Current breakdown of the integer and fractional quantum Hall effects detected by torque magnetometry

We have developed a novel technique that enables measurements of the breakdown of both the integer and fractional quantum Hall effects in a two-dimensional electron system without the need to contact the sample. The critical Hall electric fields that we measure are significantly higher than those reported by other workers, and support the quasi-elastic inter-Landau-level tunnelling model of breakdown. Comparison of the fractional quantum Hall effect results with those obtained on the integer quantum Hall effect allows the fractional quantum Hall effect energy gap to be determined and provides a test of the composite-fermion theory. The temperature dependence of the critical current gives an insight into the mechanism by which momentum may be conserved during the breakdown process.     

Affine semigroup rings that are complete intersections

This paper presents a result concerning the structure of affine semigroup rings that are complete intersections. It generalizes to arbitrary dimensions earlier results for semigroups of dimension less than four. The proof depends on a decomposition theorem for mixed dominating matrices.

     

Existence and Uniqueness Theorems for Massless Fields on a Class of Spacetimes with Closed Timelike Curves

We study the massless scalar field on asymptotically flat spacetimes with closed timelike curves (CTC’s), in which all future-directed CTC’s traverse one end of a handle (wormhole) and emerge from the other end at an earlier time. For a class of static geometries of this type, and for smooth initial data with all derivatives in L ₂ on ℐ ⁻ , we prove existence of smooth solutions which are regular at null and spatial infinity (have finite energy and finite L ₂ -norm) and have the given initial data on ℐ ⁻ . A restricted uniqueness theorem is obtained, applying to solutions that fall off in time at any fixed spatial position. For a complementary class of spacetimes in which CTC’s are confined to a compact region, we show that when solutions exist they are unique in regions exterior to the CTC’s. (We believe that more stringent uniqueness theorems hold, and that the present limitations are our own.) An extension of these results to Maxwell fields and massless spinor fields is sketched. Finally, we discuss a conjecture whose meaning is essentially that the Cauchy problem for free fields is well defined in the presence of CTC’s whenever the problem is well-posed in a geometric-optics limit. We provide some evidence in support of this conjecture, and we present counterexamples that show that neither existence nor uniqueness is guaranteed under weaker conditions. In particular, both existence and uniqueness can fail in smooth, asymptotically flat spacetimes with a compact nonchronal region. Received: 28 November 1994/Accepted: 20 May 1996     

Highly connected multicoloured subgraphs of multicoloured graphs

Suppose the edges of the complete graph on n vertices, E(K_n), are coloured using r colours; how large a k-connected subgraph are we guaranteed to find, which uses only at most s of the colours? This question is due to Bollobás, and the case s=1 was considered in Liu et al. [Highly connected monochromatic subgraphs of multicoloured graphs, J. Graph Theory, to appear]. Here we shall consider the case s2, proving in particular that when s=2 and r+1 is a power of 2 then the answer lies between 4n/(r+1)-17kr(r+2k+1) and 4n/(r+1)+4, that if r=2s+1 then the answer lies between and , and that phase transitions occur at s=r/2 and . We shall also mention some of the more glaring open problems relating to this question.     

The Spatial Dependence of Spin-Echo Signals

The signals in NMR spin echoes which are refocused by 90° pulses are spatially modulated. The spatial modulation is not normally observed in images or profiles obtained using Hahn or stimulated echoes, but may cause errors if the sample structure varies on the distance scale of the modulation. Localized spectra measured using stimulated echoes will also show errors under these conditions. Simple Fourier-transform arguments show that conditions which allow the modulation to become visible in an image or profile have the effect of introducing a second echo into the time-domain acquisition window. Phase cycling may be used to remove the spatial dependence of the signals.     

Muonium diffusion in “dirty” solid krypton

Muonium spin dynamics in solid krypton in the presence of deliberately introduced impurities has been studied in the temperature range from 2 K to 115 K. At temperatures up to 40 K, muonium transverse relaxation rate turns out to be virtually unaffected by impurities. This implies that the Mu energy level shifts in the neighboring wells induced by impurities are smaller than the characteristic Mu activation energy in the one‐phonon regime of quantum diffusion. This revised version was published online in August 2006 with corrections to the Cover Date.     

Entropy bounds for endomorphisms commuting withK actions

Shereshevsky has shown that a shift-commuting homeomorphism from the two-dimensional full shift to itself cannot be expansive, and asked if such a homeomorphism can have finite positive entropy. We formulate an algebraic analogue of this problem, and answer it in a special case by proving the following: ifT : X → X is a mixing endomorphism of a compact metrizable abelian groupX, andT commutes with a completely positive entropyZ ²-actionS onX by continuous automorphisms, thenT has infinite entropy. Dedicated to the memory of Dr. Elizabeth Mary Hartley (1923–1998) The authors gratefully acknowledge support from EPSRC award no. 9570016X, N.S.F. grant No. DMS-94-01093, and the hospitality of the Warwick Mathematics Research Institute.