A Reciprocity Theorem for Monomer-Dimer Coverings

The problem of counting monomer-dimer coverings of a lattice is a longstanding problem in statistical mechanics. It has only been exactly solved for the special case of dimer coverings in two dimensions. In earlier work, Stanley proved a reciprocity principle governing the number $N(m,n)$ of dimer coverings of an $m$ by $n$ rectangular grid (also known as perfect matchings), where $m$ is fixed and $n$ is allowed to vary. As reinterpreted by Propp, Stanley's result concerns the unique way of extending $N(m,n)$ to $n < 0$ so that the resulting bi-infinite sequence, $N(m,n)$ for $n \in {Z}$, satisfies a linear recurrence relation with constant coefficients. In particular, Stanley shows that $N(m,n)$ is always an integer satisfying the relation $N(m,-2-n) = \epsilon_{m,n}N(m,n)$ where $\epsilon_{m,n} = 1$ unless $m\equiv$ 2(mod 4) and $n$ is odd, in which case $\epsilon_{m,n} = -1$. Furthermore, Propp's method is applicable to higher-dimensional cases. This paper discusses similar investigations of the numbers $M(m,n)$, of monomer-dimer coverings, or equivalently (not necessarily perfect) matchings of an $m$ by $n$ rectangular grid. We show that for each fixed $m$ there is a unique way of extending $M(m,n)$ to $n < 0$ so that the resulting bi-infinite sequence, $M(m,n)$ for $n \in {Z}$, satisfies a linear recurrence relation with constant coefficients. We show that $M(m,n)$, a priori a rational number, is always an integer, using a generalization of the combinatorial model offered by Propp. Lastly, we give a new statement of reciprocity in terms of multivariate generating functions from which Stanley's result follows.Comment: 13 pages, 12 figures, to appear in the proceedings of the Discrete Models for Complex Systems (DMCS) 2003 conference. (v2 - some minor changes

Diffraction-limited CCD imaging with faint reference stars

By selecting short exposure images taken using a CCD with negligible readout noise we obtained essentially diffraction-limited 810 nm images of faint objects using nearby reference stars brighter than I=16 at a 2.56 m telescope. The FWHM of the isoplanatic patch for the technique is found to be 50 arcseconds, providing ~20% sky coverage around suitable reference stars.Comment: 4 page letter accepted for publication in Astronomy and Astrophysic

Forming norms: informing diagnosis and management in sports medicine

Clinicians aim to identify abnormalities, and distinguish harmful from harmless abnormalities. In sports medicine, measures of physical function such as strength, balance and joint flexibility are used as diagnostic tools to identify causes of pain and disability and monitor progression in response to an intervention. Comparing results from clinical measures against ‘normal’ values guides decision-making regarding health outcomes. Understanding ‘normal’ is therefore central to appropriate management of disease and disability. However, ‘normal’ is difficult to clarify and definitions are dependent on context. ‘Normal’ in the clinical setting is best understood as an appropriate state of physical function. Particularly as disease, pain and sickness are expected occurrences of being human, understanding ‘normal’ at each stage of the lifespan is essential to avoid the medicalisation of usual life processes. Clinicians use physical measures to assess physical function and identify disability. Accurate diagnosis hinges on access to ‘normal’ reference values for such measures. However our knowledge of ‘normal’ for many clinical measures in sports medicine is limited. Improved knowledge of normal physical function across the lifespan will assist greatly in the diagnosis and management of pain, disease and disability

A site-specific standard for comparing dynamic solar ultraviolet protection characteristics of established tree canopies

A standardised procedure for making fair and comparable assessments of the ultraviolet protection of an established tree canopy that takes into account canopy movement and the changing position of the sun is presented for use by government, planning, and environmental health authorities. The technique utilises video image capture and replaces the need for measurement by ultraviolet radiometers for surveying shade quality characteristics of trees growing in public parks, playgrounds and urban settings. The technique improves upon tree shade assessments that may be based upon single measurements of the ultraviolet irradiance observed from a fixed point of view. The presented technique demonstrates how intelligent shade audits can be conducted without the need for specialist equipment, enabling the calculation of the Shade Protection Index (SPI) and Ultraviolet Protection Factor (UPF) for any discreet time interval and over a full calendar year

Multiquantum well structure with an average electron mobility of 4.0×10^6 cm^2/V s

We report a modulation-doped multiquantum well structure which suppresses the usual ambient light effect associated with modulation doping. Ten GaAs quantum wells 300-Å wide are symmetrically modulation doped using Si δ doping at the center of 3600-Å-wide Al0.1Ga0.9As barriers. The low field mobility of each well is 4.0×10^6 cm/V s at a density of 6.4×10^10 cm^−2 measured at 0.3 K either in the dark, or during, or after, exposure to light. This mobility is an order of magnitude improvement over previous work on multiwells

Phase Diagrams for the ν\nu = 1/2 Fractional Quantum Hall Effect in Electron Systems Confined to Symmetric, Wide GaAs Quantum Wells

We report an experimental investigation of fractional quantum Hall effect (FQHE) at the even-denominator Landau level filling factor $\nu$ = 1/2 in very high quality wide GaAs quantum wells, and at very high magnetic fields up to 45 T. The quasi-two-dimensional electron systems we study are confined to GaAs quantum wells with widths $W$ ranging from 41 to 96 nm and have variable densities in the range of $\simeq 4 \times 10^{11}$ to $\simeq 4 \times 10^{10}$ cm$^{-2}$. We present several experimental phase diagrams for the stability of the $\nu=1/2$ FQHE in these quantum wells. In general, for a given $W$, the 1/2 FQHE is stable in a limited range of intermediate densities where it has a bilayer-like charge distribution; it makes a transition to a compressible phase at low densities and to an insulating phase at high densities. The densities at which the $\nu=1/2$ FQHE is stable are larger for narrower quantum wells. Moreover, even a slight charge distribution asymmetry destabilizes the $\nu=1/2$ FQHE and turns the electron system into a compressible state. We also present a plot of the symmetric-to-antisymmetric subband separation ($\Delta_{SAS}$), which characterizes the inter-layer tunneling, vs density for various $W$. This plot reveals that $\Delta_{SAS}$ at the boundary between the compressible and FQHE phases increases \textit{linearly} with density for all the samples. Finally, we summarize the experimental data in a diagram that takes into account the relative strengths of the inter-layer and intra-layer Coulomb interactions and $\Delta_{SAS}$. We conclude that, consistent with the conclusions of some of the previous studies, the $\nu=1/2$ FQHE observed in wide GaAs quantum wells with symmetric charge distribution is stabilized by a delicate balance between the inter-layer and intra-layer interactions, and is very likely described by a two-component ($\Psi_{311}$) state.Comment: Accepted for publication in Phys. Rev.

Electrovac pppp-waves

New exact solutions of the Einstein-Maxwell field equations that describe $pp$-waves are presented

Multicomponent fractional quantum Hall states with subband and spin degrees of freedom

In wide GaAs quantum wells where two electric subbands are occupied we apply a parallel magnetic field or increase the electron density to cause a crossing of the two $N=0$ Landau levels of these subbands and with opposite spins. Near the crossing, the fractional quantum Hall states in the filling factor range $1<\nu<3$ exhibit a remarkable sequence of pseudospin polarization transitions resulting from the interplay between the spin and subband degrees of freedom. The field positions of the transitions yield a new and quantitative measure of the composite Fermions' discrete energy level separations. Surprisingly, the separations are smaller when the electrons have higher spin-polarization