New Q matrices and their functional equations for the eight vertex model at elliptic roots of unity

The Q matrix invented by Baxter in 1972 to solve the eight vertex model at roots of unity exists for all values of N, the number of sites in the chain, but only for a subset of roots of unity. We show in this paper that a new Q matrix, which has recently been introduced and is non zero only for N even, exists for all roots of unity. In addition we consider the relations between all of the known Q matrices of the eight vertex model and conjecture functional equations for them.Comment: 20 pages, 2 Postscript figure

Bethe Ansatz Equations for the Broken ZNZ_{N}-Symmetric Model

We obtain the Bethe Ansatz equations for the broken ${\bf Z}_{N}$-symmetric model by constructing a functional relation of the transfer matrix of $L$-operators. This model is an elliptic off-critical extension of the Fateev-Zamolodchikov model. We calculate the free energy of this model on the basis of the string hypothesis.Comment: 43 pages, latex, 11 figure

Planar lattice gases with nearest-neighbour exclusion

We discuss the hard-hexagon and hard-square problems, as well as the corresponding problem on the honeycomb lattice. The case when the activity is unity is of interest to combinatorialists, being the problem of counting binary matrices with no two adjacent 1's. For this case we use the powerful corner transfer matrix method to numerically evaluate the partition function per site, density and some near-neighbour correlations to high accuracy. In particular for the square lattice we obtain the partition function per site to 43 decimal places.Comment: 16 pages, 2 built-in Latex figures, 4 table

Biomass-supported catalysts on Desulfovibrio desulfuricans and Rhodobacter sphaeroides

A Rhodobacter sphaeroides-supported dried, ground palladium catalyst (‘‘Rs-Pd(0)’’) was compared with a Desulfovibrio desulfuricans-supported catalyst (‘‘Dd-Pd(0)’’)and with unsupported palladium metal particles made by reduction under H2 (‘‘Chem-Pd(0)’’). Cell surface-located clusters of Pd(0) nanoparticles were detected on both D. desulfuricans and R. sphaeroides but the size and location of deposits differed among comparably loaded preparations.\ud \ud These differences may underlie the observation of different activities of Dd-Pd(0) and Rs-Pd(0) when compared with respect to their ability to promote hydrogen release from hypophosphite and to catalyze chloride release from chlorinated aromatic compounds. Dd-Pd(0) was more effective in the reductive dehalogenation of polychlorinated biphenyls (PCBs), whereas Rs-Pd(0) was more effective in the initial dehalogenation of pentachlorophenol (PCP) although the rate of chloride release from PCP was comparable with both preparations after 2 h

Numerical Renormalization Group at Criticality

We apply a recently developed numerical renormalization group, the corner-transfer-matrix renormalization group (CTMRG), to 2D classical lattice models at their critical temperatures. It is shown that the combination of CTMRG and the finite-size scaling analysis gives two independent critical exponents.Comment: 5 pages, LaTeX, 5 figures available upon reques

On kernel engineering via Paley–Wiener

A radial basis function approximation takes the form $$s(x)=\sum_{k=1}^na_k\phi(x-b_k),\quad x\in {\mathbb{R}}^d,$$ where the coefficients a 1,…,a n are real numbers, the centres b 1,…,b n are distinct points in ℝ d , and the function φ:ℝ d →ℝ is radially symmetric. Such functions are highly useful in practice and enjoy many beautiful theoretical properties. In particular, much work has been devoted to the polyharmonic radial basis functions, for which φ is the fundamental solution of some iterate of the Laplacian. In this note, we consider the construction of a rotation-invariant signed (Borel) measure μ for which the convolution ψ=μ φ is a function of compact support, and when φ is polyharmonic. The novelty of this construction is its use of the Paley–Wiener theorem to identify compact support via analysis of the Fourier transform of the new kernel ψ, so providing a new form of kernel engineering

Analysis of Three-Dimensional Protein Images

A fundamental goal of research in molecular biology is to understand protein structure. Protein crystallography is currently the most successful method for determining the three-dimensional (3D) conformation of a protein, yet it remains labor intensive and relies on an expert's ability to derive and evaluate a protein scene model. In this paper, the problem of protein structure determination is formulated as an exercise in scene analysis. A computational methodology is presented in which a 3D image of a protein is segmented into a graph of critical points. Bayesian and certainty factor approaches are described and used to analyze critical point graphs and identify meaningful substructures, such as alpha-helices and beta-sheets. Results of applying the methodologies to protein images at low and medium resolution are reported. The research is related to approaches to representation, segmentation and classification in vision, as well as to top-down approaches to protein structure prediction.Comment: See http://www.jair.org/ for any accompanying file

The existence of matrices with prescribed characteristic and permanental polynomials

AbstractLet d(λ) and p(λ) be monic polynomials of degree n⩾2 with coefficients in F, an algebraically closed field or the field of all real numbers. Necessary and sufficient conditions for the existence of an n-square matrix A over F such that det(λI−A)=d(λ) and per(λI−A=p(λ) are given in terms of the coefficients of d(λ) and p(λ)

Construction of some missing eigenvectors of the XYZ spin chain at the discrete coupling constants and the exponentially large spectral degeneracy of the transfer matrix

We discuss an algebraic method for constructing eigenvectors of the transfer matrix of the eight vertex model at the discrete coupling parameters. We consider the algebraic Bethe ansatz of the elliptic quantum group $E_{\tau, \eta}(sl_2)$ for the case where the parameter $\eta$ satisfies $2 N \eta = m_1 + m_2 \tau$ for arbitrary integers $N$, $m_1$ and $m_2$. When $m_1$ or $m_2$ is odd, the eigenvectors thus obtained have not been discussed previously. Furthermore, we construct a family of degenerate eigenvectors of the XYZ spin chain, some of which are shown to be related to the $sl_2$ loop algebra symmetry of the XXZ spin chain. We show that the dimension of some degenerate eigenspace of the XYZ spin chain on $L$ sites is given by $N 2^{L/N}$, if $L/N$ is an even integer. The construction of eigenvectors of the transfer matrices of some related IRF models is also discussed.Comment: 19 pages, no figure (revisd version with three appendices