ϵ expansion for directed animals introduces the problem of directed lattice animals and shows that they are highly anisotropic with two different correlation lengths xi sub and xi perpendicular to parallel and perpendicular to the privileged direction a field theory is introduced and the critical exponents calculated to first order in epsilon dc d where the upper critical dimension dc is 7

finite size scaling for directed bond percolation with and without cycles on a triangular lattice the authors apply finite size scaling and phenomenological renormalisation group arguments to the problems of directed acyclic directed cyclic and undirected bond percolation on a triangular lattice the results are in good agreement with known estimates and show that the phenomenological renormalisation procedure is sensitive to the difference between global and local directional biases as well as the distinction between locally directed and fully isotropic problems in addition the authors draw tentative comments on the extension to directed problems of the relation a pi eta where a is the inverse correlation length amplitude of a strip of finite width at criticality and eta is the exponent that describes the decay of correlations at the critical point known to hold for various isotropic systems

reconstruction of turbulent fluctuations using a hybrid rans les approach a hybrid rans les approach is presented and assessed considering several flows which can be interpreted as the most general case of the nlde approach as defined by morris et al p j morris l n long a bangalore and a wang j comput phys 133 56 1997 a decomposition into three parts of the exact solution of the navier stokes equations is considered mean flow resolved fluctuations and unresolved subgrid fluctuations the mean flow is computed using a classical rans method while resolved fluctuations are derived from a les method this approach is succesfully assessed on the stationary and the pulsed plane channel flow configurations the case of the flow around a low pressure turbine blade is discussed in a second step the hybrid method is also demonstrated to be robust with respect to several sources of error and provides good results in the case of a coarsened grid simulation a reduced computational domain simulation and nonconsistent mean flow fields

resistor diode percolation on the hierarchical diamond lattice the resistor diode percolation problem on the hierarchical diamond lattice is studied by using an exact real space renormalisation group transformation this problem exhibits several critical and multicritical points the flow diagram obtained is topologically equivalent to the one found by redner 1982 in the framework of an approximate renormalisation group transformation on regular lattices

upper bounds for the critical probability of oriented percolation in two dimensions we give a method for obtaining upper bounds on the critical probability in oriented bond percolation in two dimensions this method enables us to prove that the critical probability is at most 0 6863 greatly improving the best published upper bound 0 84 we also prove that our method can be used to give arbitrarily good upper bounds we also use a slight variant of our method to obtain an upper bound of 0 725 99 for the critical probability in oriented site percolation

shattering and non uniqueness in fragmentation models an analytic approach abstract fragmentation equations have features which are rather unexpected in such a relatively simple linear framework these are shattering attributed to a phase transition creating a dust of zero size particles and the existence of multiple solutions the existence of these phenomena indicates that the fragmentation equation alone does not offer a full description of the dynamics of the problem in this paper we present a survey of recent results explaining both features from a unified dynamical systems point of view we also use the developed theory to discuss a number of conjectures and observations made in the literature of the subject

numerical diagonalization study of the trimer deposition evaporation model in one dimension we study the model of deposition evaporation of trimers on a line recently introduced by barma et al the stochastic matrix of the model can be written in the form of the hamiltonian of a quantum spin 1 2 chain with three spin couplings given by h sigma i 1 sigma i sigma i 1 sigma i 2 sigma i sigma i 1sigma i 2 hc we study by exact numerical diagonalization of h the variation of the gap in the eigenvalue spectrum with the system size for rings of size up to 30 for the sector corresponding to the initial condition in which all sites are empty we find that the gap vanishes as l z where the gap exponent z is approximately 2 55 or 0 15 this model is equivalent to an interfacial roughening model where the dynamical variables at each site are matrices from our estimate for the gap exponent we conclude that the model belongs to a new universality class distinct from that studied by kardar et al

domain decomposition technique for solution of acoustic wave scattering a novel domain decomposition technique is applied that couples nonlinear and linear euler solvers to model acoustic scattering of waves in the vicinity of high velocity gradients and large amplitude waves to reduce numerical resources the linear solver is applied in domains in which the wave amplitude is small and linearization is applicable whereas the nonlinear solver is used in the vicinity of high wave amplitudes and stagnation points three numerical models are compared in this paper 1 a linear euler 2 a nonlinear euler and 3 the domain decomposition technique for all three models the finite difference dispersion relation preserving scheme is applied to all spatial derivatives and the low dissipation dispersion runge kutta scheme is applied for the temporal integration a perfectly matched layer is applied at the edges of the computational domain to absorb outgoing waves and minimize reflections to include complex bodies and geometries an overset mesh is applied after model validation t

on phase diagrams for directed percolation problems the authors propose a real space renormalisation group scheme for anisotropic directed bond percolation in two dimensions which differing from previous phenomenological or small cell real space renormalisation groups is sensitive to the existence of an infinite cluster along any lattice direction this enables them to explore new aspects of the phase diagram for the problem which were not accessible through existing treatments in particular the crossover exponent between one and two dimensional behaviour is shown to be exactly one for directed percolation numerical estimates are in agreement with known inequalities

enumeration of directed site animals on two dimensional lattices studies the problem of directed site animals on the square triangular and hexagonal lattices closed form expressions are proposed for a s the number of animals of size s on the square and triangular lattices these expressions have been checked for s or 33 and s or 10 for the square and triangular lattices respectively by explicit enumeration they imply that a s varies as lambda 2 s 0 for large s where lambda 3 for the square lattice and lambda 4 for the triangular and theta 1 2 for both for the hexagonal lattice a s is found for s or 48 the results are consistent with lambda 2 0252 or 0 0005 and theta 1 2

radiated noise from airfoils in realistic mean flows the long term objective of the research described is to use computational aeroacoustics methodology and parallel computers to increase the understanding of broadband blade noise in a systematic progression toward simulations of completely realistic configurations and conditions some simplified problems that address the important features of the flow are investigated a two dimensional navier stokes code implemented using the message passing library and fortran 90 on the ibm sp2 is used to perform the calculations results are presented for the interaction of a vortical gust and naca airfoils including nonlinear effects the influence of gust frequency and airfoil thickness is described a multigrid method is used to obtain converged steady state solutions before the gust is introduced in a source region inside the domain

directed percolation a finite size renormalisation group approach the finite size renormalisation group technique introduced by nightingale 1976 is applied to the directed percolation problem the decay of correlations is anisotropic in this model and finite size scaling is extended to treat such anisotropy precise estimates for critical exponents and percolation probabilities are obtained for site bond and site bond percolation on the square lattice with bonds directed along the positive axes both free boundary conditions for which the results converge linearly with 1 n as n to infinity and helical boundary conditions for which unexpectedly the results converge linearly with 1 n3 are considered

z 2 gauge model on fractal lattices hamiltonians with z 2 gauge symmetry are studied on a class of fractal lattices the lattices considered have fourier and hausdorf besicovitch dimensions strictly greater than 2 but they have a low degree of connectivity characterised by their connectivity index 2 it is proved that there is no phase transition at non zero temperatures

infinite range mean field percolation transfer matrix study of longitudinal correlation length two infinite range directed percolation models equivalent also to epidemic models are considered for a finite population finite number of sites n at each time directed axis step the general features of the transfer matrix spectrum evolution operator spectrum for the epidemic are studied numerically and compared with analytical predictions in the limitn one of the models is devised to allow numerical results to be obtained forn as high as nearly 800 for the largest longitudinal percolation correlation length relaxation time for epidemic the finite n behavior of this correlation length is studied in detail including scaling near the percolation transition exponential divergence withn above the percolation transition as well as other noncritical and critical point properties

a new type of kinetic critical phenomenon the authors study a new critical phenomenon in an non thermal one dimensional lattice model it is characterised by the transition from stability to instability of kinks between ordered states below the critical point the kinks are stable and move by annihilating random walks above the critical point they are unstable against creation of kink antikink pairs the spontaneous production of pairs is assumed to be absent at the critical point p pcr the density of kinks decreases like n approximately talpha with alpha 0 27 or 0 08 above the critical point the density of kinks in the stationary state is approximately p pcr beta with beta 0 06 or 0 2 possible extensions to two or more dimensions and possible applications are discussed

flory theory for directed lattice animals and directed percolation the free energies of directed lattice animals in good and theta solvents and the free energy of directed percolation are found by the use of a simple flory type approximation which accounts for the inherent anisotropy of these systems from these free energies the authors obtain the upper critical dimension below which mean field theory breaks down they also calculate closed form dimension dependent expressions for the parallel and perpendicular correlation length exponents which characterise the asymptotic cluster shapes these exponents are in excellent agreement with existing numerical data

a transfer matrix study of directed lattice animals and directed percolation on a square lattice we studied the large scale properties of directed lattice animals and directed percolation on a square lattice using a transfer matrix approach on strips of finite widths we generated relatively long sequences of estimates for effective values of critical fugacity percolation threshold and correlation length critical exponents we applied two different extrapolation methods to obtain estimates for infinite systems the precision of our final estimates is comparable to or better than the precision of the best currently available results

series expansion analysis of directed site bond percolation on the square and simple cubic lattices analysis of low density series for site bond percolation on the directed square sq and simple cubic sc lattice and related series for bond percolation on the honeycomb h and diamond di lattices is found to be consistent with pc sq site bond pc h bond 0 8228 or 0 0002 pc sc site bond pc di bond 0 637 or 0 002 and previous estimates of gamma nu sub and nu perpendicular to analysis of the square lattice series supports the validity of the scaling relation gamma 0 gamma d 1 nu perpendicular to for the two dimensional lattices site percolation on the honeycomb and diamond lattices is also considered

a hierarchical percolation model some exact results on the sierpinski gasket a hierarchical percolation model is formulated on the sierpinski gasket with the use of position space renormalisation group methods exact results are obtained that place this model in a distinct universality class from that of the geometric percolation model for the first of two cases considered pc 1 nu 1 d and for the second pc 1 2 nu 1 d 1 where d is the fractal dimension of the sierpinski gasket the hierarchical model is also formulated on the triangular lattice and some preliminary results are obtained

transfer matrix study of the chiral clock model in the hamiltonian limit the logarithm of the transfer matrix of the two dimensional three state chiral clock model is shown to be equivalent to a one dimensional non hermitian quantum hamiltonian in the region of small chirality the critical behaviour of the quantum model is investigated by a series expansion and by finite size scaling a single phase boundary between a modulated paramagnetic and a ferromagnetic phase is located with good accuracy both methods yield comparable results for the critical exponent beta of the wavevector of the modulated high temperature phase the critical exponents nu x nu tau for the correlation lengths obtained from finite size scaling point towards the existence of a lifshitz point at finite chirality the results of the series expansion for the exponent nu x however are inconsistent with finite size scaling for larger chirality the authors finite size results clearly reveal the occurrence of an incommensurate phase with algebraically decaying correlations between the ferromagnetic and the modulated paramagnetic phase they analyse the critical behaviour of this incommensurate phase at its melting line and determine the exponent eta that controls the algebraic decay of correlations

percolation and conduction in a random resistor diode network studies percolation and conduction in a randomly diluted network in which the occupied bonds may be either resistor like transmitting connectivity in both directions along a bond or diode like transmitting in only one direction this system exhibits novel phase transitions signalling the onset of infinite clusters with either isotropic or unidirectional connectivity position space renormalisation group methods are applied to map out the phase diagram and calculate exponents associated with these phase transitions

diode resistor percolation in two and three dimensions i upper bounds on critical probability obtains upper bounds to the critical probability for percolation in a random network made of oriented diodes and resistors it is shown that for the square lattice pc 0 3700 and for the simple cubic lattice pc 0 2417

nonlocal percolation and partial ordering in the dilute baxter wu model the zero temperature limit of the dilute baxter wu model is studied in the context of percolation theory the percolation model describing the propagation of order is identified and shown to possess a nonlocal mechanism for the propagation of that order a monte carlo study of this model is performed using the nonlocal cluster identification algorithm developed by fried and schick 1992 the author also shows that the mechanism responsible for nonlocal percolation allows for the possibility of the propagation of partial order at intermediate impurity concentration a partially ordered phase is found and its structure is discussed

a long range domany kinzel model of directed percolation a long range domany kinzel 1981 model proposed by wu and stanley 1982 is solved using random walk formulations in this model for every site i j in a two dimensional lattice there is a directed bond present from site i j to i 1 j with probability one there are also m 1 directed bonds present from i j to i k j 1 k 1 0 1 2 3 m 1 with respective probabilities pk 1 where m is any positive integer an exact expression is obtained to determine the critical percolation angle theta c for any distribution of pn the system percolates in the region theta c theta or 0 with probability one and zero outside this region where theta is the angle measured from the x axis if the authors let m go to infinity and pn varies as p1n s they find that when s or approximately 2 when m is large but finite theta c is also obtained for the following two distributions a pn a a n with a 0 b pn beta m and beta 0

confluent singularities in directed bond percolation existing series for directed bond percolation on the square honeycomb and face centred cubic lattices are analysed with a view to detecting their confluent singularities the authors compare our results with the confluent singularities in reggeon field theory which has been shown to be in the same universality class

partially directed site percolation on the square and triangular lattices using the phenomenological renormalisation pr method the authors obtain the critical probability for partially directed site percolation on the square and triangular lattices pc 0 6317 or 0 006 and pc 0 5468 or 0 005 respectively they discuss the corrections to scaling in the calculation of the anisotropy exponent theta nu sub nu perpendicular to and the critical probability and show that it is necessary to work with at least two correction terms this is in contrast to the usual analysis where only one correction exponent is used in pr calculations

relation between size and shape of isotropic and directed percolation clusters and lattice animals shows that the generalised lattice animal model of family and coniglio 1980 naturally leads to a unified scaling picture for percolation and lattice animals in which the fugacity for occupied elements plays the dual role of a temperature like and a field like variable within this single scaling field description of percolation there is only one independent exponent from which all others can be obtained the author defines a new set of exponents alpha beta and gamma for percolation and finds that they are all related to the cluster number exponent theta through the relation alpha gamma 1 beta 3 theta in analogy with lattice animals to relate the cluster radius exponent nu to the other exponents he uses the generalised ginzburg criteria to obtain a modified hyperscaling relation for isotropic and directed percolation and lattice animals using this relation he finds that theta 1 nu sub perpendicular to d 1 for directed percolation and theta nu perpendicular to d 1 for directed lattice animals where nu sub and nu perpendicular to are exponents characterising the parallel and perpendicular cluster radii respectively using the same approach he obtains the stauffer relation theta 1 d nu and the parisi sourlas relation theta 1 d 2 nu for isotropic percolation and lattice animals respectively the above relations give the following expressions for theta within the flory theory theta percolation 3d 2 d 2 theta directed percolation 6d 5 2d 4 theta animals 7d 6 2d 4 and theta directed animals 9 d 1 4d 8

phenomenological renormalisation of the self avoiding walk in two dimensions phenomenological renormalisation is used to calculate the exponent v and the connective constant of the self avoiding walk problem on a square lattice a transfer matrix technique is developed for the polymer problem the results indicate that flory s value v 0 75 is true in two dimensions to extremely high accuracy

convergence and extrapolation in finite size scaling renormalization abstract finite size renormalization technique is studied by examining a test model which is exactly solvable and the problem of lattice animals and self avoiding walks bulk estimates of the leading irrelevant rg eigenvalue are found and compared with independent estimates from finite size data we introduce an extrapolation technique which is applicable to non monotonic sequences of exponent approximants

mean field renormalisation group calculations for directed percolation on the square lattice a simple extrapolation technique for mean field renormalisation group calculations is described application of this method to large cell calculations for percolation on the directed square lattice yields pc 0 7067 or 0 0001 site problem and pc 0 6454 or 0 0002 bond problem

convergence of finite size scaling renormalisation techniques abstract the rate of convergence of the transfer matrix finite size scaling or phenomenological renormalisation method is studied it is shown both heuristically and numerically that the convergence of estimates for exponents etc is governed asymptotically by the leading irrelevant variable scaling exponent the more rapid apparent convergence rates observed in many practical calculations for two dimensional lattices of widths up to ten lattice spacings are attributed to cancellation between various correction terms

compact disturbance equations for aeroacoustic simulations a set of compact disturbance equations is developed for high accuracy and efficient aeroacoustic simulations the compact disturbance equations in their complete form are an exact rearrangement of the navier stokes equations but incorporate various linear and nonlinear disturbance equations such as the linearized euler equations and the linearized navier stokes equations their attractive mathematical properties facilitate the implementations of the full compact disturbance equations and the reduced equations in essentially the same form as in existing computational fluid dynamics solvers with minor modifications a high resolution unsteady simulation in a reduced domain can be embedded inside a less expensive reynolds averaged navier stokes solution of flow in a larger possibly very complex configuration this results in a hybrid reynolds averaged navier stokes large eddy simulation method with reduced meshing difficulties and computational costs but improved grid qualities and more accurate boundary

cellular automata and statistical mechanical models we elaborate on the analogy between the transfer matrix of usual lattice models and the master equation describing the time development of cellular automata transient and stationary properties of probabilistic automata are linked to surface and bulk properties respectively of restricted statistical mechanical systems it is demonstrated that methods of statistical physics can be successfully used to describe the dynamic and the stationary behavior of such automata some exact results are derived including duality transformations exact mappings disorder and linear solutions many examples are worked out in detail to demonstrate how to use statistical physics in order to construct cellular automata with desired properties this approach is considered to be a first step toward the design of fully parallel probabilistic systems whose computational abilities rely on the cooperative behavior of their components

the pair connectedness for directed percolation on the honeycomb and diamond lattices the pair connectedness for directed site percolation on the honeycomb and diamond lattices is related to that of the square and simple cubic lattices respectively in the case of bond percolation the same correspondence leads to site bond percolation on the latter pair of lattices

the collapse of directed animals directed animals on a square lattice with attractive nearest neighbour and next nearest neighbour interactions are considered using an equivalence between the collapse problem and the directed site bond percolation problem the bulk free energy per site is calculated exactly in the collapsed phase it shows no singularity as the transition is approached from the collapsed side for the three dimensional directed animals bulk free energy per site in the collapsed phase is calculated exactly if the interaction weights are related by an algebraic equation