Erratum

2-RULED CALIBRATED 4-FOLDS IN ⁷ AND ⁸ JASON LOTAY     

CALIBRATED SUBMANIFOLDS OF R7 AND R8 WITH SYMMETRIES

The principal theory of this paper comprises a technique forconstructing associative, coassociative and Cayley submanifoldsof Euclidean space with symmetries, using first-order ordinarydifferential equations. Explicit examples of U(1)-invariantassociative cones in ⁷ and SU(2)-invariant Cayley 4-folds in⁸ are then produced using this method. Further examples of associative3-folds are presented, which are ruled, and other systems ofdifferential equations defining calibrated submanifolds in ⁷and ⁸ are given.     

Spectroscopy of Na: Bridging the two-proton radioactivity of Mg

The unbound nucleus ¹⁸Na, the intermediate nucleus in the two-proton radioactivity of ¹⁹Mg, was studied by the measurement of the resonant elastic scattering reaction H(¹⁷Ne,p)¹⁷Ne performed at 4 A.MeV using a radioactive beam from the SPIRAL Facility. Spectroscopic properties of the low-lying states were obtained in an R-matrix analysis of the excitation function.     

2-Ruled Calibrated 4-Folds in R7 and R8

This article introduces the notion of 2-ruled 4-folds: submanifoldsof Rⁿ fibred over a 2-fold by affine 2-planes. This is motivatedby a paper by Joyce and previous work of the present author.A 2-ruled 4-fold M is r-framed if an oriented basis is smoothlyassigned to each fibre, and then we may write M in terms oforthogonal smooth maps ₁,₂ : S^n–1 and a smooth map : Rⁿ. We focus on 2-ruled Cayley 4-folds in R⁸ as certainother calibrated 4-folds in R⁷ and R⁸ can be considered as specialcases. The main result characterizes non-planar, r-framed, 2-ruledCayley 4-folds, using a coupled system of nonlinear, first-order,partial differential equations that ₁ and ₂ satisfy, and anothersuch equation on which is linear in . We give a means of constructing2-ruled Cayley 4-folds starting from particular 2-ruled Cayleycones using holomorphic vector fields. This is used to giveexplicit examples of U(1)-invariant 2-ruled Cayley 4-folds asymptoticto a U(1)³-invariant 2-ruled Cayley cone. Examples are alsogiven based on ruled calibrated 3-folds in C³ and R⁷ and complexcones in C⁴.     

Desingularization of Coassociative 4-Folds with Conical Singularities

Suppose that N is a compact coassociative 4-fold with a conical singularity in a 7-manifold M, with a G₂ structure given by a closed 3-form. We construct a smooth family, {N′(t) : t ∈ (0,τ)} for some τ > 0, of compact, nonsingular, coassociative 4-folds in M which converge to N in the sense of currents, in geometric measure theory, as t → 0. This realisation of desingularizations of N is achieved by gluing in an asymptotically conical coassociative 4-fold in , dilated by t, then deforming the resulting compact 4-dimensional submanifold of M to the required coassociative 4-fold. Received: March 2007 Revision: June 2007 Accepted: June 2007     

Single-neutron states in (101)Sn

The first data on the relative single-particle energies outside the doubly magic (100)Sn nucleus were obtained. A prompt 171.7(6) keV gamma-ray transition was correlated with protons emitted following the beta decay of (101)Sn and is interpreted as the transition between the single-neutron g(7/2) and d(5/2) orbitals in (101)Sn. This observation provides a stringent test of current nuclear structure models. The measured nug(7/2)-nud(5/2) energy splitting is compared with values calculated using mean-field nuclear potentials and is used to calculate low-energy excited states in light Sn isotopes in the framework of the shell model. The correlation technique used in this work offers possibilities for future, more extensive spectroscopy near (100)Sn.