critical behaviour of directed self avoiding walks a new directed self avoiding walk directed saw is proposed and studied on a square lattice the critical behaviour of such directed saws in two dimensions is established and shown to be of different universality class from that of an ordinary saw

critical exponents of directed self avoiding walks it is argued that directed self avoiding random walks are in the same universality class as directed unrestricted random walks and therefore should exhibit mean field exponents gamma 1 nu sub 1 nu perpendicular to 1 2 in all dimensionalities

directed self avoiding polygons the polygon exponent h associated with directed self avoiding walks in two dimensions has been obtained by exact enumeration and series extrapolation methods the value of h is found to be 1 40 or 0 05 and the connectivity constant associated with such polygons is seen to be less than that of the corresponding walks

directed self avoiding walks on sierpinski carpets series results using a graph counting technique suitable for self similar fractals we obtain the exact densities of partially pdsaw and fully fdsaw directed self avoiding walks on sierpinski carpets from them we calculate the root mean square transverse and longitudinal displacements of the n step walks up to n 20 for pdsaws and up to n 23 for fdsaws the critical exponents nu perpendicular to found for the pdsaw depend on the fractal dimension as well as on the lacunarity of the lattice the results indicate that pdsaws and fdsaws have different critical behaviour on the same carpet

renormalisation group study of fully directed self avoiding walks the author presents a discussion of the critical properties of fully directed self avoiding walks on hypercubic lattices a position space renormalisation group transformation is carried out analytically in two dimensions for any value of the scaling factor b the finite size scaling properties of the transformation are explicitly exhibited including the well known finite size scaling hypothesis v b v approximately 1n b 1 it is shown that the rg transformation in this case is sensitive to v sub thus giving support to existing arguments

spiral growing self avoiding walk a new universality class the author introduces the concept of a spiral growing self avoiding walk sgsaw by imposing privman s spiralling constraint ibid vol 16 p l571 1983 on the growing self avoiding walk introduced by majid et al phys rev lett vol 52 p 1257 1984 and lyklema and kremer ibid vol 17 p l691 1984 he studies the critical properties of the sgsaw using monte carlo simulation the exponent nu associated with the average radius of gyration sn is 0 8 on the other hand the critical exponent nu r associated with the end to end distance rn is much less than nu the latter is in sharp contrast to the result nu nu r for the usual saw directed gsaws are shown to behave like directed usual saws

critical exponent of a directed self avoiding walk it has been shown that the angular correlation range is infinite for the directed self avoiding walk problem on the square lattice this result implies a one dimensional like critical exponent namely nu 1

effect of anisotropic constraints on self avoiding walks the author studies the effect of four types of anisotropic constraints on self avoiding walks saws on the square lattice in which walks along one lattice direction are constrained differently from those of the other three identically constrained lattice directions by numerical studies the author indicates that all these types of anisotropically constrained saws belong to the universality class of directed self avoiding walks

size and shape of directed lattice animals uses position space renormalisation and exact enumeration methods to study configurational properties of directed lattice animals clusters of directed bonds in which the tail of a new bond must be added to the head of existing bonds exponents are found for the divergence of correlation lengths parallel and perpendicular to the preferred axis in addition an exponent is found that characterises the singularity of the directed animal generating function and critical points are located

yetter drinfeld modules under cocycle twists abstract we give an explicit formula for the correspondence between simple yetter drinfeld modules for certain finite dimensional pointed hopf algebras h and those for cocycle twists h of h this implies an equivalence between modules for their drinfeld doubles to illustrate our results we consider the restricted two parameter quantum groups u r s sl n under conditions on the parameters guaranteeing that u r s sl n is a drinfeld double of its borel subalgebra we determine explicit correspondences between u r s sl n modules for different values of r and s and provide examples where no such correspondence can exist our examples were obtained via the computer algebra system singular plural

mean field type equations for spread of epidemics the small world model abstract in the paper we study a cellular automata ca model of epidemic dynamics the effects of local spatial correlations on a temporal aggregated spread of single epidemics are studied as a function of increasing proportion of global contacts small world model we conjecture that even in the presence of high local correlations the aggregated mean field type models can be quite successful if the contact rate is treated as a free parameter the dependence of the estimated contact rate on the mixing parameter can be understood in terms of a simple probabilistic model the contact rate reflects not only a microscopic and epidemiological situation but also a complicated social pattern including short and long range contacts as well as a possibly hierarchical structure of human society

spiral self avoiding walks self avoiding walks with a spiral constraint on the square lattice are enumerated up to 40 steps numerical evidence suggests that they belong to a new universality class

on the universality class of self avoiding walks it is shown that within a two parameter cell to bond position space renormalisation group psrg theory self avoiding walks and trails on the square lattice obey the same end to end distance critical exponent nu a critical examination of this result and recent speculations on this problem suggests that one should not draw qualitative predictions based on numerical estimates of the one parameter psrg theory

some exact results from the mean field renormalization group abstract the mean field renormalization group is applied to the study of the directed self avoiding random walk in d dimensions and to the spin ferromagnetic ising model on the bethe lattice in both cases the known exact results are obtained

triviality of the critical exponents of directed self avoiding walks on sierpinski carpets the authors present an analytic argument for the critical exponents nu perpendicular to and nu sub of the fully directed self avoiding walks on a family of sierpinski carpets in contrast to the cases of random walks or isotropic self avoiding walks on a fractal lattice they find that both exponents do not depend on the fractal dimension of the underlying carpet but take a trivial value of unity only the correction to scaling exponents vary with the fractal property of the underlying lattice numerical simulations confirm the prediction

from equilibrium spin models to probabilistic cellular automata the general equivalence betweend dimensional probabilistic cellular automata pca and d 1 dimensional equilibrium spin models satisfying a disorder condition is first described in a pedagogical way and then used to analyze the phase diagrams the critical behavior and the universality classes of some automata diagrammatic representations of time dependent correlation functions of pca are introduced two important classes of pca are singled out for which these correlation functions simplify 1 quasi hamiltonian automata which have a current carrying steady state and for which some correlation functions are those of ad dimensional static model pca satisfying the detailed balance condition appear as a particular case of these rules for which the current vanishes 2 linear and more generally affine pca for which the diagrammatics reduces to a random walk problem closely related to d 1 dimensional directed saws both problems display a critical behavior with mean field exponents in any dimension the correlation length and effective velocity of propagation of excitations can be calculated for affine pca as is shown on an explicitd 1 example we conclude with some remarks on nonlinear pca for which the diagrammatics is related to reaction diffusion processes and which belong in some cases to the universality class of reggeon field theory

a transfer matrix for the backbone exponent of two dimensional percolation rephrasing the backbone of two dimensional percolation as a monochromatic path crossing problem we investigate the latter by a transfer matrix approach conformal invariance links the backbone dimension db to the highest eigenvalue of the transfer matrix t and we obtain the result db 1 6431 0 0006 for a strip of width l t is roughly of size 23l but we manage to reduce it to l we find that the value of db is stable with respect to inclusion of additional blobs tangent to the backbone in a finite number of points

solving first order initial value problems by using an explicit non standard a stable one step method in variable step size formulation this paper presents the construction of a new family of explicit schemes for the numerical solution of initial value problems of ordinary differential equations odes the one parameter family is constructed by considering a suitable rational approximation to the theoretical solution resulting a family with second order convergence and a stable imposing that the principal term in the local truncation error vanishes we obtain an expression for the parameter value in terms of the point xn yn on each step with this approach the resulting method has third order convergence maintaining the characteristic of a stability finally combining this last method with other of order two in order to get an estimation for the local truncation error an implementation in variable step size has been considered the proposed method can be used in a wide range of problems for solving numerically a scalar ode or a system of first order odes several numerical examples are given to illustrate the efficiency and performance of the proposed method in comparison with some existing methods in the literature

on the symmetry classes of planar self avoiding walks we present new results on the class of anisotropic spiral walks in two dimensions we find that these are directed problems in the sense that the usual relation v 2vperpendicular to holds between the length scale exponents in contradistinction however they do not seem to fall in the usual directed universality class v 1 motivated by this the universality classes of self avoiding walks saw on the square lattice are discussed we argue that of the 84 models that exist by restricting the possible two step configurations there are four major categories with a total of seven generic types the importance of reflection symmetry in this classification is discussed

an exactly solvable self avoiding walks model ii triangular and honeycomb lattices a modified self avoiding walks model previously proposed for the square lattice is applied to the triangular and honeycomb lattices in the model the walker is restricted not to take any turn which will put the walker in a direction rotated by more than a certain angle phi max from any of the directions previously taken the generating functions are obtained and various quantities are evaluated exactly for all three square triangular and honeycomb lattices it is found that the model exhibits the characteristics reminiscent of one dimensional self avoiding walks in all directions while strongly retaining the anisotropic effect of the direction of the first step for the honeycomb lattice the authors have studied the model for phi max pi a model and phi max 3 2 pi b model unlike other walks the b model exhibits a special property the oscillations between even and odd steps in various quantities such as the number of n step walks do not decay as the number of steps increases

on the apparent failure of a flory approximation for directed linear polymers the author discusses the apparent failure of a flory approximation for directed linear polymers commenting on the details implicit in this type of picture the relationship of a standard flory viewpoint to a simplified application of fractal theory is explored and a correspondence is drawn between aspects of both theories