Enumeration of three-dimensional convex polygons

A self-avoiding polygon (SAP) on a graph is an elementary cycle. Counting SAPs on the hypercubic lattice ℤ ^d withd≥2, is a well-known unsolved problem, which is studied both for its combinatorial and probabilistic interest and its connections with statistical mechanics. Of course, polygons on ℤ ^d are defined up to a translation, and the relevant statistic is their perimeter. A SAP on ℤ ^d is said to beconvex if its perimeter is “minimal”, that is, is exactly twice the sum of the side lengths of the smallest hyper-rectangle containing it. In 1984, Delest and Viennot enumerated convex SAPs on the square lattice [6], but no result was available in a higher dimension. We present an elementar approach to enumerate convex SAPs in any dimension. We first obtain a new proof of Delest and Viennot's result, which explains combinatorially the form of the generating function. We then compute the generating function for convex SAPs on the cubic lattice. In a dimension larger than 3, the details of the calculations become very cumbersome. However, our method suggests that the generating function for convex SAPs on ℤ ^d is always a quotient ofdifferentiably finite power series.     

Enumeration of Almost-Convex Polygons on the Square Lattice

We classify self-avoiding polygons on the square lattice according to a concavity measure, m, where 2m is the difference between the number of steps in the polygon and the perimeter of the minimal rectangle bounding the polygon. We generate series expansions for the perimeter generating functions S_m(x) for polygons of concavity m. We analyze the series S_m(x) for m = 0 to 3. If N_m,n denotes the number of polygons of perimeter 2n and concavity m, with m = o(n^1/2), we prove that N_m,n ? 2^2n?m?7n^m+1/m!, and that the radius of convergence of the series counting all polygons with m = o(n) is equal to 1/4. Our numerical data leads us to conjecture that in fact for m = o(n^1/2), a result confirmed for m = 0 and 1.     

The Susceptibility of the Square Lattice Ising Model: New Developments

We have made substantial advances in elucidating the properties of the susceptibility of the square lattice Ising model. We discuss its analyticity properties, certain closed form expressions for subsets of the coefficients, and give an algorithm of complexity O(N⁶) to determine its first N coefficients. As a result, we have generated and analyzed series with more than 300 terms in both the high- and low-temperature regime. We quantify the effect of irrelevant variables to the scaling-amplitude functions. In particular, we find and quantify the breakdown of simple scaling, in the absence of irrelevant scaling fields, arising first at order |T–T_c|^9/4, though high-low temperature symmetry is still preserved. At terms of order |T–T_c|^17/4 and beyond, this symmetry is no longer present. The short-distance terms are shown to have the form (T–T_c)^p (log |T–T_c|)^q with pq². Conjectured exact expressions for some correlation functions and series coefficients in terms of elliptic theta functions also foreshadow future developments.     

Transmission X-ray microscopy using X-ray magnetic circular dichroism

X-ray magnetic circular dichroism (X-MCD) was used as a large, element-specific and quantitative magnetic contrast mechanism in the soft X-ray microscopes at BESSY I (Berlin) and the ALS (Berkeley). The present state and potential of magnetic transmission X-ray microscopy (MTXM) is outlined. The possibility to record images in varying magnetic fields and the high spatial resolution down to 25 nm were used to image out-of-plane magnetized (4 ?Fe / 4 ?Gd)×75 systems. Magnetic domains could be studied in arrays of circular and square dots with lateral dimensions down to 180 nm. Hysteresis loops of individual dots were deduced using the direct proportionality of the X-MCD contrast to the sample magnetization. Images of a 3 nmCr / 50 nmFe / 6 nmCr film demonstrate for the first time that MTXM is also able to observe in-plane magnetized domains. In the future the possible applications of MTXM will be extended with regard to the strength of the external field, the available energy range and the sample conditions by building a dedicated transmission X-ray microscope for magnetic imaging at BESSY II. Received: 22 May 2001 / Accepted: 4 July 2001 / Published online: 5 October 2001     

Copper and iron interactions with angiotensin-converting enzyme inhibitors. A study by fast-atom bombardment tandem mass spectrometry

Fast atom bombardment, combined with high-energy collision-induced tandem mass spectrometry, has been used to investigate gas-phase metal-ion interactions with captopril, enalaprilat and lisinopril, all angiotensin-converting enzyme inhibitors.Suggestions for the location of metal-binding sites are presented. For captopril, metal binding occurs most likely at both the sulphur and the nitrogen atom. For enalaprilat and lisinopril, binding preferably occurs at the amine nitrogen. Copyright 1999 John Wiley & Sons, Ltd.