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

Shallow-water spinal injuries – devastating but preventable

Background. Shallow-water diving injuries have devastating consequences for patients and their families, requiring intensive use of resources in both the acute and rehabilitative phases of injury. With the final clinical outcome often poor, the question is raised as to whether a target group can be identified for whom to implement a preventive programme. Objective. To evaluate the demographics, clinical features and outcomes of shallow-water diving injuries in an acute spinal cord injury (ASCI) unit. Materials and methods. All patients admitted to the ASCI unit with diving-related injuries were entered into the study. Data regarding demographics, injury profile and subsequent management were collated. All case notes and X-rays were reviewed. Ethical approval was obtained. Results. Forty-six patients were reviewed from 19 April 2003 to 8 February 2009. A steady annual increase in diving injuries was noted. A very specific patient profile was identified: 91% male incidence, average age 23 years, 37% admitted alcohol use, with a summer-time prevalence. Compression-flexion type injuries were most prevalent, with an orthopaedic level of C5 and neurological level of C4 being the most common injury sites. A third of diving injuries occurred in the sea, 20% in swimming pools, 20% in rivers, 11% in tidal pools and 4% in dams. Conclusion. A very specific patient profile was identified, and the severity of shallow-water diving injuries was confirmed. No current preventive programme exists except for a single television advertisement. These data will be used to motivate further educational and preventive programmes for reducing the incidence of diving-related injuries

Temporal variability of gas seeps offshore New Zealand: Multi-frequency geoacoustic imaging of the Wairarapa area, Hikurangi margin

Cold seeps of the Wairarapa area situated around 1000 metres water depth on the Hikurangi margin offshore New Zealand’s North Island have been investigated using multibeam bathymetry, 75 and 410 kHz sidescan sonar imagery as well as 2-8 kHz Chirp sediment echosounder data. Towed video camera observations allowed groundtruthing the various geoacoustic data. At least eight different seep locations displaying various degrees of seep activity have been identified in the study area which consists of an elongated, northward-widening ridge that is part of the Hikurangi margin accretionary complex and well separated from direct terrigenous input by margin channels surrounding the ridge. The geoacoustic signature of individual cold-seep sites range from smooth areas with slightly elevated backscatter intesity resulting from high gas content or the presence of near-surface gas hydrates to rough areas with widespread patches of carbonates at the seafloor. Two cold seeps also show indications for active gas emissions in form of acoustic plumes in the water column. Repeated sidescan sonar imagery of the plumes indicates they are highly variable in intensity and directivity in the water column reflecting a likely control of gas emission by tides and currents. Although gas emission appears strongly focused in the Wairarapa area the actual extent of the cold seep structures are slightly wider in the subsurface as shown by sediment echosounder profiles. On a larger scale, however, fluid seepage at the Wairarapa area is still highly focused

Sliding mode control of quantum systems

This paper proposes a new robust control method for quantum systems with uncertainties involving sliding mode control (SMC). Sliding mode control is a widely used approach in classical control theory and industrial applications. We show that SMC is also a useful method for robust control of quantum systems. In this paper, we define two specific classes of sliding modes (i.e., eigenstates and state subspaces) and propose two novel methods combining unitary control and periodic projective measurements for the design of quantum sliding mode control systems. Two examples including a two-level system and a three-level system are presented to demonstrate the proposed SMC method. One of main features of the proposed method is that the designed control laws can guarantee desired control performance in the presence of uncertainties in the system Hamiltonian. This sliding mode control approach provides a useful control theoretic tool for robust quantum information processing with uncertainties.Comment: 18 pages, 4 figure

Beat Cepheids as Probes of Stellar and Galactic Metallicity

The mere location of a Beat Cepheid model in a Period Ratio vs. Period diagram (Petersen diagram) puts very tight constraints on its metallicity Z. The Beat Cepheid Peterson diagrams are revisited with linear nonadiabatic turbulent convective models, and their accuracy as a probe for stellar metallicity is evaluated. They are shown to be largely independent of the helium content Y, and they are also only weakly dependent on the mass-luminosity relation that is used in their construction. However, they are found to show sensitivity to the relative abundances of the elements that are lumped into the metallicity parameter Z. Rotation is estimated to have but a small effect on the 'pulsation metallicities'. A composite Petersen diagram is presented that allows one to read off upper and lower limits on the metallicity Z from the measured period P0 and period ratio P1/P0.Comment: 9 pages, 12 color figures (black and white version available from 1st author's website). With minor revisions. to appear in Ap

Microminiaturized, biopotential conditioning system (MBCS)

Multichannel, medical monitoring system allows almost complete freedom of movement for subject during monitoring periods. System comprises monitoring unit (biobelt), transmission line, and data acquisition unit. Belt, made of polybenzimidizole fabric, is wrapped around individual's waist and held in place by overlapping sections of Velcro closure material