Refined conformal spectra in the dimer model

Working with Lieb's transfer matrix for the dimer model, we point out that the full set of dimer configurations may be partitioned into disjoint subsets (sectors) closed under the action of the transfer matrix. These sectors are labelled by an integer or half-integer quantum number we call the variation index. In the continuum scaling limit, each sector gives rise to a representation of the Virasoro algebra. We determine the corresponding conformal partition functions and their finitizations, and observe an intriguing link to the Ramond and Neveu-Schwarz sectors of the critical dense polymer model as described by a conformal field theory with central charge c=-2.Comment: 44 page

Simulation of associative learning with the replaced elements model

Associative learning theories can be categorised according to whether they treat the representation of stimulus compounds in an elemental or configural manner. Since it is clear that a simple elemental approach to stimulus representation is inadequate there have been several attempts to produce more elaborate elemental models. One recent approach, the Replaced Elements Model (Wagner, 2003), reproduces many results that have until recently been uniquely predicted by Pearce’s Configural Theory (Pearce, 1994). Although it is possible to simulate the Replaced Elements Model using “standard” simulation programs the generation of the correct stimulus representation is complex. The current paper describes a method for simulation of the Replaced Elements Model and presents the results of two example simulations that show differential predictions of Replaced Elements and Pearce’s Configural Theor

W-Extended Fusion Algebra of Critical Percolation

Two-dimensional critical percolation is the member LM(2,3) of the infinite series of Yang-Baxter integrable logarithmic minimal models LM(p,p'). We consider the continuum scaling limit of this lattice model as a `rational' logarithmic conformal field theory with extended W=W_{2,3} symmetry and use a lattice approach on a strip to study the fundamental fusion rules in this extended picture. We find that the representation content of the ensuing closed fusion algebra contains 26 W-indecomposable representations with 8 rank-1 representations, 14 rank-2 representations and 4 rank-3 representations. We identify these representations with suitable limits of Yang-Baxter integrable boundary conditions on the lattice and obtain their associated W-extended characters. The latter decompose as finite non-negative sums of W-irreducible characters of which 13 are required. Implementation of fusion on the lattice allows us to read off the fusion rules governing the fusion algebra of the 26 representations and to construct an explicit Cayley table. The closure of these representations among themselves under fusion is remarkable confirmation of the proposed extended symmetry.Comment: 30 page

The Solubility and Heat of Solution of Succinic Acid in Water and the Paraffin Alcohols

The following is the report of an investigation undertaken for the purpose of collecting further information concerning the influence of solvent upon certain specific properties of solutions. The solvents chosen, including water, represent the lower homologues of the paraffin alcohol series

Saffron thistle

Saffron thistle is a declared primary noxious weed in Western Australia and is especially troublesome in the northern agricultural areas. Field trials aimed at finding the cheapest and most effective method of controlling the weed have shown that 2,4-D applied at the correct stage is the best chemical to use

Geometric Exponents, SLE and Logarithmic Minimal Models

In statistical mechanics, observables are usually related to local degrees of freedom such as the Q < 4 distinct states of the Q-state Potts models or the heights of the restricted solid-on-solid models. In the continuum scaling limit, these models are described by rational conformal field theories, namely the minimal models M(p,p') for suitable p, p'. More generally, as in stochastic Loewner evolution (SLE_kappa), one can consider observables related to nonlocal degrees of freedom such as paths or boundaries of clusters. This leads to fractal dimensions or geometric exponents related to values of conformal dimensions not found among the finite sets of values allowed by the rational minimal models. Working in the context of a loop gas with loop fugacity beta = -2 cos(4 pi/kappa), we use Monte Carlo simulations to measure the fractal dimensions of various geometric objects such as paths and the generalizations of cluster mass, cluster hull, external perimeter and red bonds. Specializing to the case where the SLE parameter kappa = 4p'/p is rational with p < p', we argue that the geometric exponents are related to conformal dimensions found in the infinitely extended Kac tables of the logarithmic minimal models LM(p,p'). These theories describe lattice systems with nonlocal degrees of freedom. We present results for critical dense polymers LM(1,2), critical percolation LM(2,3), the logarithmic Ising model LM(3,4), the logarithmic tricritical Ising model LM(4,5) as well as LM(3,5). Our results are compared with rigourous results from SLE_kappa, with predictions from theoretical physics and with other numerical experiments. Throughout, we emphasize the relationships between SLE_kappa, geometric exponents and the conformal dimensions of the underlying CFTs.Comment: Added reference

Smoothed Particle Hydrodynamics in cosmology: a comparative study of implementations

We analyse the performance of twelve different implementations of Smoothed Particle Hydrodynamics (SPH) using seven tests designed to isolate key hydrodynamic elements of cosmological simulations which are known to cause the SPH algorithm problems. In order, we consider a shock tube, spherical adiabatic collapse, cooling flow model, drag, a cosmological simulation, rotating cloud-collapse and disc stability. In the implementations special attention is given to the way in which force symmetry is enforced in the equations of motion. We study in detail how the hydrodynamics are affected by different implementations of the artificial viscosity including those with a shear-correction modification. We present an improved first-order smoothing-length update algorithm that is designed to remove instabilities that are present in the Hernquist and Katz (1989) algorithm. For all tests we find that the artificial viscosity is the most important factor distinguishing the results from the various implementations. The second most important factor is the way force symmetry is achieved in the equation of motion. Most results favour a kernel symmetrization approach. The exact method by which SPH pressure forces are included has comparatively little effect on the results. Combining the equation of motion presented in Thomas and Couchman (1992) with a modification of the Monaghan and Gingold (1983) artificial viscosity leads to an SPH scheme that is both fast and reliable.Comment: 30 pages, 26 figures and 9 tables included. Submitted to MNRAS. Postscript version available at ftp://phobos.astro.uwo.ca/pub/etittley/papers/sphtest.ps.g

Integrals of Motion for Critical Dense Polymers and Symplectic Fermions

We consider critical dense polymers ${\cal L}(1,2)$. We obtain for this model the eigenvalues of the local integrals of motion of the underlying Conformal Field Theory by means of Thermodynamic Bethe Ansatz. We give a detailed description of the relation between this model and Symplectic Fermions including the indecomposable structure of the transfer matrix. Integrals of motion are defined directly on the lattice in terms of the Temperley Lieb Algebra and their eigenvalues are obtained and expressed as an infinite sum of the eigenvalues of the continuum integrals of motion. An elegant decomposition of the transfer matrix in terms of a finite number of lattice integrals of motion is obtained thus providing a reason for their introduction.Comment: 53 pages, version accepted for publishing on JSTA