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

some ramifications of a theorem of boas and pollard concerning the completion of a set of functions in l 2 about fifty years ago r p boas and harry pollard proved that an orthonormal system that is completable by the adjunction of a finite number of functions also can be completed by multiplying the elements of the given system by a fixed bounded nonnegative measurable function in subsequent years several variations and extensions of this theorem have been given by a number of other investigators and this program is continued here a mildly surprising corollary of one of the results is that the trigonometric and walsh systems can be multiplicatively transformed into quasibases for l1 0 1 1 a curious connection between disparate mathematical ideas occurs in the theory of complete systems of real valued measurable functions the first of these notions is due to boas and pollard 3 who showed that certain incomplete systems in l may be transformed into complete systems by multiplying the elements of the system by a fixed bounded nonnegative measurable function subsequently talalyan 17 18 proved that a system is complete in measure on a set e if and only if for every positive e is complete in l ee where ee is a measurable subset of e that has measure greater than e e in addition talalyan showed that if has these properties so also does any family obtained from by deleting a finite number of its members later goffman and waterman 6 gave a new proof of this theorem and observed that it is always possible to make certain infinite deletions from a system that is complete in measure so as to leave a residual system that also satisfies this condition the somewhat surprising coincidence of these ideas was noted in 15 where the following result was established theorem a let e be a measurable set of finite positive measure and let n n 1 2 be a subset of l e then the following conditions are equivalent bp there exists a bounded measurable function m such that m n n 1 2 is complete in l e m is complete in measure on e received by the editors march 8 1995 and in revised form july 21 1995 1991 mathematics subject classification primary 42b65 42c15 46b15 41a30 41a58

stability of stochastic 2 d systems highlights l p stability of the 2 d random sequence is introduced the stability of the stochastic 2 d systems in the l p sense is studied some sufficient conditions about the stability of the stochastic 2 d systems are discussed the stability of the stochastic 2 d systems in the l p sense is studied and some sufficient conditions about the stability are also given in this paper

formulation and asymptotic properties of the bifurcation ratio in horton s law for the equiprobable binary tree model the bifurcation ratio for the equiprobable binary tree model is formulated we obtain the exact expression of the kth moment of the second order streams we also obtain a recursive equation between rth and r 1 th order streams horton s law is confirmed numerically by calculating this recursive equation and asymptotic properties of the bifurcation ratio are discussed

deterministic screened growth models abstract two deterministic modifications of the two dimensional screened growth models have been investigated in one of these models the threshold growth probability model all of those sites with growth probabilities greater than p max are filled at each stage in the growth process here p max is the maximum growth probability for any of the unoccupied perimeter sites this model leads to clusters with a power law relationship between the radius of gyration r g and the cluster size m r g m for this model the effective fractal are changed dimensionally d 1 varies continously as the parameters in the screening function are changed in the second model the continuous growth model growth is assumed to occur in each of the unoccupied surface sites at a rate proportional to their growth probabilities these sites become occupied when the growth in them reaches a fixed value of 1 this model leads to clusters with four dense symmetric arms in this model the effective dimensionality d crosses over from a value of 2 0 for small clusters or small length scales to a smaller value probably 1 0 on longer length scales similar results were obtained from three dimensional models

locally optimal adaptive control without persistent excitation it is shown that without a persistent excitation condition the least squares ls estimates will converge to some manifold when a ls estimator is combined with a controller computed from the estimated model an equation of this manifold depends upon a type of the control law applied it is proven that there exists the unique control law which is invariable in this manifold this analysis leads to designing adaptive locally optimal control without any excitation signals and modifications of the algorithm

orientation of particle attachment and local isotropy in diffusion limited aggregation dla we simulate 50 off lattice diffusion limited aggregation dla clusters of one million particles each the probability distribution of the angle of attachment of arriving particles with respect to the local radial direction is obtained numerically for increasing cluster size n the distribution crosses over extremely accurately to a cosine with an amplitude which decreases towards zero as a power law in n from this specific viewpoint asymptotically large dla clusters are locally isotropic

stochastic approximation method with gradient averaging for unconstrained problems a stochastic approximation algorithm is studied in which random gradient estimates are averaged by an auxiliary filter convergence of the algorithm is proved under a rather general noise condition a practical version of the method is described and numerical results are given

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

self similarity of harmonic measure on dla abstract the right hand side of the curve of the harmonic measure on dla is undefined this does not necessarily imply that the harmonic measure and the dla geometry are not self similar we show for off lattice dla that the right hand tail satisfies a different rescaling rule this cauchy rescaling is compatible with self similarity the analysis is done on off off lattice dla in which both the brownian motion and the laplace equation are off lattice the cluster sizes range between 32 and 50 000 atoms the square lattice used to numerically estimate the laplacian potential introduces a lower cutoff on the spatial resolution of this potential we find a dependence of the right tail of the distribution of holders on this ultraviolet cutoff whereas the shape of the tail does depend on this ultraviolet lattice cutoff the applicability of the collapse rules do not

on convergence of the recursive identification algorithms the paper is concerned with identification of a linear dynamic system through recursive algorithms of the approximate maximum likelihood aml method in all papers on the convergence of the recursive aml procedure no matter what method of analysis is used a certain rather restrictive positive real condition is imposed on the model of disturbance in this work it is shown that recursive aml algorithms converge with probability one under a fairly broad also positive real condition which significantly expands the aml application area the well known principle of averaging from physics is used as a tool of analysis a sketch of the proof of the principle of averaging for stochastic difference equations is given

topological self similarity on the random binary tree model asymptotic analysis on some statistical properties of the random binary tree model is developed we quantify a hierarchical structure of branching patterns based on the horton strahler analysis we introduce a transformation of a binary tree and derive a recursive equation about branch orders as an application of the analysis topological self similarity and its generalization is proved in an asymptotic sense also some important examples are presented

lattice induced anisotropy in a diffusion limited growth model the authors have investigated the diffusion limited aggregation process on a square lattice by solving the laplace equation numerically in contrast to previous models at each time step the occupancy probability is used directly for the growth of all of the perimeter sites they find that regardless of the initial shape of the seed particles the final pattern of the aggregate has an anisotropic shape they have interpreted the pronouncement of the lattice induced anisotropy as arising from averaging of the shape fluctuations in this growth process the values of the fractal dimension of the clusters are found to agree with the prediction of turkevich and scher 1985 implying that the cluster in this model have the same structure as large dla and dielectric breakdown clusters

local isotropy and geometry of particle flux lines in diffusion limited aggregation we present a theoretical description of the branch orientation and the geometry of walker flux lines in diffusion limited aggregation dla it is based on the self similarity and non directedness of the branch structure and explains recent numerical results on the probability distribution of the branch orientation and the angle of particle attachment our results support the idea that asymptotically large off lattice dla clusters are locally isotropic and every individual walker flux line exhibits a certain type of self similarity

a renormalisation group approach to the viscous finger fractal at finite viscosity ratio a real space renormalisation group method is applied to viscous fingering to analyse the fractal nature at a finite viscosity ratio the renormalisation group transformation is obtained for the permeability of the surface layer which plays the role of coupling constant in the renormalisation group of the phase transition a stable fixed point is found as a function of the viscosity ratio the growth probabilities on the perimeter bonds of the injected fluid are evaluated at the fixed point the multifractal scaling properties are found to be described by the generalised dimensions and the alpha f spectra it is shown that at a finite viscosity ratio the displaced area is compact with a surface fractal dimension between 1 and the dla result with increasing viscosity ratio this result is consistent with that of king s numerical simulation 1987

a remark on the uncertainty principle for hilbertian basis on considere le cas l 2 r dx a une dimension soit l 2 r et la transformee de fourier exp ix x dx on demontre que etant donne 1 2 1 2 il y a une base i pour l 2 r qui satisfait x i et i et i pour chaque i

extended least squares and their applications to adaptive control and prediction in linear systems herein strong consistency of recursive extended least squares is established under considerably weaker assumptions than previously assumed in the literature the argument used to establish consistency also leads to certain basic properties of adaptive predictors based on these recursive estimators making use of these properties of the adaptive predictors simple modifications of the astrom wittenmark self tuning regulator are proposed and shown to be asymptotically optimal

fracture propagation governed by the laplace equation we propose a model of fracture propagation when it is a two dimensional system and has displacement restricted to be perpendicular to the plane its displacement obeys the laplace equation because the diffusion limited aggregation dla is also governed by the laplace equation we can make use of many methods developed for dla to study our model monte carlo simulations are done and df of the model turns out to be about 1 55 0 05 it differs from that of dla moreover nagatani s renormalization is used to investigate the model and the multifractalities of both the growth probability and the mass distributions are found the generalized dimension dm q of the mass distribution is dm df dm 0 dm a fractal pattern is proposed to show the dependence of dm q upon q

sigma delta spl sigma spl delta quantization and finite frames the k level sigma delta spl sigma spl delta scheme with step size spl delta is introduced as a technique for quantizing finite frame expansions for spl ropf sup d error estimates for various quantized frame expansions are derived and in particular it is shown that spl sigma spl delta quantization of a unit norm finite frame expansion in spl ropf sup d achieves approximation error where n is the frame size and the frame variation spl sigma f p is a quantity which reflects the dependence of the spl sigma spl delta scheme on the frame here spl par spl middot spl par is the d dimensional euclidean 2 norm lower bounds and refined upper bounds are derived for certain specific cases as a direct consequence of these error bounds one is able to bound the mean squared error mse by an order of 1 n sup 2 when dealing with sufficiently redundant frame expansions this represents a significant improvement over classical pulse code modulation pcm quantization which only has mse of order 1 n under certain nonrigorous statistical assumptions spl sigma spl delta also achieves the optimal mse order for pcm with consistent reconstruction

a renormalisation group approach to the scaling structure of diffusion limited aggregation a renormalisation group method is presented to analyse the scaling structure of the growth probability in diffusion limited aggregation the recursion relation for the growth probability is derived under the renormalisation transformation the growth probability assigned to each growth bond is represented by a random multiplicative process the scaling of the highest growth probability is derived and the fractal dimension is found a hierarchy of generalised dimensions d q is calculated to describe the growth probability the partition of q 1 d q into a density of singularities f q with singularity strength alpha q is made and the alpha f spectrums are found

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

two theorems on lattice expansions it is shown that there is a tradeoff between the smoothness and decay properties of the dual functions occurring in the lattice expansion problem more precisely it is shown that if g and g are dual then 1 at least one of h sup 1 2 g and h sup 1 2 g is n in l sup 2 r and 2 at least one of hg and g is not in l sup 2 r here h is the operator 1 4 pi sup 2 d sup 2 dt sup 2 t sup 2 the first result is a generalization of a theorem first stated by r c balian 1987 the second result is new and relies heavily on the fact that when g in w sup 2 2 s with s 1 2 1 2 1 2 1 2 and g 0 than 1 g not in l sup 2 s

structural five fold symmetry in the fractal morphology of diffusion limited aggregates abstract the statistical self similarity of the geometry of diffusion limited aggregates and the multifractal nature of the growth probability distribution on the surface of the growing clusters are investigated using the wavelet transform this study reveals the existence of a predominant structural five fold symmetry in the internal frozen region as well as in the active outer region of the interface this observation is corroborated by a statistical analysis of the screening effects that govern diffusion limited aggregation dla growth in linear and sector shaped cells the existence of this symmetry is likely to be a clue to a hierarchichal fractal ordering we report on the discovery of fibonacci sequences in the inner extinct region of large mass off lattice dla clusters with a branching ratio which converges asymptotically to the golden mean we suggest an interpretation of the dla morphology as a quasifractal counterpart of the well ordered snowflake fractal architecture

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

central limit theorem for the bifurcation ratio of a random binary tree in order to formulate and examine the central limit theorem for a binary tree numerically a method for generating random binary trees is presented we first propose the correspondence between binary trees and a certain type of binary sequences which we call dyck sequences then the method for generating random dyck sequences is shown also we propose the method of branch ordering of a binary tree by means of only the corresponding dyck sequence we confirm that the method is in good consistency with the topological analysis of binary trees known as the horton strahler analysis two types of central limit theorem are numerically confirmed and the obtained results are expressed in simple forms furthermore the proposed method is available for a wide range of the topological analysis of binary trees