rigid animals in three dimensions the number of rigid clusters c n is enumerated for the central force model on a site diluted fcc lattice the exponent vr governing the growth of the radius of gyration of the rigid animal is also calculated it is found that c n approximately n0 333n and vr 0 32 or 0 02

exposure response relationships for transportation noise this article presents synthesis curves for the relationship between dnl and percentage highly annoyed for three transportation noise sources the results are based on all 21 datasets examined by schultz j acoust soc am 64 377 405 1978 and fidell et al j acoust soc am 89 221 233 1991 for which acceptable dnl and percentage highly annoyed measure could be derived augmented with 34 datasets separate nonidentical curves were found for aircraft road traffic and railway noise a difference between sources was found using data for all studies combined and for only those studies in which respondents evaluated two sources the latter outcome strengthens the conclusion that the differences between sources cannot be explained by differences in study methodology

branched polymer approach to the structure of lattice animals and percolation clusters the authors treat the percolation problem as a solution of randomly branched chains and argue that such chains are swollen by a three body repulsion acting only within a connected cluster this type of repulsion should not be subject to the screening effects which inhibit ordinary repulsions the repulsion becomes relevant below six dimensions they estimate its effect on the size of a cluster in analogy with flory s argument for the swelling of linear chains above six dimensions the cluster size scales as mass to the 1 4 power as with isolated branched chains in high dimensions

monte carlo renormalisation group approach to multifractal structure of growth probability distribution in dla a monte carlo renormalisation group method is presented to study the fractal structure of a dla cluster this method converges very rapidly with an increase in the size of the cell applying this technique to the dla grown on the square lattice the scaling structure of the growth probability distribution in the surface layer is calculated an infinite set of generalised dimensions d q and the alpha f spectra are found the author estimates that the maximum value of the generalised dimension and the surface fractal dimension are dinfinity 0 5 0 05 and d0 1 5 0 05 up to the scale factor b 30 in excellent agreement with the conjecture by ball 1986

geometrical crossover and self similarity of dla and viscous fingering clusters abstract diffusion limited aggregation dla clusters and the structures observed in viscous fingering experiments at high capillary numbers are tree like fractals the different branches may be assigned a branch order in a way that exhibits scaling and permits a self similar characterization by the bifurcation ratio rn and the length ratio rl of branches of different orders the fractal dimension is given by d log r n log 1 r l good agreement between experiments and simulations is found a crossover function characterizes the branch orders and we conclude that dla is in a state of geometrical crossover branches are linear up to their length l but fractally distributed on length scales much larger than l the effective fractal dimension of dla depends on how d is measured and over what part of the cluster

position space renormalisation group for directed diffusion limited aggregation the author uses a position space renormalisation group transformation to show that directed diffusion limited aggregation and directed animals are in different universality classes

geometrical arguments against the alexander orbach conjecture for lattice animals and diffusion limited aggregates geometrical arguments are given which suggest that the alexander orbach conjecture 1982 does not hold for lattice animals in d 2 and for the diffusion limited aggregates for large dimensions

two way unary automata versus logarithmic space we show that if l nl the classical logarithmic space classes then each unary 2nfa a two way nondeterministic finite automaton can be converted into an equivalent 2dfa a deterministic two way automaton keeping the number of states polynomial unlike other results of this kind here the deterministic simulation is valid for inputs of all lengths not only polynomially long ones this shows a connection between the standard logarithmic space complexity and the state complexity of two way unary automata it indicates that l could be separated from nl by proving a superpolynomial gap in the number of states for the conversion from unary 2nfas to 2dfa moreover without any unproven assumptions we show that each n state unary 2nfa can be simulated by an equivalent 2ufa an unambiguous 2nfa with a polynomial number of states

a new finite size scaling approach to random walks we present a new finite size scaling method for the random walks rw superseding a previously widely used renormalization group approach which is shown here to be inconsistent the method is valid in any dimension and is based on the exact solution for the two point correlation function and on finite size scaling as an example the phase diagram is derived for the random walk in two dimensions with a surface bulk interaction where the system has either a surface or a defect line we also discuss an initial calculation of the corresponding phase diagram for the case of a critically diluted lattice

on measuring nondeterminism in regular languages abstract it is well known that allowing nondeterminism in a finite automaton can produce in the most extreme case an exponential savings in the number of states required to recognize a regular language this paper studies situations intermediate between forbidding nondeterminism and allowing it the amount of nondeterminism used by a finite automaton is quantified so that the decrease in the size of the state space that occurs as the amount of nondeterminism that is permitted increases in increments can be studied these intermediate situations are shown always to lie between two extremes 1 there are no savings as the amount of nondeterminism increases incrementally so that savings occur only when the amount of nondeterminism becomes unlimited 2 each increment of nondeterminism results in additional savings the number s of states decreasing approximately as s 1 i until exponential savings have been achieved after about i log s log log s increments

dynamic university classes for diffusion limited aggregation and lattice animals a recent application of the real space renormalisation group to the problem of anomalous diffusion on witten sander clusters is generalised and the authors demonstrate that these clusters lie in a distinct dynamic universality class from random lattice animals

caliper diameter of branched polymers the authors report analyses of exact numerical data for the spanning diameter of two dimensional lattice animals up to size n 17 estimates of the exponent nu are consistent with previous studies the leading correction to scaling has an exponent sigma approximately nu which does not result from irrelevant variable effects interpretation of this correction as a surface term is proposed

position space renormalisation group approach for bond lattice animals the authors perform two position space renormalisation group calculations based on a centre rule weight function for bond lattice animals on the square lattice the critical exponent nu is determined within the range 0 6430 0 6528

the monadic quantifier alternation hierarchy over grids and graphs the monadic second order quantifier alternation hierarchy over the class of finite graphs is shown to be strict the proof is based on automata theoretic ideas and starts from a restricted class of graph like structures namely finite two dimensional grids considering grids where the width is a function of the height we prove that the difference between the levels k 1 and k of the monadic hierarchy is witnessed by a set of grids where this function is k 1 fold exponential we then transfer the hierarchy result to the class of directed or undirected graphs using an encoding technique called strong reduction it is notable that one can obtain sets of graphs which occur arbitrarily high in the monadic hierarchy but are already definable in the first order closure of existential monadic second order logic we also verify that these graph properties even belong to the complexity class nlog which indicates a profound difference between the monadic hierarchy and the polynomial hierarchy

on the limits of the communication complexity technique for proving lower bounds on the size of minimal nfa s in contrast to the minimization of deterministic finite automata dfa s the task of constructing a minimal nondeterministic finite automaton nfa for a given nfa is pspace complete moreover there are no polynomial approximation algorithms with a constant approximation ratio for estimating the number of states of minimal nfa s since one is unable to efficiently estimate the size of a minimal nfa in an efficient way one should ask at least for developing mathematical proof methods that help to prove good lower bounds on the size of a minimal nfa for a given regular language here we consider the robust and most successful lower bound proof technique that is based on communication complexity in this paper it is proved that even a strong generalization of this method fails for some concrete regular languages to fail is considered here in a very strong sense there is an exponential gap between the size of a minimal nfa and the achievable lower bound for a specific sequence of regular languages the generalization of the concept of communication protocols is also strong here it is shown that cutting the input word into 2 o n 1 4 pieces for a size n of a minimal nondeterministic finite automaton and investigating the necessary communication transfer between these pieces as parties of a multiparty protocol does not suffice to get good lower bounds on the size of minimal nondeterministic automata it seems that for some regular languages one cannot really abstract from the automata model that cuts the input words into particular symbols of the alphabet and reads them one by one using its input head

position space renormalisation group for aggregation fractals a simple position space renormalisation group scheme is used to find the fractal dimension of sutherland type aggregation clusters grown in low dimensional space clusters which move along paths of fractal dimension dw 0 1 2 are considered and a systematic variation in cluster fractal dimension matching that of recent computer simulation is found

superuniversal spectral dimension for dilute branched polymers the authors introduce the problem of random walks on lattice animals dilute branched polymers and calculate the spectral dimension ds of animals in two and three dimensions by using a two parameter position space renormalisation group method this problem which is a generalisation of the problem of the ant in the labyrinth has been independently proposed and investigated by wilke et al 1984 the spectral dimension is given by ds 2da dw where da is the fractal dimension of the lattice animal and dw the fractal dimension of the random walk on the lattice animal the results indicate that ds may be a superuniversal quantity i e its value is independent of dimension moreover the authors find ds to be close to 4 3 which is the spectral dimension of the largest percolation cluster at the percolation threshold pc as conjectured by alexander and orbach 1982 since both lattice animals and the largest percolation cluster at pc have a homogeneous interior structure the results suggest that the spectral dimension of all such fractals may equal 4 3

on the position space renormalisation group approach to diffusion limited cluster growth problems the authors investigate the application of position space renormalisation group methods to the witten sander 1981 model of diffusion limited cluster growth they show that this method cannot be used to investigate the asymptotic behaviour of the witten sander model in three and higher dimensions even though it yields satisfactory results in two dimensions with small cells

a monte carlo renormalisation group for aggregation processes a general monte carlo renormalisation group transformation for the static as well as the kinetic properties of aggregates is formulated and applied to different growth models particle aggregation dla cluster aggregation cici and invasion percolation ip the leading critical exponent is determined numerically for dla the first correction to scaling exponent has been estimated as well

descriptional and computational complexity of finite automata a survey finite automata are probably best known for being equivalent to right linear context free grammars and thus for capturing the lowest level of the chomsky hierarchy the family of regular languages over the last half century a vast literature documenting the importance of deterministic nondeterministic and alternating finite automata as an enormously valuable concept has been developed in the present paper we tour a fragment of this literature mostly we discuss developments relevant to finite automata related problems like for example i simulation of and by several types of finite automata ii standard automata problems such as fixed and general membership emptiness universality equivalence and related problems and iii minimization and approximation we thus come across descriptional and computational complexity issues of finite automata we do not prove these results but we merely draw attention to the big picture and some of the main ideas involved

a new simulation of branched polymers the scanning method for the simulation of linear chain is extended to general models of branched polymers without loops a branched chain grows in time namely a number of steps from the origin therefore i in contrast to other simulation techniques which are of a relaxation type the chains are statistically independent and the statistical error can reliably be estimated ii the probability of a chain is known and hence the entropy and iii the scanning construction enables one to study geometrical properties which depend on time for self avoiding trees on a square lattice the author obtains the relatively accurate estimates for the static critical exponents nu 0 640 or 0 004 and theta 1 003 or 0 02 and for the connective constant mu 5 1419 or 0 003 the author also obtains critical exponents gamma t approximately 1 26 and nu t approximately 0 83 which characterise the growth in time of the number of bonds and the gyration radius respectively application of the scanning method to more complex branched polymers is discussed

communication complexity method for measuring nondeterminism in finite automata while deterministic finite automata seem to be well understood surprisingly many important problems concerning nondeterministic finite automata nfa s remain open one such problem area is the study of different measures of nondeterminism in finite automata and the estimation of the sizes of minimal nondeterministic finite automata in this paper the concept of communication complexity is applied in order to achieve progress in this problem area the main results are as follows 1 deterministic communication complexity provides lower bounds on the size of nfa s with bounded unambiguity applying this fact the proofs of several results about nfa s with limited ambiguity can be simplified and presented in a uniform way 2 there is a family of languages konk2 with an exponential size gap between nfa s with polynomial leaf number ambiguity and nfa s with ambiguity k this partially provides an answer to the open problem posed by b ravikumar and o ibarra 1989 siam j comput 18 1263 1282 and h leung 1998 siam j comput 27 1073 1082

anomalous diffusion on regular and random models for diffusion limited aggregation families of regular fractal models for witten sander dla clusters in d dimensions are proposed resembling the computer generated aggregates the authors evaluate random walk dimensionalities exactly using resistance scaling ideas the applicability of the conflicting conjectures of alexander orbach 1982 ao dw 3df 2 and aharony stauffer 1984 as dw df 1 to dla is then examined one finds that the dendritic nature of the fractals presented ensures that as is satisfied exactly in all dimensions an rsrg fugacity transformation method is applied to the simplest of the regular fractals in d 2 the result for dw is compared with the exact resistance scaling result to obtain an estimate of the error introduced by the rsrg the rg method is then generalised to treat the random dla problem the resulting estimate for dw is in fortuitously good agreement with as the corresponding monte carlo data and the results above for the regular fractal models

scaling properties of the surface of the eden model in d 2 3 4 the surface of the eden model is investigated numerically by finite size scaling using a strip geometry three different versions are studied and it shows that the one mostly used previously exhibits strong finite size corrections in two dimensions it is found that the surface thickness simply grows as the square root of the width of the strip for infinitely long strips this result as well as the results in d 3 and d 4 suggest that the surface of the eden shares some properties with the equilibrium models used to describe the roughening transition moreover it is found that the surface thickness scales differently with the height of the cluster for infinitely large strips

on the number of lattice animals embeddable in the square lattice enumeration of lattice animals embeddable in a square lattice has been extended to 24 cell animals by redelmeier 1981 it is shown that the number of animals per site an approximately 0 317 4 0626 nn 1 exp 0 465 n 0 87 to a high degree of accuracy

separating exponentially ambiguous finite automata from polynomially ambiguous finite automata we resolve an open problem raised by ravikumar and ibarra siam j comput 18 1989 pp 1263 1282 on the succinctness of representations relating to the types of ambiguity of finite automata we show that there exists a family of nondeterministic finite automata an over a two letter alphabet such that for any positive integer n an is exponentially ambiguous and has n states whereas the smallest equivalent deterministic finite automaton has 2n states and any smallest equivalent polynomially ambiguous finite automaton has 2n 1 states

spectral dimension and the shortest path of saw s with multi neighbour interactions the spectral dimension and the shortest path of self avoiding walks saws with bridge length interaction range b 1 2 3 2 are studied numerically the spectral dimension is calculated by performing exact multi neighbour random walks on the monte carlo generated saw configurations it is found that the spectral dimension is not affected by the finite range of interactions finite bridge length and approach to ds 1 both in dimensions d 2 and 3 the shortest path length sn of the n step saw with local bridges is also investigated it is shown that sn n a n delta b c n the authors numerical simulation results indicate that the exponent delta is independent of the dimensionality and is about 3 16 in dimensions d 2 5

two dimensional diffusion limited aggregation a finite size scaling approach finite size scaling in the diffusion limited aggregation model of witten and sander 1981 1983 is studied in two dimensions via monte carlo simulations on semi infinite strips of width n 2 20 due to a crossover between self avoiding walk like growth for small n and dendritic growth for large n wide strips n 8 20 are needed to deduce the witten sander fractal dimension d approximately 5 3 from the finite size scaling analysis

universality in branched polymers on d dimensional hypercubic lattices the authors have derived series for weakly and strongly embeddable trees in d dimensional simple hypercubic lattices for arbitrary integral d for d 2 3 9 they present series evidence that such trees are in the same universality class as lattice animals in addition they have derived expansions in inverse powers of sigma 2d 1 for the growth parameters for bond and site trees and compare these with the corresponding results for animals

demographic and attitudinal factors that modify annoyance from transportation noise the effect of demographic variables sex age education level occupational status size of household homeownership dependency on the noise source and use of the noise source and two attitudinal variables noise sensitivity and fear of the noise source on noise annoyance is investigated it is found that fear and noise sensitivity have a large impact on annoyance dnl equivalent equal to at most 19 and 11 db respectively demographic factors are much less important noise annoyance is not related to gender but age has an effect dnl equivalent equal to 5 db the effects of the other demographic factors on noise annoyance are very small i e the equivalent dnl difference is equal to 1 2 db and in the case of dependency 3 db the results are based on analyses of the original data from various previous field surveys of response to noise from transportation sources number of cases depending on the variable between 15 000 and 42000

cluster renormalisation study of site lattice animals in two and three dimensions the author studies site lattice animals using the cluster renormalisation cr approach previously applied to bond animals the exponent v characterising the asymptotic behaviour of the mean square radius of gyration of site animals and the fractal dimension df 1 v are determined in two and three dimensions he finds v 0 649 or 0 009 and v approximately 0 51 in two and three dimensions respectively in addition he studies restricted valence site animals on a triangular lattice using a single parameter cr approach

community reaction to aircraft noise time of day penalty and tradeoff between levels of overflights a decrease in the level of sound events can compensate for an increase in the level of other events but noise metrics assume different tradeoffs noise metrics also differ in the penalty applied to noise in the evening and to noise in the night and in the definition of these periods these two aspects of noise metrics i e the tradeoff and the penalty for the nighttime 23 7h are investigated a general model of the relation between sels of sound events aircraft overflights and noise annoyance is presented which allows for a wide range of tradeoffs and time of day penalties the tradeoff and time of day penalty parameters of the model are fitted to the data from an aircraft noise study conducted around amsterdam airport schiphol which is especially suited for investigating the tradeoff and time of day penalties it was found that in this study the tradeoff between the levels of events in metrics based on l aeq s such as l aeq 24 h dnl and denl is approximately correct for the prediction of noise annoyance furthermore it was found that the strongest correlation with annoyance is obtained with a nighttime penalty of circa 10 db no suitable data were available for further tests of the tradeoff the result with respect to the nighttime penalty was weakly further supported by the outcome of analyses of the original data from four other aircraft noise surveys one survey conducted around british airports and three coordinated surveys carried out around paris orly amsterdam schiphol and glasgow abbotsinch c 2000 acoustical society of america

extended range ballistic aggregation exact results and scaling the authors present the exact solution of an infinite range ballistic aggregation model the density is found to decrease as n 1 2 as n to infinity where n is the number of columns in which particles move exact expressions for the correlation function and the hole size distribution are derived both decay exponentially over their entire domain using a scaling ansatz similar to that made in finite size scaling in critical phenomena they extrapolate these results to large but finite ranges computer simulations test the proposed scaling

cluster diversity and entropy on the percolation model the lattice animal identification algorithm we present an algorithm to identify and count different lattice animals la s in the site percolation model this algorithm allows a definition of clusters based on the distinction of cluster shapes in contrast with the well known hoshen kopelman algorithm in which the clusters are differentiated by their sizes it consists in coding each unit cell of a cluster according to the nearest neighbors nn and ordering the codes in a proper sequence in this manner a la is represented by a specific code sequence in addition with some modification the algorithm is capable of differentiating between fixed and free la s the enhanced hoshen kopelman algorithm j hoshen m w berry and k s minser phys rev e 56 1455 1997 is used to compose the set of nn code sequences of each cluster using monte carlo simulations on planar square lattices up to 2000x2000 we apply this algorithm to the percolation model we calculate the cluster diversity and cluster entropy of the system which leads to the determination of probabilities associated with the maximum of these functions we show that these critical probabilities are associated with the percolation transition and with the complexity of the system

diffusion on lattice animals and percolation clusters a renormalisation group approach a position space renormalisation group method is developed to study the scaling properties of random walks on fractals calculations in two dimensions for site lattice animals and site percolation clusters yield numerical results for the critical exponent defined by the asymptotic behaviour of the root mean square displacement of the walker the renormalisation group results are consistent with recent monte carlo calculations

a transfer matrix approach to random resistor networks introduces a transfer matrix formulation to compute the conductance of random resistor networks the authors apply this method to strips of width up to 40 and use finite size scaling arguments to obtain an estimate for the conductivity critical exponent in two dimensions t 1 28 or 0 03

ambiguity and structural ambiguity of symmetric difference nfas abstract okhotin showed an exponential trade off in the conversion from nondeterministic unary automata to unambiguous nondeterministic unary automata we show that the trade off in the case of unary symmetric difference automata to finitely structurally ambiguous unary symmetric difference automata is linear with constant 1 in the number of states in particular for every n state unary nondeterministic symmetric difference automaton there is an equivalent finitely structurally ambiguous n state unary symmetric difference nondeterministic automaton we also consider the complexity of deciding unambiguity for k deterministic finite automata and investigate the interplay between ambiguity and structural ambiguity

scaling and critical probabilities for cluster size and la diversity on randomly occupied square lattices using monte carlo simulations we report the behaviour of the total number of clusters the cluster size diversity and the lattice animals la diversity on randomly occupied square lattices the critical probability associated with the maximum of these variables is determined in comparison with the percolation probability pc our simulations indicate that pc and the critical probability of the maximum cluster size diversity pc d s max occur at the same point as indicated in a previous paper tsang i r and tsang i j 1997 j phys a math gen 30 l239 the probability for the maximum number of clusters is obtained at a lower value pc n max 0 27 0 01 we describe the cluster identification algorithm used to count the different la and determined the critical probability for the maximum la diversity pc d f max 0 45 0 02 we derive the exponents characterizing the relation between d s max d f max and n max and several scaling relations between the variables measured the lattice size and the probability of occupation p in addition we show the scaling behaviour of la diversity versus cluster size diversity for each value of p

state complexity of operations on two way finite automata over a unary alphabet the paper determines the number of states in two way deterministic finite automata 2dfa over a one letter alphabet sufficient and in the worst case necessary to represent the results of basic language theoretic operations on 2dfas with a certain number of states it is proved that i intersection of an m state 2dfa and an n state 2dfa requires between m n and m n 1 states ii union of an m state 2dfa and an n state 2dfa between m n and 2m n 4 states iii kleene star of an n state 2dfa g n o n 2 states where g n e 1 o 1 n l n n is the maximum value of lcm p 1 p k for p i

large cell renormalisation and systems of dimensionality larger than the upper marginal dimension a recent argument dismissing the applicability of large cell renormalisation schemes to systems of dimensionality d larger than the upper marginal dimension dc is critically discussed in this connection new large cell renormalisation results for the random walk for d 3 and 4 are presented which indicate convergence to the correct results

nondeterministic complexity in subclasses of convex languages abstract we study the nondeterministic state complexity of basic regular operations on the classes of prefix suffix factor and subword free closed and convex regular languages and on the classes of right left two sided and all sided ideal regular languages for the operations of concatenation intersection union reversal star and complementation we get tight upper bounds for all considered classes except for complementation on factor and subword convex languages most of our witnesses are described over optimal alphabets the description of a proper suffix convex language over a five letter alphabet meeting the upper bound 2 n for complementation and obtaining an asymptotically tight bound n for complementation of unary prefix free languages are among the most interesting results of this paper

relating the type of ambiguity of finite automata to the succinctness of their representation we consider the problem of how the size of a nondeterministic finite automaton nfa representing a regular language depends on the degree of ambiguity of the nfa we obtain results for the unary and bounded inputs and partial results for the unrestricted inputs one of the main results of this paper shows that for unrestricted inputs deterministic unambiguous and nondeterministic machines form a hierarchy with respect to the number of states solving an open problem of stearns and hunt we also propose a new approach to the study of the succinctness of representation through regularity preserving closure properties and obtain some results in this direction

on the relationship between the critical exponents of percolation conductivity and static exponents of percolation the author argues that the critical exponent t of random conductance networks near the percolation threshold is given by t d 1 nu for low dimensionalities and t 1 beta for high dimensionalities where nu is the correlation length exponent beta the backbone exponent and d is dimensionality the author argues that what separates the two regimes is a critical fractal dimensionality dl which equals 2 the author also argues that dl is also a critical fractal dimensionality for fractals such as lattice animals and diffusion limited aggregates the result for low dimensionalities has been also obtained by aharony and stauffer by a different argument

cell renormalisation of growth processes true self avoiding walks and growing animals the authors present a generating function formalism for growth processes in general and introduce a new cell renormalisation scheme as examples they treat the true self avoiding walk and the eden process or growing animals in two and three dimensions using small cells the results in both cases indicate a substantial increase in the fractal dimension d compared with their equilibrium counterparts the case of the eden process may however suggest a peculiar convergence behaviour as the cell size b tends to infinity

unambiguous finite automata over a unary alphabet nondeterministic finite automata nfa with at most one accepting computation on every input string are known as unambiguous finite automata ufa this paper considers ufas over a one letter alphabet and determines the exact number of states in dfas needed to represent unary languages recognized by n state ufas in terms of a new number theoretic function g the growth rate of g n and therefore of the ufa dfa tradeoff is estimated as e q n l n 2 n 3 the conversion of an n state unary nfa to a ufa requires ufas with g n o n 2 e 1 o 1 n l n n states where g n is the greatest order of a permutation of n elements known as landau s function in addition it is shown that representing the complement of n state unary ufas requires ufas with at least n 2 o 1 states in the worst case while the kleene star requires up to exactly n 1 2 1 states

a monte carlo renormalization group study of driven interface dynamics the authors have extended the monte carlo renormalization group technique to study the dynamics of driven interface systems the method allows systematic elimination of irrelevant scaling fields and thus helps in extracting the asymptotic scaling behaviour in particular it can be applied to determine scaling exponents from a matching criteria they have applied this method to interfaces evolving according to the kardar parisi zhang equation

state complexity of permutation on finite languages over a binary alphabet abstract the set of all strings parikh equivalent to a string in a language l is called the permutation of l the permutation of a finite n state dfa deterministic finite automaton language over a binary alphabet can be recognized by a dfa with n 2 n 2 2 states we show that if the language consists of equal length binary strings the bound can be improved to f n n 2 n 1 3 and for every n congruent to 1 modulo 3 there exists an n state dfa a recognizing a set of equal length strings such that the minimal dfa for the permutation of l a needs f n states

possible relations for topological and transport properties of the lattice animal model of branched polymers the author proposes that the following relations hold exactly for the lattice animal model of branched polymers the resistivity exponent zeta a is exactly equal to da the fractal dimensionality of the backbone of the anomals the random walk fractal dimensionality dw is given by dw da da where da is the fractal dimensionality of the animals the spectral dimension ds of the backbone of the animals is given by ds 1 at all dimensions i e the backbone is a highly decorated but essentially chain like object these are in contrast with earlier suggestions that one needs two three topological properties to obtain zeta a dw the author proposes that these relations may hold for all clusters for which loops are irrelevant and have a finite upper critical dimensionality the implications of these results for diffusion limited aggregates and percolation clusters are also discussed

first exit time of termites and random super normal conductor networks the resistance of a random super normal conductor network is calculated through the diffusion of a termite which performs a random walk on the normal bonds and has the same probability to exist from any point on a cluster of superconducting bonds the first time t that the termite exits at a distance r from the origin averaged over many configurations and runs is found to behave as t approximately r2k for r xi pc p s describes the divergence of the conductivity as p to pc p being the concentration of the superconductor scaling arguments are used to show that k 1 s 2 theta where 2 theta is the fractal dimension of random walks on single clusters at pc numerical simulations at two dimensions d 2 yields s 1 34 or 0 10 and k 1 34 or 0 03 in agreement with scaling the authors also show that the probability of return to the origin at time t behaves as t dk2 preliminary results at d 1 and other calculations methods are also discussed

operational state complexity of unary nfas with finite nondeterminism we study the state complexity of language operations for unary nfas with limited nondeterminism we consider the boolean operations concatenation and kleene star we give upper bounds for the state complexity of these language operations and lower bounds that are fairly close to the upper bounds our constructions rely on the fact that minimal unary nfas with limited nondeterminism can be found in chrobak normal form for most measures of nondeterminism the measures of nondeterminism which are considered here with the above property are tree width advice and trace

finite automata and unary languages abstract we prove that o e n log n states are sufficient to simulate an n state 1nfa recognizing a unary language by a 1dfa the lower bound is the same similar tight bounds are shown for the simulation of a 2dfa by a 1dfa and a 1nfa we also show that o n 2 states are sufficient and necessary to simulate an n state 1nfa recognizing a unary language by a 2dfa

optimal simulation of self verifying automata by deterministic automata self verifying automata are a special variant of finite automata with a symmetric kind of nondeterminism we study the conversion of self verifying automata to deterministic automata from a descriptional complexity point of view the main result is the exact cost in terms of the number of states of such a simulation

random walk on self avoiding walk a model for conductivity of linear polymers random walks on self avoiding walks saws are studied using monte carlo techniques on a square lattice with nearest neighbour hopping along the chain and between saw points which are nearest neighbours on the embedding lattice the average of the square of the end to end distance for random walks of t steps on saws of length n is fitted to the scaling forms rt2 varies as ndelta tk for t or approximately ntheta where theta approximately 2 nu s k nu s being the average end to end distance exponent for saws the observed value of the exponent delta is supported by the authors real space renormalisation group result for the conductivity of saw chains the exponent k has been related to the effective fractal dimension of the saw chain

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

real space renormalisation group approach to random walks on self avoiding walks the t dependence of the average of the end to end distance rt2 of t step random walks on self avoiding walks is studied by small cell real space renormalisation the results are in qualitative agreement with earlier monte carlo data

magic numbers in the state hierarchy of finite automata a number d is magic for n if there is no regular language for which an optimal nondeterministic finite state automaton nfa uses exactly n states and at the same time the optimal deterministic finite state automaton dfa uses exactly d states we show that in the case of unary regular languages the state hierarchy of dfa s for the family of languages accepted by n state nfa s is not contiguous there are some holes in the hierarchy i e magic numbers in between values that are not magic this solves for automata with a single letter input alphabet an open problem of existence of magic numbers actually most of the numbers is magic in the unary case as an additional bonus we also get a new universal lower bound for the conversion of unary d state dfa s into equivalent nfa s nondeterminism does not reduce the number of states below log 2d not even in the best case

self avoiding walk model for proteins a real space renormalisation group treatment diffusion on the model for proteins introduced by helman coniglio and tsallis 1984 hct is discussed using real space renormalisation group ideas the authors perform cell to cell transformations in two dimensions for both the hct model and the self avoiding walk the latter system is quasi linear and so they expect trivial diffusive behaviour given by dw 2df 8 3 diffusion on the hct model however is complicated by the possibility of jumps across nearest neighbour hydrogen bond bridges the authors results are not consistent with the prediction that dw 2 for the hct model rather they support the assertion that dwhct is close to but slightly greater than dwsaw

walks trials and polymers with loops lattice models interpolating between free and self avoiding random walks are investigated generating functions are constructed for k tolerant walks various trail problems etc for trail problems the effective field theory describing the global behaviour is analysed in the vicinity of the upper critical dimension dc 4 their asymptotic large scale behaviours are the same as those of the self avoiding walk arguments are presented to support the same conclusion for much more general classes of walks including k tolerant walks one of the models exhibits a new tricritical point of order epsilon 1 2 if the fugacity for crossing is increased

the growing self avoiding walk the authors introduce a new self avoiding walk with one step probabilities which depend on the local environment as a consequence this walk is irreversible and models the growth process of a linear polymer in a good solvent to calculate its properties they have performed exact enumerations up to 22 steps on the square lattice and on the diamond lattice this gives for the critical indices the values v 0 68 gamma 1 16 in two dimensions and v 0 525 and gamma 1 in three dimensions

critical exponents and corrections to scaling for bond trees in two dimensions the author has analysed the newly obtained series of the radius of gyration rn and the number of clusters nn for n bond trees i e branch polymers without loops on the square n or 14 and triangular n or 11 lattices to estimate the critical parameters respective estimates of the exponents nu and theta for rn and nn are consistent with the corresponding values for lattice animals while the correction to scaling exponent delta 1 is inconsistent with the animal value in addition delta 1 has different values for rn delta 1 0 635 and nn delta 1 1 3 for bond trees he has also estimated an exponent delta characterising the density distribution delta 2 69 for the triangular lattice

a new monte carlo simulation for two models of self avoiding lattice trees in two dimensions by means of a new monte carlo sampling of a grand canonical ensemble we verify universality for the critical exponents and of two models of lattice trees constrained to be self avoiding on sites or on bonds the attrition constants are also obtained this algorithm a generalization of that recently proposed by berretti and sokal for random walks appears to optimize the critical slowing down in the scaling region systematic and statistical errors are carefully estimated

the backbone and conductivity of random clusters the author studies the properties of the backbone of and conduction in random clusters which have homogeneous interior structure these include the largest percolation cluster at pc lattice animals and the witten sander ws aggregates for lattice animals the fractal dimension da of the backbone is estimated the for first time and is found to be about 1 14 at d 2 and about 1 39 at d 3 these values are much lower than the corresponding values for the animals themselves the fractal and spectral dimensions of the backbone of the ws aggregates are estimated to be about 1 25 and 1 06 at d 2 respectively a recent hypothesis which relates the fractal dimension of random walks on these clusters to those of the clusters and their backbones is also discussed it is shown that for the ws aggregates ds ds 2 on a bethe lattice i e at d infinity in contrast with ds 4 3 and ds 1 for percolation clusters and lattice animals the conductivity exponent ta of lattice animals is found to be ta d 2 approximately 0 73 and ta d 3 approximately 1 19 and ta d or 8 2 whereas for the ws aggregates the author finds tws d 2 approximately 0 67 and tws d 3 approximately 0 94 and tws d to infinity 1

the k th prime is greater than k ln k ln ln k 1 for k 2 rosser and schoenfeld have used the fact that the first 3 500 000 zeros of the riemann zeta function lie on the critical line to give estimates on x and x with an improvement of the above result by brent et al we are able to improve these estimates and to show that the k th prime is greater than k ln k ln ln k 1 for k 2 we give further results without proof

self similarity and structure of dla and viscous fingering clusters diffusion limited aggregation clusters and the structures observed in viscous fingering experiments at high capillary numbers are tree like fractals the different branches may be assigned a branch order in a way that exhibits scaling and permits a self similar characterisation in terms of the bifurcation ratio r and the length ratio r of branches of different orders the fractal dimension is given by d ln rn ln l rl good agreement between experiments and simulations is found a crossover function characterises the branch orders

self avoiding walks on percolation clusters the author studies self avoiding walks saws on percolation clusters a scaling function representation for r the mean end to end distance is proposed which describes a crossover from ordinary saws to saws on fractals he distinguishes between saws on a single cluster for which r approximately nx where x nu 1 and saws on all clusters for which r approximately nz where z nu 2 and n is the number of monomers in the walk he estimates nu 1 d 2 approximately 1 285 and the correction to scaling exponent omega d 2 approximately 1 3 and nu 1 d 3 approximately 1 38 two plausible generalisations of the flory 1953 approximation for nu 1 are investigated and it is argued that none of them provides a satisfactory approximation for nu 1 at all dimensions

renormalisation group for dla and fixed point distribution a renormalization group method is presented to analyse the multifractal structure of the growth probability in the diffusion limited aggregation dla a renormalisation group transformation is derived for the probability distribution of the growth bond s conductance after repeated scaling an invariant distribution is reached as a fixed point distribution the growth probability assigned to each growth bond is represented by a random multiplicative process of the cell s growth probability with fixed point distribution a hierarchy of generalised dimensions d q is calculated and the alpha f spectrum is found

diffusion on random clusters and the parasite problem in the parasite problem a particle ant diffuses randomly on a random percolation cluster in the limit of concentration to 0 lattice animal monte carlo simulations and scaling arguments show that for large animals the distance r travelled by this parasite increases as t 1 za with time t the authors find za approximately 3 4 on the simple cubic lattice and za approximately 2 6 on the square lattice this anomalous diffusion is in rough agreement with a generalisation of a suggestion by alexander and orbach 1982 heuristic arguments in favour of this suggestion are given also they look at corrections to scaling for concentrations equal to the percolation threshold

nondeterministic state complexity of star free languages we investigate the nondeterministic state complexity of several operations on finite automata accepting star free and unary star free languages it turns out that in most cases exactly the same tight bounds as for general regular languages are reached this nicely complements the results recently obtained by brzozowski and liu 2011 8 for the operation problem of star free and unary star free languages accepted by deterministic finite automata

an algorithm for enumerating rigid clusters a more efficient algorithm for enumerating rigid clusters is presented using this algorithm the author has extended the series of site rigid animals enumerated by prunet and blanc 1986 by five more terms and has also calculated for the first time the exponent governing the growth of the radius of gyration for rigid animals

evolution criterion and renormalisation group for dla a renormalisation group method is presented to derive the fractal dimension for the diffusion limited aggregation dla in a non equilibrium steady state the dissipation energy and the conductivity of the surface layer play the roles of the free energy and the coupling constant in an equilibrium state the renormalisation group transformation of the conductivity of the surface layer is obtained in the replacement of the coupling constant the renormalisation group equation has a non trivial solution where the derivative at the fixed point has a positive value of less than one this is consistent with the theorem of minimum entropy production the fractal dimension is expressed in terms of the derivative of the renormalisation group equation at the fixed point the fractal dimension df 1 628 is found by using the 2 2 small cell renormalisation

diffusivity and radius of random animals percolation clusters and compact clusters defines a cluster diffusivity ds by introducing a random rearrangement between arbitrary cluster and perimeter sites such that the cluster remains connected monte carlo simulation is used on the square and simple cubic lattices to determine the dependence of ds and the radius of gyration rs on s the number of cluster sites three limiting cases are considered random animals p 0 percolation clusters p pc and compact clusters p pc the results for the exponent rho of rs are consistent with the best experimental and theoretical values the authors develop an elementary scaling theory for the observed power law dependence of ds in terms of different mechanisms dominant for small and large clusters a position space renormalisation group calculation in two dimensions yields corrections to the elementary theory for random animals and percolation clusters the predictions are consistent with the monte carlo results for ds

homogeneity and spectral dimension of aggregation fractals the fracton or spectral dimension ds of two sutherland type ghost aggregation fractals is obtained for high space dimension by a scaling argument if as seems probable these fractals are homogeneous the results ds 2ln2 ln3 and ds 4 square root 17 1 support recent conjectures that ds is close to 4 3 for all homogeneous fractals however they are not consistent with the stronger conjecture that ds is exactly 4 3 for all such fractals