Existence of extremal metrics on almost homogeneous manifolds of cohomogeneity one —IV

This paper is the first of a series of papers in which we generalize our results in (Asian J. of Math. 4, 817–830 (2000); J. Geom. Anal. 12, 63–79 (2002); Intern. J. Math. 14, 259–287 (2003)) to the general complex compact almost homogeneous manifolds of real cohomogeneity one. In this paper we deal with the exceptional case of the G ₂ action (Cf. Intern. J. Math. 14, 259–287 (2003), p. 285). In particular, we prove the existence of Kähler-Einstein metric on this manifold.     

A high power hybrid mid-IR laser source

An efficient hybrid mid-IR laser system comprising a thulium fibre laser, Ho:YAG solid state laser and a zinc germanium phosphide optical parametric oscillator is presented. A 790 nm diode pumped 1908 nm thulium fibre laser operating at 30 W pumps an RTP q-switched Ho:YAG laser emitting 17 W at 40 kHz and 2090 nm. The zinc germanium phosphide optical parametric oscillator efficiently converts this into the 3-5 μm region producing 10.1 W with 59% optical conversion efficiency and an M² = 1.5.     

The action of the etale fundamental group scheme on the connected component of the essentially finite one

We follow the pattern in [14, Section 4] to define an action of the etale fundamental group scheme on the local component of the essentially finite fundamental group scheme of Nori. We show that the associated representation is faithful when X is a curve of genus ⩾2.     

Twisted analytic torsion

We review the Reidemeister, Ray-Singer’s analytic torsion and the Cheeger-Mller theorem. We describe the analytic torsion of the de Rham complex twisted by a flux form introduced by the current authors and recall its properties. We define a new twisted analytic torsion for the complex of invariant differential forms on the total space of a principal circle bundle twisted by an invariant flux form. We show that when the dimension is even, such a torsion is invariant under certain deformation of the metric and the flux form. Under T-duality which exchanges the topology of the bundle and the flux form and the radius of the circular fiber with its inverse, the twisted torsion of invariant forms are inverse to each other for any dimension.     

Quaternionic structures

Any oriented 4-dimensional real vector bundle is naturally a line bundle over a bundle of quaternion algebras. In this paper we give an account of modules over bundles of quaternion algebras, discussing Morita equivalence, characteristic classes and K-theory. The results have been used to describe obstructions for the existence of almost quaternionic structures on 8-dimensional Spin^c manifolds in ?adek et al. (2008) [5] and may be of some interest, also, in quaternionic and algebraic geometry.     

Certain algebraic surfaces with Eisenbud‐Harris general fibration of genus 4

We classify the algebraic surfaces with Eisenbud‐Harris general fibration of genus 4 over a rational curve or an elliptic curve whose slope attains the lower bound. The classification of our surfaces is strongly related to the result of the classification for certain relative quadric hypersurfaces in 3‐dimensional projective space bundles over a rational curve and an elliptic curve. We further prove some results about the canonical maps, the quadric hulls of the canonical images and the deformation for these surfaces.     

A Sufficient Condition for Splitting of Arithmetically Cohen–Macaulay Bundles on General Hypersurfaces

A vector bundle on a hypersurface is arithmetically Cohen–Macaulay if its intermediate cohomologies vanish. On projective spaces, such bundles coincide with those which split into a direct sum of line bundles, but this fails on hypersurfaces of higher degree in general. In this article, we give an inequality which gives a sufficient condition for splitting of arithmetically Cohen–Macaulay bundles on general hypersurfaces.     

Existence of Indecomposable Rank Two Vector Bundles on Higher Dimensional Toric Varieties

In the mid 1970s, Hartshorne conjectured that, for all n > 7, any rank 2 vector bundles on ? ⁿ is a direct sum of line bundles. This conjecture remains still open. In this paper, we construct indecomposable rank two vector bundles on a large class of Fano toric varieties. Unfortunately, this class does not contain ? ⁿ .     

Testing of a scintillator and fibre optic based radiation sensor

We describe the optimisation of RadLine^®; a small, real time, remotely operated radiation detector, which consists of an inorganic scintillation crystal coupled to a fibre optic cable transporting produced photons to a CCD camera some distance away. RadLine^® is tested in a beta and gamma narrow radiation field of 2.4 GBq, from a Caesium-137 (662 KeV) source, at doses rates between 0.125 mSvhr⁻¹ and 10 mSvhr⁻¹. Our results establish that the lower limit of the device corresponds to a dose rate of 0.2 mSvhr⁻¹, constrained by the signal to noise ratio of the instrument. We also demonstrate the process of characterising the RadLine^® for utilisation underwater due to its partial electrical inactiveness; and to consider how the instrument might perform in aquatic environments and ultimately in a First Generation Magnox Storage Ponds (FGMSP). The RadLine^® brings a marked difference to actual underwater radiation monitoring devices such as; HPGe, CZT and GM detectors, which not only incorporate the whole electronics within and are more bulky, only perform over a short range. The RadLine^®’s design offers signification value for intermediate (>100 m) and long range detection.