Zeroes of the Jones polynomial

We study the distribution of zeroes of the Jones polynomial $V_K(t)$ for a knot $K$. We have computed numerically the roots of the Jones polynomial for all prime knots with $N\leq 10$ crossings, and found the zeroes scattered about the unit circle $|t|=1$ with the average distance to the circle approaching a nonzero value as $N$ increases. For torus knots of the type $(m,n)$ we show that all zeroes lie on the unit circle with a uniform density in the limit of either $m$ or $n\to \infty$, a fact confirmed by our numerical findings. We have also elucidated the relation connecting the Jones polynomial with the Potts model, and used this relation to derive the Jones polynomial for a repeating chain knot with $3n$ crossings for general $n$. It is found that zeroes of its Jones polynomial lie on three closed curves centered about the points $1, i$ and $-i$. In addition, there are two isolated zeroes located one each near the points $t_\pm = e^{\pm 2\pi i/3}$ at a distance of the order of $3^{-(n+2)/2}$. Closed-form expressions are deduced for the closed curves in the limit of $n\to \infty$.Comment: 12 pages, 5 figure

Solvable RSOS models based on the dilute BWM algebra

In this paper we present representations of the recently introduced dilute Birman-Wenzl-Murakami algebra. These representations, labelled by the level-$l$ B$^{(1)}_n$, C$^{(1)}_n$ and D$^{(1)}_n$ affine Lie algebras, are Baxterized to yield solutions to the Yang-Baxter equation. The thus obtained critical solvable models are RSOS counterparts of the, respectively, D$^{(2)}_{n+1}$, $A^{(2)}_{2n}$ and B$^{(1)}_n$ $R$-matrices of Bazhanov and Jimbo. For the D$^{(2)}_{n+1}$ and B$^{(1)}_n$ algebras the RSOS models are new. An elliptic extension which solves the Yang-Baxter equation is given for all three series of dilute RSOS models.Comment: 25 pages, uuencoded compressed PostScript file, Amsterdam preprint ITFA-94-2

Faddeev-Volkov solution of the Yang-Baxter Equation and Discrete Conformal Symmetry

The Faddeev-Volkov solution of the star-triangle relation is connected with the modular double of the quantum group U_q(sl_2). It defines an Ising-type lattice model with positive Boltzmann weights where the spin variables take continuous values on the real line. The free energy of the model is exactly calculated in the thermodynamic limit. The model describes quantum fluctuations of circle patterns and the associated discrete conformal transformations connected with the Thurston's discrete analogue of the Riemann mappings theorem. In particular, in the quasi-classical limit the model precisely describe the geometry of integrable circle patterns with prescribed intersection angles.Comment: 26 pages, 18 color figures, minor correction

Reaction-Diffusion Processes as Physical Realizations of Hecke Algebras

We show that the master equation governing the dynamics of simple diffusion and certain chemical reaction processes in one dimension give time evolution operators (Hamiltonians) which are realizations of Hecke algebras. In the case of simple diffusion one obtains, after similarity transformations, reducible hermitian representations while in the other cases they are non-hermitian and correspond to supersymmetric quotients of Hecke algebras.Comment: Latex, 6 pages, BONN-HE-93.1

On spherical averages of radial basis functions

A radial basis function (RBF) has the general form $$s(x)=\sum_{k=1}^{n}a_{k}\phi(x-b_{k}),\quad x\in\mathbb{R}^{d},$$ where the coefficients a 1,…,a n are real numbers, the points, or centres, b 1,…,b n lie in ℝ d , and φ:ℝ d →ℝ is a radially symmetric function. Such approximants are highly useful and enjoy rich theoretical properties; see, for instance (Buhmann, Radial Basis Functions: Theory and Implementations, [2003]; Fasshauer, Meshfree Approximation Methods with Matlab, [2007]; Light and Cheney, A Course in Approximation Theory, [2000]; or Wendland, Scattered Data Approximation, [2004]). The important special case of polyharmonic splines results when φ is the fundamental solution of the iterated Laplacian operator, and this class includes the Euclidean norm φ(x)=‖x‖ when d is an odd positive integer, the thin plate spline φ(x)=‖x‖2log  ‖x‖ when d is an even positive integer, and univariate splines. Now B-splines generate a compactly supported basis for univariate spline spaces, but an analyticity argument implies that a nontrivial polyharmonic spline generated by (1.1) cannot be compactly supported when d>1. However, a pioneering paper of Jackson (Constr. Approx. 4:243–264, [1988]) established that the spherical average of a radial basis function generated by the Euclidean norm can be compactly supported when the centres and coefficients satisfy certain moment conditions; Jackson then used this compactly supported spherical average to construct approximate identities, with which he was then able to derive some of the earliest uniform convergence results for a class of radial basis functions. Our work extends this earlier analysis, but our technique is entirely novel, and applies to all polyharmonic splines. Furthermore, we observe that the technique provides yet another way to generate compactly supported, radially symmetric, positive definite functions. Specifically, we find that the spherical averaging operator commutes with the Fourier transform operator, and we are then able to identify Fourier transforms of compactly supported functions using the Paley–Wiener theorem. Furthermore, the use of Haar measure on compact Lie groups would not have occurred without frequent exposure to Iserles’s study of geometric integration

Nitric oxide treatments as adjuncts to reperfusion in acute myocardial infarction: a systematic review of experimental and clinical studies

Unmodified reperfusion therapy for acute myocardial infarction (AMI) is associated with irreversible myocardial injury beyond that sustained during ischemia. Studies in experimental models of ischemia/reperfusion and in humans undergoing reperfusion therapy for AMI have examined potential beneficial effects of nitric oxide (NO) supplemented at the time of reperfusion. Using a rigorous systematic search approach, we have identified and critically evaluated all the relevant experimental and clinical literature to assess whether exogenous NO given at reperfusion can limit infarct size. An inclusive search strategy was undertaken to identify all in vivo experimental animal and clinical human studies published in the period 1990–2014 where NO gas, nitrite, nitrate or NO donors were given to ameliorate reperfusion injury. Articles were screened at title and subsequently at abstract level, followed by objective full text analysis using a critical appraisal tool. In twenty-one animal studies, all NO treatments except nitroglycerin afforded protection against measures of reperfusion injury, including infarct size, creatinine kinase release, neutrophil accumulation and cardiac dysfunction. In three human AMI RCT’s, there was no consistent evidence of infarct limitation associated with NO treatment as an adjunct to reperfusion. Despite experimental evidence that most NO treatments can reduce infarct size when given as adjuncts to reperfusion, the value of these interventions in clinical AMI is unproven. Our study raises issues for the design of further clinical studies and emphasises the need for improved design of animal studies to reflect more accurately the comorbidities and other confounding factors seen in clinical AMI

Using a cognitive architecture to examine what develops

Different theories of development propose alternative mechanisms by which development occurs. Cognitive architectures can be used to examine the influence of each proposed mechanism of development while keeping all other mechanisms constant. An ACT-R computational model that matched adult behavior in solving a 21-block pyramid puzzle was created. The model was modified in three ways that corresponded to mechanisms of development proposed by developmental theories. The results showed that all the modifications (two of capacity and one of strategy choice) could approximate the behavior of 7-year-old children on the task. The strategy-choice modification provided the closest match on the two central measures of task behavior (time taken per layer, r = .99, and construction attempts per layer, r = .73). Modifying cognitive architectures is a fruitful way to compare and test potential developmental mechanisms, and can therefore help in specifying “what develops.

Baxterization, dynamical systems, and the symmetries of integrability

We resolve the `baxterization' problem with the help of the automorphism group of the Yang-Baxter (resp. star-triangle, tetrahedron, \dots) equations. This infinite group of symmetries is realized as a non-linear (birational) Coxeter group acting on matrices, and exists as such, {\em beyond the narrow context of strict integrability}. It yields among other things an unexpected elliptic parametrization of the non-integrable sixteen-vertex model. It provides us with a class of discrete dynamical systems, and we address some related problems, such as characterizing the complexity of iterations.Comment: 25 pages, Latex file (epsf style). WARNING: Postscript figures are BIG (600kB compressed, 4.3MB uncompressed). If necessary request hardcopy to [email protected] and give your postal mail addres

Unsigned state models for the Jones polynomial

It is well a known and fundamental result that the Jones polynomial can be expressed as Potts and vertex partition functions of signed plane graphs. Here we consider constructions of the Jones polynomial as state models of unsigned graphs and show that the Jones polynomial of any link can be expressed as a vertex model of an unsigned embedded graph. In the process of deriving this result, we show that for every diagram of a link in the 3-sphere there exists a diagram of an alternating link in a thickened surface (and an alternating virtual link) with the same Kauffman bracket. We also recover two recent results in the literature relating the Jones and Bollobas-Riordan polynomials and show they arise from two different interpretations of the same embedded graph.Comment: Minor corrections. To appear in Annals of Combinatoric