Continuations of Hermitian indefinite functions and corresponding canonical systems : an example

M. G. Krein established a close connection between the continuation problem of positive definite functions from a finite interval to the real axis and the inverse spectral problem for differential operators. In this note we study such a connection for the function f(t) = 1 − |t|, t - R, which is not positive definite on R: its restrictions fa := f|(−2a,2a) are positive definite if a ≤ 1 and have one negative square if a > 1. We show that with f a canonical differential equation or a Sturm-Liouville equation can be associated which have a singularity

Lifting zero-dimensional schemes and divided powers

We study divided power structures on finitely generated $k$-algebras, where $k$ is a field of positive characteristic $p$. As an application we show examples of $0$-dimensional Gorenstein $k$-schemes that do not lift to a fixed noetherian local ring of non-equal characteristic. We also show that Frobenius neighbourhoods of a singular point of a general hypersurface of large dimension have no liftings to mildly ramified rings of non-equal characteristic.Comment: v2: various changes, added interesting Example 4.3, 12 pages, to appear in Bull. Lond. Math. So

Moduli spaces of principal bundles on singular varieties

Let k be an algebraically closed field of characteristic zero. Let f:X-->S be a flat, projective morphism of k-schemes of finite type with integral geometric fibers. We prove existence of a projective relative moduli space for semistable singular principal bundles on the fibres of f. This generalizes the result of A. Schmitt who studied the case when X is a nodal curve.Comment: 25 pages; dedicated to the memory of Professor Masaki Maruyam

Pre-Supernova Evolution of Massive Single and Binary Stars

Massive stars are essential to understand a variety of branches of astronomy including galaxy and star cluster evolution, nucleosynthesis and supernovae, pulsars and black holes. It has become evident that massive star evolution is very diverse, being sensitive to metallicity, binarity, rotation, and possibly magnetic fields. While the problem to obtain a good statistical observational database is alleviated by current large spectroscopic surveys, it remains a challenge to model these diverse paths of massive stars towards their violent end stage. We show that the main sequence stage offers the best opportunity to gauge the relevance of the various possible evolutionary scenarios. This also allows to sketch the post-main sequence evolution of massive stars, for which observations of Wolf-Rayet stars give essential clues. Recent supernova discoveries due to the current boost in transient searches allow tentative mappings of progenitor models with supernova types, including pair instability supernovae and gamma-ray bursts.Comment: 57 pages, 10 figures; free link to complete published paper: http://www.annualreviews.org/eprint/kf2CFBEBKi93fbcGJCBW/full/10.1146/annurev-astro-081811-12553

Semistable principal G-bundles in positive characteristic

Let $X$ be a normal projective variety defined over an algebraically closed field $k$ of positive characteristic. Let $G$ be a connected reductive group defined over $k$. We prove that some Frobenius pull back of a principal $G$-bundle admits the canonical reduction $E_P$ such that its extension by $P\to P/R_u(P)$ is strongly semistable. Then we show that there is only a small difference between semistability of a principal $G$-bundle and semistability of its Frobenius pull back. This and the boundedness of the family of semistable torsion free sheaves imply the boundedness of semistable principal $G$-bundles.Comment: 23 pages; The final version of this article will be published in the Duke Mathematical Journal, published by Duke University Pres