a series test of the scaling limit of self avoiding walks it is widely believed that the scaling limit of self avoiding walks saws at the critical temperature is conformally invariant and consequently describable by schramm loewner evolution with parameter 8 3 we consider saws in a rectangle which originate at its centre and end at the boundary we assume that the boundary density transforms covariantly in a way that depends precisely on as conjectured by lawler schramm and werner 2004 fractal geometry and applications a jubilee of benoit mandelbrot part 2 pp 339 64 it has previously been shown by guttmann and kennedy 2013 j eng math at press that in the limit of an infinitely large rectangle the ratio of the fraction of saws hitting the side of the rectangle to the fraction that hit the end of the rectangle can be calculated by considering rectangles of fixed aspect ratio 2 and also rectangles of aspect ratio 10 we calculate this ratio exactly for larger and larger rectangles by extrapolating this data to infinite rectangle size and invoking the above conjectures we obtain the estimate 2 666 64 0 000 07 for rectangles of aspect ratio 2 and 2 666 75 0 000 15 for rectangles of aspect ratio 10 we also provide numerical evidence supporting the conjectured distribution of saws striking the boundary at various points in the case of rectangles with aspect ratio 2

universality class of trails in two dimensions a trail is a walk on a lattice that may visit a site more than once but a bond at most once we have carried out transfer matrix studies of trails on the square lattice and of hybrid walks that interpolate between self avoiding walks and trails the results are in agreement with the same universal exponents as self avoiding walks however the finite size corrections are much larger than for self avoiding walks an explanation in terms of an irrelevant variable with scaling index is given

the regularizing properties of anisotropic radial basis functions in the present work we consider the problem of interpolating scattered data using radial basis functions rbf in general it is well known that this leads to a discrete linear inverse problem that needs to be regularized in order to provide a meaningful solution the work focuses on a metric regularization approach based on a new class of rbf called anisotropic rbf the work provides theoretical justifications for the regularization approach and it considers a suitable proposal for the metric supporting it by numerical examples

self avoiding polygons on the square l and manhattan lattices transfer matrix techniques are used to extend the self avoiding polygon generating function on the square lattice to terms in x46 corresponding to 46 step polygons these techniques are then extended to apply to directed square lattices such as the l and manhattan lattice and the self avoiding polygon generating function to x48 is found for these lattices series analysis confirms that the specific heat exponent alpha 1 2 for the self avoiding walk problem and gives the following estimates for the connective constants mu sq 2 638155 or 0 000025 mu l 1 5657 or 0 0019 and mu man 1 7328 or 0 0005 some evidence for a correction to scaling exponent delta approximately 0 84 is found from square lattice series

a numerical adaptation of self avoiding walk identities from the honeycomb to other 2d lattices recently duminil copin and smirnov proved a long standing conjecture of nienhuis that the connective constant of self avoiding walks saws on the honeycomb lattice is

polygons on the honeycomb lattice the authors enumerated the number of self avoiding polygons on the honeycomb lattice to 82 steps analysis of the resulting series gives for the connective constant mu 1 8477594 or 6 10 6 in agreement with nienhuis value 1984 of mu 2 2 12 1 84775906 the critical exponent alpha 0 4999 or 10 4 unbiased or alpha 0 49998 or 2 10 0 5 biased a non physical singularity on the negative real axis is accurately located and some curious results concerning the confluent singularity are obtained

multilevel sparse kernel based interpolation a multilevel kernel based interpolation method suitable for moderately high dimensional function interpolation problems is proposed the method termed multilevel sparse kernel based interpolation mlski for short uses both levelwise and directionwise multilevel decomposition of structured or mildly unstructured interpolation data sites in conjunction with the application of kernel based interpolants with different scaling in each direction the multilevel interpolation algorithm is based on a hierarchical decomposition of the data sites whereby at each level the detail is added to the interpolant by interpolating the resulting residual of the previous level on each level anisotropic radial basis functions are used for solving a number of small interpolation problems which are subsequently linearly combined to produce the interpolant mlski can be viewed as an extension of d boolean interpolation which is closely related to ideas in sparse grid and hyperbolic crosses literature to kernel ba

self avoiding walks on the simple cubic lattice we have substantially extended the series for the number of self avoiding walks and the mean square end to end distance on the simple cubic lattice our analysis of the series gives refined estimates for the critical point and critical exponents our estimates of the exponents and are in good agreement with recent high precision monte carlo estimates and also with recent renormalization group estimates critical amplitude estimates are also given a new improved rigorous upper bound for the connective constant 4 7114 is obtained

universal amplitude combinations for self avoiding walks polygons and trails the authors give exact relations for a number of amplitude combinations that occur in the study of self avoiding walks polygons and lattice trails in particular they 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 they also survey numerical results for a wide range of amplitudes on a number of oriented and regular lattices and provide some new ones

the size and number of rings on the square lattice the authors have calculated the number of self avoiding polygons on the square lattice to 56 steps and the caliper size to 54 steps analysis of the generating function permits estimates of the connective constant mu 2 638 1585 or 10 6 and the critical exponents alpha 0 500 06 or 0 000 06 and v 0 753 or 0 007 the singularity structure of the polygon generating function is found to be consistent with a correction to the scaling exponent delta 1 5 as predicted by nienhuis 1982 1984 the confluent part however maps into the additive analytic term due to the value of the exponent alpha

representing implicit surfaces satisfying lipschitz conditions by 4 dimensional point sets abstract for any given implicit surface satisfying two lipschitz conditions this work triangulates the surface using a set of 4 dimensional points with small hausdorff distances every 4 dimensional point is a 3 dimensional point with a weight compared to traditional triangulation approaches our method does not explicitly require the storage of connectivity information instead the connectivity information can be implicitly generated using a concept called the alpha shape with two calculated bounds the density of the 4 dimensional points adapts to the local curvature moreover the generated triangulation has been mathematically proven to be homeomorphic to the original implicit surface the hausdorff distance between the triangulation and the original surface is demonstrated to be small enough according to the experimental results

oriented self avoiding walks with orientation dependent interactions we consider oriented self avoiding walks on the square lattice with different interactions between steps depending on their relative parallel or anti parallel orientations we derive rigorous bounds on the free energy and prove the existence of a phase transition by means of exact enumeration and monte carlo simulation we study the phase diagram and the mean number of contacts we show that the mean number of anti parallel contacts increases linearly with the number of steps while the mean number of parallel contacts asymptotically approach a constant

enumeration of self avoiding walks on the square lattice we describe a new algorithm for the enumeration of self avoiding walks on the square lattice using up to 128 processors on a hp alpha server cluster we have enumerated the number of self avoiding walks on the square lattice to length 71 series for the metric properties of mean square end to end distance mean square radius of gyration and mean square distance of monomers from the end points have been derived to length 59 an analysis of the resulting series yields accurate estimates of the critical exponents and confirming predictions of their exact values likewise we obtain accurate amplitude estimates yielding precise values for certain universal amplitude combinations finally we report on an analysis giving compelling evidence that the leading non analytic correction to scaling exponent 1 3 2

polygons in restricted geometries subjected to infinite forces we consider self avoiding polygons in a restricted geometry namely an infinite l m tube in these polygons are subjected to a force f parallel to the infinite axis of the tube when the force stretches the polygons while when the force is compressive we obtain and prove the asymptotic form of the free energy in both limits we conjecture that the asymptote is the same as the limiting free energy of hamiltonian polygons polygons which visit every vertex in a box we investigate such polygons and in particular use a transfer matrix methodology to establish that the conjecture is true for some small tube sizes

the extension of self avoiding random walk series in two dimensions the authors have extended the enumeration of square lattice and triangular lattice self avoiding walks and their end to end distance by two terms they have also calculated 17 terms in the series for the resistance of saws with bridges on the square lattice analysis of the new data allows the connective constants to be more accurately estimated and affirms the values gamma 43 32 and v 3 4 and 1 gamma for the exponent of the resistance series

an average weight distance enumerator for binary expansions of reed solomon codes an average hamming weight enumerator is derived for the codewords at each hamming distance from a received pattern in the set of all possible binary expansions of a reed solomon code since these codes may be decoded by list decoders such as those studied by sudan 1997 the enumerator can be used to estimate the average number of codewords in the list returned by such a decoder

hard and soft decision decoding beyond the half minimum distance an algorithm for linear codes corresp a decoding algorithm for linear codes that uses the minimum weight words of the dual code as parity checks is defined this algorithm is able to correct beyond the half minimum distance and has the capability of including soft decision decoding results on applying this algorithm to quadratic residue qr codes bch codes and the golay codes with and without soft decision decoding are presented

self avoiding walks in two to five dimensions exact enumerations and series study the method of concatenation the addition of precomputed shorter chains to the ends of a centrally generated longer chain has permitted the extension of the exact series for cn the number of distinct configurations for self avoiding walks of length n the authors report on the leading exponent y and xc the reciprocal of the connectivity constant for the 2d honeycomb lattice 42 terms 1 3437 0 541 1968 the 2d square lattice 30 terms 1 3436 0 379 0520 the 3d simple cubic lattice 23 terms 1 161 932 0 213 4987 the 4d hypercubic 18 terms y approximately 1 0 147 60 and the 5d hypercubic lattice 13 terms y or 1 025 0 113 05 in addition they have also evaluated the leading correction terms honeycomb delta approximately 1 square delta approximately 0 85 simple cubic delta approximately 1 0 and the 4d hypercubic logarithmic correction with delta approximately 0 25

determination of the exponent for saws on the two dimensional manhattan lattice we present a high statistics monte carlo determination of the exponent for self avoiding walks on a manhattan lattice in two dimensions a conservative estimate is in agreement with the universal value on regular lattices but in conflict with predictions from conformal field theory and with a recent estimate from exact enumerations we find strong corrections to scaling that seem to indicate the presence of a non analytic exponent if we assume we find where the error is purely statistical

self avoiding polygons on the square lattice we have developed an improved algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 90 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 529 27 1 biased and the critical exponent 0 500 0005 10 unbiased the critical point is indistinguishable from a root of the polynomial 581x4 7x2 13 0 an asymptotic expansion for the coefficients is given for all n there is strong evidence for the absence of any non analytic correction to scaling exponent

radial basis functions for the multivariate interpolation of large scattered data sets an efficient method for the multivariate interpolation of very large scattered data sets is presented it is based on the local use of radial basis functions and represents a further improvement of the well known shepard s method although the latter is simple and well suited for multivariate interpolation it does not share the good reproduction quality of other methods widely used for bivariate interpolation on the other band radial basis functions which have proven to be highly useful for multivariate scattered data interpolation have a severe drawback they are unable to interpolate large sets in an efficient and numerically stable way and maintain a good level of reproduction quality at the same time both problems have been circumvented using radial basis functions to evaluate the nodal function of the modified shepard s method this approach exploits the flexibility of the method and improves its reproduction quality the proposed algorithm has been implemented and numerical results confirm its efficiency

polyharmonic splines an approximation method for noisy scattered data of extra large size the aim of this paper is to provide a fast method with a good quality of reproduction to recover functions from very large and irregularly scattered samples of noisy data which may present outliers to the given sample of size n we associate a uniform grid and around each grid point we condense the local information given by the noisy data by a suitable estimator the recovering is then performed by a stable interpolation based on isotropic polyharmonic b splines due to the good approximation rate we need only m n degrees of freedom to recover the phenomenon faithfully

synergistic development of differential approximants and the finite lattice method in lattice statistics several decades of parallel developments in the calculation and analysis of series expansions for lattice statistics have led to many new insights into critical phenomena these studies have centered on the use of the finite lattice method for series expansions in lattice statistics and the use of differential approximants in analysing such series one of these strands of research ultimately led to the result that a number of unsolved lattice statistics problems cannot be expressed as d finite functions somewhat ironically given power and success of differential approximants in analysing series neither the assumed functional form nor any finite generalisation thereof can fit such cases exactly

self avoiding walks on a two layer square lattice abstract we have calculated exactly the number s n and the root mean square end to end distance r n of self avoiding walks with n steps on a two layer square lattice up to 24 steps by computer we estimated the connective constant and the exponents by the ratio method of zinn justin our results indicate that the self avoiding walk on the two layer lattice belongs to the same universality class as that of self avoiding walks on the two dimensional lattice

an extension of the two dimensional self avoiding walk series on the square lattice the authors have adapted the dimerization algorithm for enumerating self avoiding walks to exploit the parallelism of the connection machine system by this approach they have extended the series for the number of self avoiding walks on the 2d square lattice by five terms these data may permit more accurate estimates of critical properties

a parallel algorithm for the enumeration of self avoiding polygons on the square lattice we have developed a parallel algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 110 we have also extended the series for the first 10 area weighted moments and the radius of gyration to 100 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 530 31 3 biased and the critical exponent 0 500 000 1 2 unbiased in addition we obtain very accurate estimates for the leading amplitudes confirming to a high degree of accuracy various predictions for universal amplitude combinations

self avoiding rings on the triangular lattice the finite lattice method of series expansion is used to extend the enumeration of self avoiding polygons on the triangular lattice through to rings of 35 steps the authors also give the enumeration of these triangular polygons grouped by perimeter and area of up to 21 unit triangles for perimeters up to 24 steps

self avoiding walks crossing a square we study a restricted class of self avoiding walks saws which start at the origin 0 0 end at l l and are entirely contained in the square 0 l 0 l on the square lattice z 2 the number of distinct walks is known to grow as l 2 o l 2 we estimate 1 744 550 0 000 005 as well as obtaining strict upper and lower bounds 1 628 1 it grows as l 2 h ere is the growth constant of unconstrained saws in z 2 f orx 1 we provide numerical evidence but no proof that the average walk length grows as l 4 3 another problem we study is that of saws as described above that pass through the central vertex of the square we estimate the proportion of such walks as a fraction of the total and find it to be just below 80 of the total number of saws we also consider hamiltonian walks under the same restriction they are known to grow as l 2 o l2

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

self avoiding walk enumeration via the lace expansion we introduce a new method for the enumeration of self avoiding walks based on the lace expansion we also introduce an algorithmic improvement called the two step method for self avoiding walk enumeration problems we obtain significant extensions of existing series on the cubic and hypercubic lattices in all dimensions d 3 we enumerate 32 step self avoiding polygons in d 3 26 step self avoiding polygons in d 4 30 step self avoiding walks in d 3 and 24 step self avoiding walks and polygons in all dimensions d 4 we analyze these series to obtain estimates for the connective constant and various critical exponents and amplitudes in dimensions 3 d 8 we also provide major extensions of 1 d expansions for the connective constant and for two critical amplitudes

compressed self avoiding walks bridges and polygons we study various self avoiding walks saws which are constrained to lie in the upper half plane and are subjected to a compressive force this force is applied to the vertex or vertices of the walk located at the maximum distance above the boundary of the half space in the case of bridges this is the unique end point in the case of saws or self avoiding polygons this corresponds to all vertices of maximal height we first use the conjectured relation with the schramm loewner evolution to predict the form of the partition function including the values of the exponents and then we use series analysis to test these predictions

shape preserving surface reconstruction using locally anisotropic radial basis function interpolants in this paper we deal with the problem of reconstructing surfaces from unorganized sets of points while capturing the significant geometry details of the modelled surface such as edges flat regions and corners this is obtained by exploiting the good approximation capabilities of the radial basis functions rbf the local nature of the method proposed in 1 and introducing information on shape features and data anisotropies detected from the given surface points the result is a shape preserving reconstruction given by a weighted combination of locally aniso tropic interpolants for each local interpolant the anisotropy is obtained by replacing the euclidean norm with a suitable metric which takes into account the local distribution of the points thus hyperellipsoid basis functions named anisotropic rbfs are defined results from the application of the method to the reconstruction of object surfaces in 3 are presented confirming the effectiveness of the approach

improved lower bounds on the connective constants for two dimensional self avoiding walks we calculate improved lower bounds for the connective constants for self avoiding walks on the square hexagonal triangular 4 82 and 3 122 lattices the bound is found by kesten s method of irreducible bridges this involves using transfer matrix techniques to exactly enumerate the number of bridges of a given span to very many steps upper bounds are obtained from recent exact enumeration data for the number of self avoiding walks and compared to current best available upper bounds from other methods

enumeration of self avoiding trails on a square lattice using a transfer matrix technique the authors describe a new algebraic technique utilizing transfer matrices for enumerating self avoiding trails on the square lattice they have enumerated trails to 31 steps and find increased evidence that trails are in the self avoiding walk universality class assuming that trails behave like a lambda nn1132 they find lambda 2 72062 or 0 000006 and a 1 272 or 0 002

iterative list decoding of some ldpc codes we present an iterative list decoding algorithm for low density parity check ldpc codes in particular we apply this decoder to a class of ldpc codes from finite geometries and show that the 73 45 10 projective geometry code can be maximum likelihood ml decoded with low complexity moreover the list decoding approach enables us to give a theoretical analysis of the performance we also consider list bit flipping bf decoding of longer ldpc codes

two bit flipping decoding algorithms for low density parity check codes in this letter a low complexity decoding algorithm for binary linear block codes is applied to low density paritycheck ldpc codes and improvements are described namely an extension to soft decision decoding and a loop detection mechanism for soft decoding only one real valued addition per code symbol is needed while the remaining operations are only binary as in the hard decision case the decoding performance is considerably increased by the loop detection simulation results are used to compare the performance with other known decoding strategies for ldpc codes with the result that the presented algorithms offer excellent performances at smaller complexity

two dimensional oriented self avoiding walks with parallel contacts two closely related models of oriented self avoiding walks osaws on a square lattice are studied we use the pruned enriched rosenbluth method to determine numerically the phase diagram both models have three phases a tight spiral phase in which the binding of parallel steps dominates a collapsed phase when the binding of antiparallel steps dominates and a free open coil phase we show that the system features a first order phase transition from the free phase to the tight spiral phase while both other transitions are continuous the location of the phases is determined accurately we also study turning numbers and gamma exponents in various regions of the phase diagram

algebraic techniques for enumerating self avoiding walks on the square lattice the authors describe a new algebraic technique for enumerating self avoiding walks on the rectangular lattice the computational complexity of enumerating walks of n steps is of order 3n 4 times a polynomial in n and so the approach is greatly superior to direct counting techniques they have enumerated walks of up to 39 steps as a consequence they are able to accurately estimate the critical point critical exponent and critical amplitude

nonuniform star polymers in two dimensions we present simulation results for nonuniform 2d star polymers using a recursive implementation of the enrichment method we show that the choice of parameters in this method is not trivial and that a good choice can lead to extremely efficient code we find that the exponents gamma f governing the increase of the f star partition function with n are compatible with the prediction gamma funiform f 1 by duplantier 1986

lower bound on the connective constant for square lattice self avoiding walks by enumerating irreducible bridges exactly up to 40 steps and by obtaining a lower bound to the number of bridges with less than 125 steps a lower bound for the connective constant for square lattice self avoiding walks of 2 62 is obtained

new lower bounds on the self avoiding walk connective constant we give an elementary new method for obtaining rigorous lower bounds on the connective constant for self avoiding walks on the hypercubic lattice d the method is based on loop erasure and restoration and does not require exact enumeration data our bounds are best for highd and in fact agree with the first four terms of the 1 d expansion for the connective constant the bounds are the best to date for dimensionsd 3 but do not produce good results in two dimensions ford 3 4 5 and 6 respectively our lower bound is within 2 4 0 43 0 12 and 0 044 of the value estimated by series extrapolation

the number and size of branched polymers in high dimensions we consider two models of branched polymers lattice trees on thed dimensional hypercubic lattice i the nearest neighbor model in sufficiently high dimensions and ii a spread out or long range model ford 8 in which trees are constructed from bonds of length less than or equal to a large parameterl we prove that for either model the critical exponent for the number of branched polymers exists and equals 5 2 and that the critical exponentv for the radius of gyration exists and equals 1 4 this improves our earlier results for the corresponding generating functions the proof uses the lace expansion together with an analysis involving fractional derivatives which has been applied previously to the self avoiding walk in a similar context

self avoiding walks and polygons on quasiperiodic tilings we enumerate self avoiding walks and polygons counted by perimeter on the quasiperiodic rhombic penrose and ammann beenker tilings thereby considerably extending previous results in contrast to similar problems on regular lattices these numbers depend on the chosen initial vertex we compare different ways of counting and demonstrate that suitable averaging improves convergence to the asymptotic regime this leads to improved estimates for critical points and exponents which support the conjecture that self avoiding walks on quasiperiodic tilings belong to the same universality class as self avoiding walks on the square lattice for polygons the obtained enumeration data do not allow us to draw decisive conclusions about the exponent

square lattice self avoiding walks and biased differential approximants the model of self avoiding lattice walks and the asymptotic analysis of power series have been two of the major research themes of tony guttmann in this paper we bring the two together and perform a new analysis of the generating functions for the number of square lattice self avoiding walks and some of their metric properties such as the mean square end to end distance the critical point x c for self avoiding walks is known to a high degree of accuracy and we utilise this knowledge to undertake a new numerical analysis of the series using biased differential approximants the new method is major advance in asymptotic power series analysis in that it allows us to bias differential approximants to have a singularity of order q at x c when biasing at x c with the analysis yields a very accurate estimate for the critical exponent thus confirming the conjectured exact value to eight significant digits and removing a long standing minor discrepancy between exact and numerical results the analysis of the mean square end to end distance yields thus confirming the exact value to seven significant digits

mean field critical behaviour for percolation in high dimensions the triangle condition for percolation states that sum limits x y tau 0 x tau 0 y cdot tau y 0 is finite at the critical point where x y is the probability that the sitesx andy are connected we use an expansion related to the lace expansion for a self avoiding walk to prove that the triangle condition is satisfied in two situations i for nearest neighbour independent bond percolation on thed dimensional hypercubic lattice ifd is sufficiently large and ii in more than six dimensions for a class of spread out models of independent bond percolation which are believed to be in the same universality class as the nearest neighbour model the class of models in ii includes the case where the bond occupation probability is constant for bonds of length less than some large number and is zero otherwise in the course of the proof an infrared bound is obtained the triangle condition is known to imply that various critical exponents take their mean field bethe lattice values gamma beta 1 delta delta t 2 t underset raise0 3em hbox smash scriptscriptstyle geqslant 2 and that the percolation density is continuous at the critical point we also prove thatv2 in i and ii wherev2 is the critical exponent for the correlation length

on the area of square lattice polygons we consider the generating function of self avoiding square lattice polygons grouped by both area and perimeter the generating function for polygons of arean is found to diverge atx c 0 251834 with an exponent of zero the mean perimeter of polygons with arean is found to be proportional ton while the mean area of polygons with perimetern is found to be proportional ton 1 5

self avoiding walk in five or more dimensions i the critical behaviour we use the lace expansion to study the standard self avoiding walk in thed dimensional hypercubic lattice ford 5 we prove that the numbercn ofn step self avoiding walks satisfiescn a n where is the connective constant i e 1 and that the mean square displacement is asymptotically linear in the number of steps i e v 1 2 a bound is obtained forcn x the number ofn step self avoiding walks ending atx the correlation length is shown to diverge asymptotically like z 1 2 the critical two point function is shown to decay at least as fast as x 2 and its fourier transform is shown to be asymptotic to a multiple ofk 2 ask 0 i e 0 we also prove that the scaling limit is gaussian in the sense of convergence in distribution to brownian motion the infinite self avoiding walk is constructed in this paper we prove these results assuming convergence of the lace expansion the convergence of the lace expansion is proved in a companion paper

logarithmic correction for the susceptibility of the 4 dimensional weakly self avoiding walk a renormalisation group analysis we prove that the susceptibility of the continuous time weakly self avoiding walk on mathbb z d in the critical dimension d 4 has a logarithmic correction to mean field scaling behaviour as the critical point is approached with exponent frac 1 4 for the logarithm the susceptibility has been well understood previously for dimensions d 5 using the lace expansion but the lace expansion does not apply when d 4 the proof begins by rewriting the walk two point function as the two point function of a supersymmetric field theory the field theory is then analysed via a rigorous renormalisation group method developed in a companion series of papers by providing a setting where the methods of the companion papers are applied together the proof also serves as an example of how to assemble the various ingredients of the general renormalisation group method in a coordinated manner