nonperturbative tricritical exponents of trails i exact enumeration on a triangular lattice as the fugacity of intersections in trails intersecting but non overlapping lattice walks is increased the configurations change from swollen ones with the scaling exponents of self avoiding walks to compact ones separating the two regimes is a potentially new tricritical point with no perturbative renormalisation fixed point associated with it supporting evidence for the existence of a tricritical point its likely location and exponents are computed for the first time from exact enumeration of all trails up to length of fifteen lattice constants on the triangular lattice the divergence of the specific heat indicates the location of the tricritical point generalised ratio and pade methods are used to extract the scaling exponents for the number of configurations and their end to end distance

self avoiding path walks on lattices a new universality class the trails problem self avoiding path walks is reconsidered by the conventional series expansion approach an exact enumeration on the square triangular and simple cubic lattices is done by computer up to 18 12 and 11 steps respectively the more self consistent and refined results between the connective constant and the corresponding critical exponent indicate the possibility of there being a different universality class from that of the self avoiding walk an average approach based on stolz s theorem is proposed which often appears to be smoother and better than the traditional one

specific heat and crossover exponents at the tricritical collapse of trails from the exact enumeration on a triangular lattice of trail configurations at the tricritical regime as the fugacity for intersections is increased the authors find phi t 0 68 8 for the crossover exponent and hence alpha t 0 5 2 for the specific heat exponent it indicates the collapse transition of trails to differ from that of polymer chains at the theta point in two dimensions

critical exponents for square lattice trails with a fixed number of vertices of degree 4 we prove several previously conjectured results about the number of n edge trails and n edge embeddings of eulerian graphs each with a fixed number k of degree 4 vertices in the lattice 2 in particular under the assumption that the relevant critical exponents exist we prove that the difference between the critical exponent for closed trails eulerian graph embeddings and that for self avoiding circuits polygons is exactly k the number of degree 4 vertices similarly we prove that the difference between the critical exponent for either open trails or open eulerian graph embeddings and that for self avoiding walks is also k these results are proved by establishing upper and lower bounds for the number of n edge embeddings of closed open eulerian graphs with k vertices of degree 4 in terms of the number of n edge self avoiding polygons walks the lower bounds are proved using a kesten pattern theorem argument and the upper bounds are established by developing based on a detailed case analysis a method for removing vertices of degree 4 from an embedding by altering at most a constant independent of n number of vertices and edges of the embedding the work presented here extends and improves the arguments first given in the work of zhao and lookman 1993 j phys a math gen 26 1067 76

asymptotic properties of lattice trails the trail problem a variant of the self avoiding walk in which excluded volume is associated with lattice bonds rather than sites is studied numerically in two and three dimensions monte carlo and exact series methods are used but the results obtained lack the consistency of similar results for self avoiding walks the evidence in support of the two models belonging to the same universality class is suggestive though not entirely convincing but slowly decaying corrections to scaling could account for apparent exponent differences

the geometrical properties of planar lattice trails the authors investigate the winding angle and mean square end to end distance of lattice trails on a square lattice by both exact enumeration and monte carlo methods they find similar behaviour to that of self avoiding walks to the extent that the complete distribution is found to agree with the recent exact calculation of the winding angle distribution function for self avoiding walks by duplantier and saleur 1988 who found a gaussian distribution the exponent nu was found to be indistinguishable from 3 4

lattice trails i exact results considers the problem of lattice trails introduced by malakis 1975 1976 the author proves the existence of a finite connective constant and establishes a result for the growth of n step trails tn analogous to the best known result for self avoiding walks tn lambda n expt o n for the honeycomb d 2 and lave s lattice d 3 the author establishes a counting theorem from which one can deduce the exact value of the connective constant lambda for the honeycomb lattice lambda 2 2 square root 2 further it follows that the trail problem is in the same universality class as the self avoiding walk problem for those lattices an exact amplitude relation between trails and self avoiding walks and between dumb bells and trails and self avoiding walks is established the non existence of a counting theorem for arbitrary lattices is established an inequality for the triangular lattice connective constant is proved a high density expansion for lambda for the d dimensional hybercubic lattice is also obtained the author argues that contrary to recent suggestions the model is in the same universality class as the self avoiding walk model

scaling of self avoiding walks and self avoiding trails in three dimensions motivated by recent claims of a proof that the length scale exponent for the end to end distance scaling of self avoiding walks is precisely 7 12 0 5833 we present results of large scale simulations of self avoiding walks and self avoiding trails with repulsive contact interactions on the simple cubic lattice we find no evidence to support this claim our estimate 0 5874 2 is in accord with the best previous results from simulations

four dimensional polymer collapse ii interacting self avoiding trails we have simulated four dimensional interacting self avoiding trails isat on the hypercubic lattice with standard interactions at a wide range of temperatures up to length 4096 and at some temperatures up to length 16384 the results con3rm the earlier prediction using data from a non standard model at a single temperature of a collapse phase transition occurring at 3nite temperature moreover they are in accordwith the phenomenological theory originally proposed by lifshitz grosberg andkhokhlov in three d andrecently given new impetus by its use in the description of simulational results for four dimensional interacting self avoiding walks isaw in fact we argue that the available data is consistent with the conclusion that the collapse transitions of isat andisaw lie in the same universality class in contrad iction with long standing predictions we deduce that there exists a pseudo 3rst order transition for isat in four dimensions at 3nite lengths while the thermodynamic limit is described by the standard polymer mean 3eld theory giving a second order transition in contradiction to the prediction that the upper critical dimension for isat is du 4 c 2001 elsevier science b v all rights reserved

on the universality classes of growing self avoiding walks and trails it is shown that within a two parameter 3 3 cell to bond position space renormalisation group theory growing self avoiding walks and growing self avoiding trails on the square lattice are in the same universality class

lattice trails ii numerical results for pt i see ibid vol 18 p 567 1985 a numerical study of the properties of lattice trails on the honeycomb square triangular and simple cubic lattices is made critical points are estimated for all lattices and upper and lower bounds established extensive series have been obtained and series analysis of both trail generating functions and mean square end to end distance series are not inconsistent with the conclusion that the problem is in the same universality class as the self avoiding walk problem a pseudo star triangle transformation is defined and the analyticity properties of that function coupled with previous exact results clearly supports that conclusion for the triangular lattice as well as providing excellent unbiased critical point estimates the author also shows that the connective constant for d or 2 dimensional hypercubic trails is strictly greater than the corresponding quantity for saws

self avoiding walks neighbour avoiding walks and trails on semiregular lattices we study self avoiding and neighbour avoiding walks and lattice trails on two semiregular lattices the lattice and the lattice for the lattice we find the exact connective constant for both self avoiding walks neighbour avoiding walks and trails for the lattice we generate long series which permit the accurate estimation of the connective constant for self avoiding walks and trails

evaluating and approximating fir filters an approach based on functions of matrices most of the controllers designed for systems with timedelays involve finite impulse response fir filters in their structure in particular prediction based controllers for systems with input output delays we present theory algorithms and software to evaluate the frequency response to approximate the filters by stable linear timeinvariant systems and to compute their induced cal l 2 gain the approach is based on the theory of functions of matrices in the approach pole zero cancelations in frequency domain representations are explicitly taken into account

universal amplitudes ratio at the collapse transition of polymers in two dimensions first values of the universal ratio of the mean square radius of gyration and the mean square end to end distance for 2d linear polymers at their collapse transition are calculated by the scanning simulation method two models are considered i self avoiding walks saws with nearest neighbour attractions for which this ratio is found to be 0 179 or 0 003 at the theta point ii self intersecting trails at their tricritical point for which the value 0 166 or 0 002 is obtained this provides yet another indication that these models may belong to different tricritical universality classes the authors have also computed this ratio for trails at the high temperature limit with non interacting intersections and found 0 141 or 0 001 in excellent agreement with earlier estimates of this ratio for saws in their swollen phase

non perturbative tricritical exponents of trails ii exact enumerations on square and simple cubic lattices for pt i see ibid vol 21 p 773 1988 the authors present exact enumerations of trails self intersecting but non overlapping lattice walks tabulated according to their length l number of intersection i and end to end distance for the square lattice up to l 21 i 7 and for the simple cubic lattice up to l 15 i 5 they introduce a fugacity for intersection to explore the transition to a collapsed phase through a novel tricritical point which is unaccessible by the renormalisation group approach the existence of tricritical points in both lattices is manifested by the divergence of the specific heat the values of the tricritical couplings and exponents are extracted by a dlogpade analysis they find for the three dimensional lattice nu t approximately 0 48 and gamma t approximately 0 43 nu t approximately 0 52 and gamma t approximately 1 25 for the square lattice

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

the 3 server problem in the plane in the k server problem we wish to minimize in an online fashion the movement cost of k servers in response to a sequence of requests we assume that k 2 the request issued at each step is specified by a point r in a given metric space m to serve this request one of the k servers must move to r it is known that if m has at least k 1 points then no online algorithm for the k server problem in m has competitive ratio smaller than k the best known upper bound on the competitive ratio in arbitrary metric spaces by koutsoupias and papadimitriou j acm 42 1995 971 is 2k 1 there are only a few special cases for which k competitive algorithms are known for k 2 when m is a tree or when m has at most k 2 points we prove that the work function algorithm is 3 competitive for the 3 server problem in the manhattan plane as a corollary we obtain a 4 243 competitive algorithm for 3 servers in the euclidean plane the best previously known competitive ratio for 3 servers in these metric spaces was 5

a monte carlo study of lattice trails the authors use the recently developed monte carlo algorithm of berretti and sokal j stat phys vol 40 p 483 1985 to study the critical behaviour of lattice trails on the square and triangular lattices they find support for the earlier belief that this problem is in the same universality class as self avoiding walks accepting this they obtain the most precise estimates to date of the connective constants it is argued that this problem is particularly well suited to study by monte carlo methods rather than by series analysis

the statistics of collapsing square lattice trails with a fixed number of vertices of degree 4 a trail on the square lattice with a fixed number k of vertices of degree 4 is called a k trail we model polymer collapse using k trails by incorporating an interaction energy which is proportional to the number of nearest neighbour contact edges of the trail it is known that the number of square lattice n edge closed open k trails can be bounded above and below to o nk by the number of n step self avoiding circuits walks this along with pattern theorems for self interacting self avoiding circuits and walks are used herein to establish upper and lower bounds to o nk for the collapsing free energy of k trails in terms of self avoiding circuits or walks as appropriate we also use pattern theorems to obtain bounds on the limiting nearest neighbour contact density for collapsing k trails finally we investigate k trails with a fixed density of nearest neighbour contacts and show that their limiting entropy per monomer is independent of k

the growing self avoiding trail the author introduces a new random walk which can be used as a model for the theta polymer the walk belongs to a different universality class from the usual saw extensive monte carlo calculations have been performed to calculate the exponents gamma and nu the author finds in two dimensions nu 0 535 and gamma 1 025 and in three dimensions nu 0 50 and gamma 1 0 this result strongly suggests that the upper critical dimension is equal to three

bounds on connective constants for self avoiding walks new lower bounds on the connective constant of the square and simple cubic lattice self avoiding walks are obtained by enumerating a particular subset of self avoiding walks and using a result of kesten the author finds mu sq 2 5680 and mu sc 4 352

critical exponents for simple non uniform polymer networks the authors study a number of non uniform specified topologies and show rigorously that for certain topologies with cut edges the critical exponent gamma t is in agreement with a conjecture given by gaunt et al 1984 and that the exponent nu t nu the exponent for self avoiding walks the authors also find that the scaling relations gamma t gamma t1 and gamma t gamma t11 are the same as for self avoiding walks previously conjectured only for uniform networks by assigning an interaction energy to a nearest neighbour contact they prove that the collapse transition for these topologies is the same as that for self avoiding walks

on two dimensional self avoiding random walks following nienhuis 1982 exact evaluation of the connective constant of the honeycomb lattice self avoiding walk model and the exact exponent values he has conjectured the author has re examined the available data on all three regular two dimensional lattices in the case of the triangular lattice the author has additionally corrected and extended the extant series the author finds support for nienhuis value gamma 111 32 for all three lattices and further finds that alpha 1 2 for all three lattices as given by nienhuis result nu 3 4 and the hyperscaling relation d nu 2 alpha the author is unable to find any consistent evidence of a correction to scaling exponent delta 1 1 cannot be ruled out

existence of four dimensional polymer collapse i kinetic growth trails abstract we present the results of simulations of kinetic growth trails kgt bond avoiding walks in four dimensions we use a mapping from a kinetic growth model to a static model of self interacting trails isat at a particular temperature to argue that this temperature is precisely the collapse temperature of four dimensional interacting trails to do this we show that the kinetic growth trails behave neither like static non interacting trails which should behave as excluded volume dominated four dimensional polymers that is self avoiding walks or collapsed four dimensional polymers but rather show an intermediate behaviour this is the first indication of collapse in any four dimensional lattice polymer model and so may be helpful in deciding which of the competing models of polymers is a good model in lower dimensions we have calculated various exponents of the kgt model and identified them with certain critical exponents of the static isat problem