Geometrical Properties of Two-Dimensional Interacting Self-Avoiding Walks at the Theta-Point

We perform a Monte Carlo simulation of two-dimensional N-step interacting self-avoiding walks at the theta point, with lengths up to N=3200. We compute the critical exponents, verifying the Coulomb-gas predictions, the theta-point temperature T_theta = 1.4986(11), and several invariant size ratios. Then, we focus on the geometrical features of the walks, computing the instantaneous shape ratios, the average asphericity, and the end-to-end distribution function. For the latter quantity, we verify in detail the theoretical predictions for its small- and large-distance behavior.Comment: 23 pages, 4 figure

Exact Solution of the Discrete (1+1)-dimensional RSOS Model with Field and Surface Interactions

We present the solution of a linear Restricted Solid--on--Solid (RSOS) model in a field. Aside from the origins of this model in the context of describing the phase boundary in a magnet, interest also comes from more recent work on the steady state of non-equilibrium models of molecular motors. While similar to a previously solved (non-restricted) SOS model in its physical behaviour, mathematically the solution is more complex. Involving basic hypergeometric functions ${}_3\phi_2$, it introduces a new form of solution to the lexicon of directed lattice path generating functions.Comment: 10 pages, 2 figure

Exact Results for Hamiltonian Walks from the Solution of the Fully Packed Loop Model on the Honeycomb Lattice

We derive the nested Bethe Ansatz solution of the fully packed O($n$) loop model on the honeycomb lattice. From this solution we derive the bulk free energy per site along with the central charge and geometric scaling dimensions describing the critical behaviour. In the $n=0$ limit we obtain the exact compact exponents $\gamma=1$ and $\nu=1/2$ for Hamiltonian walks, along with the exact value $\kappa^2 = 3 \sqrt 3 /4$ for the connective constant (entropy). Although having sets of scaling dimensions in common, our results indicate that Hamiltonian walks on the honeycomb and Manhattan lattices lie in different universality classes.Comment: 12 pages, RevTeX, 3 figures supplied on request, ANU preprint MRR-050-9

Six-minute walk test on a special treadmill: Primary results in healthy volunteers

Background: The guidelines approved by the American Thoracic Society in 2002 definitely recognize the six-minute walk test (6MWT) as a useful tool for the evaluation of physical efficiency in individuals with at least moderate chronic obstructive pulmonary disease, heart failure and intermittent dysbasia. So far, the American Thoracic Society has not approved the use of a treadmill to determine the six-minute walking distance (6MWD) because patients are unable to pace themselves on a treadmill. The purpose of our work was to prove that these problems could be avoided if physical efficiency is evaluated with the use of a modified treadmill. Methods: The work evaluates the function of a treadmill able to adjust its speed to the walking speed of healthy volunteers. The evaluation is based on a comparison of the distance covered by the healthy volunteers and the comfort of the test on the treadmill during six minutes with the distance covered and comfort during the same period in a 22-metre-long hallway in 29 healthy volunteers. Non-invasive blood pressure and pulse measurements were taken immediately before and after the test. Results: The average distance covered during the six-minute period on the treadmill was 57.1 m longer than in the hallway. The comfort of the treadmill test was indicated to be better by 18 subjects, worse by 4 subjects and identical by 7 subjects. Conclusions: The tests confirm that the speed of the modified treadmill adjusts properly to the walking speed of the healthy volunteers. The hemodynamic effects were identical for the healthy volunteers both in the hallway and treadmill tests. The distance differences were caused by turnarounds in the corridor test. The results obtained with the special treadmill allow us to develop a new method and, at present, provide a basis for a second stage of research comprising subjects with diagnosed heart failure. (Cardiol J 2007; 14: 447-452

Scaling of Self-Avoiding Walks in High Dimensions

We examine self-avoiding walks in dimensions 4 to 8 using high-precision Monte-Carlo simulations up to length N=16384, providing the first such results in dimensions $d > 4$ on which we concentrate our analysis. We analyse the scaling behaviour of the partition function and the statistics of nearest-neighbour contacts, as well as the average geometric size of the walks, and compare our results to $1/d$-expansions and to excellent rigorous bounds that exist. In particular, we obtain precise values for the connective constants, $\mu_5=8.838544(3)$, $\mu_6=10.878094(4)$, $\mu_7=12.902817(3)$, $\mu_8=14.919257(2)$ and give a revised estimate of $\mu_4=6.774043(5)$. All of these are by at least one order of magnitude more accurate than those previously given (from other approaches in $d>4$ and all approaches in $d=4$). Our results are consistent with most theoretical predictions, though in $d=5$ we find clear evidence of anomalous $N^{-1/2}$-corrections for the scaling of the geometric size of the walks, which we understand as a non-analytic correction to scaling of the general form $N^{(4-d)/2}$ (not present in pure Gaussian random walks).Comment: 14 pages, 2 figure

Generalized Noiseless Quantum Codes utilizing Quantum Enveloping Algebras

A generalization of the results of Rasetti and Zanardi concerning avoiding errors in quantum computers by using states preserved by evolution is presented. The concept of dynamical symmetry is generalized from the level of classical Lie algebras and groups to the level of dynamical symmetry based on quantum Lie algebras and quantum groups (in the sense of Woronowicz). A natural connection is proved between states preserved by representations of a quantum group and states preserved by evolution with dynamical symmetry of the appropriate universal enveloping algebra. Illustrative examples are discussed.Comment: 10 pages, LaTeX, 2 figures Postscrip

Chebyshev type lattice path weight polynomials by a constant term method

We prove a constant term theorem which is useful for finding weight polynomials for Ballot/Motzkin paths in a strip with a fixed number of arbitrary `decorated' weights as well as an arbitrary `background' weight. Our CT theorem, like Viennot's lattice path theorem from which it is derived primarily by a change of variable lemma, is expressed in terms of orthogonal polynomials which in our applications of interest often turn out to be non-classical. Hence we also present an efficient method for finding explicit closed form polynomial expressions for these non-classical orthogonal polynomials. Our method for finding the closed form polynomial expressions relies on simple combinatorial manipulations of Viennot's diagrammatic representation for orthogonal polynomials. In the course of the paper we also provide a new proof of Viennot's original orthogonal polynomial lattice path theorem. The new proof is of interest because it uses diagonalization of the transfer matrix, but gets around difficulties that have arisen in past attempts to use this approach. In particular we show how to sum over a set of implicitly defined zeros of a given orthogonal polynomial, either by using properties of residues or by using partial fractions. We conclude by applying the method to two lattice path problems important in the study of polymer physics as models of steric stabilization and sensitized flocculation.Comment: 27 pages, 14 figure

Scaling prediction for self-avoiding polygons revisited

We analyse new exact enumeration data for self-avoiding polygons, counted by perimeter and area on the square, triangular and hexagonal lattices. In extending earlier analyses, we focus on the perimeter moments in the vicinity of the bicritical point. We also consider the shape of the critical curve near the bicritical point, which describes the crossover to the branched polymer phase. Our recently conjectured expression for the scaling function of rooted self-avoiding polygons is further supported. For (unrooted) self-avoiding polygons, the analysis reveals the presence of an additional additive term with a new universal amplitude. We conjecture the exact value of this amplitude.Comment: 17 pages, 3 figure

Two dimensional self-avoiding walk with hydrogen-like bonding: Phase diagram and critical behaviour

The phase diagram for a two-dimensional self-avoiding walk model on the square lattice incorporating attractive short-ranged interactions between parallel sections of walk is derived using numerical transfer matrix techniques. The model displays a collapse transition. In contrast to the standard $\theta$-point model, the transition is first order. The phase diagram in the full fugacity-temperature plane displays an additional transition line, when compared to the $\theta$-point model, as well as a critical transition at finite temperature in the hamiltonian walk limit.Comment: 22 pages, 13 figures. To appear in Journal of Physics

Dry and Humid Periods Reconstructed from Tree Rings in the Former Territory of Sogdiana (Central Asia) and Their Socio-economic Consequences over the Last Millennium

One of the richest societies along the Silk Road developed in Sogdiana, located in present-day Tajikistan, Uzbekistan, and Kyrgyzstan. This urban civilisation reached its greatest prosperity during the golden age of the Silk Road (sixth to ninth century ce). Rapid political and economic changes, accelerated by climatic variations, were observed during last millennium in this region. The newly developed tree-ring-based reconstruction of precipitation for the pastmillennium revealed a series of dry and wet stages. During the Medieval Climate Anomaly (MCA), two dry periods occurred (900–1000 and 1200–1250), interrupted by a phase of wetter conditions. Distinct dry periods occurred around 1510–1650, 1750–1850, and 1920–1970, respectively. The juniper tree-ring record of moisture changes revealed that major dry and pluvial episodes were consistent with those indicated by hydroclimatic proxy data from adjacent areas. These climate fluctuations have had longand short term consequences for human history in the territory of former Sogdiana