Continuity and change in arable land management in the Northern Isles : evidence from anthropogenic soils

Human activity can affect the soil in ways which are traceable long after the land has been given over to other uses, and past land management practices can be reconstructed by investigation of these relict characteristics. In some regions the addition of fertilising materials to the arable soils has created artificially deepened anthropogenic topsoils which can be over 1m thick. Such relict soils are found all over the world, and are widespread in north-western Europe. This work focuses on the anthropogenic soils in the Northern Isles, which were formed from the Neolithic period up until the 20th century. Three multi-period sites were investigated using thin section micromorphology, organic/inorganic phosphate analysis, soil magnetism, particle size distribution, loss on ignition and soil pH. Current views of anthropogenic soil formation, based on pedological investigation and historical documentary sources, are that they are formed as a result of the addition of domestic animal manures and turf used as animal bedding to arable areas. This project sets out to test the hypothesis that in fact anthropogenic soils are the result of a wide range of formation processes which took place over extended periods of time. The hypothesis has been tested by analysing soils and associated middens of different dates, which have been sealed and protected by blown sand deposits. The results have shown that in the Neolithic period arable soils were created by cultivating the settlement's midden heaps as well as by adding midden material to the surrounding soils. In the Bronze Age human manure, ash and domestic waste were spread onto the fields around the settlements to create arable topsoils up to 35cm thick. In the Iron Age arable agriculture was intensified by selective use of organic manures on one of the sites investigated, but organic waste material was not used as efficiently as it was in later periods, and on both sites it was allowed to accumulate within the settlements. In the Norse period, when the intensive system used in historical times appears to have originated, organic waste may have been used more efficiently. These changes appear to reflect a greater organisation of land resources and manuring strategies and increased demand for arable production over time

A new transfer-matrix algorithm for exact enumerations: Self-avoiding polygons on the square lattice

We present a new and more efficient implementation of transfer-matrix methods for exact enumerations of lattice objects. The new method is illustrated by an application to the enumeration of self-avoiding polygons on the square lattice. A detailed comparison with the previous best algorithm shows significant improvement in the running time of the algorithm. The new algorithm is used to extend the enumeration of polygons to length 130 from the previous record of 110.Comment: 17 pages, 8 figures, IoP style file

Critical behaviour of the two-dimensional Ising susceptibility

We report computations of the short-distance and the long-distance (scaling) contributions to the square-lattice Ising susceptibility in zero field close to T_c. Both computations rely on the use of nonlinear partial difference equations for the correlation functions. By summing the correlation functions, we give an algorithm of complexity O(N^6) for the determination of the first N series coefficients. Consequently, we have generated and analysed series of length several hundred terms, generated in about 100 hours on an obsolete workstation. In terms of a temperature variable, \tau, linear in T/T_c-1, the short-distance terms are shown to have the form \tau^p(ln|\tau|)^q with p>=q^2. To O(\tau^14) the long-distance part divided by the leading \tau^{-7/4} singularity contains only integer powers of \tau. The presence of irrelevant variables in the scaling function is clearly evident, with contributions of distinct character at leading orders |\tau|^{9/4} and |\tau|^{17/4} being identified.Comment: 11 pages, REVTex

Universal Amplitude Combinations for Self-Avoiding Walks, Polygons and Trails

We give exact relations for a number of amplitude combinations that occur in the study of self-avoiding walks, polygons and lattice trails. In particular, we elucidate the lattice-dependent factors which occur in those combinations which are otherwise universal, show how these are modified for oriented lattices, and give new results for amplitude ratios involving even moments of the area of polygons. We also survey numerical results for a wide range of amplitudes on a number of oriented and regular lattices, and provide some new ones.Comment: 20 pages, NI 92016, OUTP 92-54S, UCSBTH-92-5

Perimeter Generating Functions For The Mean-Squared Radius Of Gyration Of Convex Polygons

We have derived long series expansions for the perimeter generating functions of the radius of gyration of various polygons with a convexity constraint. Using the series we numerically find simple (algebraic) exact solutions for the generating functions. In all cases the size exponent $\nu=1$.Comment: 8 pages, 1 figur

Statistics of lattice animals (polyominoes) and polygons

We have developed an improved algorithm that allows us to enumerate the number of site animals (polyominoes) on the square lattice up to size 46. Analysis of the resulting series yields an improved estimate, $\tau = 4.062570(8)$, for the growth constant of lattice animals and confirms to a very high degree of certainty that the generating function has a logarithmic divergence. We prove the bound $\tau > 3.90318.$ We also calculate the radius of gyration of both lattice animals and polygons enumerated by area. The analysis of the radius of gyration series yields the estimate $\nu = 0.64115(5)$, for both animals and polygons enumerated by area. The mean perimeter of polygons of area $n$ is also calculated. A number of new amplitude estimates are given.Comment: 10 pages, 2 eps figure

Size and area of square lattice polygons

We use the finite lattice method to calculate the radius of gyration, the first and second area-weighted moments of self-avoiding polygons on the square lattice. The series have been calculated for polygons up to perimeter 82. Analysis of the series yields high accuracy estimates confirming theoretical predictions for the value of the size exponent, $\nu=3/4$, and certain universal amplitude combinations. Furthermore, a detailed analysis of the asymptotic form of the series coefficients provide the firmest evidence to date for the existence of a correction-to-scaling exponent, $\Delta = 3/2$.Comment: 12 pages 3 figure

Low temperature series expansions for the square lattice Ising model with spin S > 1

We derive low-temperature series (in the variable $u = \exp[-\beta J/S^2]$) for the spontaneous magnetisation, susceptibility and specific heat of the spin-$S$ Ising model on the square lattice for $S=\frac32$, 2, $\frac52$, and 3. We determine the location of the physical critical point and non-physical singularities. The number of non-physical singularities closer to the origin than the physical critical point grows quite rapidly with $S$. The critical exponents at the singularities which are closest to the origin and for which we have reasonably accurate estimates are independent of $S$. Due to the many non-physical singularities, the estimates for the physical critical point and exponents are poor for higher values of $S$, though consistent with universality.Comment: 14 pages, LaTeX with IOP style files (ioplppt.sty), epic.sty and eepic.sty. To appear in J. Phys.

Self-avoiding walks and polygons on the triangular lattice

We use new algorithms, based on the finite lattice method of series expansion, to extend the enumeration of self-avoiding walks and polygons on the triangular lattice to length 40 and 60, respectively. For self-avoiding walks to length 40 we also calculate series for the metric properties of mean-square end-to-end distance, mean-square radius of gyration and the mean-square distance of a monomer from the end points. For self-avoiding polygons to length 58 we calculate series for the mean-square radius of gyration and the first 10 moments of the area. Analysis of the series yields accurate estimates for the connective constant of triangular self-avoiding walks, $\mu=4.150797226(26)$, and confirms to a high degree of accuracy several theoretical predictions for universal critical exponents and amplitude combinations.Comment: 24 pages, 6 figure