooth stable planes have been introduced in [4]. We show that every continuous collineation between two smooth stable planes is in fact a smooth collineation. This implies that the group Γ of all continuous collineations of a smooth stable plane is a Lie transformation group on both the set P of points and the set ℒ of lines. In particular, this shows that the point and line sets of a (topological) stable plane ℐ admit at most one smooth structure such that ℐ becomes a smooth stable plane. The investigation of central and axial collineations in the case of (topological) stable planes due to R. Löwen ([25], [26], [27]) is continued for smooth stable planes. Many results of [26] which are only proved for low dimensional planes (dim ℐ ≤ 4) are transferred to smooth stable planes of arbitrary finite dimension. As an application of these transfers we show that the stabilizers Γ[c,c] 1 and Γ[A,A] 1 (see (3.2) Notation) are closed, simply connected, solvable subgroups of Aut(ℐ) (Corollary (4.17)). Moreover, we show that Γ[c,c] is even abelian (Theorem (4.18)). In the closing section we investigate the behaviour of reflections

An alternative to anti-branes and O-planes?

In this paper we consider type IIA compactifications in the isotropic Z2 x Z2 orbifold with a flux-induced perturbative superpotential combined with non-perturbative effects. Without requiring the presence of O-planes, and simply having D6-branes as local sources, we demonstrate the existence of de Sitter (dS) critical points, where the non-perturbative contributions to the cosmological constant have negligible size. We note, however, that these solutions generically have tachyons. By means of a more systematic search, we are able to find two examples of stable dS vacua with no need for anti-branes or O-planes, which, however, exhibit important non-perturbative corrections. The examples that we present turn out to remain stable even after opening up the fourteen non-isotropic moduli.Comment: 12 pages, 4 tables; v2: typos corrected, published versio

Stable planes

Stable planes are a special kind of topological linear spaces. In particular, there is a 'planarity condition' that excludes spaces of geometrical dimension greater than 2. Embeddability problems are posed and answered, and an outline of the classification program is given

Causes of high-temperature superconductivity in the hydrogen sulfide electron-phonon system

The electron and phonon spectra, as well as the density of electron and phonon states of the stable orthorhombic structure of hydrogen sulfide (SH2) at pressures 100-180 GPa have been calculated. It is found that the set of parallel planes of hydrogen atoms is formed at pressure ~175 GPa as a result of structural changes in the unit cell of the crystal under pressure. There should be complete concentration of hydrogen atoms in these planes. As a result the electron properties of the system acquire a quasi-two-dimensional character. The features of in phase and antiphase oscillations of hydrogen atoms in these planes leading to two narrow high-energy peaks in the phonon density of states are investigated

On the Classical Stability of Orientifold Cosmologies

We analyze the classical stability of string cosmologies driven by the dynamics of orientifold planes. These models are related to time-dependent orbifolds, and resolve the orbifold singularities which are otherwise problematic by introducing orientifold planes. In particular, we show that the instability discussed by Horowitz and Polchinski for pure orbifold models is resolved by the presence of the orientifolds. Moreover, we discuss the issue of stability of the cosmological Cauchy horizon, and we show that it is stable to small perturbations due to in-falling matter.Comment: 40 pages, 13 figures. Reference and conclusion added. Published versio

Smooth stable planes and the moduli spaces of locally compact translation planes

Erworben im Rahmen der Schweizer Nationallizenzen (http://www.nationallizenzen.ch

Disorder and interaction effects in two dimensional graphene sheets

The interplay between different types of disorder and electron-electron interactions in graphene planes is studied by means of Renormalization Group techniques. The low temperature properties of the system are determined by fixed points where the strength of the interactions remains finite, as in one dimensional Luttinger liquids. These fixed points can be either stable (attractive), when the disorder is associated to topological defects in the lattice or to a random mass term, or unstable (repulsive) when the disorder is induced by impurities outside the graphene planes. In addition, we analyze mid-gap states which can arise near interfaces or vacancies.Comment: 4 pages, 3 figure